Research Article Fractal Oscillations of Chirp Functions and ...Fractal Oscillations of Chirp...
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Hindawi Publishing Corporation International Journal of Differential Equations Volume 2013, Article ID 857410, 11 pages http://dx.doi.org/10.1155/2013/857410 Research Article Fractal Oscillations of Chirp Functions and Applications to Second-Order Linear Differential Equations Mervan PašiT 1 and Satoshi Tanaka 2 1 Department of Applied Mathematics, Faculty of Science, University of Zagreb, 10000 Zagreb, Croatia 2 Okayama University of Science, Okayama 700-0005, Japan Correspondence should be addressed to Satoshi Tanaka; [email protected] Received 7 December 2012; Accepted 8 January 2013 Academic Editor: Norio Yoshida Copyright © 2013 M. Paˇ si´ c and S. Tanaka. is is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. We derive some simple sufficient conditions on the amplitude (), the phase (), and the instantaneous frequency () such that the so-called chirp function () = ()(()) is fractal oscillatory near a point = 0 , where () = () and = () is a periodic function on R. It means that () oscillates near = 0 , and its graph Γ() is a fractal curve in R 2 such that its box- counting dimension equals a prescribed real number ∈ [1, 2) and the -dimensional upper and lower Minkowski contents of Γ() are strictly positive and finite. It numerically determines the order of concentration of oscillations of () near = 0 . Next, we give some applications of the main results to the fractal oscillations of solutions of linear differential equations which are generated by the chirp functions taken as the fundamental system of all solutions. 1. Introduction e brilliant heuristic approach of Tricot [1] to the fractal curves such as the graph of functions () = sin − and () = cos − gave the main motivation for studying the fractal properties near =0 of graph of oscillatory solutions of various types of differential equations: linear Euler-type equation + − =0 (see [2]), general second-order linear equation + () = 0 (see [3]) where () satisfies the Hartman-Wintner asymptotic condition near =0, half- linear equation (| | −2 ) + ()|| −2 =0 (see [4]), linear self-adjoint equation (() ) + () = 0 (see [5]), and - Laplace differential equations in an annular domain (see [6]). A function () is said to be a chirp function if it possesses the form () = ()(()), where () and () denote, respectively, the amplitude and phase of (), and () is a periodic function on R. In all previously mentioned papers [2–5], authors are dealing with the fractal oscillations of second-order differential equations and are deriving some sufficient conditions on the coefficients of considered equa- tions such that all their solutions () together with the first derivative () admit asymptotic behaviour near =0. It is formally written in the form of a chirp function, that is, () = () sin(()) and () = () cos(()) near = 0. According to it, one can say that the asymptotic formula for solutions of considered equations satisfies the chirp-like behaviour near =0 (on the asymptotic formula for solu- tions near = ∞, see [7, 8]). en, in the dependence of a prescribed real number ∈ [1, 2), authors give some asymptotic conditions on (), (), and () such that all solutions () are fractal oscillatory near =0 with the fractal dimension . In this paper, independently of the asymptotic theory of differential equations, we firstly study the fractal oscillations of a chirp function; see eorems 8 and 11. Second, taking two linearly independent chirp functions 1 () and 2 (), we generate some new classes of fractal oscillatory linear differential equations which are not considered in [2–5] and have the general solution in the form of () = 1 1 () + 2 2 (); see eorems 16 and 17 (on some detailed description of the solution space of the second-order linear differential equations and on their constructions, we refer the reader to [9, 10]). Finally, we suggest that the reader considers the fractal oscillations near an arbitrary real point = 0 instead of =0 and studies the fractal oscillations near = 0
Transcript of Research Article Fractal Oscillations of Chirp Functions and ...Fractal Oscillations of Chirp...
Hindawi Publishing CorporationInternational Journal of Differential EquationsVolume 2013 Article ID 857410 11 pageshttpdxdoiorg1011552013857410
Research ArticleFractal Oscillations of Chirp Functions and Applications toSecond-Order Linear Differential Equations
Mervan PašiT1 and Satoshi Tanaka2
1 Department of Applied Mathematics Faculty of Science University of Zagreb 10000 Zagreb Croatia2 Okayama University of Science Okayama 700-0005 Japan
Correspondence should be addressed to Satoshi Tanaka tanakaxmathousacjp
Received 7 December 2012 Accepted 8 January 2013
Academic Editor Norio Yoshida
Copyright copy 2013 M Pasic and S Tanaka This is an open access article distributed under the Creative Commons AttributionLicense which permits unrestricted use distribution and reproduction in any medium provided the original work is properlycited
We derive some simple sufficient conditions on the amplitude 119886(119909) the phase 120593(119909) and the instantaneous frequency 120596(119909) suchthat the so-called chirp function 119910(119909) = 119886(119909)119878(120593(119909)) is fractal oscillatory near a point 119909 = 119909
0 where 1205931015840(119909) = 120596(119909) and 119878 = 119878(119905)
is a periodic function on R It means that 119910(119909) oscillates near 119909 = 1199090 and its graph Γ(119910) is a fractal curve in R2 such that its box-
counting dimension equals a prescribed real number 119904 isin [1 2) and the 119904-dimensional upper and lower Minkowski contents of Γ(119910)are strictly positive and finite It numerically determines the order of concentration of oscillations of 119910(119909) near 119909 = 119909
0 Next we
give some applications of the main results to the fractal oscillations of solutions of linear differential equations which are generatedby the chirp functions taken as the fundamental system of all solutions
1 Introduction
The brilliant heuristic approach of Tricot [1] to the fractalcurves such as the graph of functions 119910(119909) = 119909120572 sin119909minus120573 and119910(119909) = 119909
120572 cos119909minus120573 gave the main motivation for studying thefractal properties near 119909 = 0 of graph of oscillatory solutionsof various types of differential equations linear Euler-typeequation 11991010158401015840 + 120582119909minus120590119910 = 0 (see [2]) general second-orderlinear equation 11991010158401015840 + 119891(119909)119910 = 0 (see [3]) where 119891(119909) satisfiesthe Hartman-Wintner asymptotic condition near 119909 = 0 half-linear equation (|1199101015840|119901minus21199101015840)1015840+119891(119909)|119910|119901minus2119910 = 0 (see [4]) linearself-adjoint equation (119901(119909)1199101015840)1015840 + 119902(119909)119910 = 0 (see [5]) and 119901-Laplace differential equations in an annular domain (see [6])
A function119910(119909) is said to be a chirp function if it possessesthe form 119910(119909) = 119886(119909)119878(120593(119909)) where 119886(119909) and 120593(119909) denoterespectively the amplitude and phase of 119910(119909) and 119878(119905) is aperiodic function on R In all previously mentioned papers[2ndash5] authors are dealing with the fractal oscillations ofsecond-order differential equations and are deriving somesufficient conditions on the coefficients of considered equa-tions such that all their solutions 119910(119909) together with the firstderivative 1199101015840(119909) admit asymptotic behaviour near 119909 = 0 It
is formally written in the form of a chirp function that is119910(119909) = 119886(119909) sin(120593(119909)) and 1199101015840(119909) = 119887(119909) cos(120593(119909)) near 119909 =0 According to it one can say that the asymptotic formulafor solutions of considered equations satisfies the chirp-likebehaviour near 119909 = 0 (on the asymptotic formula for solu-tions near 119909 = infin see [7 8]) Then in the dependenceof a prescribed real number 119904 isin [1 2) authors give someasymptotic conditions on 119886(119909) 119887(119909) and 120593(119909) such that allsolutions 119910(119909) are fractal oscillatory near 119909 = 0 with thefractal dimension 119904
In this paper independently of the asymptotic theory ofdifferential equations we firstly study the fractal oscillationsof a chirp function see Theorems 8 and 11 Second takingtwo linearly independent chirp functions 119910
1(119909) and 119910
2(119909)
we generate some new classes of fractal oscillatory lineardifferential equations which are not considered in [2ndash5] andhave the general solution in the form of 119910(119909) = 119888
11199101(119909) +
11988821199102(119909) seeTheorems 16 and 17 (on some detailed description
of the solution space of the second-order linear differentialequations and on their constructions we refer the readerto [9 10]) Finally we suggest that the reader considers thefractal oscillations near an arbitrary real point 119909 = 119909
0instead
of 119909 = 0 and studies the fractal oscillations near 119909 = 1199090
2 International Journal of Differential Equations
from the left side and from both sides seeTheorem 24 Manyexamples are considered to show the originality of obtainedresults
The chirp functions are also appearing in the time-frequency analysis see for instance [11ndash15] as well as inseveral applications of the time-frequency analysis see forinstance [16ndash20]
2 Statement of the Main Results
We study some local asymptotic behaviours of fractal typesfor the so-called chirp function as follows
119910 (119909) = 119886 (119909) 119878 (120593 (119909)) 119909 isin (0 1199050] (1)
where 1199050gt 0 the amplitude 119886 = 119886(119909) is a nontrivial function
on (0 1199050] the phase 120593 = 120593(119909) is singular at 119909 = 0 and 119878 = 119878(119905)
is a periodic function on R such as cos(119905) and sin(119905)The chirp function (1) is sometimes written in the follow-
ing form in which the so-called instantaneous frequency 120596(119909)appears instead of phase 120593(119909)
119910 (119909) = 119886 (119909) 119878 (int
1199050
119909
120596 (120585) 119889120585) 119909 isin (0 1199050] (2)
Let for some 1199050gt 0 the functions 119886 isin 119862((0 119905
0]) 120593 isin
1198622((0 1199050]) and 119878 isin 1198621(R) satisfy the following basic struc-
tural conditions
119886 (119909) gt 0 119886 (119909) is bounded on (0 1199050] (3)
lim119909rarr+0
120593 (119909) = infin 120593 (119909) gt 0 1205931015840(119909) lt 0 119909 isin (0 119905
0]
(4)
for some 120591 gt 0 |119878 (119905 + 120591)| = |119878 (119905)| forall119905 isin R
for some 1205910isin R 119878 (120591
0) = 0 119878 (119905) = 0
forall119905 isin (1205910 1205910+ 120591)
(5)
Definition 1 A function 119910 isin 119862((0 1199050]) is oscillatory near
119909 = 0 if there is a decreasing sequence 119909119899isin (0 119905
0] such that
119910(119909119899) = 0 for 119899 isin N and 119909
119899 0 as 119899 rarr infin see Figure 1
It is easy to prove the next proposition
Proposition 2 Let 119886 isin 119862((0 1199050]) satisfy 119886(119909) gt 0 on (0 119905
0] let
120593 isin 119862((0 1199050]) be strictly decreasing on (0 119905
0] and let 119878 isin 119862(R)
satisfy (5) Then the following two conditions are equivalent
(i) the chirp function 119910(119909) = 119886(119909)119878(120593(119909)) is bounded andoscillatory near 119909 = 0
(ii) the amplitude 119886(119909) is bounded and lim119909rarr+0
120593(119909) = infin
On the qualitative and oscillatory behaviours of solutionsof differential equations of several types we refer the readerto [7 8]
minus05
minus1
1
05
0201 025015005 03
Figure 1 119910 is continuous bounded and oscillatory near 119909 = 0
Example 3 The following threemain types of chirp functionssatisfy the conditions (3) (4) and (5)
(i) the so-called (120572 120573)-chirps
1199101(119909) = 119909
120572 cos (119909minus120573) 1199102(119909) = 119909
120572 sin (119909minus120573) (6)
where 120572 ge 0 and 120573 gt 0 the first studies on the (120572 120573)-chirps appeared in [1 13 14] from different point ofviews
(ii) the so-called logarithmic chirps
1199101(119909) = 119909
120574 cos (120588 log119909) 1199102(119909) = 119909
120574 sin (120588 log119909) (7)
where 120574 gt 0 and 120588 isin R this type of chirps appearsin definition of the Lamperti transform (see [11]) andin the fundamental system of solutions of the famousEuler equation 11991010158401015840 + 120582119909minus2119910 = 0 for 120574 = 12 and 120588 =radic120582 minus 14 (see [21])
(iii) the chirp function of exponential type
1199101(119909) = 119909119890
minus1(2119909) cos (1198901119909)
1199102(119909) = 119909119890
minus1(2119909) sin (1198901119909) (8)
Example 4 Let 1199101(119909) = 119886(119909) cos(120593(119909)) and 119910
2(119909) =
119886(119909) sin(120593(119909)) 119909 isin (0 1199050] where the amplitude 119886 and phase
120593 satisfy (3) and (4) respectively Then 119910(119909) = 11988811199101(119909) +
11988821199102(119909) 11988821+1198882
2gt 0 is also a chirp function which is bounded
and oscillatory near 119909 = 0 Indeed if 1198881= 0 or 119888
2= 0 then
119910(119909) = 11988821199102(119909) or 119910(119909) = 119888
11199101(119909) since 119878(119905) = cos 119905 and
119878(119905) = sin 119905 satisfy (5) both cases 119910(119909) are chirp functionsotherwise we have 119910(119909) = 119860119886(119909) sin(120593(119909) + 119861) where 119860 =(1198882
1+ 1198882
2)12 and 119861 = arctan(119888
11198882) obviously the amplitude
119860119886(119909) and the phase 120593(119909) + 119861 satisfy the required conditions(3) and (4) respectively
Next let
Γ (119910) = (1199051 1199052) isin R2 1199051isin (0 119905
0] 1199052= 119910 (119905
1)
(9)
denotes the graph of a function 119910 isin 119862((0 1199050]) In this paper
we show that the graph Γ(119910) of chirp function (1) is a fractal
International Journal of Differential Equations 3
curve in R2 in respect to the Minkowski-Bouligand dimen-sion dim
119872Γ(119910) = 119904 (box-counting dimension) and the 119904-
dimensional upper and lowerMinkowski contents119872lowast119904(Γ(119910))and119872119904
lowast(Γ(119910)) defined by
dim119872Γ (119910) = lim
120576rarr+0
(2 minus
log 1003816100381610038161003816Γ120576(119910)1003816100381610038161003816
log 120576)
119872lowast119904(Γ (119910)) = lim sup
120576rarr+0
(2120576)119904minus2 1003816100381610038161003816Γ120576(119910)1003816100381610038161003816
119872119904
lowast(Γ (119910)) = lim inf
120576rarr+0
(2120576)119904minus2 1003816100381610038161003816Γ120576(119910)1003816100381610038161003816
(10)
Here Γ120576(119910) denotes the 120576-neighbourhood of graph Γ(119910)
defined by
Γ120576(119910) = (119905
1 1199052) isin R2 119889 ((119905
1 1199052) Γ (119910)) le 120576 120576 gt 0
(11)
and 119889((1199051 1199052) Γ(119910)) denotes the distance from (119905
1 1199052) to Γ(119910)
and |Γ120576(119910)| denotes the Lebesgue measure of Γ
120576(119910) On the
box-counting dimension and the 119904-dimensional Minkowskicontent we refer the reader to [22ndash27]
The main fractal properties considered in the paper aregiven in the next definitions
Definition 5 For a given real number 119904 isin [1 2) and a function119910 isin 119862((0 119905
0]) which is bounded on (0 119905
0] and oscillatory
near 119909 = 0 it is said that 119910(119909) is fractal oscillatory near 119909 = 0with the fractal dimension 119904 if
dim119872Γ (119910) = 119904 0 lt 119872
119904
lowast(Γ (119910)) le 119872
lowast119904(Γ (119910)) lt infin
(12)
On the contrary if there is no any 119904 isin [1 2) such that Γ(119910)satisfies (12) then 119910(119909) is not fractal oscillatory near 119909 = 0
Fractal oscillations can be understood also as a refine-ment of rectifiable and nonrectifiable oscillations They arerecently studied in [3 4 21 28ndash30]
Example 6 The chirp functions
119910 (119909) = 119909 cos( 1119909
) 119910 (119909) = 119909 sin( 1119909
) (13)
are not fractal oscillatory near 119909 = 0 In fact119872lowast1(Γ(119910)) = infinand dim
119872Γ(119910) = 1 (see [2]) It also implies that119872lowast119904(Γ(119910)) =
0 for all 119904 isin (1 2) (see [1 23]) and thus the statement (12) isnot satisfied for any 119904 isin [1 2) Hence 119910
1and 1199102are not fractal
oscillatory near 119909 = 0
Example 7 Let 120588 gt 0 It is clear that the chirp functions
119910 (119909) = radic119909 cos (120588 log119909) 119910 (119909) = radic119909 sin (120588 log119909)(14)
are oscillatory near 119909 = 0 Moreover the length of Γ(119910)is finite (see [21]) and therefore we observe that 0 lt
1198721
lowast(Γ(119910)) le 119872
lowast1(Γ(119910)) lt infin and dim
119872Γ(119910) = 1 (see [22])
Thus such chirp functions are fractal oscillatory near 119909 = 0with the fractal dimension 1
In order to show that the chirp function (1) is fractaloscillatory near 119909 = 0 we need to impose on amplitude 119886(119909)the following additional structural condition
119886 isin 1198621((0 1199050]) 119886
1015840(119909) ge 0 for 119909 isin (0 119905
0] (15)
Now we are able to state the first main result of the paper
Theorem 8 Let the functions 119886(119909) 120593(119909) and 119878(119905) satisfy thestructural conditions (3) (4) (5) (15) and
12059310158401015840(119909) gt 0 on (0 119905
0] lim sup
119909rarr0
(
1
minus1205931015840(119909)
)
1015840
lt infin (16)
Let the amplitude 119886(119909) and the phase 120593(119909) satisfy the followingasymptotic conditions near 119909 = 0
lim inf119909rarr0
([minus1205931015840(119909)]
2minus119904
int
119909
0
119886 (120585) 119889120585) gt 0 (17)
lim sup119909rarr0
[119909119886 (119909)](119904minus1)(2minus119904)
int
1199050
119909
119886 (120585) [minus1205931015840(120585)] 119889120585 lt infin (18)
where 119904 isin (1 2) is a given real numberThen the chirp function(1) is fractal oscillatory near 119909 = 0with the fractal dimension 119904
Remark 9 Let 119904 isin (1 2) Assume that (15) and (17) holdThen
lim sup119909rarr0
(119909119886 (119909) [minus1205931015840(119909)]
2minus119904
) lt infin (19)
implies
lim sup119909rarr0
[
1
minus1205931015840(119909)
]
119904minus1
int
1199050
119909
119886 (120585) [minus1205931015840(120585)] 119889120585 lt infin (20)
and (18) Therefore condition (18) is better than (19) inTheorem 8
The proof of Remark 9 is presented in the AppendixTheorem 8 can be rewritten in the term of 120596(119909) instead of
120593(119909) as follows
Corollary 10 Let the functions 119886(119909)120596(119909) and 119878(119905) satisfy thestructural conditions (3) (5) (15) and
120596 isin 1198621((0 1199050]) 120596 (119909) gt 0 120596
1015840(119909) lt 0
119900119899 (0 1199050] 120596 notin 119871
1((0 1199050])
lim sup119909rarr0
(
1
120596 (119909)
)
1015840
lt infin
(21)
Let the amplitude 119886(119909) and the instantaneous frequency 120596(119909)satisfy the following asymptotic conditions near 119909 = 0
lim inf119909rarr0
([120596 (119909)]2minus119904int
119909
0
119886 (120585) 119889120585) gt 0
lim sup119909rarr0
[119909119886 (119909)](119904minus1)(2minus119904)
int
1199050
119909
119886 (120585) 120596 (120585) 119889120585 lt infin
(22)
where 119904 isin (1 2) is a given real numberThen the chirp function(2) is fractal oscillatory near 119909 = 0with the fractal dimension 119904
4 International Journal of Differential Equations
Now we state analogous result to Theorem 8 in the caseof 119904 = 1 which will be proved in Section 3
Theorem 11 Assume that the function 119878(119905) satisfies (5) 119886 isin119862([0 119905
0])cap1198621((0 1199050])120593 isin 1198622((0 119905
0]) and lim
119909rarr+0120593(119909) = infin
If 1198861015840 isin 1198711((0 1199050]) and1205931015840119886 isin 1198711((0 119905
0]) then the chirp function
(1) is fractal oscillatory near 119909 = 0with the fractal dimension 1
In respect to some existing results on the fractal dimen-sion of graph of chirp functions previous theorems are themost simple and general It is because in [31 32] authorsrequire some extra conditions on the chirp function 119910(119909)which are not easy to be satisfied in the application forinstance the rapid convex-concave properties of 119910(119909) as in[31] and a condition on the curvature of 119910(119909) as in [32]
According to previous theorems we can show the fractaloscillations of the so-called (120572 120573)-chirp as well as logarithmicchirp functions
Example 12 We consider the (120572 120573)-chirp 119910(119909) = 119909120572119878(119909minus120573)where 119878(119905) = cos 119905 or 119878(119905) = sin 119905 and 120573 gt 120572 ge 0 It is fractaloscillatory near 119909 = 0 with the fractal dimension 119904 = 2 minus(1 + 120572)(1 + 120573) In fact it is easy to see that 119886(119909) = 119909120572 119878(119905)and 120593(119909) = 119909minus120573 satisfy (3) (4) (5) (15) and (16) When 119904 =2 minus (1 + 120572)(1 + 120573) we see that
[minus1205931015840(119909)]
2minus119904
int
119909
0
119886 (120585) 119889120585 =
120573(1+120572)(1+120573)
1 + 120572
gt 0
[119909119886 (119909)](119904minus1)(2minus119904)
int
1199050
119909
119886 (120585) [minus1205931015840(120585)] 119889120585
=
120573
120573 minus 120572
(1 minus 119909120573minus120572119905120572minus120573
0) le
120573
120573 minus 120572
(23)
FromTheorem 8 it follows that 119910(119909) = 119909120572119878(119909minus120573) 120573 gt 120572 ge 0is fractal oscillatory near 119909 = 0 with the fractal dimension119904 = 2 minus (1 + 120572)(1 + 120573)
Example 13 We consider the (120572 120573)-chirp 119910(119909) = 119909120572119878(119909minus120573)again where 119878(119905) = cos 119905 or 119878(119905) = sin 119905 Now we assume that0 lt 120573 lt 120572 ApplyingTheorem 8 we easily see that it is fractaloscillatory near 119909 = 0 with the fractal dimension 1
Example 14 We consider the logarithmic chirp functions
1199101(119909) = 119909
120574 cos (120588 log119909) 1199102(119909) = 119909
120574 sin (120588 log119909) (24)
where 120574 gt 0 and 120588 gt 0 Put 119878(119905) = cos(minus119905) or 119878(119905) = sin(minus119905)119886(119909) = 119909
120574 and 120593(119909) = minus120588 log119909 Then we easily see that119878(119905) satisfies (5) and that 1198861015840(119909) = 120574119909
120574minus1isin 1198711((0 1199050]) and
119886(119909)1205931015840(119909) = minus120588119909
120574minus1isin 1198711((0 1199050]) Theorem 11 implies that
1199101(119909) and 119910
2(119909) are fractal oscillatory near 119909 = 0 with the
fractal dimension 1
Question 1 Is it possible to applyTheorem 8 on the exponen-tial chirp given in Example 3(iii)
Example 15 Let 1199101(119909) = 119886(119909) cos(120593(119909)) and 119910
2(119909) =
119886(119909) sin(120593(119909)) 119909 isin (0 1199050] where the amplitude 119886(119909) and
the phase 120593(119909) satisfy all assumptions of Theorem 8 for anarbitrary given 119904 isin (1 2) Then the chirp function 119910(119909) =11988811199101(119909) + 119888
21199102(119909) 11988821+ 1198882
2gt 0 is also fractal oscillatory
near 119909 = 0 with the fractal dimension 119904 Indeed similarlyas in Example 4 119910(119909) can be rewritten in the form 119910(119909) =
1198891sin(120593(119909) + 119889
2) where 119889
1= 0 and 119889
2isin R It is clear that the
function 119878(119905) = 1198891sin(119905 + 119889
2) satisfies the required condition
(5) and henceTheorem 8 proves that119910(119909) is fractal oscillatorynear 119909 = 0 with the fractal dimension 119904
Next we pay attention to the fractal oscillations ofsolutions of linear differential equations generated by thesystem of functions as follows
119910 (119909) = 11988811199101(119909) + 119888
21199102(119909) 119888
2
1+ 1198882
2gt 0
where
1199101 (119909) = 119886 (119909) cos (120593 (119909)) 1199102 (119909) = 119886 (119909) sin (120593 (119909))
119909 isin (0 1199050]
(25)
It is not difficult to check that (25) is the fundamental systemof all solutions of the following linear differential equation
11991010158401015840+ (minus2119877
119886minus 1198771205931015840) 1199101015840+ (minus119877
1015840
119886+ 1198772
119886+ 1198771198861198771205931015840 + (120593
1015840)
2
) 119910 = 0
119909 isin (0 1199050]
(26)
where 119877119891= 119877119891(119909) = 119891
1015840(119909)119891(119909) for some 1198621-function 119891 =
119891(119909)
Theorem 16 Let the functions 119886 120593 isin 1198622((0 1199050]) satisfy struc-
tural conditions (3) (4) and (15) as well as the conditions (16)(17) and (18) in respect to a given real number 119904 isin (1 2)Then every nontrivial solution 119910 isin 1198622((0 119905
0]) of (26) is fractal
oscillatory near 119909 = 0 with the fractal dimension 119904
With the help ofTheorem 11 we can state analogous resultto Theorem 16 in the case of 119904 = 1
Theorem 17 Assume that 119886 isin 119862([0 1199050])cap1198622((0 1199050]) 119886(119909) = 0
for 119909 isin (0 1199050] 120593 isin 119862
2((0 1199050]) lim
119909rarr+0120593(119909) = infin 1198861015840 isin
1198711((0 1199050]) and 1205931015840119886 isin 1198711((0 119905
0]) Then every nontrivial solu-
tion 119910 isin 1198622((0 1199050]) of (26) is fractal oscillatory near 119909 = 0
with the fractal dimension 1
The previous two theorems will be proved in Section 4Assumptions on the coefficients of (26) in general are dif-ferent to those considered in [2ndash5]
When 1198861015840119886 = minus(12)120593101584010158401205931015840 then (26) becomes the un-damped equation
11991010158401015840+ (
1
2
119878der (1205931015840) + (120593
1015840)
2
)119910 = 0 119909 isin (0 1199050] (27)
International Journal of Differential Equations 5
where 119878der(119891) denotes the Schwarzian derivative of119891 definedby
119878der (119891) =11989110158401015840
119891
minus
3
2
(
1198911015840
119891
)
2
(28)
Hence from Theorem 16 we obtain the following conse-quence
Corollary 18 Let the functions 119886 isin 1198622((0 1199050]) and 120593 isin
1198623((0 1199050]) satisfy structural conditions (3) (4) and (15) as
well as the conditions (16) (17) and (18) in respect to a givenreal number 119904 isin (1 2) Let 1198861015840119886 = minus(12)120593101584010158401205931015840 Then everynontrivial solution 119910 isin 1198622((0 119905
0]) of (27) is fractal oscillatory
near 119909 = 0 with the fractal dimension 119904
As a consequence of Theorem 16 and Corollary 18 wederive the following examples for linear differential equationsof second order having all the solutions to be fractal oscilla-tory near 119909 = 0
Example 19 The so-called damped chirp equation
11991010158401015840+
120573 minus 2120572 + 1
119909
1199101015840+ (
1205732
1199092120573+2
minus
120572 (120573 minus 120572)
1199092
)119910 = 0
119909 isin (0 1199050]
(29)
is fractal oscillatory near 119909 = 0 with the fractal dimension2 minus (1 + 120572)(1 + 120573) where 120573 gt 120572 ge 0 When 119886(119909) = 119909120572 and120593(119909) = 119909
minus120573 (26) becomes (29) It is easy to see that (3) (4)(15) and (16) are satisfied In the same as in Example 12 wesee that (17) and (18) hold for 119904 = 2 minus (1 + 120572)(1 + 120573) HenceTheorem 16 proves that every nontrivial solution of (29) isfractal oscillatory near 119909 = 0 with the fractal dimension 2 minus(1 + 120572)(1 + 120573)
Nowwe assume that 0 lt 120573 lt 120572ThenTheorem 17 impliesthat (29) is fractal oscillatory near 119909 = 0 with the fractaldimension 1
Example 20 The following equation
11991010158401015840+
1 minus 2120574
119909
1199101015840+
1205742+ 1205882
1199092119910 = 0 119909 isin (0 119905
0] (30)
is fractal oscillatory near 119909 = 0 with the fractal dimension1 where 120574 gt 0 and 120588 gt 0 In the case where 119886(119909) = 119909
120574
and 120593(119909) = minus120588 log119909 (26) becomes (30) We see that 119886 isin119862([0 119905
0]) cap 119862
2((0 1199050]) 120593 isin 1198622((0 119905
0]) lim
119909rarr+0120593(119909) = infin
1198861015840isin 1198711((0 1199050]) and 1205931015840119886 isin 1198711((0 119905
0]) Therefore Theorem 17
implies that every nontrivial solution of (30) is fractaloscillatory near 119909 = 0 with the fractal dimension 1
Question 2 What can we say about the application of Theo-rem 16 on the case of 120593(119909) given in Example 3(iii)
At the end of this section we suggest that the readerstudies some invariant properties of fractal oscillations of thechirp function (1) in respect to the translation and reflexionAnalogously to Definitions 1 and 5 one can define the fractaloscillations near an arbitrary real point 119909 = 119909
0as follows
minus05
minus025 minus02 minus015 minus01
minus01
minus03
05
1
minus005
Figure 2 119910 is oscillatory near 119909 = 0 from the left side
Definition 21 Let 1199090isin R and 120575 gt 0 A function 119910 isin
119862((1199090 1199090+ 120575]) is oscillatory near 119909 = 119909
0 if there is a
decreasing sequence 119887119899isin (1199090 1199090+ 120575] such that 119910(119887
119899) = 0
for 119899 isin N and 119887119899 1199090as 119899 rarr infin Moreover if the graph
Γ(119910) satisfies the condition (12) for some 119904 isin [1 2) then 119910(119909)is said to be fractal oscillatory near 119909 = 119909
0with the fractal
dimension 119904
Definition 22 Let 1199090isin R and 120575 gt 0 It is said that a function
119910 isin 119862([1199090minus 120575 119909
0)) is fractal oscillatory near 119909 = 0 from
the left side with the fractal dimension 119904 isin [1 2) if there is anincreasing sequence 119886
119899isin [1199090minus 120575 119909
0) such that 119910(119886
119899) = 0 for
119899 isin N 119886119899 1199090as 119899 rarr infin and the graph Γ(119910) satisfies the
condition (12) see Figure 2
Definition 23 Let 1199090isin R and let 120575 gt 0 It is said that a
function 119910 isin 119862([1199090minus120575 1199090) cup (1199090 1199090+120575]) is two-sided fractal
oscillatory near 119909 = 0 with the fractal dimension 119904 isin [1 2)if there is an increasing sequence 119886
119899isin [1199090minus 120575 119909
0) and a
decreasing sequence 119887119899isin (1199090 1199090+ 120575] such that 119910(119886
119899) =
119910(119887119899) = 0 for 119899 isin N 119886
119899 1199090and 119887119899 1199090as 119899 rarr infin
and the graph Γ(119910) satisfies the condition (12) see Figure 3
With the help ofTheorem 8 we state the following result
Theorem 24 Let 1199090isin R and let 120575 gt 0 Let the functions
119886 120596 isin 1198621((0 120575]) satisfy structural conditions (3) (4) (5) (15)
and (16) as well as conditions (17) and (18) in respect to a givenreal number 119904 isin (1 2) Then one has
(i) the chirp function 119910(119909) = 119886(119909 minus 1199090)119878(120593(119909 minus 119909
0)) 119909 isin
(1199090 1199090+ 120575] is fractal oscillatory near 119909 = 119909
0with the
fractal dimension 119904
(ii) the chirp function 119910(119909) = 119886(1199090minus 119909)119878(120593(119909
0minus 119909)) 119909 isin
[1199090minus 120575 119909
0) is fractal oscillatory near 119909 = 119909
0from the
left side with the fractal dimension 119904
(iii) the chirp function 119910(119909) = 119886(|119909minus1199090|)119878(120593(|119909minus119909
0|)) 119909 isin
[1199090minus120575 1199090)cup(1199090 1199090+120575] is two-sided fractal oscillatory
near 119909 = 1199090with the fractal dimension 119904
6 International Journal of Differential Equations
minus03 minus02 minus01
minus1
01 02 03
minus05
1
05
Figure 3 119910 is two-sided oscillatory near 119909 = 0
3 Proof for the Fractal Oscillations ofChirp Functions
In this section we give the proofs of the main results dealingwith the fractal oscillations of chirp functions
By Definition 5 it follows that if for a prescribed realnumber 119904 isin (1 2) there are two positive constants 119888
1and 1198882
such that
11988811205762minus119904le1003816100381610038161003816Γ120576(119910)1003816100381610038161003816le 11988821205762minus119904 120576 isin (0 120576
0) (31)
for some 1205760gt 0 then a function 119910 isin 119862((0 119905
0]) is fractal
oscillatory near 119909 = 0 with the fractal dimension 119904 Thefollowing lemma plays the essential role in the proof of (31)
Lemma 25 Let 119910 isin 119862((0 1199050]) be a bounded function on
(0 1199050] and let 119886
119899isin (0 119905
0] be a decreasing sequence of con-
secutive zeros of 119910(119909) such that 119886119899rarr 0 Let 120576
0gt 0 and let
119896 = 119896(120576) be an index function satisfying1003816100381610038161003816119886119899minus 119886119899+1
1003816100381610038161003816le 120576 forall119899 ge 119896 (120576) 120576 isin (0 120576
0) (32)
Then1003816100381610038161003816Γ120576(119910)1003816100381610038161003816ge sum
119899ge119896(120576)
max119909isin[119886119899+1119886119899]
1003816100381610038161003816119910 (119909)
1003816100381610038161003816(119886119899minus 119886119899+1)
forall120576 isin (0 1205760)
(33)
Proof It is exactly the same as [3 pp 2350]
Let us remark that for some 1205760gt 0 we say that a function
119896 = 119896(120576) is an index function on (0 1205760) if 119896 (0 120576
0) rarr N and
lim120576rarr0
119896(120576) = infin
Lemma 26 Let 119910 isin 119862[119886 119887] Then
1003816100381610038161003816Γ120576(119910)1003816100381610038161003816le 4120587120576length (119910) + 41205871205762 for every 120576 gt 0 (34)
Proof Let 120576 gt 0 Set 1199091= 119886 and set
119909119894+1= max 119909 isin [119909
119894 119887] 119889 ((119905 119910 (119905)) (119909
119894 119910 (119909119894)))
le 120576 for 119905 isin [119909119894 119909]
(35)
for 119894 = 1 2 Then there exists 119899 ge 2 such that 119909119899= 119887 Set
119873 = max119899 isin N 119909119899lt 119887 We see that119873 ge 1
119886 = 1199091lt 1199092lt sdot sdot sdot lt 119909
119894lt 119909119894+1lt sdot sdot sdot lt 119909
119873lt 119909119873+1
= 119887 (36)
and if119873 ge 2 then
119889 ((119909119894 119910 (119909119894)) (119909
119894+1 119910 (119909119894+1))) = 120576 119894 = 1 2 119873 minus 1
(37)
We will show that
Γ120576(119910) sub
119873
⋃
119894=1
1198612120576(119909119894 119910 (119909119894)) (38)
where
1198612120576(1199051 1199052) = (120591
1 1205912) isin R2 119889 ((119905
1 1199052) (1205911 1205912)) le 2120576
(39)
Let (1205911 1205912) isin Γ
120576(119910) Then there exists 120585 isin [119886 119887] such that
119889((1205911 1205912) (120585 119910(120585))) le 120576 Because of the definition of 119909
119894+1 we
find that 120585 isin [119909119894 119909119894+1] for some 119894 isin 1 2 119873 so that
119889 ((120585 119910 (120585)) (119909119894 119910 (119909119894))) le 120576 (40)
Hence it follows that
119889 ((1205911 1205912) (119909119894 119910 (119909119894))) le 119889 ((120591
1 1205912) (120585 119910 (120585)))
+ 119889 ((120585 119910 (120585)) (119909119894 119910 (119909119894))) le 2120576
(41)
which means that (1205911 1205912) isin 119861
2120576(119909119894 119910(119909119894)) Therefore we
obtain (38) By (38) we conclude that
1003816100381610038161003816Γ120576(119910)1003816100381610038161003816le
119873
sum
119894=1
10038161003816100381610038161198612120576(119909119894 119910 (119909119894))1003816100381610038161003816= 4119873120587120576
2 (42)
When119873 = 1 from (42) it follows that1003816100381610038161003816Γ120576(119910)1003816100381610038161003816le 4120587120576
2le 4120587120576length (119910) + 41205871205762 (43)
Now we assume that119873 ge 2 We observe that
length (119910) =119873
sum
119894=1
length (119910|[119909119894 119909119894+1]
)
ge
119873
sum
119894=1
119889 ((119909119894 119910 (119909119894)) (119909
119894+1 119910 (119909119894+1)))
ge
119873minus1
sum
119894=1
119889 ((119909119894 119910 (119909119894)) (119909
119894+1 119910 (119909119894+1)))
= (119873 minus 1) 120576
(44)
that is
119873120576 le length (119910) + 120576 (45)
Combining (42) with (45) we obtain1003816100381610038161003816Γ120576(119910)1003816100381610038161003816le 4120587120576length (119910) + 41205871205762 (46)
International Journal of Differential Equations 7
Lemma 27 Let 119910 isin 119862(0 1199050] be bounded on (0 119905
0] and let
120576 gt 0 Then there exists 119862 gt 0 such that for every 120576 gt 0 and119905 isin (0 119905
0]
1003816100381610038161003816Γ120576(119910)1003816100381610038161003816le 119862[119905 sup
119909isin(0119905]
1003816100381610038161003816119910 (119909)
1003816100381610038161003816+ 120576length (119910|
[1199051199050]) + 120576 + 120576
2]
(47)
Proof Let 120576 gt 0 and let 119905 isin (0 1199050] From the geometry point
of view it is clear that1003816100381610038161003816Γ120576(119910|(0119905])1003816100381610038161003816sub 119877 (48)
where 119877 is a rectangle in R2 defined by
119877 = (1199051 1199052) 1199051isin [minus120576 119905 + 120576]
1199052isin [minus120576 minus sup
119909isin(0119905]
1003816100381610038161003816119910 (119909)
1003816100381610038161003816 120576 + sup119909isin(0119905]
1003816100381610038161003816119910 (119909)
1003816100381610038161003816]
(49)
Hence we have
1003816100381610038161003816Γ120576(119910(0119905])1003816100381610038161003816le (2120576 + 119905) (2120576 + 2 sup
119909isin(0119905]
1003816100381610038161003816119910 (119909)
1003816100381610038161003816)
= 41205762+ 2(119905 + 2 sup
119909isin(0119905]
1003816100381610038161003816119910 (119909)
1003816100381610038161003816) 120576
+ 2119905 sup119909isin(0119905]
1003816100381610038161003816119910 (119909)
1003816100381610038161003816
le 41205762+ 1198621120576 + 2119905 sup
119909isin(0119905]
1003816100381610038161003816119910 (119909)
1003816100381610038161003816
(50)
where 1198621= 2(1199050+ 2sup
119909isin(01199050]|119910(119909)|) Lemma 26 implies that
10038161003816100381610038161003816Γ120576(119910|[1199051199050])
10038161003816100381610038161003816le 4120587120576length (119910|
[1199051199050]) + 4120587120576
2 (51)
Consequently we see that1003816100381610038161003816Γ120576(119910)1003816100381610038161003816le1003816100381610038161003816Γ120576(119910|(0119905])1003816100381610038161003816+
10038161003816100381610038161003816Γ120576(119910|[1199051199050])
10038161003816100381610038161003816
le 41205762+ 1198621120576 + 2119905 sup
119909isin(0119905]
1003816100381610038161003816119910 (119909)
1003816100381610038161003816
+ 4120587120576length (119910|[1199051199050]) + 4120587120576
2
le 119862[119905 sup119909isin(0119905]
1003816100381610038161003816119910 (119909)
1003816100381610038161003816+ 120576length (119910|
[1199051199050]) + 120576 + 120576
2]
(52)
where 119862 = max1198621 4(120587 + 1)
Lemma 28 Let 119910 isin 1198621(0 1199050] be bounded on (0 119905
0] Assume
that
int
1199050
0
100381610038161003816100381610038161199101015840(119909)
10038161003816100381610038161003816119889119909 = infin (53)
lim sup119909rarr0
[119909 sup120585isin(0119909]
1003816100381610038161003816119910 (120585)
1003816100381610038161003816]
(119904minus1)(2minus119904)
int
1199050
119909
100381610038161003816100381610038161199101015840(120585)
10038161003816100381610038161003816119889120585 lt infin (54)
for some 119904 isin (1 2) Then there exists 1198882gt 0 such that
1003816100381610038161003816Γ120576(119910)1003816100381610038161003816le 11988821205762minus119904 120576 isin (0 1) (55)
Proof Let 120576 isin (0 1) and let 119904 isin (1 2) be satisfy (54) By (53)there exists 119909
1isin (0 119905
0) which depends on 120576 and 119904 such that
int
1199050
1199091
100381610038161003816100381610038161199101015840(119905)
10038161003816100381610038161003816119889119905 = 120576
1minus119904 (56)
Therefore we have
length (119910|[1199091 1199050]
) = int
1199050
1199091
radic1 +10038161003816100381610038161199101015840(119909)1003816100381610038161003816
2119889119909
le int
1199050
1199091
(1 +
100381610038161003816100381610038161199101015840(119909)
10038161003816100381610038161003816) 119889119909
= 1199050minus 1199091+ 1205761minus119904
le 1199050+ 1205761minus119904
(57)
By (54) there exists 1198621gt 0 such that
[119909 sup120585isin(0119909]
1003816100381610038161003816119910 (120585)
1003816100381610038161003816]
(119904minus1)(2minus119904)
int
1199050
119909
100381610038161003816100381610038161199101015840(120585)
10038161003816100381610038161003816119889120585 le 119862
1 119909 isin (0 119905
0]
(58)
which implies that
[
[
1199091sup120585isin(01199091]
1003816100381610038161003816119910 (120585)
1003816100381610038161003816]
]
(119904minus1)(2minus119904)
int
1199050
1199091
100381610038161003816100381610038161199101015840(120585)
10038161003816100381610038161003816119889120585 le 119862
1 (59)
that is
1199091sup120585isin(01199091]
1003816100381610038161003816119910 (120585)
1003816100381610038161003816le 11986221205762minus119904 (60)
where 1198622= 119862(2minus119904)(119904minus1)
1 From Lemma 27 (57) and (60) it
follows that1003816100381610038161003816Γ120576(119910)1003816100381610038161003816le 119862 [119862
21205762minus119904+ 120576 (1199050+ 1205761minus119904) + 120576 + 120576
2]
= 119862 [(1198622+ 1) + (119905
0+ 1) 120576
119904minus1+ 120576119904] 1205762minus119904
le 119862 (1198622+ 1199050+ 3) 120576
2minus119904 120576 isin (0 1)
(61)
In order to showTheorems 8 and 11 we need the followingtwo geometric lemmas
Lemma 29 (see [1]) If Γ sube R2 is a simple curve (ie itsparameterization is a bijection) and length(Γ) lt infin then
length (Γ) = lim120576rarr0
1003816100381610038161003816Γ120576
1003816100381610038161003816
2120576
(62)
where Γ120576denotes the 120576-neighborhood of the graph Γ
8 International Journal of Differential Equations
Now we are able to proveTheorem 8
Proof of Theorem 8 Let 119904 isin (1 2) and let 119910(119909) be a chirpfunction given by (1) We note here that it is enough to showthat 119910(119909) satisfies (31)
At the first let 119909119899be a sequence defined by 119909
119899= 120593minus1(1205910+
119899120591) for all sufficiently large 119899 isin N From (4) it follows that120593minus1(119905) is decreasing Hence 119909
119899is decreasing as well as 119909
119899rarr
0 as 119899 rarr infin because of lim119909rarr+0
120593(119909) = infin (see (4))We notethat 119910(119909
119899) = 0 and 119910(119909) = 0 on (119909
119899+1 119909119899) for all sufficiently
large 119899 isin N Also minus11205931015840(119909) is an increasing function becauseof (16) The mean value theorem shows that
120591
minus1205931015840(119909119899+1)
le 119909119899minus 119909119899+1le
120591
minus1205931015840(119909119899)
(63)
Now let 1205760isin (0 1) Let 119896(120576) be the smallest natural number
satisfying120591
minus1205931015840(119909119896(120576))
le 120576 forall120576 isin (0 1205760) (64)
Such 119896(120576) exists for every 120576 isin (0 1205760) since 119909
119899rarr 0 as 119899 rarr
infin and lim119909rarr+0
1205931015840(119909) = minusinfin (this equality is true because
1205931015840notin 1198711(0 1199050) since lim
119909rarr+0120593(119909) = infin) Moreover since 119909
119899
is decreasing and minus11205931015840(119909) is increasing we obtain
minus1205931015840(119909119899) ge 120591120576
minus1forall119899 ge 119896 (120576) (65)
Combining (63) and (65) it is easy to deduce that suchdefined 119896(120576) satisfies condition (32)
By (16) there exists 119871 gt 0 such that (1(minus1205931015840(119909)))1015840 le 119871 for119909 isin (0 119905
0] which means that
minus
minus12059310158401015840(119909)
minus1205931015840(119909)
le minus1198711205931015840(119909) 119909 isin (0 119905
0] (66)
Integrating (66) on [120593minus1(119905 + 2120591) 120593minus1(119905)] we have
logminus1205931015840(120593minus1(119905 + 2120591))
minus1205931015840(120593minus1(119905))
le 2119871120591 119905 isin [120593 (1199050) infin) (67)
that is
minus1205931015840(120593minus1(119905)) ge minus119890
minus21198711205911205931015840(120593minus1(119905 + 120591)) 119905 isin [120593 (119905
0) infin)
(68)
By the definition of 119896(120576) and (64) we see that120591
minus1205931015840(119909119896(120576)minus1
)
gt 120576 (69)
Hence from (68) it follows that
120591120576minus1gt minus1205931015840(119909119896(120576)minus1
)
= minus1205931015840(120593minus1(1205910+ (119896 (120576) minus 1) 120591))
ge minus119890minus2119871120591
1205931015840(120593minus1(1205910+ (119896 (120576) minus 1) 120591 + 2120591))
= minus119890minus2119871120591
1205931015840(120593minus1(1205910+ (119896 (120576) + 1) 120591))
= minus119890minus2119871120591
1205931015840(119909119896(120576)+1
)
(70)
which implies that
minus1205931015840(119909119896(120576)+1
) le 119872120576minus1 120576 gt 0 (71)
where119872 = 1205911198902119871120591
Next it is clear that the assumption (5) ensures a realnumber119872
0gt 0 such that
1198720= max119905isinR
|119878 (119905)| = max119905isin[0120591]
|119878 (119905)| (72)
Hence from (15) we have
max119909isin[119909119899+1 119909119899]
1003816100381610038161003816119910 (119909)
1003816100381610038161003816= max119909isin[119909119899+1119909119899]
(119886 (119909)1003816100381610038161003816119878 (120593 (119909))
1003816100381610038161003816)
ge 119886 (119909119899+1) max119909isin[119909119899+1119909119899]
1003816100381610038161003816119878 (120593 (119909))
1003816100381610038161003816
= 1198720119886 (119909119899+1)
(73)
Now from (63) and (73) it follows that
max119909isin[119909119899+1 119909119899]
1003816100381610038161003816119910 (119909)
1003816100381610038161003816(119909119899minus 119909119899+1) ge 119872
0119886 (119909119899+1) (119909119899minus 119909119899+1)
ge 1205911198720
119886 (119909119899+1)
minus1205931015840(119909119899+1)
= 1205911198720
119886 (120593minus1(1205910+(119899+1) 120591))
minus1205931015840(120593minus1(1205910+(119899+1) 120591))
(74)
Since 120593minus1(119905) is decreasing as well as minus11205931015840(119909) is increasingand 119886(119909) is nondecreasing and positive we conclude that thefunction
119886 (120593minus1(119905))
minus1205931015840(120593minus1(119905))
is nonincreasing (75)
Hence we observe that
sum
119899ge119896(120576)
max119909isin[119909119899+1119909119899]
1003816100381610038161003816119910 (119909)
1003816100381610038161003816(119909119899minus 119909119899+1)
ge 1205911198720sum
119899ge119896(120576)
119886 (120593minus1(1205910+ (119899 + 1) 120591))
minus1205931015840(120593minus1(1205910+ (119899 + 1) 120591))
ge 1205911198720int
infin
119896(120576)
119886 (120593minus1(1205910+ (119905 + 1) 120591))
minus1205931015840(120593minus1(1205910+ (119905 + 1) 120591))
119889119905
= 1198720int
120593minus1(1205910+(119896(120576)+1)120591)
0
119886 (120585) 119889120585
= 1198720int
119909119896(120576)+1
0
119886 (120585) 119889120585
(76)
Next from assumption (17) we get some1198721gt 0 such that
int
119909
0
119886 (120585) 119889120585 ge 1198721[minus1205931015840(119909)]
minus(2minus119904)
119909 isin (0 1199050] (77)
International Journal of Differential Equations 9
Now from (76) (77) and (71) we obtain
sum
119899ge119896(120576)
max119909isin[119909119899+1119909119899]
1003816100381610038161003816119910 (119909)
1003816100381610038161003816(119909119899minus 119909119899+1)
ge 11987201198721[minus1205931015840(119909119896(120576)+1
)]
minus(2minus119904)
ge 11987201198721(119872120576minus1)
minus(2minus119904)
= 11987201198721119872minus(2minus119904)
1205762minus119904
(78)
Thus using Lemma 25 we have
1003816100381610038161003816Γ120576(119910)1003816100381610038161003816ge 11988811205762minus119904 120576 isin (0 120576
0) (79)
for some 1198881gt 0
Next we prove that 119910(119909) satisfies (53) Assume to thecontrary that
int
1199050
0
100381610038161003816100381610038161199101015840(119909)
10038161003816100381610038161003816119889119909 lt infin (80)
Then we have
length (119910) = int1199050
0
radic1 +10038161003816100381610038161199101015840(119909)1003816100381610038161003816
2119889119909
le int
1199050
0
(1 +
100381610038161003816100381610038161199101015840(119909)
10038161003816100381610038161003816) 119889119909 lt infin
(81)
From Lemma 29 it follows that
length (119910) = lim120576rarr+0
1003816100381610038161003816Γ120576(119910)1003816100381610038161003816
2120576
(82)
On the other hand by (79) we see that
lim120576rarr+0
1003816100381610038161003816Γ120576(119910)1003816100381610038161003816
2120576
= infin (83)
This is a contaradiction and hence 119910(119909) satisfies (53)Finally we show that 119910(119909) also satisfies inequality (54)
Because of (5) we obtain a constant 119888 gt 0 such that
max119905isinR
|119878 (119905)| = max119905isin[0120591]
|119878 (119905)| le 119888
max119905isinR
100381610038161003816100381610038161198781015840(119905)
10038161003816100381610038161003816= max119905isin[0120591]
100381610038161003816100381610038161198781015840(119905)
10038161003816100381610038161003816le 119888
(84)
By (15) we find that
sup119905isin(0119909]
1003816100381610038161003816119910 (119905)
1003816100381610038161003816= sup119905isin(0119909]
|119886 (119905)|1003816100381610038161003816119878 (120593 (119905))
1003816100381610038161003816le 119888119886 (119909) (85)
Hence from (1) and (15) it follows that100381610038161003816100381610038161199101015840(119909)
10038161003816100381610038161003816le 119888 (119886
1015840(119909) + 119886 (119909) [minus120593
1015840(119909)]) (86)
and therefore
int
1199050
119909
100381610038161003816100381610038161199101015840(120585)
10038161003816100381610038161003816119889120585 le 119888119886 (119905
0) + 119888 int
1199050
119909
119886 (120585) [minus1205931015840(120585)] 119889120585 (87)
Using (85) and (87) we see that
[119909 sup120585isin(0119909]
1003816100381610038161003816119910 (120585)
1003816100381610038161003816]
(119904minus1)(2minus119904)
int
1199050
119909
100381610038161003816100381610038161199101015840(120585)
10038161003816100381610038161003816119889120585
le 119888119886 (1199050) [119888119909119886 (119909)]
(119904minus1)(2minus119904)
+ 119888[119888119909119886 (119909)](119904minus1)(2minus119904)
int
1199050
119909
119886 (120585) [minus1205931015840(120585)] 119889120585
le 119888119886 (1199050) [1198881199050119886 (1199050)](119904minus1)(2minus119904)
+ 1198881(2minus119904)
[119909119886 (119909)](119904minus1)(2minus119904)
int
1199050
119909
119886 (120585) [minus1205931015840(120585)] 119889120585
(88)
By (18) we conclude that (54) holdsBy Lemma 28 there exists 119888
2gt 0 such that
1003816100381610038161003816Γ120576(119910)1003816100381610038161003816le 11988821205762minus119904 120576 isin (0 120576
0) (89)
Thus we have proved that the chirp function119910(119909) given by (1)satisfies the desired inequality (31) This completes the proofof Theorem 8
Proof of Theorem 11 Let 119910(119909) be the chirp function (1) It iseasy to see that 119910(119909) is oscillatory near 119909 = 0 By (5) thereexists119872 gt 0 such that
|119878 (119905)| le 119872
100381610038161003816100381610038161198781015840(119905)
10038161003816100381610038161003816le 119872 119905 isin R (90)
Hence we observe that100381610038161003816100381610038161199101015840(119909)
10038161003816100381610038161003816le
100381610038161003816100381610038161198861015840(119909) 119878 (120593 (119909))
10038161003816100381610038161003816+
10038161003816100381610038161003816119886 (119909) 119878 (120593 (119909)) 120593
1015840(119909)
10038161003816100381610038161003816
le 119872
100381610038161003816100381610038161198861015840(119909)
10038161003816100381610038161003816+ 119872
10038161003816100381610038161003816119886 (119909) 120593
1015840(119909)
10038161003816100381610038161003816
(91)
for 119909 isin (0 1199050] From (81) it follows that length(119910) lt infin
Lemma 29 shows that
length (119910) = lim120576rarr0
1003816100381610038161003816Γ120576(119910)1003816100381610038161003816
2120576
(92)
which implies that 120576length(Γ) le |Γ120576(119910)| le 4120576length(Γ) 120576 isin
(0 1205760) for some 120576
0gt 0 Therefore 119910(119909) is fractal oscillatory
near 119909 = 0 with the fractal dimension 1
4 Proof for the Fractal Oscillations of (26)In this section we give the proofs for the fractal oscillations ofthe linear second-order differential equation (26) consideredas an application of the main results on the fractal oscillationof chirp functions
Before we present the proofs of Theorems 16 and 17 wemake the following observation Since
1199101(119909) = 119886 (119909) cos (120593 (119909)) 119910
2(119909) = 119886 (119909) sin (120593 (119909))
(93)
are solutions of (26) we see that 119910(119909) = 11988811199101(119909) + 119888
21199102(119909)
is a fundamental system of all solutions of (26) Assume that1198882
1+ 1198882
2gt 0 and set 119878(119905) = 119888
1sin 119905 + 119888
2cos 119905 Then 119878(119905) clearly
satisfies (5)
10 International Journal of Differential Equations
Proof of Theorem 16 Applying Theorem 8 on 119910(119909) =
11988811199101(119909) + 119888
21199102(119909) we conclude that 119910(119909) is fractal oscillatory
with the fractal dimension 119904
Proof of Theorem 17 Theorem 11 implies that 119910(119909) =
11988811199101(119909) + 119888
21199102(119909) is fractal oscillatory near 119909 = 0 with the
fractal dimension 1
Appendix
Proof of Remark 9 By (17) there exists 1198881gt 0 such that
1198881le [minus120593
1015840(119909)]
2minus119904
int
119909
0
119886 (120585) 119889120585 119909 isin (0 1199050] (A1)
From (15) it follows that
1198881le [minus120593
1015840(119909)]
2minus119904
119886 (119909) int
119909
0
119889120585 = 119909119886 (119909) [minus1205931015840(119909)]
2minus119904
119909 isin (0 1199050]
(A2)
which implies that1
minus1205931015840(119909)
le 119888minus1(2minus119904)
11199091(2minus119904)
[119886 (119909)]1(2minus119904)
119909 isin (0 1199050] (A3)
By (19) there exists 1198882gt 0 such that
119909119886 (119909) [minus1205931015840(119909)]
2minus119904
le 1198882 119909 isin (0 119905
0] (A4)
and hence
minus1205931015840(119909) le 119888
1(2minus119904)
2119909minus1(2minus119904)
[119886 (119909)]minus1(2minus119904)
119909 isin (0 1199050]
(A5)Therefore we see that
[
1
minus1205931015840(119909)
]
119904minus1
int
1199050
119909
119886 (120585) [minus1205931015840(120585)] 119889120585
le 119888minus(119904minus1)(2minus119904)
1119909(119904minus1)(2minus119904)
[119886 (119909)](119904minus1)(2minus119904)
1198881(2minus119904)
2
times int
1199050
119909
120585minus1(2minus119904)
[119886 (120585)]minus(119904minus1)(2minus119904)
119889120585
le 1198883119909(119904minus1)(2minus119904)
[119886 (119909)](119904minus1)(2minus119904)
[119886 (119909)]minus(119904minus1)(2minus119904)
times int
1199050
119909
120585minus1(2minus119904)
119889120585
= 1198883
2 minus 119904
119904 minus 1
(1 minus 119909(119904minus1)(2minus119904)
119905minus(119904minus1)(2minus119904)
0)
le 1198883
2 minus 119904
119904 minus 1
119909 isin (0 1199050]
(A6)
so that (20) holds We conclude that
[119909119886 (119909)](119904minus1)(2minus119904)
int
1199050
119909
119886 (120585) [minus1205931015840(120585)] 119889120585
= (119909119886 (119909) [minus1205931015840(119909)]
2minus119904
)
(119904minus1)(2minus119904)
times ([
1
minus1205931015840(119909)
]
119904minus1
int
1199050
119909
119886 (120585) [minus1205931015840(120585)] 119889120585)
(A7)
By (19) and (20) we conclude that (18) is satisfied The proofof Remark 9 is complete
References
[1] C Tricot Curves and Fractal Dimension Springer New YorkNY USA 1995
[2] M Pasic ldquoFractal oscillations for a class of second order lineardifferential equations of Euler typerdquo Journal of MathematicalAnalysis and Applications vol 341 no 1 pp 211ndash223 2008
[3] M K Kwong M Pasic and J S W Wong ldquoRectifiable oscil-lations in second-order linear differential equationsrdquo Journal ofDifferential Equations vol 245 no 8 pp 2333ndash2351 2008
[4] M Pasic and J S W Wong ldquoRectifiable oscillations in second-order half-linear differential equationsrdquo Annali di MatematicaPura ed Applicata Series 4 vol 188 no 3 pp 517ndash541 2009
[5] M Pasic and S Tanaka ldquoFractal oscillations of self-adjoint anddamped linear differential equations of second-orderrdquo AppliedMathematics and Computation vol 218 no 5 pp 2281ndash22932011
[6] Y Naito M Pasic S Tanaka and D Zubrinic ldquoFractal oscilla-tions near domain boundary of radially symmetric solutions ofp-Laplace equations fractal geometry and dynamical systemsin pure and applied mathematicsrdquo Contemporary MathematicsAmerican Mathematical Society In press
[7] W A Coppel Stability and Asymptotic Behavior of DifferentialEquations D C Heath andCompany BostonMass USA 1965
[8] P Hartman Ordinary Differential Equations BirkhauserBoston Mass USA 2nd edition 1982
[9] F Neuman ldquoA general construction of linear differentialequations with solutions of prescribed propertiesrdquo AppliedMathematics Letters vol 17 no 1 pp 71ndash76 2004
[10] F Neuman ldquoStructure of solution spaces via transformationrdquoApplied Mathematics Letters vol 21 no 5 pp 529ndash533 2008
[11] P Borgnat and P Flandrin ldquoOn the chirp decomposition ofWeierstrass-Mandelbrot functions and their time-frequencyinterpretationrdquo Applied and Computational Harmonic Analysisvol 15 no 2 pp 134ndash146 2003
[12] E J Candes P R Charlton and H Helgason ldquoDetectinghighly oscillatory signals by chirplet path pursuitrdquo Applied andComputational Harmonic Analysis vol 24 no 1 pp 14ndash402008
[13] S Jaffard andYMeyer ldquoWaveletmethods for pointwise regular-ity and local oscillations of functionsrdquoMemoirs of the AmericanMathematical Society vol 123 no 587 pp 1ndash110 1996
[14] Y Meyer and H Xu ldquoWavelet analysis and chirpsrdquo Applied andComputational Harmonic Analysis vol 4 no 4 pp 366ndash3791997
[15] G RenQChen P Cerejeiras andUKaehle ldquoChirp transformsand chirp seriesrdquo Journal of Mathematical Analysis and Applica-tions vol 373 no 2 pp 356ndash369 2011
[16] M Kepesi and L Weruaga ldquoAdaptive chirp-based time-frequency analysis of speech signalsrdquo Speech Communicationvol 48 no 5 pp 474ndash492 2006
[17] L Weruaga and M Kepesi ldquoThe fan-chirp transform for non-stationary harmonic signalsrdquo Signal Processing vol 87 no 6 pp1504ndash1522 2007
[18] E Barlow A J Mulholland A Nordon and A GachaganldquoTheoretical analysis of chirp excitation of contrast agentsrdquoPhysics Procedia vol 3 no 1 pp 743ndash747 2009
International Journal of Differential Equations 11
[19] M H Pedersen T X Misaridis and J A Jensen ldquoClinicalevaluation of chirp-coded excitation in medical ultrasoundrdquoUltrasound in Medicine and Biology vol 29 no 6 pp 895ndash9052003
[20] T Paavle M Min and T Parve ldquoUsing of chirp excitation forbioimpedance estimation theoretical aspects andmodelingrdquo inProceedings of the 11th International Biennial Baltic ElectronicsConference (BECrsquo08) pp 325ndash328 Tallinn Estonia October2008
[21] M Pasic ldquoRectifiable and unrectifiable oscillations for a class ofsecond-order linear differential equations of Euler typerdquo Journalof Mathematical Analysis and Applications vol 335 no 1 pp724ndash738 2007
[22] K Falconer Fractal Geometry Mathematical Foundations andApplications John Wiley amp Sons Hoboken NJ USA 1999
[23] P Mattila Geometry of Sets and Measures in Euclidean SpacesFractals and Rectifiability vol 44 of Cambridge Studies inAdvanced Mathematics Cambridge University Press Cam-bridge UK 1995
[24] K J Falconer ldquoOn the Minkowski measurability of fractalsrdquoProceedings of the American Mathematical Society vol 123 no4 pp 1115ndash1124 1995
[25] C Q He and M L Lapidus Generalized Minkowski ContentSpectrum of Fractal Drums Fractal Strings and the RiemannZeta-Function vol 127 of Memoirs of the American Mathemat-ical Society American Mathematical Society Providence RIUSA 1997
[26] M L Lapidus and M van Frankenhuijsen Fractal geometryComplex Dimensions and Zeta Functions Geometry and Spec-tra of Fractal Strings Springer Monographs in MathematicsSpringer New York NY USA 2006
[27] M Pasic ldquoMinkowski-Bouligand dimension of solutions ofthe one-dimensional 119901-Laplacianrdquo Journal of Differential Equa-tions vol 190 no 1 pp 268ndash305 2003
[28] J S W Wong ldquoOn rectifiable oscillation of Euler type secondorder linear differential equationsrdquo Electronic Journal of Quali-tativeTheory of Differential Equations vol 2007 no 20 pp 1ndash122007
[29] M Pasic ldquoRectifiable and unrectifiable oscillations for a gen-eralization of the Riemann-Weber version of Euler differentialequationrdquo Georgian Mathematical Journal vol 15 no 4 pp759ndash774 2008
[30] M Pasic and S Tanaka ldquoRectifiable oscillations of self-adjointand damped linear differential equations of second-orderrdquoJournal of Mathematical Analysis and Applications vol 381 no1 pp 27ndash42 2011
[31] M Pasic D Zubrinic and V Zupanovic ldquoOscillatory andphase dimensions of solutions of some second-order differentialequationsrdquo Bulletin des Sciences Mathematiques vol 133 no 8pp 859ndash874 2009
[32] L Korkut and M Resman ldquoFractal oscillations of chirp-likefunctionsrdquo Georgian Mathematical Journal vol 19 no 4 pp705ndash720 2012
Submit your manuscripts athttpwwwhindawicom
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Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
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Discrete Dynamics in Nature and Society
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Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Stochastic AnalysisInternational Journal of
2 International Journal of Differential Equations
from the left side and from both sides seeTheorem 24 Manyexamples are considered to show the originality of obtainedresults
The chirp functions are also appearing in the time-frequency analysis see for instance [11ndash15] as well as inseveral applications of the time-frequency analysis see forinstance [16ndash20]
2 Statement of the Main Results
We study some local asymptotic behaviours of fractal typesfor the so-called chirp function as follows
119910 (119909) = 119886 (119909) 119878 (120593 (119909)) 119909 isin (0 1199050] (1)
where 1199050gt 0 the amplitude 119886 = 119886(119909) is a nontrivial function
on (0 1199050] the phase 120593 = 120593(119909) is singular at 119909 = 0 and 119878 = 119878(119905)
is a periodic function on R such as cos(119905) and sin(119905)The chirp function (1) is sometimes written in the follow-
ing form in which the so-called instantaneous frequency 120596(119909)appears instead of phase 120593(119909)
119910 (119909) = 119886 (119909) 119878 (int
1199050
119909
120596 (120585) 119889120585) 119909 isin (0 1199050] (2)
Let for some 1199050gt 0 the functions 119886 isin 119862((0 119905
0]) 120593 isin
1198622((0 1199050]) and 119878 isin 1198621(R) satisfy the following basic struc-
tural conditions
119886 (119909) gt 0 119886 (119909) is bounded on (0 1199050] (3)
lim119909rarr+0
120593 (119909) = infin 120593 (119909) gt 0 1205931015840(119909) lt 0 119909 isin (0 119905
0]
(4)
for some 120591 gt 0 |119878 (119905 + 120591)| = |119878 (119905)| forall119905 isin R
for some 1205910isin R 119878 (120591
0) = 0 119878 (119905) = 0
forall119905 isin (1205910 1205910+ 120591)
(5)
Definition 1 A function 119910 isin 119862((0 1199050]) is oscillatory near
119909 = 0 if there is a decreasing sequence 119909119899isin (0 119905
0] such that
119910(119909119899) = 0 for 119899 isin N and 119909
119899 0 as 119899 rarr infin see Figure 1
It is easy to prove the next proposition
Proposition 2 Let 119886 isin 119862((0 1199050]) satisfy 119886(119909) gt 0 on (0 119905
0] let
120593 isin 119862((0 1199050]) be strictly decreasing on (0 119905
0] and let 119878 isin 119862(R)
satisfy (5) Then the following two conditions are equivalent
(i) the chirp function 119910(119909) = 119886(119909)119878(120593(119909)) is bounded andoscillatory near 119909 = 0
(ii) the amplitude 119886(119909) is bounded and lim119909rarr+0
120593(119909) = infin
On the qualitative and oscillatory behaviours of solutionsof differential equations of several types we refer the readerto [7 8]
minus05
minus1
1
05
0201 025015005 03
Figure 1 119910 is continuous bounded and oscillatory near 119909 = 0
Example 3 The following threemain types of chirp functionssatisfy the conditions (3) (4) and (5)
(i) the so-called (120572 120573)-chirps
1199101(119909) = 119909
120572 cos (119909minus120573) 1199102(119909) = 119909
120572 sin (119909minus120573) (6)
where 120572 ge 0 and 120573 gt 0 the first studies on the (120572 120573)-chirps appeared in [1 13 14] from different point ofviews
(ii) the so-called logarithmic chirps
1199101(119909) = 119909
120574 cos (120588 log119909) 1199102(119909) = 119909
120574 sin (120588 log119909) (7)
where 120574 gt 0 and 120588 isin R this type of chirps appearsin definition of the Lamperti transform (see [11]) andin the fundamental system of solutions of the famousEuler equation 11991010158401015840 + 120582119909minus2119910 = 0 for 120574 = 12 and 120588 =radic120582 minus 14 (see [21])
(iii) the chirp function of exponential type
1199101(119909) = 119909119890
minus1(2119909) cos (1198901119909)
1199102(119909) = 119909119890
minus1(2119909) sin (1198901119909) (8)
Example 4 Let 1199101(119909) = 119886(119909) cos(120593(119909)) and 119910
2(119909) =
119886(119909) sin(120593(119909)) 119909 isin (0 1199050] where the amplitude 119886 and phase
120593 satisfy (3) and (4) respectively Then 119910(119909) = 11988811199101(119909) +
11988821199102(119909) 11988821+1198882
2gt 0 is also a chirp function which is bounded
and oscillatory near 119909 = 0 Indeed if 1198881= 0 or 119888
2= 0 then
119910(119909) = 11988821199102(119909) or 119910(119909) = 119888
11199101(119909) since 119878(119905) = cos 119905 and
119878(119905) = sin 119905 satisfy (5) both cases 119910(119909) are chirp functionsotherwise we have 119910(119909) = 119860119886(119909) sin(120593(119909) + 119861) where 119860 =(1198882
1+ 1198882
2)12 and 119861 = arctan(119888
11198882) obviously the amplitude
119860119886(119909) and the phase 120593(119909) + 119861 satisfy the required conditions(3) and (4) respectively
Next let
Γ (119910) = (1199051 1199052) isin R2 1199051isin (0 119905
0] 1199052= 119910 (119905
1)
(9)
denotes the graph of a function 119910 isin 119862((0 1199050]) In this paper
we show that the graph Γ(119910) of chirp function (1) is a fractal
International Journal of Differential Equations 3
curve in R2 in respect to the Minkowski-Bouligand dimen-sion dim
119872Γ(119910) = 119904 (box-counting dimension) and the 119904-
dimensional upper and lowerMinkowski contents119872lowast119904(Γ(119910))and119872119904
lowast(Γ(119910)) defined by
dim119872Γ (119910) = lim
120576rarr+0
(2 minus
log 1003816100381610038161003816Γ120576(119910)1003816100381610038161003816
log 120576)
119872lowast119904(Γ (119910)) = lim sup
120576rarr+0
(2120576)119904minus2 1003816100381610038161003816Γ120576(119910)1003816100381610038161003816
119872119904
lowast(Γ (119910)) = lim inf
120576rarr+0
(2120576)119904minus2 1003816100381610038161003816Γ120576(119910)1003816100381610038161003816
(10)
Here Γ120576(119910) denotes the 120576-neighbourhood of graph Γ(119910)
defined by
Γ120576(119910) = (119905
1 1199052) isin R2 119889 ((119905
1 1199052) Γ (119910)) le 120576 120576 gt 0
(11)
and 119889((1199051 1199052) Γ(119910)) denotes the distance from (119905
1 1199052) to Γ(119910)
and |Γ120576(119910)| denotes the Lebesgue measure of Γ
120576(119910) On the
box-counting dimension and the 119904-dimensional Minkowskicontent we refer the reader to [22ndash27]
The main fractal properties considered in the paper aregiven in the next definitions
Definition 5 For a given real number 119904 isin [1 2) and a function119910 isin 119862((0 119905
0]) which is bounded on (0 119905
0] and oscillatory
near 119909 = 0 it is said that 119910(119909) is fractal oscillatory near 119909 = 0with the fractal dimension 119904 if
dim119872Γ (119910) = 119904 0 lt 119872
119904
lowast(Γ (119910)) le 119872
lowast119904(Γ (119910)) lt infin
(12)
On the contrary if there is no any 119904 isin [1 2) such that Γ(119910)satisfies (12) then 119910(119909) is not fractal oscillatory near 119909 = 0
Fractal oscillations can be understood also as a refine-ment of rectifiable and nonrectifiable oscillations They arerecently studied in [3 4 21 28ndash30]
Example 6 The chirp functions
119910 (119909) = 119909 cos( 1119909
) 119910 (119909) = 119909 sin( 1119909
) (13)
are not fractal oscillatory near 119909 = 0 In fact119872lowast1(Γ(119910)) = infinand dim
119872Γ(119910) = 1 (see [2]) It also implies that119872lowast119904(Γ(119910)) =
0 for all 119904 isin (1 2) (see [1 23]) and thus the statement (12) isnot satisfied for any 119904 isin [1 2) Hence 119910
1and 1199102are not fractal
oscillatory near 119909 = 0
Example 7 Let 120588 gt 0 It is clear that the chirp functions
119910 (119909) = radic119909 cos (120588 log119909) 119910 (119909) = radic119909 sin (120588 log119909)(14)
are oscillatory near 119909 = 0 Moreover the length of Γ(119910)is finite (see [21]) and therefore we observe that 0 lt
1198721
lowast(Γ(119910)) le 119872
lowast1(Γ(119910)) lt infin and dim
119872Γ(119910) = 1 (see [22])
Thus such chirp functions are fractal oscillatory near 119909 = 0with the fractal dimension 1
In order to show that the chirp function (1) is fractaloscillatory near 119909 = 0 we need to impose on amplitude 119886(119909)the following additional structural condition
119886 isin 1198621((0 1199050]) 119886
1015840(119909) ge 0 for 119909 isin (0 119905
0] (15)
Now we are able to state the first main result of the paper
Theorem 8 Let the functions 119886(119909) 120593(119909) and 119878(119905) satisfy thestructural conditions (3) (4) (5) (15) and
12059310158401015840(119909) gt 0 on (0 119905
0] lim sup
119909rarr0
(
1
minus1205931015840(119909)
)
1015840
lt infin (16)
Let the amplitude 119886(119909) and the phase 120593(119909) satisfy the followingasymptotic conditions near 119909 = 0
lim inf119909rarr0
([minus1205931015840(119909)]
2minus119904
int
119909
0
119886 (120585) 119889120585) gt 0 (17)
lim sup119909rarr0
[119909119886 (119909)](119904minus1)(2minus119904)
int
1199050
119909
119886 (120585) [minus1205931015840(120585)] 119889120585 lt infin (18)
where 119904 isin (1 2) is a given real numberThen the chirp function(1) is fractal oscillatory near 119909 = 0with the fractal dimension 119904
Remark 9 Let 119904 isin (1 2) Assume that (15) and (17) holdThen
lim sup119909rarr0
(119909119886 (119909) [minus1205931015840(119909)]
2minus119904
) lt infin (19)
implies
lim sup119909rarr0
[
1
minus1205931015840(119909)
]
119904minus1
int
1199050
119909
119886 (120585) [minus1205931015840(120585)] 119889120585 lt infin (20)
and (18) Therefore condition (18) is better than (19) inTheorem 8
The proof of Remark 9 is presented in the AppendixTheorem 8 can be rewritten in the term of 120596(119909) instead of
120593(119909) as follows
Corollary 10 Let the functions 119886(119909)120596(119909) and 119878(119905) satisfy thestructural conditions (3) (5) (15) and
120596 isin 1198621((0 1199050]) 120596 (119909) gt 0 120596
1015840(119909) lt 0
119900119899 (0 1199050] 120596 notin 119871
1((0 1199050])
lim sup119909rarr0
(
1
120596 (119909)
)
1015840
lt infin
(21)
Let the amplitude 119886(119909) and the instantaneous frequency 120596(119909)satisfy the following asymptotic conditions near 119909 = 0
lim inf119909rarr0
([120596 (119909)]2minus119904int
119909
0
119886 (120585) 119889120585) gt 0
lim sup119909rarr0
[119909119886 (119909)](119904minus1)(2minus119904)
int
1199050
119909
119886 (120585) 120596 (120585) 119889120585 lt infin
(22)
where 119904 isin (1 2) is a given real numberThen the chirp function(2) is fractal oscillatory near 119909 = 0with the fractal dimension 119904
4 International Journal of Differential Equations
Now we state analogous result to Theorem 8 in the caseof 119904 = 1 which will be proved in Section 3
Theorem 11 Assume that the function 119878(119905) satisfies (5) 119886 isin119862([0 119905
0])cap1198621((0 1199050])120593 isin 1198622((0 119905
0]) and lim
119909rarr+0120593(119909) = infin
If 1198861015840 isin 1198711((0 1199050]) and1205931015840119886 isin 1198711((0 119905
0]) then the chirp function
(1) is fractal oscillatory near 119909 = 0with the fractal dimension 1
In respect to some existing results on the fractal dimen-sion of graph of chirp functions previous theorems are themost simple and general It is because in [31 32] authorsrequire some extra conditions on the chirp function 119910(119909)which are not easy to be satisfied in the application forinstance the rapid convex-concave properties of 119910(119909) as in[31] and a condition on the curvature of 119910(119909) as in [32]
According to previous theorems we can show the fractaloscillations of the so-called (120572 120573)-chirp as well as logarithmicchirp functions
Example 12 We consider the (120572 120573)-chirp 119910(119909) = 119909120572119878(119909minus120573)where 119878(119905) = cos 119905 or 119878(119905) = sin 119905 and 120573 gt 120572 ge 0 It is fractaloscillatory near 119909 = 0 with the fractal dimension 119904 = 2 minus(1 + 120572)(1 + 120573) In fact it is easy to see that 119886(119909) = 119909120572 119878(119905)and 120593(119909) = 119909minus120573 satisfy (3) (4) (5) (15) and (16) When 119904 =2 minus (1 + 120572)(1 + 120573) we see that
[minus1205931015840(119909)]
2minus119904
int
119909
0
119886 (120585) 119889120585 =
120573(1+120572)(1+120573)
1 + 120572
gt 0
[119909119886 (119909)](119904minus1)(2minus119904)
int
1199050
119909
119886 (120585) [minus1205931015840(120585)] 119889120585
=
120573
120573 minus 120572
(1 minus 119909120573minus120572119905120572minus120573
0) le
120573
120573 minus 120572
(23)
FromTheorem 8 it follows that 119910(119909) = 119909120572119878(119909minus120573) 120573 gt 120572 ge 0is fractal oscillatory near 119909 = 0 with the fractal dimension119904 = 2 minus (1 + 120572)(1 + 120573)
Example 13 We consider the (120572 120573)-chirp 119910(119909) = 119909120572119878(119909minus120573)again where 119878(119905) = cos 119905 or 119878(119905) = sin 119905 Now we assume that0 lt 120573 lt 120572 ApplyingTheorem 8 we easily see that it is fractaloscillatory near 119909 = 0 with the fractal dimension 1
Example 14 We consider the logarithmic chirp functions
1199101(119909) = 119909
120574 cos (120588 log119909) 1199102(119909) = 119909
120574 sin (120588 log119909) (24)
where 120574 gt 0 and 120588 gt 0 Put 119878(119905) = cos(minus119905) or 119878(119905) = sin(minus119905)119886(119909) = 119909
120574 and 120593(119909) = minus120588 log119909 Then we easily see that119878(119905) satisfies (5) and that 1198861015840(119909) = 120574119909
120574minus1isin 1198711((0 1199050]) and
119886(119909)1205931015840(119909) = minus120588119909
120574minus1isin 1198711((0 1199050]) Theorem 11 implies that
1199101(119909) and 119910
2(119909) are fractal oscillatory near 119909 = 0 with the
fractal dimension 1
Question 1 Is it possible to applyTheorem 8 on the exponen-tial chirp given in Example 3(iii)
Example 15 Let 1199101(119909) = 119886(119909) cos(120593(119909)) and 119910
2(119909) =
119886(119909) sin(120593(119909)) 119909 isin (0 1199050] where the amplitude 119886(119909) and
the phase 120593(119909) satisfy all assumptions of Theorem 8 for anarbitrary given 119904 isin (1 2) Then the chirp function 119910(119909) =11988811199101(119909) + 119888
21199102(119909) 11988821+ 1198882
2gt 0 is also fractal oscillatory
near 119909 = 0 with the fractal dimension 119904 Indeed similarlyas in Example 4 119910(119909) can be rewritten in the form 119910(119909) =
1198891sin(120593(119909) + 119889
2) where 119889
1= 0 and 119889
2isin R It is clear that the
function 119878(119905) = 1198891sin(119905 + 119889
2) satisfies the required condition
(5) and henceTheorem 8 proves that119910(119909) is fractal oscillatorynear 119909 = 0 with the fractal dimension 119904
Next we pay attention to the fractal oscillations ofsolutions of linear differential equations generated by thesystem of functions as follows
119910 (119909) = 11988811199101(119909) + 119888
21199102(119909) 119888
2
1+ 1198882
2gt 0
where
1199101 (119909) = 119886 (119909) cos (120593 (119909)) 1199102 (119909) = 119886 (119909) sin (120593 (119909))
119909 isin (0 1199050]
(25)
It is not difficult to check that (25) is the fundamental systemof all solutions of the following linear differential equation
11991010158401015840+ (minus2119877
119886minus 1198771205931015840) 1199101015840+ (minus119877
1015840
119886+ 1198772
119886+ 1198771198861198771205931015840 + (120593
1015840)
2
) 119910 = 0
119909 isin (0 1199050]
(26)
where 119877119891= 119877119891(119909) = 119891
1015840(119909)119891(119909) for some 1198621-function 119891 =
119891(119909)
Theorem 16 Let the functions 119886 120593 isin 1198622((0 1199050]) satisfy struc-
tural conditions (3) (4) and (15) as well as the conditions (16)(17) and (18) in respect to a given real number 119904 isin (1 2)Then every nontrivial solution 119910 isin 1198622((0 119905
0]) of (26) is fractal
oscillatory near 119909 = 0 with the fractal dimension 119904
With the help ofTheorem 11 we can state analogous resultto Theorem 16 in the case of 119904 = 1
Theorem 17 Assume that 119886 isin 119862([0 1199050])cap1198622((0 1199050]) 119886(119909) = 0
for 119909 isin (0 1199050] 120593 isin 119862
2((0 1199050]) lim
119909rarr+0120593(119909) = infin 1198861015840 isin
1198711((0 1199050]) and 1205931015840119886 isin 1198711((0 119905
0]) Then every nontrivial solu-
tion 119910 isin 1198622((0 1199050]) of (26) is fractal oscillatory near 119909 = 0
with the fractal dimension 1
The previous two theorems will be proved in Section 4Assumptions on the coefficients of (26) in general are dif-ferent to those considered in [2ndash5]
When 1198861015840119886 = minus(12)120593101584010158401205931015840 then (26) becomes the un-damped equation
11991010158401015840+ (
1
2
119878der (1205931015840) + (120593
1015840)
2
)119910 = 0 119909 isin (0 1199050] (27)
International Journal of Differential Equations 5
where 119878der(119891) denotes the Schwarzian derivative of119891 definedby
119878der (119891) =11989110158401015840
119891
minus
3
2
(
1198911015840
119891
)
2
(28)
Hence from Theorem 16 we obtain the following conse-quence
Corollary 18 Let the functions 119886 isin 1198622((0 1199050]) and 120593 isin
1198623((0 1199050]) satisfy structural conditions (3) (4) and (15) as
well as the conditions (16) (17) and (18) in respect to a givenreal number 119904 isin (1 2) Let 1198861015840119886 = minus(12)120593101584010158401205931015840 Then everynontrivial solution 119910 isin 1198622((0 119905
0]) of (27) is fractal oscillatory
near 119909 = 0 with the fractal dimension 119904
As a consequence of Theorem 16 and Corollary 18 wederive the following examples for linear differential equationsof second order having all the solutions to be fractal oscilla-tory near 119909 = 0
Example 19 The so-called damped chirp equation
11991010158401015840+
120573 minus 2120572 + 1
119909
1199101015840+ (
1205732
1199092120573+2
minus
120572 (120573 minus 120572)
1199092
)119910 = 0
119909 isin (0 1199050]
(29)
is fractal oscillatory near 119909 = 0 with the fractal dimension2 minus (1 + 120572)(1 + 120573) where 120573 gt 120572 ge 0 When 119886(119909) = 119909120572 and120593(119909) = 119909
minus120573 (26) becomes (29) It is easy to see that (3) (4)(15) and (16) are satisfied In the same as in Example 12 wesee that (17) and (18) hold for 119904 = 2 minus (1 + 120572)(1 + 120573) HenceTheorem 16 proves that every nontrivial solution of (29) isfractal oscillatory near 119909 = 0 with the fractal dimension 2 minus(1 + 120572)(1 + 120573)
Nowwe assume that 0 lt 120573 lt 120572ThenTheorem 17 impliesthat (29) is fractal oscillatory near 119909 = 0 with the fractaldimension 1
Example 20 The following equation
11991010158401015840+
1 minus 2120574
119909
1199101015840+
1205742+ 1205882
1199092119910 = 0 119909 isin (0 119905
0] (30)
is fractal oscillatory near 119909 = 0 with the fractal dimension1 where 120574 gt 0 and 120588 gt 0 In the case where 119886(119909) = 119909
120574
and 120593(119909) = minus120588 log119909 (26) becomes (30) We see that 119886 isin119862([0 119905
0]) cap 119862
2((0 1199050]) 120593 isin 1198622((0 119905
0]) lim
119909rarr+0120593(119909) = infin
1198861015840isin 1198711((0 1199050]) and 1205931015840119886 isin 1198711((0 119905
0]) Therefore Theorem 17
implies that every nontrivial solution of (30) is fractaloscillatory near 119909 = 0 with the fractal dimension 1
Question 2 What can we say about the application of Theo-rem 16 on the case of 120593(119909) given in Example 3(iii)
At the end of this section we suggest that the readerstudies some invariant properties of fractal oscillations of thechirp function (1) in respect to the translation and reflexionAnalogously to Definitions 1 and 5 one can define the fractaloscillations near an arbitrary real point 119909 = 119909
0as follows
minus05
minus025 minus02 minus015 minus01
minus01
minus03
05
1
minus005
Figure 2 119910 is oscillatory near 119909 = 0 from the left side
Definition 21 Let 1199090isin R and 120575 gt 0 A function 119910 isin
119862((1199090 1199090+ 120575]) is oscillatory near 119909 = 119909
0 if there is a
decreasing sequence 119887119899isin (1199090 1199090+ 120575] such that 119910(119887
119899) = 0
for 119899 isin N and 119887119899 1199090as 119899 rarr infin Moreover if the graph
Γ(119910) satisfies the condition (12) for some 119904 isin [1 2) then 119910(119909)is said to be fractal oscillatory near 119909 = 119909
0with the fractal
dimension 119904
Definition 22 Let 1199090isin R and 120575 gt 0 It is said that a function
119910 isin 119862([1199090minus 120575 119909
0)) is fractal oscillatory near 119909 = 0 from
the left side with the fractal dimension 119904 isin [1 2) if there is anincreasing sequence 119886
119899isin [1199090minus 120575 119909
0) such that 119910(119886
119899) = 0 for
119899 isin N 119886119899 1199090as 119899 rarr infin and the graph Γ(119910) satisfies the
condition (12) see Figure 2
Definition 23 Let 1199090isin R and let 120575 gt 0 It is said that a
function 119910 isin 119862([1199090minus120575 1199090) cup (1199090 1199090+120575]) is two-sided fractal
oscillatory near 119909 = 0 with the fractal dimension 119904 isin [1 2)if there is an increasing sequence 119886
119899isin [1199090minus 120575 119909
0) and a
decreasing sequence 119887119899isin (1199090 1199090+ 120575] such that 119910(119886
119899) =
119910(119887119899) = 0 for 119899 isin N 119886
119899 1199090and 119887119899 1199090as 119899 rarr infin
and the graph Γ(119910) satisfies the condition (12) see Figure 3
With the help ofTheorem 8 we state the following result
Theorem 24 Let 1199090isin R and let 120575 gt 0 Let the functions
119886 120596 isin 1198621((0 120575]) satisfy structural conditions (3) (4) (5) (15)
and (16) as well as conditions (17) and (18) in respect to a givenreal number 119904 isin (1 2) Then one has
(i) the chirp function 119910(119909) = 119886(119909 minus 1199090)119878(120593(119909 minus 119909
0)) 119909 isin
(1199090 1199090+ 120575] is fractal oscillatory near 119909 = 119909
0with the
fractal dimension 119904
(ii) the chirp function 119910(119909) = 119886(1199090minus 119909)119878(120593(119909
0minus 119909)) 119909 isin
[1199090minus 120575 119909
0) is fractal oscillatory near 119909 = 119909
0from the
left side with the fractal dimension 119904
(iii) the chirp function 119910(119909) = 119886(|119909minus1199090|)119878(120593(|119909minus119909
0|)) 119909 isin
[1199090minus120575 1199090)cup(1199090 1199090+120575] is two-sided fractal oscillatory
near 119909 = 1199090with the fractal dimension 119904
6 International Journal of Differential Equations
minus03 minus02 minus01
minus1
01 02 03
minus05
1
05
Figure 3 119910 is two-sided oscillatory near 119909 = 0
3 Proof for the Fractal Oscillations ofChirp Functions
In this section we give the proofs of the main results dealingwith the fractal oscillations of chirp functions
By Definition 5 it follows that if for a prescribed realnumber 119904 isin (1 2) there are two positive constants 119888
1and 1198882
such that
11988811205762minus119904le1003816100381610038161003816Γ120576(119910)1003816100381610038161003816le 11988821205762minus119904 120576 isin (0 120576
0) (31)
for some 1205760gt 0 then a function 119910 isin 119862((0 119905
0]) is fractal
oscillatory near 119909 = 0 with the fractal dimension 119904 Thefollowing lemma plays the essential role in the proof of (31)
Lemma 25 Let 119910 isin 119862((0 1199050]) be a bounded function on
(0 1199050] and let 119886
119899isin (0 119905
0] be a decreasing sequence of con-
secutive zeros of 119910(119909) such that 119886119899rarr 0 Let 120576
0gt 0 and let
119896 = 119896(120576) be an index function satisfying1003816100381610038161003816119886119899minus 119886119899+1
1003816100381610038161003816le 120576 forall119899 ge 119896 (120576) 120576 isin (0 120576
0) (32)
Then1003816100381610038161003816Γ120576(119910)1003816100381610038161003816ge sum
119899ge119896(120576)
max119909isin[119886119899+1119886119899]
1003816100381610038161003816119910 (119909)
1003816100381610038161003816(119886119899minus 119886119899+1)
forall120576 isin (0 1205760)
(33)
Proof It is exactly the same as [3 pp 2350]
Let us remark that for some 1205760gt 0 we say that a function
119896 = 119896(120576) is an index function on (0 1205760) if 119896 (0 120576
0) rarr N and
lim120576rarr0
119896(120576) = infin
Lemma 26 Let 119910 isin 119862[119886 119887] Then
1003816100381610038161003816Γ120576(119910)1003816100381610038161003816le 4120587120576length (119910) + 41205871205762 for every 120576 gt 0 (34)
Proof Let 120576 gt 0 Set 1199091= 119886 and set
119909119894+1= max 119909 isin [119909
119894 119887] 119889 ((119905 119910 (119905)) (119909
119894 119910 (119909119894)))
le 120576 for 119905 isin [119909119894 119909]
(35)
for 119894 = 1 2 Then there exists 119899 ge 2 such that 119909119899= 119887 Set
119873 = max119899 isin N 119909119899lt 119887 We see that119873 ge 1
119886 = 1199091lt 1199092lt sdot sdot sdot lt 119909
119894lt 119909119894+1lt sdot sdot sdot lt 119909
119873lt 119909119873+1
= 119887 (36)
and if119873 ge 2 then
119889 ((119909119894 119910 (119909119894)) (119909
119894+1 119910 (119909119894+1))) = 120576 119894 = 1 2 119873 minus 1
(37)
We will show that
Γ120576(119910) sub
119873
⋃
119894=1
1198612120576(119909119894 119910 (119909119894)) (38)
where
1198612120576(1199051 1199052) = (120591
1 1205912) isin R2 119889 ((119905
1 1199052) (1205911 1205912)) le 2120576
(39)
Let (1205911 1205912) isin Γ
120576(119910) Then there exists 120585 isin [119886 119887] such that
119889((1205911 1205912) (120585 119910(120585))) le 120576 Because of the definition of 119909
119894+1 we
find that 120585 isin [119909119894 119909119894+1] for some 119894 isin 1 2 119873 so that
119889 ((120585 119910 (120585)) (119909119894 119910 (119909119894))) le 120576 (40)
Hence it follows that
119889 ((1205911 1205912) (119909119894 119910 (119909119894))) le 119889 ((120591
1 1205912) (120585 119910 (120585)))
+ 119889 ((120585 119910 (120585)) (119909119894 119910 (119909119894))) le 2120576
(41)
which means that (1205911 1205912) isin 119861
2120576(119909119894 119910(119909119894)) Therefore we
obtain (38) By (38) we conclude that
1003816100381610038161003816Γ120576(119910)1003816100381610038161003816le
119873
sum
119894=1
10038161003816100381610038161198612120576(119909119894 119910 (119909119894))1003816100381610038161003816= 4119873120587120576
2 (42)
When119873 = 1 from (42) it follows that1003816100381610038161003816Γ120576(119910)1003816100381610038161003816le 4120587120576
2le 4120587120576length (119910) + 41205871205762 (43)
Now we assume that119873 ge 2 We observe that
length (119910) =119873
sum
119894=1
length (119910|[119909119894 119909119894+1]
)
ge
119873
sum
119894=1
119889 ((119909119894 119910 (119909119894)) (119909
119894+1 119910 (119909119894+1)))
ge
119873minus1
sum
119894=1
119889 ((119909119894 119910 (119909119894)) (119909
119894+1 119910 (119909119894+1)))
= (119873 minus 1) 120576
(44)
that is
119873120576 le length (119910) + 120576 (45)
Combining (42) with (45) we obtain1003816100381610038161003816Γ120576(119910)1003816100381610038161003816le 4120587120576length (119910) + 41205871205762 (46)
International Journal of Differential Equations 7
Lemma 27 Let 119910 isin 119862(0 1199050] be bounded on (0 119905
0] and let
120576 gt 0 Then there exists 119862 gt 0 such that for every 120576 gt 0 and119905 isin (0 119905
0]
1003816100381610038161003816Γ120576(119910)1003816100381610038161003816le 119862[119905 sup
119909isin(0119905]
1003816100381610038161003816119910 (119909)
1003816100381610038161003816+ 120576length (119910|
[1199051199050]) + 120576 + 120576
2]
(47)
Proof Let 120576 gt 0 and let 119905 isin (0 1199050] From the geometry point
of view it is clear that1003816100381610038161003816Γ120576(119910|(0119905])1003816100381610038161003816sub 119877 (48)
where 119877 is a rectangle in R2 defined by
119877 = (1199051 1199052) 1199051isin [minus120576 119905 + 120576]
1199052isin [minus120576 minus sup
119909isin(0119905]
1003816100381610038161003816119910 (119909)
1003816100381610038161003816 120576 + sup119909isin(0119905]
1003816100381610038161003816119910 (119909)
1003816100381610038161003816]
(49)
Hence we have
1003816100381610038161003816Γ120576(119910(0119905])1003816100381610038161003816le (2120576 + 119905) (2120576 + 2 sup
119909isin(0119905]
1003816100381610038161003816119910 (119909)
1003816100381610038161003816)
= 41205762+ 2(119905 + 2 sup
119909isin(0119905]
1003816100381610038161003816119910 (119909)
1003816100381610038161003816) 120576
+ 2119905 sup119909isin(0119905]
1003816100381610038161003816119910 (119909)
1003816100381610038161003816
le 41205762+ 1198621120576 + 2119905 sup
119909isin(0119905]
1003816100381610038161003816119910 (119909)
1003816100381610038161003816
(50)
where 1198621= 2(1199050+ 2sup
119909isin(01199050]|119910(119909)|) Lemma 26 implies that
10038161003816100381610038161003816Γ120576(119910|[1199051199050])
10038161003816100381610038161003816le 4120587120576length (119910|
[1199051199050]) + 4120587120576
2 (51)
Consequently we see that1003816100381610038161003816Γ120576(119910)1003816100381610038161003816le1003816100381610038161003816Γ120576(119910|(0119905])1003816100381610038161003816+
10038161003816100381610038161003816Γ120576(119910|[1199051199050])
10038161003816100381610038161003816
le 41205762+ 1198621120576 + 2119905 sup
119909isin(0119905]
1003816100381610038161003816119910 (119909)
1003816100381610038161003816
+ 4120587120576length (119910|[1199051199050]) + 4120587120576
2
le 119862[119905 sup119909isin(0119905]
1003816100381610038161003816119910 (119909)
1003816100381610038161003816+ 120576length (119910|
[1199051199050]) + 120576 + 120576
2]
(52)
where 119862 = max1198621 4(120587 + 1)
Lemma 28 Let 119910 isin 1198621(0 1199050] be bounded on (0 119905
0] Assume
that
int
1199050
0
100381610038161003816100381610038161199101015840(119909)
10038161003816100381610038161003816119889119909 = infin (53)
lim sup119909rarr0
[119909 sup120585isin(0119909]
1003816100381610038161003816119910 (120585)
1003816100381610038161003816]
(119904minus1)(2minus119904)
int
1199050
119909
100381610038161003816100381610038161199101015840(120585)
10038161003816100381610038161003816119889120585 lt infin (54)
for some 119904 isin (1 2) Then there exists 1198882gt 0 such that
1003816100381610038161003816Γ120576(119910)1003816100381610038161003816le 11988821205762minus119904 120576 isin (0 1) (55)
Proof Let 120576 isin (0 1) and let 119904 isin (1 2) be satisfy (54) By (53)there exists 119909
1isin (0 119905
0) which depends on 120576 and 119904 such that
int
1199050
1199091
100381610038161003816100381610038161199101015840(119905)
10038161003816100381610038161003816119889119905 = 120576
1minus119904 (56)
Therefore we have
length (119910|[1199091 1199050]
) = int
1199050
1199091
radic1 +10038161003816100381610038161199101015840(119909)1003816100381610038161003816
2119889119909
le int
1199050
1199091
(1 +
100381610038161003816100381610038161199101015840(119909)
10038161003816100381610038161003816) 119889119909
= 1199050minus 1199091+ 1205761minus119904
le 1199050+ 1205761minus119904
(57)
By (54) there exists 1198621gt 0 such that
[119909 sup120585isin(0119909]
1003816100381610038161003816119910 (120585)
1003816100381610038161003816]
(119904minus1)(2minus119904)
int
1199050
119909
100381610038161003816100381610038161199101015840(120585)
10038161003816100381610038161003816119889120585 le 119862
1 119909 isin (0 119905
0]
(58)
which implies that
[
[
1199091sup120585isin(01199091]
1003816100381610038161003816119910 (120585)
1003816100381610038161003816]
]
(119904minus1)(2minus119904)
int
1199050
1199091
100381610038161003816100381610038161199101015840(120585)
10038161003816100381610038161003816119889120585 le 119862
1 (59)
that is
1199091sup120585isin(01199091]
1003816100381610038161003816119910 (120585)
1003816100381610038161003816le 11986221205762minus119904 (60)
where 1198622= 119862(2minus119904)(119904minus1)
1 From Lemma 27 (57) and (60) it
follows that1003816100381610038161003816Γ120576(119910)1003816100381610038161003816le 119862 [119862
21205762minus119904+ 120576 (1199050+ 1205761minus119904) + 120576 + 120576
2]
= 119862 [(1198622+ 1) + (119905
0+ 1) 120576
119904minus1+ 120576119904] 1205762minus119904
le 119862 (1198622+ 1199050+ 3) 120576
2minus119904 120576 isin (0 1)
(61)
In order to showTheorems 8 and 11 we need the followingtwo geometric lemmas
Lemma 29 (see [1]) If Γ sube R2 is a simple curve (ie itsparameterization is a bijection) and length(Γ) lt infin then
length (Γ) = lim120576rarr0
1003816100381610038161003816Γ120576
1003816100381610038161003816
2120576
(62)
where Γ120576denotes the 120576-neighborhood of the graph Γ
8 International Journal of Differential Equations
Now we are able to proveTheorem 8
Proof of Theorem 8 Let 119904 isin (1 2) and let 119910(119909) be a chirpfunction given by (1) We note here that it is enough to showthat 119910(119909) satisfies (31)
At the first let 119909119899be a sequence defined by 119909
119899= 120593minus1(1205910+
119899120591) for all sufficiently large 119899 isin N From (4) it follows that120593minus1(119905) is decreasing Hence 119909
119899is decreasing as well as 119909
119899rarr
0 as 119899 rarr infin because of lim119909rarr+0
120593(119909) = infin (see (4))We notethat 119910(119909
119899) = 0 and 119910(119909) = 0 on (119909
119899+1 119909119899) for all sufficiently
large 119899 isin N Also minus11205931015840(119909) is an increasing function becauseof (16) The mean value theorem shows that
120591
minus1205931015840(119909119899+1)
le 119909119899minus 119909119899+1le
120591
minus1205931015840(119909119899)
(63)
Now let 1205760isin (0 1) Let 119896(120576) be the smallest natural number
satisfying120591
minus1205931015840(119909119896(120576))
le 120576 forall120576 isin (0 1205760) (64)
Such 119896(120576) exists for every 120576 isin (0 1205760) since 119909
119899rarr 0 as 119899 rarr
infin and lim119909rarr+0
1205931015840(119909) = minusinfin (this equality is true because
1205931015840notin 1198711(0 1199050) since lim
119909rarr+0120593(119909) = infin) Moreover since 119909
119899
is decreasing and minus11205931015840(119909) is increasing we obtain
minus1205931015840(119909119899) ge 120591120576
minus1forall119899 ge 119896 (120576) (65)
Combining (63) and (65) it is easy to deduce that suchdefined 119896(120576) satisfies condition (32)
By (16) there exists 119871 gt 0 such that (1(minus1205931015840(119909)))1015840 le 119871 for119909 isin (0 119905
0] which means that
minus
minus12059310158401015840(119909)
minus1205931015840(119909)
le minus1198711205931015840(119909) 119909 isin (0 119905
0] (66)
Integrating (66) on [120593minus1(119905 + 2120591) 120593minus1(119905)] we have
logminus1205931015840(120593minus1(119905 + 2120591))
minus1205931015840(120593minus1(119905))
le 2119871120591 119905 isin [120593 (1199050) infin) (67)
that is
minus1205931015840(120593minus1(119905)) ge minus119890
minus21198711205911205931015840(120593minus1(119905 + 120591)) 119905 isin [120593 (119905
0) infin)
(68)
By the definition of 119896(120576) and (64) we see that120591
minus1205931015840(119909119896(120576)minus1
)
gt 120576 (69)
Hence from (68) it follows that
120591120576minus1gt minus1205931015840(119909119896(120576)minus1
)
= minus1205931015840(120593minus1(1205910+ (119896 (120576) minus 1) 120591))
ge minus119890minus2119871120591
1205931015840(120593minus1(1205910+ (119896 (120576) minus 1) 120591 + 2120591))
= minus119890minus2119871120591
1205931015840(120593minus1(1205910+ (119896 (120576) + 1) 120591))
= minus119890minus2119871120591
1205931015840(119909119896(120576)+1
)
(70)
which implies that
minus1205931015840(119909119896(120576)+1
) le 119872120576minus1 120576 gt 0 (71)
where119872 = 1205911198902119871120591
Next it is clear that the assumption (5) ensures a realnumber119872
0gt 0 such that
1198720= max119905isinR
|119878 (119905)| = max119905isin[0120591]
|119878 (119905)| (72)
Hence from (15) we have
max119909isin[119909119899+1 119909119899]
1003816100381610038161003816119910 (119909)
1003816100381610038161003816= max119909isin[119909119899+1119909119899]
(119886 (119909)1003816100381610038161003816119878 (120593 (119909))
1003816100381610038161003816)
ge 119886 (119909119899+1) max119909isin[119909119899+1119909119899]
1003816100381610038161003816119878 (120593 (119909))
1003816100381610038161003816
= 1198720119886 (119909119899+1)
(73)
Now from (63) and (73) it follows that
max119909isin[119909119899+1 119909119899]
1003816100381610038161003816119910 (119909)
1003816100381610038161003816(119909119899minus 119909119899+1) ge 119872
0119886 (119909119899+1) (119909119899minus 119909119899+1)
ge 1205911198720
119886 (119909119899+1)
minus1205931015840(119909119899+1)
= 1205911198720
119886 (120593minus1(1205910+(119899+1) 120591))
minus1205931015840(120593minus1(1205910+(119899+1) 120591))
(74)
Since 120593minus1(119905) is decreasing as well as minus11205931015840(119909) is increasingand 119886(119909) is nondecreasing and positive we conclude that thefunction
119886 (120593minus1(119905))
minus1205931015840(120593minus1(119905))
is nonincreasing (75)
Hence we observe that
sum
119899ge119896(120576)
max119909isin[119909119899+1119909119899]
1003816100381610038161003816119910 (119909)
1003816100381610038161003816(119909119899minus 119909119899+1)
ge 1205911198720sum
119899ge119896(120576)
119886 (120593minus1(1205910+ (119899 + 1) 120591))
minus1205931015840(120593minus1(1205910+ (119899 + 1) 120591))
ge 1205911198720int
infin
119896(120576)
119886 (120593minus1(1205910+ (119905 + 1) 120591))
minus1205931015840(120593minus1(1205910+ (119905 + 1) 120591))
119889119905
= 1198720int
120593minus1(1205910+(119896(120576)+1)120591)
0
119886 (120585) 119889120585
= 1198720int
119909119896(120576)+1
0
119886 (120585) 119889120585
(76)
Next from assumption (17) we get some1198721gt 0 such that
int
119909
0
119886 (120585) 119889120585 ge 1198721[minus1205931015840(119909)]
minus(2minus119904)
119909 isin (0 1199050] (77)
International Journal of Differential Equations 9
Now from (76) (77) and (71) we obtain
sum
119899ge119896(120576)
max119909isin[119909119899+1119909119899]
1003816100381610038161003816119910 (119909)
1003816100381610038161003816(119909119899minus 119909119899+1)
ge 11987201198721[minus1205931015840(119909119896(120576)+1
)]
minus(2minus119904)
ge 11987201198721(119872120576minus1)
minus(2minus119904)
= 11987201198721119872minus(2minus119904)
1205762minus119904
(78)
Thus using Lemma 25 we have
1003816100381610038161003816Γ120576(119910)1003816100381610038161003816ge 11988811205762minus119904 120576 isin (0 120576
0) (79)
for some 1198881gt 0
Next we prove that 119910(119909) satisfies (53) Assume to thecontrary that
int
1199050
0
100381610038161003816100381610038161199101015840(119909)
10038161003816100381610038161003816119889119909 lt infin (80)
Then we have
length (119910) = int1199050
0
radic1 +10038161003816100381610038161199101015840(119909)1003816100381610038161003816
2119889119909
le int
1199050
0
(1 +
100381610038161003816100381610038161199101015840(119909)
10038161003816100381610038161003816) 119889119909 lt infin
(81)
From Lemma 29 it follows that
length (119910) = lim120576rarr+0
1003816100381610038161003816Γ120576(119910)1003816100381610038161003816
2120576
(82)
On the other hand by (79) we see that
lim120576rarr+0
1003816100381610038161003816Γ120576(119910)1003816100381610038161003816
2120576
= infin (83)
This is a contaradiction and hence 119910(119909) satisfies (53)Finally we show that 119910(119909) also satisfies inequality (54)
Because of (5) we obtain a constant 119888 gt 0 such that
max119905isinR
|119878 (119905)| = max119905isin[0120591]
|119878 (119905)| le 119888
max119905isinR
100381610038161003816100381610038161198781015840(119905)
10038161003816100381610038161003816= max119905isin[0120591]
100381610038161003816100381610038161198781015840(119905)
10038161003816100381610038161003816le 119888
(84)
By (15) we find that
sup119905isin(0119909]
1003816100381610038161003816119910 (119905)
1003816100381610038161003816= sup119905isin(0119909]
|119886 (119905)|1003816100381610038161003816119878 (120593 (119905))
1003816100381610038161003816le 119888119886 (119909) (85)
Hence from (1) and (15) it follows that100381610038161003816100381610038161199101015840(119909)
10038161003816100381610038161003816le 119888 (119886
1015840(119909) + 119886 (119909) [minus120593
1015840(119909)]) (86)
and therefore
int
1199050
119909
100381610038161003816100381610038161199101015840(120585)
10038161003816100381610038161003816119889120585 le 119888119886 (119905
0) + 119888 int
1199050
119909
119886 (120585) [minus1205931015840(120585)] 119889120585 (87)
Using (85) and (87) we see that
[119909 sup120585isin(0119909]
1003816100381610038161003816119910 (120585)
1003816100381610038161003816]
(119904minus1)(2minus119904)
int
1199050
119909
100381610038161003816100381610038161199101015840(120585)
10038161003816100381610038161003816119889120585
le 119888119886 (1199050) [119888119909119886 (119909)]
(119904minus1)(2minus119904)
+ 119888[119888119909119886 (119909)](119904minus1)(2minus119904)
int
1199050
119909
119886 (120585) [minus1205931015840(120585)] 119889120585
le 119888119886 (1199050) [1198881199050119886 (1199050)](119904minus1)(2minus119904)
+ 1198881(2minus119904)
[119909119886 (119909)](119904minus1)(2minus119904)
int
1199050
119909
119886 (120585) [minus1205931015840(120585)] 119889120585
(88)
By (18) we conclude that (54) holdsBy Lemma 28 there exists 119888
2gt 0 such that
1003816100381610038161003816Γ120576(119910)1003816100381610038161003816le 11988821205762minus119904 120576 isin (0 120576
0) (89)
Thus we have proved that the chirp function119910(119909) given by (1)satisfies the desired inequality (31) This completes the proofof Theorem 8
Proof of Theorem 11 Let 119910(119909) be the chirp function (1) It iseasy to see that 119910(119909) is oscillatory near 119909 = 0 By (5) thereexists119872 gt 0 such that
|119878 (119905)| le 119872
100381610038161003816100381610038161198781015840(119905)
10038161003816100381610038161003816le 119872 119905 isin R (90)
Hence we observe that100381610038161003816100381610038161199101015840(119909)
10038161003816100381610038161003816le
100381610038161003816100381610038161198861015840(119909) 119878 (120593 (119909))
10038161003816100381610038161003816+
10038161003816100381610038161003816119886 (119909) 119878 (120593 (119909)) 120593
1015840(119909)
10038161003816100381610038161003816
le 119872
100381610038161003816100381610038161198861015840(119909)
10038161003816100381610038161003816+ 119872
10038161003816100381610038161003816119886 (119909) 120593
1015840(119909)
10038161003816100381610038161003816
(91)
for 119909 isin (0 1199050] From (81) it follows that length(119910) lt infin
Lemma 29 shows that
length (119910) = lim120576rarr0
1003816100381610038161003816Γ120576(119910)1003816100381610038161003816
2120576
(92)
which implies that 120576length(Γ) le |Γ120576(119910)| le 4120576length(Γ) 120576 isin
(0 1205760) for some 120576
0gt 0 Therefore 119910(119909) is fractal oscillatory
near 119909 = 0 with the fractal dimension 1
4 Proof for the Fractal Oscillations of (26)In this section we give the proofs for the fractal oscillations ofthe linear second-order differential equation (26) consideredas an application of the main results on the fractal oscillationof chirp functions
Before we present the proofs of Theorems 16 and 17 wemake the following observation Since
1199101(119909) = 119886 (119909) cos (120593 (119909)) 119910
2(119909) = 119886 (119909) sin (120593 (119909))
(93)
are solutions of (26) we see that 119910(119909) = 11988811199101(119909) + 119888
21199102(119909)
is a fundamental system of all solutions of (26) Assume that1198882
1+ 1198882
2gt 0 and set 119878(119905) = 119888
1sin 119905 + 119888
2cos 119905 Then 119878(119905) clearly
satisfies (5)
10 International Journal of Differential Equations
Proof of Theorem 16 Applying Theorem 8 on 119910(119909) =
11988811199101(119909) + 119888
21199102(119909) we conclude that 119910(119909) is fractal oscillatory
with the fractal dimension 119904
Proof of Theorem 17 Theorem 11 implies that 119910(119909) =
11988811199101(119909) + 119888
21199102(119909) is fractal oscillatory near 119909 = 0 with the
fractal dimension 1
Appendix
Proof of Remark 9 By (17) there exists 1198881gt 0 such that
1198881le [minus120593
1015840(119909)]
2minus119904
int
119909
0
119886 (120585) 119889120585 119909 isin (0 1199050] (A1)
From (15) it follows that
1198881le [minus120593
1015840(119909)]
2minus119904
119886 (119909) int
119909
0
119889120585 = 119909119886 (119909) [minus1205931015840(119909)]
2minus119904
119909 isin (0 1199050]
(A2)
which implies that1
minus1205931015840(119909)
le 119888minus1(2minus119904)
11199091(2minus119904)
[119886 (119909)]1(2minus119904)
119909 isin (0 1199050] (A3)
By (19) there exists 1198882gt 0 such that
119909119886 (119909) [minus1205931015840(119909)]
2minus119904
le 1198882 119909 isin (0 119905
0] (A4)
and hence
minus1205931015840(119909) le 119888
1(2minus119904)
2119909minus1(2minus119904)
[119886 (119909)]minus1(2minus119904)
119909 isin (0 1199050]
(A5)Therefore we see that
[
1
minus1205931015840(119909)
]
119904minus1
int
1199050
119909
119886 (120585) [minus1205931015840(120585)] 119889120585
le 119888minus(119904minus1)(2minus119904)
1119909(119904minus1)(2minus119904)
[119886 (119909)](119904minus1)(2minus119904)
1198881(2minus119904)
2
times int
1199050
119909
120585minus1(2minus119904)
[119886 (120585)]minus(119904minus1)(2minus119904)
119889120585
le 1198883119909(119904minus1)(2minus119904)
[119886 (119909)](119904minus1)(2minus119904)
[119886 (119909)]minus(119904minus1)(2minus119904)
times int
1199050
119909
120585minus1(2minus119904)
119889120585
= 1198883
2 minus 119904
119904 minus 1
(1 minus 119909(119904minus1)(2minus119904)
119905minus(119904minus1)(2minus119904)
0)
le 1198883
2 minus 119904
119904 minus 1
119909 isin (0 1199050]
(A6)
so that (20) holds We conclude that
[119909119886 (119909)](119904minus1)(2minus119904)
int
1199050
119909
119886 (120585) [minus1205931015840(120585)] 119889120585
= (119909119886 (119909) [minus1205931015840(119909)]
2minus119904
)
(119904minus1)(2minus119904)
times ([
1
minus1205931015840(119909)
]
119904minus1
int
1199050
119909
119886 (120585) [minus1205931015840(120585)] 119889120585)
(A7)
By (19) and (20) we conclude that (18) is satisfied The proofof Remark 9 is complete
References
[1] C Tricot Curves and Fractal Dimension Springer New YorkNY USA 1995
[2] M Pasic ldquoFractal oscillations for a class of second order lineardifferential equations of Euler typerdquo Journal of MathematicalAnalysis and Applications vol 341 no 1 pp 211ndash223 2008
[3] M K Kwong M Pasic and J S W Wong ldquoRectifiable oscil-lations in second-order linear differential equationsrdquo Journal ofDifferential Equations vol 245 no 8 pp 2333ndash2351 2008
[4] M Pasic and J S W Wong ldquoRectifiable oscillations in second-order half-linear differential equationsrdquo Annali di MatematicaPura ed Applicata Series 4 vol 188 no 3 pp 517ndash541 2009
[5] M Pasic and S Tanaka ldquoFractal oscillations of self-adjoint anddamped linear differential equations of second-orderrdquo AppliedMathematics and Computation vol 218 no 5 pp 2281ndash22932011
[6] Y Naito M Pasic S Tanaka and D Zubrinic ldquoFractal oscilla-tions near domain boundary of radially symmetric solutions ofp-Laplace equations fractal geometry and dynamical systemsin pure and applied mathematicsrdquo Contemporary MathematicsAmerican Mathematical Society In press
[7] W A Coppel Stability and Asymptotic Behavior of DifferentialEquations D C Heath andCompany BostonMass USA 1965
[8] P Hartman Ordinary Differential Equations BirkhauserBoston Mass USA 2nd edition 1982
[9] F Neuman ldquoA general construction of linear differentialequations with solutions of prescribed propertiesrdquo AppliedMathematics Letters vol 17 no 1 pp 71ndash76 2004
[10] F Neuman ldquoStructure of solution spaces via transformationrdquoApplied Mathematics Letters vol 21 no 5 pp 529ndash533 2008
[11] P Borgnat and P Flandrin ldquoOn the chirp decomposition ofWeierstrass-Mandelbrot functions and their time-frequencyinterpretationrdquo Applied and Computational Harmonic Analysisvol 15 no 2 pp 134ndash146 2003
[12] E J Candes P R Charlton and H Helgason ldquoDetectinghighly oscillatory signals by chirplet path pursuitrdquo Applied andComputational Harmonic Analysis vol 24 no 1 pp 14ndash402008
[13] S Jaffard andYMeyer ldquoWaveletmethods for pointwise regular-ity and local oscillations of functionsrdquoMemoirs of the AmericanMathematical Society vol 123 no 587 pp 1ndash110 1996
[14] Y Meyer and H Xu ldquoWavelet analysis and chirpsrdquo Applied andComputational Harmonic Analysis vol 4 no 4 pp 366ndash3791997
[15] G RenQChen P Cerejeiras andUKaehle ldquoChirp transformsand chirp seriesrdquo Journal of Mathematical Analysis and Applica-tions vol 373 no 2 pp 356ndash369 2011
[16] M Kepesi and L Weruaga ldquoAdaptive chirp-based time-frequency analysis of speech signalsrdquo Speech Communicationvol 48 no 5 pp 474ndash492 2006
[17] L Weruaga and M Kepesi ldquoThe fan-chirp transform for non-stationary harmonic signalsrdquo Signal Processing vol 87 no 6 pp1504ndash1522 2007
[18] E Barlow A J Mulholland A Nordon and A GachaganldquoTheoretical analysis of chirp excitation of contrast agentsrdquoPhysics Procedia vol 3 no 1 pp 743ndash747 2009
International Journal of Differential Equations 11
[19] M H Pedersen T X Misaridis and J A Jensen ldquoClinicalevaluation of chirp-coded excitation in medical ultrasoundrdquoUltrasound in Medicine and Biology vol 29 no 6 pp 895ndash9052003
[20] T Paavle M Min and T Parve ldquoUsing of chirp excitation forbioimpedance estimation theoretical aspects andmodelingrdquo inProceedings of the 11th International Biennial Baltic ElectronicsConference (BECrsquo08) pp 325ndash328 Tallinn Estonia October2008
[21] M Pasic ldquoRectifiable and unrectifiable oscillations for a class ofsecond-order linear differential equations of Euler typerdquo Journalof Mathematical Analysis and Applications vol 335 no 1 pp724ndash738 2007
[22] K Falconer Fractal Geometry Mathematical Foundations andApplications John Wiley amp Sons Hoboken NJ USA 1999
[23] P Mattila Geometry of Sets and Measures in Euclidean SpacesFractals and Rectifiability vol 44 of Cambridge Studies inAdvanced Mathematics Cambridge University Press Cam-bridge UK 1995
[24] K J Falconer ldquoOn the Minkowski measurability of fractalsrdquoProceedings of the American Mathematical Society vol 123 no4 pp 1115ndash1124 1995
[25] C Q He and M L Lapidus Generalized Minkowski ContentSpectrum of Fractal Drums Fractal Strings and the RiemannZeta-Function vol 127 of Memoirs of the American Mathemat-ical Society American Mathematical Society Providence RIUSA 1997
[26] M L Lapidus and M van Frankenhuijsen Fractal geometryComplex Dimensions and Zeta Functions Geometry and Spec-tra of Fractal Strings Springer Monographs in MathematicsSpringer New York NY USA 2006
[27] M Pasic ldquoMinkowski-Bouligand dimension of solutions ofthe one-dimensional 119901-Laplacianrdquo Journal of Differential Equa-tions vol 190 no 1 pp 268ndash305 2003
[28] J S W Wong ldquoOn rectifiable oscillation of Euler type secondorder linear differential equationsrdquo Electronic Journal of Quali-tativeTheory of Differential Equations vol 2007 no 20 pp 1ndash122007
[29] M Pasic ldquoRectifiable and unrectifiable oscillations for a gen-eralization of the Riemann-Weber version of Euler differentialequationrdquo Georgian Mathematical Journal vol 15 no 4 pp759ndash774 2008
[30] M Pasic and S Tanaka ldquoRectifiable oscillations of self-adjointand damped linear differential equations of second-orderrdquoJournal of Mathematical Analysis and Applications vol 381 no1 pp 27ndash42 2011
[31] M Pasic D Zubrinic and V Zupanovic ldquoOscillatory andphase dimensions of solutions of some second-order differentialequationsrdquo Bulletin des Sciences Mathematiques vol 133 no 8pp 859ndash874 2009
[32] L Korkut and M Resman ldquoFractal oscillations of chirp-likefunctionsrdquo Georgian Mathematical Journal vol 19 no 4 pp705ndash720 2012
Submit your manuscripts athttpwwwhindawicom
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
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Mathematical Problems in Engineering
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Differential EquationsInternational Journal of
Volume 2014
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OptimizationJournal of
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International Journal of
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Operations ResearchAdvances in
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Function Spaces
Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of Mathematics and Mathematical Sciences
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Algebra
Discrete Dynamics in Nature and Society
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
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Decision SciencesAdvances in
Discrete MathematicsJournal of
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Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Stochastic AnalysisInternational Journal of
International Journal of Differential Equations 3
curve in R2 in respect to the Minkowski-Bouligand dimen-sion dim
119872Γ(119910) = 119904 (box-counting dimension) and the 119904-
dimensional upper and lowerMinkowski contents119872lowast119904(Γ(119910))and119872119904
lowast(Γ(119910)) defined by
dim119872Γ (119910) = lim
120576rarr+0
(2 minus
log 1003816100381610038161003816Γ120576(119910)1003816100381610038161003816
log 120576)
119872lowast119904(Γ (119910)) = lim sup
120576rarr+0
(2120576)119904minus2 1003816100381610038161003816Γ120576(119910)1003816100381610038161003816
119872119904
lowast(Γ (119910)) = lim inf
120576rarr+0
(2120576)119904minus2 1003816100381610038161003816Γ120576(119910)1003816100381610038161003816
(10)
Here Γ120576(119910) denotes the 120576-neighbourhood of graph Γ(119910)
defined by
Γ120576(119910) = (119905
1 1199052) isin R2 119889 ((119905
1 1199052) Γ (119910)) le 120576 120576 gt 0
(11)
and 119889((1199051 1199052) Γ(119910)) denotes the distance from (119905
1 1199052) to Γ(119910)
and |Γ120576(119910)| denotes the Lebesgue measure of Γ
120576(119910) On the
box-counting dimension and the 119904-dimensional Minkowskicontent we refer the reader to [22ndash27]
The main fractal properties considered in the paper aregiven in the next definitions
Definition 5 For a given real number 119904 isin [1 2) and a function119910 isin 119862((0 119905
0]) which is bounded on (0 119905
0] and oscillatory
near 119909 = 0 it is said that 119910(119909) is fractal oscillatory near 119909 = 0with the fractal dimension 119904 if
dim119872Γ (119910) = 119904 0 lt 119872
119904
lowast(Γ (119910)) le 119872
lowast119904(Γ (119910)) lt infin
(12)
On the contrary if there is no any 119904 isin [1 2) such that Γ(119910)satisfies (12) then 119910(119909) is not fractal oscillatory near 119909 = 0
Fractal oscillations can be understood also as a refine-ment of rectifiable and nonrectifiable oscillations They arerecently studied in [3 4 21 28ndash30]
Example 6 The chirp functions
119910 (119909) = 119909 cos( 1119909
) 119910 (119909) = 119909 sin( 1119909
) (13)
are not fractal oscillatory near 119909 = 0 In fact119872lowast1(Γ(119910)) = infinand dim
119872Γ(119910) = 1 (see [2]) It also implies that119872lowast119904(Γ(119910)) =
0 for all 119904 isin (1 2) (see [1 23]) and thus the statement (12) isnot satisfied for any 119904 isin [1 2) Hence 119910
1and 1199102are not fractal
oscillatory near 119909 = 0
Example 7 Let 120588 gt 0 It is clear that the chirp functions
119910 (119909) = radic119909 cos (120588 log119909) 119910 (119909) = radic119909 sin (120588 log119909)(14)
are oscillatory near 119909 = 0 Moreover the length of Γ(119910)is finite (see [21]) and therefore we observe that 0 lt
1198721
lowast(Γ(119910)) le 119872
lowast1(Γ(119910)) lt infin and dim
119872Γ(119910) = 1 (see [22])
Thus such chirp functions are fractal oscillatory near 119909 = 0with the fractal dimension 1
In order to show that the chirp function (1) is fractaloscillatory near 119909 = 0 we need to impose on amplitude 119886(119909)the following additional structural condition
119886 isin 1198621((0 1199050]) 119886
1015840(119909) ge 0 for 119909 isin (0 119905
0] (15)
Now we are able to state the first main result of the paper
Theorem 8 Let the functions 119886(119909) 120593(119909) and 119878(119905) satisfy thestructural conditions (3) (4) (5) (15) and
12059310158401015840(119909) gt 0 on (0 119905
0] lim sup
119909rarr0
(
1
minus1205931015840(119909)
)
1015840
lt infin (16)
Let the amplitude 119886(119909) and the phase 120593(119909) satisfy the followingasymptotic conditions near 119909 = 0
lim inf119909rarr0
([minus1205931015840(119909)]
2minus119904
int
119909
0
119886 (120585) 119889120585) gt 0 (17)
lim sup119909rarr0
[119909119886 (119909)](119904minus1)(2minus119904)
int
1199050
119909
119886 (120585) [minus1205931015840(120585)] 119889120585 lt infin (18)
where 119904 isin (1 2) is a given real numberThen the chirp function(1) is fractal oscillatory near 119909 = 0with the fractal dimension 119904
Remark 9 Let 119904 isin (1 2) Assume that (15) and (17) holdThen
lim sup119909rarr0
(119909119886 (119909) [minus1205931015840(119909)]
2minus119904
) lt infin (19)
implies
lim sup119909rarr0
[
1
minus1205931015840(119909)
]
119904minus1
int
1199050
119909
119886 (120585) [minus1205931015840(120585)] 119889120585 lt infin (20)
and (18) Therefore condition (18) is better than (19) inTheorem 8
The proof of Remark 9 is presented in the AppendixTheorem 8 can be rewritten in the term of 120596(119909) instead of
120593(119909) as follows
Corollary 10 Let the functions 119886(119909)120596(119909) and 119878(119905) satisfy thestructural conditions (3) (5) (15) and
120596 isin 1198621((0 1199050]) 120596 (119909) gt 0 120596
1015840(119909) lt 0
119900119899 (0 1199050] 120596 notin 119871
1((0 1199050])
lim sup119909rarr0
(
1
120596 (119909)
)
1015840
lt infin
(21)
Let the amplitude 119886(119909) and the instantaneous frequency 120596(119909)satisfy the following asymptotic conditions near 119909 = 0
lim inf119909rarr0
([120596 (119909)]2minus119904int
119909
0
119886 (120585) 119889120585) gt 0
lim sup119909rarr0
[119909119886 (119909)](119904minus1)(2minus119904)
int
1199050
119909
119886 (120585) 120596 (120585) 119889120585 lt infin
(22)
where 119904 isin (1 2) is a given real numberThen the chirp function(2) is fractal oscillatory near 119909 = 0with the fractal dimension 119904
4 International Journal of Differential Equations
Now we state analogous result to Theorem 8 in the caseof 119904 = 1 which will be proved in Section 3
Theorem 11 Assume that the function 119878(119905) satisfies (5) 119886 isin119862([0 119905
0])cap1198621((0 1199050])120593 isin 1198622((0 119905
0]) and lim
119909rarr+0120593(119909) = infin
If 1198861015840 isin 1198711((0 1199050]) and1205931015840119886 isin 1198711((0 119905
0]) then the chirp function
(1) is fractal oscillatory near 119909 = 0with the fractal dimension 1
In respect to some existing results on the fractal dimen-sion of graph of chirp functions previous theorems are themost simple and general It is because in [31 32] authorsrequire some extra conditions on the chirp function 119910(119909)which are not easy to be satisfied in the application forinstance the rapid convex-concave properties of 119910(119909) as in[31] and a condition on the curvature of 119910(119909) as in [32]
According to previous theorems we can show the fractaloscillations of the so-called (120572 120573)-chirp as well as logarithmicchirp functions
Example 12 We consider the (120572 120573)-chirp 119910(119909) = 119909120572119878(119909minus120573)where 119878(119905) = cos 119905 or 119878(119905) = sin 119905 and 120573 gt 120572 ge 0 It is fractaloscillatory near 119909 = 0 with the fractal dimension 119904 = 2 minus(1 + 120572)(1 + 120573) In fact it is easy to see that 119886(119909) = 119909120572 119878(119905)and 120593(119909) = 119909minus120573 satisfy (3) (4) (5) (15) and (16) When 119904 =2 minus (1 + 120572)(1 + 120573) we see that
[minus1205931015840(119909)]
2minus119904
int
119909
0
119886 (120585) 119889120585 =
120573(1+120572)(1+120573)
1 + 120572
gt 0
[119909119886 (119909)](119904minus1)(2minus119904)
int
1199050
119909
119886 (120585) [minus1205931015840(120585)] 119889120585
=
120573
120573 minus 120572
(1 minus 119909120573minus120572119905120572minus120573
0) le
120573
120573 minus 120572
(23)
FromTheorem 8 it follows that 119910(119909) = 119909120572119878(119909minus120573) 120573 gt 120572 ge 0is fractal oscillatory near 119909 = 0 with the fractal dimension119904 = 2 minus (1 + 120572)(1 + 120573)
Example 13 We consider the (120572 120573)-chirp 119910(119909) = 119909120572119878(119909minus120573)again where 119878(119905) = cos 119905 or 119878(119905) = sin 119905 Now we assume that0 lt 120573 lt 120572 ApplyingTheorem 8 we easily see that it is fractaloscillatory near 119909 = 0 with the fractal dimension 1
Example 14 We consider the logarithmic chirp functions
1199101(119909) = 119909
120574 cos (120588 log119909) 1199102(119909) = 119909
120574 sin (120588 log119909) (24)
where 120574 gt 0 and 120588 gt 0 Put 119878(119905) = cos(minus119905) or 119878(119905) = sin(minus119905)119886(119909) = 119909
120574 and 120593(119909) = minus120588 log119909 Then we easily see that119878(119905) satisfies (5) and that 1198861015840(119909) = 120574119909
120574minus1isin 1198711((0 1199050]) and
119886(119909)1205931015840(119909) = minus120588119909
120574minus1isin 1198711((0 1199050]) Theorem 11 implies that
1199101(119909) and 119910
2(119909) are fractal oscillatory near 119909 = 0 with the
fractal dimension 1
Question 1 Is it possible to applyTheorem 8 on the exponen-tial chirp given in Example 3(iii)
Example 15 Let 1199101(119909) = 119886(119909) cos(120593(119909)) and 119910
2(119909) =
119886(119909) sin(120593(119909)) 119909 isin (0 1199050] where the amplitude 119886(119909) and
the phase 120593(119909) satisfy all assumptions of Theorem 8 for anarbitrary given 119904 isin (1 2) Then the chirp function 119910(119909) =11988811199101(119909) + 119888
21199102(119909) 11988821+ 1198882
2gt 0 is also fractal oscillatory
near 119909 = 0 with the fractal dimension 119904 Indeed similarlyas in Example 4 119910(119909) can be rewritten in the form 119910(119909) =
1198891sin(120593(119909) + 119889
2) where 119889
1= 0 and 119889
2isin R It is clear that the
function 119878(119905) = 1198891sin(119905 + 119889
2) satisfies the required condition
(5) and henceTheorem 8 proves that119910(119909) is fractal oscillatorynear 119909 = 0 with the fractal dimension 119904
Next we pay attention to the fractal oscillations ofsolutions of linear differential equations generated by thesystem of functions as follows
119910 (119909) = 11988811199101(119909) + 119888
21199102(119909) 119888
2
1+ 1198882
2gt 0
where
1199101 (119909) = 119886 (119909) cos (120593 (119909)) 1199102 (119909) = 119886 (119909) sin (120593 (119909))
119909 isin (0 1199050]
(25)
It is not difficult to check that (25) is the fundamental systemof all solutions of the following linear differential equation
11991010158401015840+ (minus2119877
119886minus 1198771205931015840) 1199101015840+ (minus119877
1015840
119886+ 1198772
119886+ 1198771198861198771205931015840 + (120593
1015840)
2
) 119910 = 0
119909 isin (0 1199050]
(26)
where 119877119891= 119877119891(119909) = 119891
1015840(119909)119891(119909) for some 1198621-function 119891 =
119891(119909)
Theorem 16 Let the functions 119886 120593 isin 1198622((0 1199050]) satisfy struc-
tural conditions (3) (4) and (15) as well as the conditions (16)(17) and (18) in respect to a given real number 119904 isin (1 2)Then every nontrivial solution 119910 isin 1198622((0 119905
0]) of (26) is fractal
oscillatory near 119909 = 0 with the fractal dimension 119904
With the help ofTheorem 11 we can state analogous resultto Theorem 16 in the case of 119904 = 1
Theorem 17 Assume that 119886 isin 119862([0 1199050])cap1198622((0 1199050]) 119886(119909) = 0
for 119909 isin (0 1199050] 120593 isin 119862
2((0 1199050]) lim
119909rarr+0120593(119909) = infin 1198861015840 isin
1198711((0 1199050]) and 1205931015840119886 isin 1198711((0 119905
0]) Then every nontrivial solu-
tion 119910 isin 1198622((0 1199050]) of (26) is fractal oscillatory near 119909 = 0
with the fractal dimension 1
The previous two theorems will be proved in Section 4Assumptions on the coefficients of (26) in general are dif-ferent to those considered in [2ndash5]
When 1198861015840119886 = minus(12)120593101584010158401205931015840 then (26) becomes the un-damped equation
11991010158401015840+ (
1
2
119878der (1205931015840) + (120593
1015840)
2
)119910 = 0 119909 isin (0 1199050] (27)
International Journal of Differential Equations 5
where 119878der(119891) denotes the Schwarzian derivative of119891 definedby
119878der (119891) =11989110158401015840
119891
minus
3
2
(
1198911015840
119891
)
2
(28)
Hence from Theorem 16 we obtain the following conse-quence
Corollary 18 Let the functions 119886 isin 1198622((0 1199050]) and 120593 isin
1198623((0 1199050]) satisfy structural conditions (3) (4) and (15) as
well as the conditions (16) (17) and (18) in respect to a givenreal number 119904 isin (1 2) Let 1198861015840119886 = minus(12)120593101584010158401205931015840 Then everynontrivial solution 119910 isin 1198622((0 119905
0]) of (27) is fractal oscillatory
near 119909 = 0 with the fractal dimension 119904
As a consequence of Theorem 16 and Corollary 18 wederive the following examples for linear differential equationsof second order having all the solutions to be fractal oscilla-tory near 119909 = 0
Example 19 The so-called damped chirp equation
11991010158401015840+
120573 minus 2120572 + 1
119909
1199101015840+ (
1205732
1199092120573+2
minus
120572 (120573 minus 120572)
1199092
)119910 = 0
119909 isin (0 1199050]
(29)
is fractal oscillatory near 119909 = 0 with the fractal dimension2 minus (1 + 120572)(1 + 120573) where 120573 gt 120572 ge 0 When 119886(119909) = 119909120572 and120593(119909) = 119909
minus120573 (26) becomes (29) It is easy to see that (3) (4)(15) and (16) are satisfied In the same as in Example 12 wesee that (17) and (18) hold for 119904 = 2 minus (1 + 120572)(1 + 120573) HenceTheorem 16 proves that every nontrivial solution of (29) isfractal oscillatory near 119909 = 0 with the fractal dimension 2 minus(1 + 120572)(1 + 120573)
Nowwe assume that 0 lt 120573 lt 120572ThenTheorem 17 impliesthat (29) is fractal oscillatory near 119909 = 0 with the fractaldimension 1
Example 20 The following equation
11991010158401015840+
1 minus 2120574
119909
1199101015840+
1205742+ 1205882
1199092119910 = 0 119909 isin (0 119905
0] (30)
is fractal oscillatory near 119909 = 0 with the fractal dimension1 where 120574 gt 0 and 120588 gt 0 In the case where 119886(119909) = 119909
120574
and 120593(119909) = minus120588 log119909 (26) becomes (30) We see that 119886 isin119862([0 119905
0]) cap 119862
2((0 1199050]) 120593 isin 1198622((0 119905
0]) lim
119909rarr+0120593(119909) = infin
1198861015840isin 1198711((0 1199050]) and 1205931015840119886 isin 1198711((0 119905
0]) Therefore Theorem 17
implies that every nontrivial solution of (30) is fractaloscillatory near 119909 = 0 with the fractal dimension 1
Question 2 What can we say about the application of Theo-rem 16 on the case of 120593(119909) given in Example 3(iii)
At the end of this section we suggest that the readerstudies some invariant properties of fractal oscillations of thechirp function (1) in respect to the translation and reflexionAnalogously to Definitions 1 and 5 one can define the fractaloscillations near an arbitrary real point 119909 = 119909
0as follows
minus05
minus025 minus02 minus015 minus01
minus01
minus03
05
1
minus005
Figure 2 119910 is oscillatory near 119909 = 0 from the left side
Definition 21 Let 1199090isin R and 120575 gt 0 A function 119910 isin
119862((1199090 1199090+ 120575]) is oscillatory near 119909 = 119909
0 if there is a
decreasing sequence 119887119899isin (1199090 1199090+ 120575] such that 119910(119887
119899) = 0
for 119899 isin N and 119887119899 1199090as 119899 rarr infin Moreover if the graph
Γ(119910) satisfies the condition (12) for some 119904 isin [1 2) then 119910(119909)is said to be fractal oscillatory near 119909 = 119909
0with the fractal
dimension 119904
Definition 22 Let 1199090isin R and 120575 gt 0 It is said that a function
119910 isin 119862([1199090minus 120575 119909
0)) is fractal oscillatory near 119909 = 0 from
the left side with the fractal dimension 119904 isin [1 2) if there is anincreasing sequence 119886
119899isin [1199090minus 120575 119909
0) such that 119910(119886
119899) = 0 for
119899 isin N 119886119899 1199090as 119899 rarr infin and the graph Γ(119910) satisfies the
condition (12) see Figure 2
Definition 23 Let 1199090isin R and let 120575 gt 0 It is said that a
function 119910 isin 119862([1199090minus120575 1199090) cup (1199090 1199090+120575]) is two-sided fractal
oscillatory near 119909 = 0 with the fractal dimension 119904 isin [1 2)if there is an increasing sequence 119886
119899isin [1199090minus 120575 119909
0) and a
decreasing sequence 119887119899isin (1199090 1199090+ 120575] such that 119910(119886
119899) =
119910(119887119899) = 0 for 119899 isin N 119886
119899 1199090and 119887119899 1199090as 119899 rarr infin
and the graph Γ(119910) satisfies the condition (12) see Figure 3
With the help ofTheorem 8 we state the following result
Theorem 24 Let 1199090isin R and let 120575 gt 0 Let the functions
119886 120596 isin 1198621((0 120575]) satisfy structural conditions (3) (4) (5) (15)
and (16) as well as conditions (17) and (18) in respect to a givenreal number 119904 isin (1 2) Then one has
(i) the chirp function 119910(119909) = 119886(119909 minus 1199090)119878(120593(119909 minus 119909
0)) 119909 isin
(1199090 1199090+ 120575] is fractal oscillatory near 119909 = 119909
0with the
fractal dimension 119904
(ii) the chirp function 119910(119909) = 119886(1199090minus 119909)119878(120593(119909
0minus 119909)) 119909 isin
[1199090minus 120575 119909
0) is fractal oscillatory near 119909 = 119909
0from the
left side with the fractal dimension 119904
(iii) the chirp function 119910(119909) = 119886(|119909minus1199090|)119878(120593(|119909minus119909
0|)) 119909 isin
[1199090minus120575 1199090)cup(1199090 1199090+120575] is two-sided fractal oscillatory
near 119909 = 1199090with the fractal dimension 119904
6 International Journal of Differential Equations
minus03 minus02 minus01
minus1
01 02 03
minus05
1
05
Figure 3 119910 is two-sided oscillatory near 119909 = 0
3 Proof for the Fractal Oscillations ofChirp Functions
In this section we give the proofs of the main results dealingwith the fractal oscillations of chirp functions
By Definition 5 it follows that if for a prescribed realnumber 119904 isin (1 2) there are two positive constants 119888
1and 1198882
such that
11988811205762minus119904le1003816100381610038161003816Γ120576(119910)1003816100381610038161003816le 11988821205762minus119904 120576 isin (0 120576
0) (31)
for some 1205760gt 0 then a function 119910 isin 119862((0 119905
0]) is fractal
oscillatory near 119909 = 0 with the fractal dimension 119904 Thefollowing lemma plays the essential role in the proof of (31)
Lemma 25 Let 119910 isin 119862((0 1199050]) be a bounded function on
(0 1199050] and let 119886
119899isin (0 119905
0] be a decreasing sequence of con-
secutive zeros of 119910(119909) such that 119886119899rarr 0 Let 120576
0gt 0 and let
119896 = 119896(120576) be an index function satisfying1003816100381610038161003816119886119899minus 119886119899+1
1003816100381610038161003816le 120576 forall119899 ge 119896 (120576) 120576 isin (0 120576
0) (32)
Then1003816100381610038161003816Γ120576(119910)1003816100381610038161003816ge sum
119899ge119896(120576)
max119909isin[119886119899+1119886119899]
1003816100381610038161003816119910 (119909)
1003816100381610038161003816(119886119899minus 119886119899+1)
forall120576 isin (0 1205760)
(33)
Proof It is exactly the same as [3 pp 2350]
Let us remark that for some 1205760gt 0 we say that a function
119896 = 119896(120576) is an index function on (0 1205760) if 119896 (0 120576
0) rarr N and
lim120576rarr0
119896(120576) = infin
Lemma 26 Let 119910 isin 119862[119886 119887] Then
1003816100381610038161003816Γ120576(119910)1003816100381610038161003816le 4120587120576length (119910) + 41205871205762 for every 120576 gt 0 (34)
Proof Let 120576 gt 0 Set 1199091= 119886 and set
119909119894+1= max 119909 isin [119909
119894 119887] 119889 ((119905 119910 (119905)) (119909
119894 119910 (119909119894)))
le 120576 for 119905 isin [119909119894 119909]
(35)
for 119894 = 1 2 Then there exists 119899 ge 2 such that 119909119899= 119887 Set
119873 = max119899 isin N 119909119899lt 119887 We see that119873 ge 1
119886 = 1199091lt 1199092lt sdot sdot sdot lt 119909
119894lt 119909119894+1lt sdot sdot sdot lt 119909
119873lt 119909119873+1
= 119887 (36)
and if119873 ge 2 then
119889 ((119909119894 119910 (119909119894)) (119909
119894+1 119910 (119909119894+1))) = 120576 119894 = 1 2 119873 minus 1
(37)
We will show that
Γ120576(119910) sub
119873
⋃
119894=1
1198612120576(119909119894 119910 (119909119894)) (38)
where
1198612120576(1199051 1199052) = (120591
1 1205912) isin R2 119889 ((119905
1 1199052) (1205911 1205912)) le 2120576
(39)
Let (1205911 1205912) isin Γ
120576(119910) Then there exists 120585 isin [119886 119887] such that
119889((1205911 1205912) (120585 119910(120585))) le 120576 Because of the definition of 119909
119894+1 we
find that 120585 isin [119909119894 119909119894+1] for some 119894 isin 1 2 119873 so that
119889 ((120585 119910 (120585)) (119909119894 119910 (119909119894))) le 120576 (40)
Hence it follows that
119889 ((1205911 1205912) (119909119894 119910 (119909119894))) le 119889 ((120591
1 1205912) (120585 119910 (120585)))
+ 119889 ((120585 119910 (120585)) (119909119894 119910 (119909119894))) le 2120576
(41)
which means that (1205911 1205912) isin 119861
2120576(119909119894 119910(119909119894)) Therefore we
obtain (38) By (38) we conclude that
1003816100381610038161003816Γ120576(119910)1003816100381610038161003816le
119873
sum
119894=1
10038161003816100381610038161198612120576(119909119894 119910 (119909119894))1003816100381610038161003816= 4119873120587120576
2 (42)
When119873 = 1 from (42) it follows that1003816100381610038161003816Γ120576(119910)1003816100381610038161003816le 4120587120576
2le 4120587120576length (119910) + 41205871205762 (43)
Now we assume that119873 ge 2 We observe that
length (119910) =119873
sum
119894=1
length (119910|[119909119894 119909119894+1]
)
ge
119873
sum
119894=1
119889 ((119909119894 119910 (119909119894)) (119909
119894+1 119910 (119909119894+1)))
ge
119873minus1
sum
119894=1
119889 ((119909119894 119910 (119909119894)) (119909
119894+1 119910 (119909119894+1)))
= (119873 minus 1) 120576
(44)
that is
119873120576 le length (119910) + 120576 (45)
Combining (42) with (45) we obtain1003816100381610038161003816Γ120576(119910)1003816100381610038161003816le 4120587120576length (119910) + 41205871205762 (46)
International Journal of Differential Equations 7
Lemma 27 Let 119910 isin 119862(0 1199050] be bounded on (0 119905
0] and let
120576 gt 0 Then there exists 119862 gt 0 such that for every 120576 gt 0 and119905 isin (0 119905
0]
1003816100381610038161003816Γ120576(119910)1003816100381610038161003816le 119862[119905 sup
119909isin(0119905]
1003816100381610038161003816119910 (119909)
1003816100381610038161003816+ 120576length (119910|
[1199051199050]) + 120576 + 120576
2]
(47)
Proof Let 120576 gt 0 and let 119905 isin (0 1199050] From the geometry point
of view it is clear that1003816100381610038161003816Γ120576(119910|(0119905])1003816100381610038161003816sub 119877 (48)
where 119877 is a rectangle in R2 defined by
119877 = (1199051 1199052) 1199051isin [minus120576 119905 + 120576]
1199052isin [minus120576 minus sup
119909isin(0119905]
1003816100381610038161003816119910 (119909)
1003816100381610038161003816 120576 + sup119909isin(0119905]
1003816100381610038161003816119910 (119909)
1003816100381610038161003816]
(49)
Hence we have
1003816100381610038161003816Γ120576(119910(0119905])1003816100381610038161003816le (2120576 + 119905) (2120576 + 2 sup
119909isin(0119905]
1003816100381610038161003816119910 (119909)
1003816100381610038161003816)
= 41205762+ 2(119905 + 2 sup
119909isin(0119905]
1003816100381610038161003816119910 (119909)
1003816100381610038161003816) 120576
+ 2119905 sup119909isin(0119905]
1003816100381610038161003816119910 (119909)
1003816100381610038161003816
le 41205762+ 1198621120576 + 2119905 sup
119909isin(0119905]
1003816100381610038161003816119910 (119909)
1003816100381610038161003816
(50)
where 1198621= 2(1199050+ 2sup
119909isin(01199050]|119910(119909)|) Lemma 26 implies that
10038161003816100381610038161003816Γ120576(119910|[1199051199050])
10038161003816100381610038161003816le 4120587120576length (119910|
[1199051199050]) + 4120587120576
2 (51)
Consequently we see that1003816100381610038161003816Γ120576(119910)1003816100381610038161003816le1003816100381610038161003816Γ120576(119910|(0119905])1003816100381610038161003816+
10038161003816100381610038161003816Γ120576(119910|[1199051199050])
10038161003816100381610038161003816
le 41205762+ 1198621120576 + 2119905 sup
119909isin(0119905]
1003816100381610038161003816119910 (119909)
1003816100381610038161003816
+ 4120587120576length (119910|[1199051199050]) + 4120587120576
2
le 119862[119905 sup119909isin(0119905]
1003816100381610038161003816119910 (119909)
1003816100381610038161003816+ 120576length (119910|
[1199051199050]) + 120576 + 120576
2]
(52)
where 119862 = max1198621 4(120587 + 1)
Lemma 28 Let 119910 isin 1198621(0 1199050] be bounded on (0 119905
0] Assume
that
int
1199050
0
100381610038161003816100381610038161199101015840(119909)
10038161003816100381610038161003816119889119909 = infin (53)
lim sup119909rarr0
[119909 sup120585isin(0119909]
1003816100381610038161003816119910 (120585)
1003816100381610038161003816]
(119904minus1)(2minus119904)
int
1199050
119909
100381610038161003816100381610038161199101015840(120585)
10038161003816100381610038161003816119889120585 lt infin (54)
for some 119904 isin (1 2) Then there exists 1198882gt 0 such that
1003816100381610038161003816Γ120576(119910)1003816100381610038161003816le 11988821205762minus119904 120576 isin (0 1) (55)
Proof Let 120576 isin (0 1) and let 119904 isin (1 2) be satisfy (54) By (53)there exists 119909
1isin (0 119905
0) which depends on 120576 and 119904 such that
int
1199050
1199091
100381610038161003816100381610038161199101015840(119905)
10038161003816100381610038161003816119889119905 = 120576
1minus119904 (56)
Therefore we have
length (119910|[1199091 1199050]
) = int
1199050
1199091
radic1 +10038161003816100381610038161199101015840(119909)1003816100381610038161003816
2119889119909
le int
1199050
1199091
(1 +
100381610038161003816100381610038161199101015840(119909)
10038161003816100381610038161003816) 119889119909
= 1199050minus 1199091+ 1205761minus119904
le 1199050+ 1205761minus119904
(57)
By (54) there exists 1198621gt 0 such that
[119909 sup120585isin(0119909]
1003816100381610038161003816119910 (120585)
1003816100381610038161003816]
(119904minus1)(2minus119904)
int
1199050
119909
100381610038161003816100381610038161199101015840(120585)
10038161003816100381610038161003816119889120585 le 119862
1 119909 isin (0 119905
0]
(58)
which implies that
[
[
1199091sup120585isin(01199091]
1003816100381610038161003816119910 (120585)
1003816100381610038161003816]
]
(119904minus1)(2minus119904)
int
1199050
1199091
100381610038161003816100381610038161199101015840(120585)
10038161003816100381610038161003816119889120585 le 119862
1 (59)
that is
1199091sup120585isin(01199091]
1003816100381610038161003816119910 (120585)
1003816100381610038161003816le 11986221205762minus119904 (60)
where 1198622= 119862(2minus119904)(119904minus1)
1 From Lemma 27 (57) and (60) it
follows that1003816100381610038161003816Γ120576(119910)1003816100381610038161003816le 119862 [119862
21205762minus119904+ 120576 (1199050+ 1205761minus119904) + 120576 + 120576
2]
= 119862 [(1198622+ 1) + (119905
0+ 1) 120576
119904minus1+ 120576119904] 1205762minus119904
le 119862 (1198622+ 1199050+ 3) 120576
2minus119904 120576 isin (0 1)
(61)
In order to showTheorems 8 and 11 we need the followingtwo geometric lemmas
Lemma 29 (see [1]) If Γ sube R2 is a simple curve (ie itsparameterization is a bijection) and length(Γ) lt infin then
length (Γ) = lim120576rarr0
1003816100381610038161003816Γ120576
1003816100381610038161003816
2120576
(62)
where Γ120576denotes the 120576-neighborhood of the graph Γ
8 International Journal of Differential Equations
Now we are able to proveTheorem 8
Proof of Theorem 8 Let 119904 isin (1 2) and let 119910(119909) be a chirpfunction given by (1) We note here that it is enough to showthat 119910(119909) satisfies (31)
At the first let 119909119899be a sequence defined by 119909
119899= 120593minus1(1205910+
119899120591) for all sufficiently large 119899 isin N From (4) it follows that120593minus1(119905) is decreasing Hence 119909
119899is decreasing as well as 119909
119899rarr
0 as 119899 rarr infin because of lim119909rarr+0
120593(119909) = infin (see (4))We notethat 119910(119909
119899) = 0 and 119910(119909) = 0 on (119909
119899+1 119909119899) for all sufficiently
large 119899 isin N Also minus11205931015840(119909) is an increasing function becauseof (16) The mean value theorem shows that
120591
minus1205931015840(119909119899+1)
le 119909119899minus 119909119899+1le
120591
minus1205931015840(119909119899)
(63)
Now let 1205760isin (0 1) Let 119896(120576) be the smallest natural number
satisfying120591
minus1205931015840(119909119896(120576))
le 120576 forall120576 isin (0 1205760) (64)
Such 119896(120576) exists for every 120576 isin (0 1205760) since 119909
119899rarr 0 as 119899 rarr
infin and lim119909rarr+0
1205931015840(119909) = minusinfin (this equality is true because
1205931015840notin 1198711(0 1199050) since lim
119909rarr+0120593(119909) = infin) Moreover since 119909
119899
is decreasing and minus11205931015840(119909) is increasing we obtain
minus1205931015840(119909119899) ge 120591120576
minus1forall119899 ge 119896 (120576) (65)
Combining (63) and (65) it is easy to deduce that suchdefined 119896(120576) satisfies condition (32)
By (16) there exists 119871 gt 0 such that (1(minus1205931015840(119909)))1015840 le 119871 for119909 isin (0 119905
0] which means that
minus
minus12059310158401015840(119909)
minus1205931015840(119909)
le minus1198711205931015840(119909) 119909 isin (0 119905
0] (66)
Integrating (66) on [120593minus1(119905 + 2120591) 120593minus1(119905)] we have
logminus1205931015840(120593minus1(119905 + 2120591))
minus1205931015840(120593minus1(119905))
le 2119871120591 119905 isin [120593 (1199050) infin) (67)
that is
minus1205931015840(120593minus1(119905)) ge minus119890
minus21198711205911205931015840(120593minus1(119905 + 120591)) 119905 isin [120593 (119905
0) infin)
(68)
By the definition of 119896(120576) and (64) we see that120591
minus1205931015840(119909119896(120576)minus1
)
gt 120576 (69)
Hence from (68) it follows that
120591120576minus1gt minus1205931015840(119909119896(120576)minus1
)
= minus1205931015840(120593minus1(1205910+ (119896 (120576) minus 1) 120591))
ge minus119890minus2119871120591
1205931015840(120593minus1(1205910+ (119896 (120576) minus 1) 120591 + 2120591))
= minus119890minus2119871120591
1205931015840(120593minus1(1205910+ (119896 (120576) + 1) 120591))
= minus119890minus2119871120591
1205931015840(119909119896(120576)+1
)
(70)
which implies that
minus1205931015840(119909119896(120576)+1
) le 119872120576minus1 120576 gt 0 (71)
where119872 = 1205911198902119871120591
Next it is clear that the assumption (5) ensures a realnumber119872
0gt 0 such that
1198720= max119905isinR
|119878 (119905)| = max119905isin[0120591]
|119878 (119905)| (72)
Hence from (15) we have
max119909isin[119909119899+1 119909119899]
1003816100381610038161003816119910 (119909)
1003816100381610038161003816= max119909isin[119909119899+1119909119899]
(119886 (119909)1003816100381610038161003816119878 (120593 (119909))
1003816100381610038161003816)
ge 119886 (119909119899+1) max119909isin[119909119899+1119909119899]
1003816100381610038161003816119878 (120593 (119909))
1003816100381610038161003816
= 1198720119886 (119909119899+1)
(73)
Now from (63) and (73) it follows that
max119909isin[119909119899+1 119909119899]
1003816100381610038161003816119910 (119909)
1003816100381610038161003816(119909119899minus 119909119899+1) ge 119872
0119886 (119909119899+1) (119909119899minus 119909119899+1)
ge 1205911198720
119886 (119909119899+1)
minus1205931015840(119909119899+1)
= 1205911198720
119886 (120593minus1(1205910+(119899+1) 120591))
minus1205931015840(120593minus1(1205910+(119899+1) 120591))
(74)
Since 120593minus1(119905) is decreasing as well as minus11205931015840(119909) is increasingand 119886(119909) is nondecreasing and positive we conclude that thefunction
119886 (120593minus1(119905))
minus1205931015840(120593minus1(119905))
is nonincreasing (75)
Hence we observe that
sum
119899ge119896(120576)
max119909isin[119909119899+1119909119899]
1003816100381610038161003816119910 (119909)
1003816100381610038161003816(119909119899minus 119909119899+1)
ge 1205911198720sum
119899ge119896(120576)
119886 (120593minus1(1205910+ (119899 + 1) 120591))
minus1205931015840(120593minus1(1205910+ (119899 + 1) 120591))
ge 1205911198720int
infin
119896(120576)
119886 (120593minus1(1205910+ (119905 + 1) 120591))
minus1205931015840(120593minus1(1205910+ (119905 + 1) 120591))
119889119905
= 1198720int
120593minus1(1205910+(119896(120576)+1)120591)
0
119886 (120585) 119889120585
= 1198720int
119909119896(120576)+1
0
119886 (120585) 119889120585
(76)
Next from assumption (17) we get some1198721gt 0 such that
int
119909
0
119886 (120585) 119889120585 ge 1198721[minus1205931015840(119909)]
minus(2minus119904)
119909 isin (0 1199050] (77)
International Journal of Differential Equations 9
Now from (76) (77) and (71) we obtain
sum
119899ge119896(120576)
max119909isin[119909119899+1119909119899]
1003816100381610038161003816119910 (119909)
1003816100381610038161003816(119909119899minus 119909119899+1)
ge 11987201198721[minus1205931015840(119909119896(120576)+1
)]
minus(2minus119904)
ge 11987201198721(119872120576minus1)
minus(2minus119904)
= 11987201198721119872minus(2minus119904)
1205762minus119904
(78)
Thus using Lemma 25 we have
1003816100381610038161003816Γ120576(119910)1003816100381610038161003816ge 11988811205762minus119904 120576 isin (0 120576
0) (79)
for some 1198881gt 0
Next we prove that 119910(119909) satisfies (53) Assume to thecontrary that
int
1199050
0
100381610038161003816100381610038161199101015840(119909)
10038161003816100381610038161003816119889119909 lt infin (80)
Then we have
length (119910) = int1199050
0
radic1 +10038161003816100381610038161199101015840(119909)1003816100381610038161003816
2119889119909
le int
1199050
0
(1 +
100381610038161003816100381610038161199101015840(119909)
10038161003816100381610038161003816) 119889119909 lt infin
(81)
From Lemma 29 it follows that
length (119910) = lim120576rarr+0
1003816100381610038161003816Γ120576(119910)1003816100381610038161003816
2120576
(82)
On the other hand by (79) we see that
lim120576rarr+0
1003816100381610038161003816Γ120576(119910)1003816100381610038161003816
2120576
= infin (83)
This is a contaradiction and hence 119910(119909) satisfies (53)Finally we show that 119910(119909) also satisfies inequality (54)
Because of (5) we obtain a constant 119888 gt 0 such that
max119905isinR
|119878 (119905)| = max119905isin[0120591]
|119878 (119905)| le 119888
max119905isinR
100381610038161003816100381610038161198781015840(119905)
10038161003816100381610038161003816= max119905isin[0120591]
100381610038161003816100381610038161198781015840(119905)
10038161003816100381610038161003816le 119888
(84)
By (15) we find that
sup119905isin(0119909]
1003816100381610038161003816119910 (119905)
1003816100381610038161003816= sup119905isin(0119909]
|119886 (119905)|1003816100381610038161003816119878 (120593 (119905))
1003816100381610038161003816le 119888119886 (119909) (85)
Hence from (1) and (15) it follows that100381610038161003816100381610038161199101015840(119909)
10038161003816100381610038161003816le 119888 (119886
1015840(119909) + 119886 (119909) [minus120593
1015840(119909)]) (86)
and therefore
int
1199050
119909
100381610038161003816100381610038161199101015840(120585)
10038161003816100381610038161003816119889120585 le 119888119886 (119905
0) + 119888 int
1199050
119909
119886 (120585) [minus1205931015840(120585)] 119889120585 (87)
Using (85) and (87) we see that
[119909 sup120585isin(0119909]
1003816100381610038161003816119910 (120585)
1003816100381610038161003816]
(119904minus1)(2minus119904)
int
1199050
119909
100381610038161003816100381610038161199101015840(120585)
10038161003816100381610038161003816119889120585
le 119888119886 (1199050) [119888119909119886 (119909)]
(119904minus1)(2minus119904)
+ 119888[119888119909119886 (119909)](119904minus1)(2minus119904)
int
1199050
119909
119886 (120585) [minus1205931015840(120585)] 119889120585
le 119888119886 (1199050) [1198881199050119886 (1199050)](119904minus1)(2minus119904)
+ 1198881(2minus119904)
[119909119886 (119909)](119904minus1)(2minus119904)
int
1199050
119909
119886 (120585) [minus1205931015840(120585)] 119889120585
(88)
By (18) we conclude that (54) holdsBy Lemma 28 there exists 119888
2gt 0 such that
1003816100381610038161003816Γ120576(119910)1003816100381610038161003816le 11988821205762minus119904 120576 isin (0 120576
0) (89)
Thus we have proved that the chirp function119910(119909) given by (1)satisfies the desired inequality (31) This completes the proofof Theorem 8
Proof of Theorem 11 Let 119910(119909) be the chirp function (1) It iseasy to see that 119910(119909) is oscillatory near 119909 = 0 By (5) thereexists119872 gt 0 such that
|119878 (119905)| le 119872
100381610038161003816100381610038161198781015840(119905)
10038161003816100381610038161003816le 119872 119905 isin R (90)
Hence we observe that100381610038161003816100381610038161199101015840(119909)
10038161003816100381610038161003816le
100381610038161003816100381610038161198861015840(119909) 119878 (120593 (119909))
10038161003816100381610038161003816+
10038161003816100381610038161003816119886 (119909) 119878 (120593 (119909)) 120593
1015840(119909)
10038161003816100381610038161003816
le 119872
100381610038161003816100381610038161198861015840(119909)
10038161003816100381610038161003816+ 119872
10038161003816100381610038161003816119886 (119909) 120593
1015840(119909)
10038161003816100381610038161003816
(91)
for 119909 isin (0 1199050] From (81) it follows that length(119910) lt infin
Lemma 29 shows that
length (119910) = lim120576rarr0
1003816100381610038161003816Γ120576(119910)1003816100381610038161003816
2120576
(92)
which implies that 120576length(Γ) le |Γ120576(119910)| le 4120576length(Γ) 120576 isin
(0 1205760) for some 120576
0gt 0 Therefore 119910(119909) is fractal oscillatory
near 119909 = 0 with the fractal dimension 1
4 Proof for the Fractal Oscillations of (26)In this section we give the proofs for the fractal oscillations ofthe linear second-order differential equation (26) consideredas an application of the main results on the fractal oscillationof chirp functions
Before we present the proofs of Theorems 16 and 17 wemake the following observation Since
1199101(119909) = 119886 (119909) cos (120593 (119909)) 119910
2(119909) = 119886 (119909) sin (120593 (119909))
(93)
are solutions of (26) we see that 119910(119909) = 11988811199101(119909) + 119888
21199102(119909)
is a fundamental system of all solutions of (26) Assume that1198882
1+ 1198882
2gt 0 and set 119878(119905) = 119888
1sin 119905 + 119888
2cos 119905 Then 119878(119905) clearly
satisfies (5)
10 International Journal of Differential Equations
Proof of Theorem 16 Applying Theorem 8 on 119910(119909) =
11988811199101(119909) + 119888
21199102(119909) we conclude that 119910(119909) is fractal oscillatory
with the fractal dimension 119904
Proof of Theorem 17 Theorem 11 implies that 119910(119909) =
11988811199101(119909) + 119888
21199102(119909) is fractal oscillatory near 119909 = 0 with the
fractal dimension 1
Appendix
Proof of Remark 9 By (17) there exists 1198881gt 0 such that
1198881le [minus120593
1015840(119909)]
2minus119904
int
119909
0
119886 (120585) 119889120585 119909 isin (0 1199050] (A1)
From (15) it follows that
1198881le [minus120593
1015840(119909)]
2minus119904
119886 (119909) int
119909
0
119889120585 = 119909119886 (119909) [minus1205931015840(119909)]
2minus119904
119909 isin (0 1199050]
(A2)
which implies that1
minus1205931015840(119909)
le 119888minus1(2minus119904)
11199091(2minus119904)
[119886 (119909)]1(2minus119904)
119909 isin (0 1199050] (A3)
By (19) there exists 1198882gt 0 such that
119909119886 (119909) [minus1205931015840(119909)]
2minus119904
le 1198882 119909 isin (0 119905
0] (A4)
and hence
minus1205931015840(119909) le 119888
1(2minus119904)
2119909minus1(2minus119904)
[119886 (119909)]minus1(2minus119904)
119909 isin (0 1199050]
(A5)Therefore we see that
[
1
minus1205931015840(119909)
]
119904minus1
int
1199050
119909
119886 (120585) [minus1205931015840(120585)] 119889120585
le 119888minus(119904minus1)(2minus119904)
1119909(119904minus1)(2minus119904)
[119886 (119909)](119904minus1)(2minus119904)
1198881(2minus119904)
2
times int
1199050
119909
120585minus1(2minus119904)
[119886 (120585)]minus(119904minus1)(2minus119904)
119889120585
le 1198883119909(119904minus1)(2minus119904)
[119886 (119909)](119904minus1)(2minus119904)
[119886 (119909)]minus(119904minus1)(2minus119904)
times int
1199050
119909
120585minus1(2minus119904)
119889120585
= 1198883
2 minus 119904
119904 minus 1
(1 minus 119909(119904minus1)(2minus119904)
119905minus(119904minus1)(2minus119904)
0)
le 1198883
2 minus 119904
119904 minus 1
119909 isin (0 1199050]
(A6)
so that (20) holds We conclude that
[119909119886 (119909)](119904minus1)(2minus119904)
int
1199050
119909
119886 (120585) [minus1205931015840(120585)] 119889120585
= (119909119886 (119909) [minus1205931015840(119909)]
2minus119904
)
(119904minus1)(2minus119904)
times ([
1
minus1205931015840(119909)
]
119904minus1
int
1199050
119909
119886 (120585) [minus1205931015840(120585)] 119889120585)
(A7)
By (19) and (20) we conclude that (18) is satisfied The proofof Remark 9 is complete
References
[1] C Tricot Curves and Fractal Dimension Springer New YorkNY USA 1995
[2] M Pasic ldquoFractal oscillations for a class of second order lineardifferential equations of Euler typerdquo Journal of MathematicalAnalysis and Applications vol 341 no 1 pp 211ndash223 2008
[3] M K Kwong M Pasic and J S W Wong ldquoRectifiable oscil-lations in second-order linear differential equationsrdquo Journal ofDifferential Equations vol 245 no 8 pp 2333ndash2351 2008
[4] M Pasic and J S W Wong ldquoRectifiable oscillations in second-order half-linear differential equationsrdquo Annali di MatematicaPura ed Applicata Series 4 vol 188 no 3 pp 517ndash541 2009
[5] M Pasic and S Tanaka ldquoFractal oscillations of self-adjoint anddamped linear differential equations of second-orderrdquo AppliedMathematics and Computation vol 218 no 5 pp 2281ndash22932011
[6] Y Naito M Pasic S Tanaka and D Zubrinic ldquoFractal oscilla-tions near domain boundary of radially symmetric solutions ofp-Laplace equations fractal geometry and dynamical systemsin pure and applied mathematicsrdquo Contemporary MathematicsAmerican Mathematical Society In press
[7] W A Coppel Stability and Asymptotic Behavior of DifferentialEquations D C Heath andCompany BostonMass USA 1965
[8] P Hartman Ordinary Differential Equations BirkhauserBoston Mass USA 2nd edition 1982
[9] F Neuman ldquoA general construction of linear differentialequations with solutions of prescribed propertiesrdquo AppliedMathematics Letters vol 17 no 1 pp 71ndash76 2004
[10] F Neuman ldquoStructure of solution spaces via transformationrdquoApplied Mathematics Letters vol 21 no 5 pp 529ndash533 2008
[11] P Borgnat and P Flandrin ldquoOn the chirp decomposition ofWeierstrass-Mandelbrot functions and their time-frequencyinterpretationrdquo Applied and Computational Harmonic Analysisvol 15 no 2 pp 134ndash146 2003
[12] E J Candes P R Charlton and H Helgason ldquoDetectinghighly oscillatory signals by chirplet path pursuitrdquo Applied andComputational Harmonic Analysis vol 24 no 1 pp 14ndash402008
[13] S Jaffard andYMeyer ldquoWaveletmethods for pointwise regular-ity and local oscillations of functionsrdquoMemoirs of the AmericanMathematical Society vol 123 no 587 pp 1ndash110 1996
[14] Y Meyer and H Xu ldquoWavelet analysis and chirpsrdquo Applied andComputational Harmonic Analysis vol 4 no 4 pp 366ndash3791997
[15] G RenQChen P Cerejeiras andUKaehle ldquoChirp transformsand chirp seriesrdquo Journal of Mathematical Analysis and Applica-tions vol 373 no 2 pp 356ndash369 2011
[16] M Kepesi and L Weruaga ldquoAdaptive chirp-based time-frequency analysis of speech signalsrdquo Speech Communicationvol 48 no 5 pp 474ndash492 2006
[17] L Weruaga and M Kepesi ldquoThe fan-chirp transform for non-stationary harmonic signalsrdquo Signal Processing vol 87 no 6 pp1504ndash1522 2007
[18] E Barlow A J Mulholland A Nordon and A GachaganldquoTheoretical analysis of chirp excitation of contrast agentsrdquoPhysics Procedia vol 3 no 1 pp 743ndash747 2009
International Journal of Differential Equations 11
[19] M H Pedersen T X Misaridis and J A Jensen ldquoClinicalevaluation of chirp-coded excitation in medical ultrasoundrdquoUltrasound in Medicine and Biology vol 29 no 6 pp 895ndash9052003
[20] T Paavle M Min and T Parve ldquoUsing of chirp excitation forbioimpedance estimation theoretical aspects andmodelingrdquo inProceedings of the 11th International Biennial Baltic ElectronicsConference (BECrsquo08) pp 325ndash328 Tallinn Estonia October2008
[21] M Pasic ldquoRectifiable and unrectifiable oscillations for a class ofsecond-order linear differential equations of Euler typerdquo Journalof Mathematical Analysis and Applications vol 335 no 1 pp724ndash738 2007
[22] K Falconer Fractal Geometry Mathematical Foundations andApplications John Wiley amp Sons Hoboken NJ USA 1999
[23] P Mattila Geometry of Sets and Measures in Euclidean SpacesFractals and Rectifiability vol 44 of Cambridge Studies inAdvanced Mathematics Cambridge University Press Cam-bridge UK 1995
[24] K J Falconer ldquoOn the Minkowski measurability of fractalsrdquoProceedings of the American Mathematical Society vol 123 no4 pp 1115ndash1124 1995
[25] C Q He and M L Lapidus Generalized Minkowski ContentSpectrum of Fractal Drums Fractal Strings and the RiemannZeta-Function vol 127 of Memoirs of the American Mathemat-ical Society American Mathematical Society Providence RIUSA 1997
[26] M L Lapidus and M van Frankenhuijsen Fractal geometryComplex Dimensions and Zeta Functions Geometry and Spec-tra of Fractal Strings Springer Monographs in MathematicsSpringer New York NY USA 2006
[27] M Pasic ldquoMinkowski-Bouligand dimension of solutions ofthe one-dimensional 119901-Laplacianrdquo Journal of Differential Equa-tions vol 190 no 1 pp 268ndash305 2003
[28] J S W Wong ldquoOn rectifiable oscillation of Euler type secondorder linear differential equationsrdquo Electronic Journal of Quali-tativeTheory of Differential Equations vol 2007 no 20 pp 1ndash122007
[29] M Pasic ldquoRectifiable and unrectifiable oscillations for a gen-eralization of the Riemann-Weber version of Euler differentialequationrdquo Georgian Mathematical Journal vol 15 no 4 pp759ndash774 2008
[30] M Pasic and S Tanaka ldquoRectifiable oscillations of self-adjointand damped linear differential equations of second-orderrdquoJournal of Mathematical Analysis and Applications vol 381 no1 pp 27ndash42 2011
[31] M Pasic D Zubrinic and V Zupanovic ldquoOscillatory andphase dimensions of solutions of some second-order differentialequationsrdquo Bulletin des Sciences Mathematiques vol 133 no 8pp 859ndash874 2009
[32] L Korkut and M Resman ldquoFractal oscillations of chirp-likefunctionsrdquo Georgian Mathematical Journal vol 19 no 4 pp705ndash720 2012
Submit your manuscripts athttpwwwhindawicom
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
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Mathematical Problems in Engineering
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Differential EquationsInternational Journal of
Volume 2014
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Mathematical PhysicsAdvances in
Complex AnalysisJournal of
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OptimizationJournal of
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CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of
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Operations ResearchAdvances in
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Function Spaces
Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
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The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014
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Algebra
Discrete Dynamics in Nature and Society
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
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Decision SciencesAdvances in
Discrete MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom
Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Stochastic AnalysisInternational Journal of
4 International Journal of Differential Equations
Now we state analogous result to Theorem 8 in the caseof 119904 = 1 which will be proved in Section 3
Theorem 11 Assume that the function 119878(119905) satisfies (5) 119886 isin119862([0 119905
0])cap1198621((0 1199050])120593 isin 1198622((0 119905
0]) and lim
119909rarr+0120593(119909) = infin
If 1198861015840 isin 1198711((0 1199050]) and1205931015840119886 isin 1198711((0 119905
0]) then the chirp function
(1) is fractal oscillatory near 119909 = 0with the fractal dimension 1
In respect to some existing results on the fractal dimen-sion of graph of chirp functions previous theorems are themost simple and general It is because in [31 32] authorsrequire some extra conditions on the chirp function 119910(119909)which are not easy to be satisfied in the application forinstance the rapid convex-concave properties of 119910(119909) as in[31] and a condition on the curvature of 119910(119909) as in [32]
According to previous theorems we can show the fractaloscillations of the so-called (120572 120573)-chirp as well as logarithmicchirp functions
Example 12 We consider the (120572 120573)-chirp 119910(119909) = 119909120572119878(119909minus120573)where 119878(119905) = cos 119905 or 119878(119905) = sin 119905 and 120573 gt 120572 ge 0 It is fractaloscillatory near 119909 = 0 with the fractal dimension 119904 = 2 minus(1 + 120572)(1 + 120573) In fact it is easy to see that 119886(119909) = 119909120572 119878(119905)and 120593(119909) = 119909minus120573 satisfy (3) (4) (5) (15) and (16) When 119904 =2 minus (1 + 120572)(1 + 120573) we see that
[minus1205931015840(119909)]
2minus119904
int
119909
0
119886 (120585) 119889120585 =
120573(1+120572)(1+120573)
1 + 120572
gt 0
[119909119886 (119909)](119904minus1)(2minus119904)
int
1199050
119909
119886 (120585) [minus1205931015840(120585)] 119889120585
=
120573
120573 minus 120572
(1 minus 119909120573minus120572119905120572minus120573
0) le
120573
120573 minus 120572
(23)
FromTheorem 8 it follows that 119910(119909) = 119909120572119878(119909minus120573) 120573 gt 120572 ge 0is fractal oscillatory near 119909 = 0 with the fractal dimension119904 = 2 minus (1 + 120572)(1 + 120573)
Example 13 We consider the (120572 120573)-chirp 119910(119909) = 119909120572119878(119909minus120573)again where 119878(119905) = cos 119905 or 119878(119905) = sin 119905 Now we assume that0 lt 120573 lt 120572 ApplyingTheorem 8 we easily see that it is fractaloscillatory near 119909 = 0 with the fractal dimension 1
Example 14 We consider the logarithmic chirp functions
1199101(119909) = 119909
120574 cos (120588 log119909) 1199102(119909) = 119909
120574 sin (120588 log119909) (24)
where 120574 gt 0 and 120588 gt 0 Put 119878(119905) = cos(minus119905) or 119878(119905) = sin(minus119905)119886(119909) = 119909
120574 and 120593(119909) = minus120588 log119909 Then we easily see that119878(119905) satisfies (5) and that 1198861015840(119909) = 120574119909
120574minus1isin 1198711((0 1199050]) and
119886(119909)1205931015840(119909) = minus120588119909
120574minus1isin 1198711((0 1199050]) Theorem 11 implies that
1199101(119909) and 119910
2(119909) are fractal oscillatory near 119909 = 0 with the
fractal dimension 1
Question 1 Is it possible to applyTheorem 8 on the exponen-tial chirp given in Example 3(iii)
Example 15 Let 1199101(119909) = 119886(119909) cos(120593(119909)) and 119910
2(119909) =
119886(119909) sin(120593(119909)) 119909 isin (0 1199050] where the amplitude 119886(119909) and
the phase 120593(119909) satisfy all assumptions of Theorem 8 for anarbitrary given 119904 isin (1 2) Then the chirp function 119910(119909) =11988811199101(119909) + 119888
21199102(119909) 11988821+ 1198882
2gt 0 is also fractal oscillatory
near 119909 = 0 with the fractal dimension 119904 Indeed similarlyas in Example 4 119910(119909) can be rewritten in the form 119910(119909) =
1198891sin(120593(119909) + 119889
2) where 119889
1= 0 and 119889
2isin R It is clear that the
function 119878(119905) = 1198891sin(119905 + 119889
2) satisfies the required condition
(5) and henceTheorem 8 proves that119910(119909) is fractal oscillatorynear 119909 = 0 with the fractal dimension 119904
Next we pay attention to the fractal oscillations ofsolutions of linear differential equations generated by thesystem of functions as follows
119910 (119909) = 11988811199101(119909) + 119888
21199102(119909) 119888
2
1+ 1198882
2gt 0
where
1199101 (119909) = 119886 (119909) cos (120593 (119909)) 1199102 (119909) = 119886 (119909) sin (120593 (119909))
119909 isin (0 1199050]
(25)
It is not difficult to check that (25) is the fundamental systemof all solutions of the following linear differential equation
11991010158401015840+ (minus2119877
119886minus 1198771205931015840) 1199101015840+ (minus119877
1015840
119886+ 1198772
119886+ 1198771198861198771205931015840 + (120593
1015840)
2
) 119910 = 0
119909 isin (0 1199050]
(26)
where 119877119891= 119877119891(119909) = 119891
1015840(119909)119891(119909) for some 1198621-function 119891 =
119891(119909)
Theorem 16 Let the functions 119886 120593 isin 1198622((0 1199050]) satisfy struc-
tural conditions (3) (4) and (15) as well as the conditions (16)(17) and (18) in respect to a given real number 119904 isin (1 2)Then every nontrivial solution 119910 isin 1198622((0 119905
0]) of (26) is fractal
oscillatory near 119909 = 0 with the fractal dimension 119904
With the help ofTheorem 11 we can state analogous resultto Theorem 16 in the case of 119904 = 1
Theorem 17 Assume that 119886 isin 119862([0 1199050])cap1198622((0 1199050]) 119886(119909) = 0
for 119909 isin (0 1199050] 120593 isin 119862
2((0 1199050]) lim
119909rarr+0120593(119909) = infin 1198861015840 isin
1198711((0 1199050]) and 1205931015840119886 isin 1198711((0 119905
0]) Then every nontrivial solu-
tion 119910 isin 1198622((0 1199050]) of (26) is fractal oscillatory near 119909 = 0
with the fractal dimension 1
The previous two theorems will be proved in Section 4Assumptions on the coefficients of (26) in general are dif-ferent to those considered in [2ndash5]
When 1198861015840119886 = minus(12)120593101584010158401205931015840 then (26) becomes the un-damped equation
11991010158401015840+ (
1
2
119878der (1205931015840) + (120593
1015840)
2
)119910 = 0 119909 isin (0 1199050] (27)
International Journal of Differential Equations 5
where 119878der(119891) denotes the Schwarzian derivative of119891 definedby
119878der (119891) =11989110158401015840
119891
minus
3
2
(
1198911015840
119891
)
2
(28)
Hence from Theorem 16 we obtain the following conse-quence
Corollary 18 Let the functions 119886 isin 1198622((0 1199050]) and 120593 isin
1198623((0 1199050]) satisfy structural conditions (3) (4) and (15) as
well as the conditions (16) (17) and (18) in respect to a givenreal number 119904 isin (1 2) Let 1198861015840119886 = minus(12)120593101584010158401205931015840 Then everynontrivial solution 119910 isin 1198622((0 119905
0]) of (27) is fractal oscillatory
near 119909 = 0 with the fractal dimension 119904
As a consequence of Theorem 16 and Corollary 18 wederive the following examples for linear differential equationsof second order having all the solutions to be fractal oscilla-tory near 119909 = 0
Example 19 The so-called damped chirp equation
11991010158401015840+
120573 minus 2120572 + 1
119909
1199101015840+ (
1205732
1199092120573+2
minus
120572 (120573 minus 120572)
1199092
)119910 = 0
119909 isin (0 1199050]
(29)
is fractal oscillatory near 119909 = 0 with the fractal dimension2 minus (1 + 120572)(1 + 120573) where 120573 gt 120572 ge 0 When 119886(119909) = 119909120572 and120593(119909) = 119909
minus120573 (26) becomes (29) It is easy to see that (3) (4)(15) and (16) are satisfied In the same as in Example 12 wesee that (17) and (18) hold for 119904 = 2 minus (1 + 120572)(1 + 120573) HenceTheorem 16 proves that every nontrivial solution of (29) isfractal oscillatory near 119909 = 0 with the fractal dimension 2 minus(1 + 120572)(1 + 120573)
Nowwe assume that 0 lt 120573 lt 120572ThenTheorem 17 impliesthat (29) is fractal oscillatory near 119909 = 0 with the fractaldimension 1
Example 20 The following equation
11991010158401015840+
1 minus 2120574
119909
1199101015840+
1205742+ 1205882
1199092119910 = 0 119909 isin (0 119905
0] (30)
is fractal oscillatory near 119909 = 0 with the fractal dimension1 where 120574 gt 0 and 120588 gt 0 In the case where 119886(119909) = 119909
120574
and 120593(119909) = minus120588 log119909 (26) becomes (30) We see that 119886 isin119862([0 119905
0]) cap 119862
2((0 1199050]) 120593 isin 1198622((0 119905
0]) lim
119909rarr+0120593(119909) = infin
1198861015840isin 1198711((0 1199050]) and 1205931015840119886 isin 1198711((0 119905
0]) Therefore Theorem 17
implies that every nontrivial solution of (30) is fractaloscillatory near 119909 = 0 with the fractal dimension 1
Question 2 What can we say about the application of Theo-rem 16 on the case of 120593(119909) given in Example 3(iii)
At the end of this section we suggest that the readerstudies some invariant properties of fractal oscillations of thechirp function (1) in respect to the translation and reflexionAnalogously to Definitions 1 and 5 one can define the fractaloscillations near an arbitrary real point 119909 = 119909
0as follows
minus05
minus025 minus02 minus015 minus01
minus01
minus03
05
1
minus005
Figure 2 119910 is oscillatory near 119909 = 0 from the left side
Definition 21 Let 1199090isin R and 120575 gt 0 A function 119910 isin
119862((1199090 1199090+ 120575]) is oscillatory near 119909 = 119909
0 if there is a
decreasing sequence 119887119899isin (1199090 1199090+ 120575] such that 119910(119887
119899) = 0
for 119899 isin N and 119887119899 1199090as 119899 rarr infin Moreover if the graph
Γ(119910) satisfies the condition (12) for some 119904 isin [1 2) then 119910(119909)is said to be fractal oscillatory near 119909 = 119909
0with the fractal
dimension 119904
Definition 22 Let 1199090isin R and 120575 gt 0 It is said that a function
119910 isin 119862([1199090minus 120575 119909
0)) is fractal oscillatory near 119909 = 0 from
the left side with the fractal dimension 119904 isin [1 2) if there is anincreasing sequence 119886
119899isin [1199090minus 120575 119909
0) such that 119910(119886
119899) = 0 for
119899 isin N 119886119899 1199090as 119899 rarr infin and the graph Γ(119910) satisfies the
condition (12) see Figure 2
Definition 23 Let 1199090isin R and let 120575 gt 0 It is said that a
function 119910 isin 119862([1199090minus120575 1199090) cup (1199090 1199090+120575]) is two-sided fractal
oscillatory near 119909 = 0 with the fractal dimension 119904 isin [1 2)if there is an increasing sequence 119886
119899isin [1199090minus 120575 119909
0) and a
decreasing sequence 119887119899isin (1199090 1199090+ 120575] such that 119910(119886
119899) =
119910(119887119899) = 0 for 119899 isin N 119886
119899 1199090and 119887119899 1199090as 119899 rarr infin
and the graph Γ(119910) satisfies the condition (12) see Figure 3
With the help ofTheorem 8 we state the following result
Theorem 24 Let 1199090isin R and let 120575 gt 0 Let the functions
119886 120596 isin 1198621((0 120575]) satisfy structural conditions (3) (4) (5) (15)
and (16) as well as conditions (17) and (18) in respect to a givenreal number 119904 isin (1 2) Then one has
(i) the chirp function 119910(119909) = 119886(119909 minus 1199090)119878(120593(119909 minus 119909
0)) 119909 isin
(1199090 1199090+ 120575] is fractal oscillatory near 119909 = 119909
0with the
fractal dimension 119904
(ii) the chirp function 119910(119909) = 119886(1199090minus 119909)119878(120593(119909
0minus 119909)) 119909 isin
[1199090minus 120575 119909
0) is fractal oscillatory near 119909 = 119909
0from the
left side with the fractal dimension 119904
(iii) the chirp function 119910(119909) = 119886(|119909minus1199090|)119878(120593(|119909minus119909
0|)) 119909 isin
[1199090minus120575 1199090)cup(1199090 1199090+120575] is two-sided fractal oscillatory
near 119909 = 1199090with the fractal dimension 119904
6 International Journal of Differential Equations
minus03 minus02 minus01
minus1
01 02 03
minus05
1
05
Figure 3 119910 is two-sided oscillatory near 119909 = 0
3 Proof for the Fractal Oscillations ofChirp Functions
In this section we give the proofs of the main results dealingwith the fractal oscillations of chirp functions
By Definition 5 it follows that if for a prescribed realnumber 119904 isin (1 2) there are two positive constants 119888
1and 1198882
such that
11988811205762minus119904le1003816100381610038161003816Γ120576(119910)1003816100381610038161003816le 11988821205762minus119904 120576 isin (0 120576
0) (31)
for some 1205760gt 0 then a function 119910 isin 119862((0 119905
0]) is fractal
oscillatory near 119909 = 0 with the fractal dimension 119904 Thefollowing lemma plays the essential role in the proof of (31)
Lemma 25 Let 119910 isin 119862((0 1199050]) be a bounded function on
(0 1199050] and let 119886
119899isin (0 119905
0] be a decreasing sequence of con-
secutive zeros of 119910(119909) such that 119886119899rarr 0 Let 120576
0gt 0 and let
119896 = 119896(120576) be an index function satisfying1003816100381610038161003816119886119899minus 119886119899+1
1003816100381610038161003816le 120576 forall119899 ge 119896 (120576) 120576 isin (0 120576
0) (32)
Then1003816100381610038161003816Γ120576(119910)1003816100381610038161003816ge sum
119899ge119896(120576)
max119909isin[119886119899+1119886119899]
1003816100381610038161003816119910 (119909)
1003816100381610038161003816(119886119899minus 119886119899+1)
forall120576 isin (0 1205760)
(33)
Proof It is exactly the same as [3 pp 2350]
Let us remark that for some 1205760gt 0 we say that a function
119896 = 119896(120576) is an index function on (0 1205760) if 119896 (0 120576
0) rarr N and
lim120576rarr0
119896(120576) = infin
Lemma 26 Let 119910 isin 119862[119886 119887] Then
1003816100381610038161003816Γ120576(119910)1003816100381610038161003816le 4120587120576length (119910) + 41205871205762 for every 120576 gt 0 (34)
Proof Let 120576 gt 0 Set 1199091= 119886 and set
119909119894+1= max 119909 isin [119909
119894 119887] 119889 ((119905 119910 (119905)) (119909
119894 119910 (119909119894)))
le 120576 for 119905 isin [119909119894 119909]
(35)
for 119894 = 1 2 Then there exists 119899 ge 2 such that 119909119899= 119887 Set
119873 = max119899 isin N 119909119899lt 119887 We see that119873 ge 1
119886 = 1199091lt 1199092lt sdot sdot sdot lt 119909
119894lt 119909119894+1lt sdot sdot sdot lt 119909
119873lt 119909119873+1
= 119887 (36)
and if119873 ge 2 then
119889 ((119909119894 119910 (119909119894)) (119909
119894+1 119910 (119909119894+1))) = 120576 119894 = 1 2 119873 minus 1
(37)
We will show that
Γ120576(119910) sub
119873
⋃
119894=1
1198612120576(119909119894 119910 (119909119894)) (38)
where
1198612120576(1199051 1199052) = (120591
1 1205912) isin R2 119889 ((119905
1 1199052) (1205911 1205912)) le 2120576
(39)
Let (1205911 1205912) isin Γ
120576(119910) Then there exists 120585 isin [119886 119887] such that
119889((1205911 1205912) (120585 119910(120585))) le 120576 Because of the definition of 119909
119894+1 we
find that 120585 isin [119909119894 119909119894+1] for some 119894 isin 1 2 119873 so that
119889 ((120585 119910 (120585)) (119909119894 119910 (119909119894))) le 120576 (40)
Hence it follows that
119889 ((1205911 1205912) (119909119894 119910 (119909119894))) le 119889 ((120591
1 1205912) (120585 119910 (120585)))
+ 119889 ((120585 119910 (120585)) (119909119894 119910 (119909119894))) le 2120576
(41)
which means that (1205911 1205912) isin 119861
2120576(119909119894 119910(119909119894)) Therefore we
obtain (38) By (38) we conclude that
1003816100381610038161003816Γ120576(119910)1003816100381610038161003816le
119873
sum
119894=1
10038161003816100381610038161198612120576(119909119894 119910 (119909119894))1003816100381610038161003816= 4119873120587120576
2 (42)
When119873 = 1 from (42) it follows that1003816100381610038161003816Γ120576(119910)1003816100381610038161003816le 4120587120576
2le 4120587120576length (119910) + 41205871205762 (43)
Now we assume that119873 ge 2 We observe that
length (119910) =119873
sum
119894=1
length (119910|[119909119894 119909119894+1]
)
ge
119873
sum
119894=1
119889 ((119909119894 119910 (119909119894)) (119909
119894+1 119910 (119909119894+1)))
ge
119873minus1
sum
119894=1
119889 ((119909119894 119910 (119909119894)) (119909
119894+1 119910 (119909119894+1)))
= (119873 minus 1) 120576
(44)
that is
119873120576 le length (119910) + 120576 (45)
Combining (42) with (45) we obtain1003816100381610038161003816Γ120576(119910)1003816100381610038161003816le 4120587120576length (119910) + 41205871205762 (46)
International Journal of Differential Equations 7
Lemma 27 Let 119910 isin 119862(0 1199050] be bounded on (0 119905
0] and let
120576 gt 0 Then there exists 119862 gt 0 such that for every 120576 gt 0 and119905 isin (0 119905
0]
1003816100381610038161003816Γ120576(119910)1003816100381610038161003816le 119862[119905 sup
119909isin(0119905]
1003816100381610038161003816119910 (119909)
1003816100381610038161003816+ 120576length (119910|
[1199051199050]) + 120576 + 120576
2]
(47)
Proof Let 120576 gt 0 and let 119905 isin (0 1199050] From the geometry point
of view it is clear that1003816100381610038161003816Γ120576(119910|(0119905])1003816100381610038161003816sub 119877 (48)
where 119877 is a rectangle in R2 defined by
119877 = (1199051 1199052) 1199051isin [minus120576 119905 + 120576]
1199052isin [minus120576 minus sup
119909isin(0119905]
1003816100381610038161003816119910 (119909)
1003816100381610038161003816 120576 + sup119909isin(0119905]
1003816100381610038161003816119910 (119909)
1003816100381610038161003816]
(49)
Hence we have
1003816100381610038161003816Γ120576(119910(0119905])1003816100381610038161003816le (2120576 + 119905) (2120576 + 2 sup
119909isin(0119905]
1003816100381610038161003816119910 (119909)
1003816100381610038161003816)
= 41205762+ 2(119905 + 2 sup
119909isin(0119905]
1003816100381610038161003816119910 (119909)
1003816100381610038161003816) 120576
+ 2119905 sup119909isin(0119905]
1003816100381610038161003816119910 (119909)
1003816100381610038161003816
le 41205762+ 1198621120576 + 2119905 sup
119909isin(0119905]
1003816100381610038161003816119910 (119909)
1003816100381610038161003816
(50)
where 1198621= 2(1199050+ 2sup
119909isin(01199050]|119910(119909)|) Lemma 26 implies that
10038161003816100381610038161003816Γ120576(119910|[1199051199050])
10038161003816100381610038161003816le 4120587120576length (119910|
[1199051199050]) + 4120587120576
2 (51)
Consequently we see that1003816100381610038161003816Γ120576(119910)1003816100381610038161003816le1003816100381610038161003816Γ120576(119910|(0119905])1003816100381610038161003816+
10038161003816100381610038161003816Γ120576(119910|[1199051199050])
10038161003816100381610038161003816
le 41205762+ 1198621120576 + 2119905 sup
119909isin(0119905]
1003816100381610038161003816119910 (119909)
1003816100381610038161003816
+ 4120587120576length (119910|[1199051199050]) + 4120587120576
2
le 119862[119905 sup119909isin(0119905]
1003816100381610038161003816119910 (119909)
1003816100381610038161003816+ 120576length (119910|
[1199051199050]) + 120576 + 120576
2]
(52)
where 119862 = max1198621 4(120587 + 1)
Lemma 28 Let 119910 isin 1198621(0 1199050] be bounded on (0 119905
0] Assume
that
int
1199050
0
100381610038161003816100381610038161199101015840(119909)
10038161003816100381610038161003816119889119909 = infin (53)
lim sup119909rarr0
[119909 sup120585isin(0119909]
1003816100381610038161003816119910 (120585)
1003816100381610038161003816]
(119904minus1)(2minus119904)
int
1199050
119909
100381610038161003816100381610038161199101015840(120585)
10038161003816100381610038161003816119889120585 lt infin (54)
for some 119904 isin (1 2) Then there exists 1198882gt 0 such that
1003816100381610038161003816Γ120576(119910)1003816100381610038161003816le 11988821205762minus119904 120576 isin (0 1) (55)
Proof Let 120576 isin (0 1) and let 119904 isin (1 2) be satisfy (54) By (53)there exists 119909
1isin (0 119905
0) which depends on 120576 and 119904 such that
int
1199050
1199091
100381610038161003816100381610038161199101015840(119905)
10038161003816100381610038161003816119889119905 = 120576
1minus119904 (56)
Therefore we have
length (119910|[1199091 1199050]
) = int
1199050
1199091
radic1 +10038161003816100381610038161199101015840(119909)1003816100381610038161003816
2119889119909
le int
1199050
1199091
(1 +
100381610038161003816100381610038161199101015840(119909)
10038161003816100381610038161003816) 119889119909
= 1199050minus 1199091+ 1205761minus119904
le 1199050+ 1205761minus119904
(57)
By (54) there exists 1198621gt 0 such that
[119909 sup120585isin(0119909]
1003816100381610038161003816119910 (120585)
1003816100381610038161003816]
(119904minus1)(2minus119904)
int
1199050
119909
100381610038161003816100381610038161199101015840(120585)
10038161003816100381610038161003816119889120585 le 119862
1 119909 isin (0 119905
0]
(58)
which implies that
[
[
1199091sup120585isin(01199091]
1003816100381610038161003816119910 (120585)
1003816100381610038161003816]
]
(119904minus1)(2minus119904)
int
1199050
1199091
100381610038161003816100381610038161199101015840(120585)
10038161003816100381610038161003816119889120585 le 119862
1 (59)
that is
1199091sup120585isin(01199091]
1003816100381610038161003816119910 (120585)
1003816100381610038161003816le 11986221205762minus119904 (60)
where 1198622= 119862(2minus119904)(119904minus1)
1 From Lemma 27 (57) and (60) it
follows that1003816100381610038161003816Γ120576(119910)1003816100381610038161003816le 119862 [119862
21205762minus119904+ 120576 (1199050+ 1205761minus119904) + 120576 + 120576
2]
= 119862 [(1198622+ 1) + (119905
0+ 1) 120576
119904minus1+ 120576119904] 1205762minus119904
le 119862 (1198622+ 1199050+ 3) 120576
2minus119904 120576 isin (0 1)
(61)
In order to showTheorems 8 and 11 we need the followingtwo geometric lemmas
Lemma 29 (see [1]) If Γ sube R2 is a simple curve (ie itsparameterization is a bijection) and length(Γ) lt infin then
length (Γ) = lim120576rarr0
1003816100381610038161003816Γ120576
1003816100381610038161003816
2120576
(62)
where Γ120576denotes the 120576-neighborhood of the graph Γ
8 International Journal of Differential Equations
Now we are able to proveTheorem 8
Proof of Theorem 8 Let 119904 isin (1 2) and let 119910(119909) be a chirpfunction given by (1) We note here that it is enough to showthat 119910(119909) satisfies (31)
At the first let 119909119899be a sequence defined by 119909
119899= 120593minus1(1205910+
119899120591) for all sufficiently large 119899 isin N From (4) it follows that120593minus1(119905) is decreasing Hence 119909
119899is decreasing as well as 119909
119899rarr
0 as 119899 rarr infin because of lim119909rarr+0
120593(119909) = infin (see (4))We notethat 119910(119909
119899) = 0 and 119910(119909) = 0 on (119909
119899+1 119909119899) for all sufficiently
large 119899 isin N Also minus11205931015840(119909) is an increasing function becauseof (16) The mean value theorem shows that
120591
minus1205931015840(119909119899+1)
le 119909119899minus 119909119899+1le
120591
minus1205931015840(119909119899)
(63)
Now let 1205760isin (0 1) Let 119896(120576) be the smallest natural number
satisfying120591
minus1205931015840(119909119896(120576))
le 120576 forall120576 isin (0 1205760) (64)
Such 119896(120576) exists for every 120576 isin (0 1205760) since 119909
119899rarr 0 as 119899 rarr
infin and lim119909rarr+0
1205931015840(119909) = minusinfin (this equality is true because
1205931015840notin 1198711(0 1199050) since lim
119909rarr+0120593(119909) = infin) Moreover since 119909
119899
is decreasing and minus11205931015840(119909) is increasing we obtain
minus1205931015840(119909119899) ge 120591120576
minus1forall119899 ge 119896 (120576) (65)
Combining (63) and (65) it is easy to deduce that suchdefined 119896(120576) satisfies condition (32)
By (16) there exists 119871 gt 0 such that (1(minus1205931015840(119909)))1015840 le 119871 for119909 isin (0 119905
0] which means that
minus
minus12059310158401015840(119909)
minus1205931015840(119909)
le minus1198711205931015840(119909) 119909 isin (0 119905
0] (66)
Integrating (66) on [120593minus1(119905 + 2120591) 120593minus1(119905)] we have
logminus1205931015840(120593minus1(119905 + 2120591))
minus1205931015840(120593minus1(119905))
le 2119871120591 119905 isin [120593 (1199050) infin) (67)
that is
minus1205931015840(120593minus1(119905)) ge minus119890
minus21198711205911205931015840(120593minus1(119905 + 120591)) 119905 isin [120593 (119905
0) infin)
(68)
By the definition of 119896(120576) and (64) we see that120591
minus1205931015840(119909119896(120576)minus1
)
gt 120576 (69)
Hence from (68) it follows that
120591120576minus1gt minus1205931015840(119909119896(120576)minus1
)
= minus1205931015840(120593minus1(1205910+ (119896 (120576) minus 1) 120591))
ge minus119890minus2119871120591
1205931015840(120593minus1(1205910+ (119896 (120576) minus 1) 120591 + 2120591))
= minus119890minus2119871120591
1205931015840(120593minus1(1205910+ (119896 (120576) + 1) 120591))
= minus119890minus2119871120591
1205931015840(119909119896(120576)+1
)
(70)
which implies that
minus1205931015840(119909119896(120576)+1
) le 119872120576minus1 120576 gt 0 (71)
where119872 = 1205911198902119871120591
Next it is clear that the assumption (5) ensures a realnumber119872
0gt 0 such that
1198720= max119905isinR
|119878 (119905)| = max119905isin[0120591]
|119878 (119905)| (72)
Hence from (15) we have
max119909isin[119909119899+1 119909119899]
1003816100381610038161003816119910 (119909)
1003816100381610038161003816= max119909isin[119909119899+1119909119899]
(119886 (119909)1003816100381610038161003816119878 (120593 (119909))
1003816100381610038161003816)
ge 119886 (119909119899+1) max119909isin[119909119899+1119909119899]
1003816100381610038161003816119878 (120593 (119909))
1003816100381610038161003816
= 1198720119886 (119909119899+1)
(73)
Now from (63) and (73) it follows that
max119909isin[119909119899+1 119909119899]
1003816100381610038161003816119910 (119909)
1003816100381610038161003816(119909119899minus 119909119899+1) ge 119872
0119886 (119909119899+1) (119909119899minus 119909119899+1)
ge 1205911198720
119886 (119909119899+1)
minus1205931015840(119909119899+1)
= 1205911198720
119886 (120593minus1(1205910+(119899+1) 120591))
minus1205931015840(120593minus1(1205910+(119899+1) 120591))
(74)
Since 120593minus1(119905) is decreasing as well as minus11205931015840(119909) is increasingand 119886(119909) is nondecreasing and positive we conclude that thefunction
119886 (120593minus1(119905))
minus1205931015840(120593minus1(119905))
is nonincreasing (75)
Hence we observe that
sum
119899ge119896(120576)
max119909isin[119909119899+1119909119899]
1003816100381610038161003816119910 (119909)
1003816100381610038161003816(119909119899minus 119909119899+1)
ge 1205911198720sum
119899ge119896(120576)
119886 (120593minus1(1205910+ (119899 + 1) 120591))
minus1205931015840(120593minus1(1205910+ (119899 + 1) 120591))
ge 1205911198720int
infin
119896(120576)
119886 (120593minus1(1205910+ (119905 + 1) 120591))
minus1205931015840(120593minus1(1205910+ (119905 + 1) 120591))
119889119905
= 1198720int
120593minus1(1205910+(119896(120576)+1)120591)
0
119886 (120585) 119889120585
= 1198720int
119909119896(120576)+1
0
119886 (120585) 119889120585
(76)
Next from assumption (17) we get some1198721gt 0 such that
int
119909
0
119886 (120585) 119889120585 ge 1198721[minus1205931015840(119909)]
minus(2minus119904)
119909 isin (0 1199050] (77)
International Journal of Differential Equations 9
Now from (76) (77) and (71) we obtain
sum
119899ge119896(120576)
max119909isin[119909119899+1119909119899]
1003816100381610038161003816119910 (119909)
1003816100381610038161003816(119909119899minus 119909119899+1)
ge 11987201198721[minus1205931015840(119909119896(120576)+1
)]
minus(2minus119904)
ge 11987201198721(119872120576minus1)
minus(2minus119904)
= 11987201198721119872minus(2minus119904)
1205762minus119904
(78)
Thus using Lemma 25 we have
1003816100381610038161003816Γ120576(119910)1003816100381610038161003816ge 11988811205762minus119904 120576 isin (0 120576
0) (79)
for some 1198881gt 0
Next we prove that 119910(119909) satisfies (53) Assume to thecontrary that
int
1199050
0
100381610038161003816100381610038161199101015840(119909)
10038161003816100381610038161003816119889119909 lt infin (80)
Then we have
length (119910) = int1199050
0
radic1 +10038161003816100381610038161199101015840(119909)1003816100381610038161003816
2119889119909
le int
1199050
0
(1 +
100381610038161003816100381610038161199101015840(119909)
10038161003816100381610038161003816) 119889119909 lt infin
(81)
From Lemma 29 it follows that
length (119910) = lim120576rarr+0
1003816100381610038161003816Γ120576(119910)1003816100381610038161003816
2120576
(82)
On the other hand by (79) we see that
lim120576rarr+0
1003816100381610038161003816Γ120576(119910)1003816100381610038161003816
2120576
= infin (83)
This is a contaradiction and hence 119910(119909) satisfies (53)Finally we show that 119910(119909) also satisfies inequality (54)
Because of (5) we obtain a constant 119888 gt 0 such that
max119905isinR
|119878 (119905)| = max119905isin[0120591]
|119878 (119905)| le 119888
max119905isinR
100381610038161003816100381610038161198781015840(119905)
10038161003816100381610038161003816= max119905isin[0120591]
100381610038161003816100381610038161198781015840(119905)
10038161003816100381610038161003816le 119888
(84)
By (15) we find that
sup119905isin(0119909]
1003816100381610038161003816119910 (119905)
1003816100381610038161003816= sup119905isin(0119909]
|119886 (119905)|1003816100381610038161003816119878 (120593 (119905))
1003816100381610038161003816le 119888119886 (119909) (85)
Hence from (1) and (15) it follows that100381610038161003816100381610038161199101015840(119909)
10038161003816100381610038161003816le 119888 (119886
1015840(119909) + 119886 (119909) [minus120593
1015840(119909)]) (86)
and therefore
int
1199050
119909
100381610038161003816100381610038161199101015840(120585)
10038161003816100381610038161003816119889120585 le 119888119886 (119905
0) + 119888 int
1199050
119909
119886 (120585) [minus1205931015840(120585)] 119889120585 (87)
Using (85) and (87) we see that
[119909 sup120585isin(0119909]
1003816100381610038161003816119910 (120585)
1003816100381610038161003816]
(119904minus1)(2minus119904)
int
1199050
119909
100381610038161003816100381610038161199101015840(120585)
10038161003816100381610038161003816119889120585
le 119888119886 (1199050) [119888119909119886 (119909)]
(119904minus1)(2minus119904)
+ 119888[119888119909119886 (119909)](119904minus1)(2minus119904)
int
1199050
119909
119886 (120585) [minus1205931015840(120585)] 119889120585
le 119888119886 (1199050) [1198881199050119886 (1199050)](119904minus1)(2minus119904)
+ 1198881(2minus119904)
[119909119886 (119909)](119904minus1)(2minus119904)
int
1199050
119909
119886 (120585) [minus1205931015840(120585)] 119889120585
(88)
By (18) we conclude that (54) holdsBy Lemma 28 there exists 119888
2gt 0 such that
1003816100381610038161003816Γ120576(119910)1003816100381610038161003816le 11988821205762minus119904 120576 isin (0 120576
0) (89)
Thus we have proved that the chirp function119910(119909) given by (1)satisfies the desired inequality (31) This completes the proofof Theorem 8
Proof of Theorem 11 Let 119910(119909) be the chirp function (1) It iseasy to see that 119910(119909) is oscillatory near 119909 = 0 By (5) thereexists119872 gt 0 such that
|119878 (119905)| le 119872
100381610038161003816100381610038161198781015840(119905)
10038161003816100381610038161003816le 119872 119905 isin R (90)
Hence we observe that100381610038161003816100381610038161199101015840(119909)
10038161003816100381610038161003816le
100381610038161003816100381610038161198861015840(119909) 119878 (120593 (119909))
10038161003816100381610038161003816+
10038161003816100381610038161003816119886 (119909) 119878 (120593 (119909)) 120593
1015840(119909)
10038161003816100381610038161003816
le 119872
100381610038161003816100381610038161198861015840(119909)
10038161003816100381610038161003816+ 119872
10038161003816100381610038161003816119886 (119909) 120593
1015840(119909)
10038161003816100381610038161003816
(91)
for 119909 isin (0 1199050] From (81) it follows that length(119910) lt infin
Lemma 29 shows that
length (119910) = lim120576rarr0
1003816100381610038161003816Γ120576(119910)1003816100381610038161003816
2120576
(92)
which implies that 120576length(Γ) le |Γ120576(119910)| le 4120576length(Γ) 120576 isin
(0 1205760) for some 120576
0gt 0 Therefore 119910(119909) is fractal oscillatory
near 119909 = 0 with the fractal dimension 1
4 Proof for the Fractal Oscillations of (26)In this section we give the proofs for the fractal oscillations ofthe linear second-order differential equation (26) consideredas an application of the main results on the fractal oscillationof chirp functions
Before we present the proofs of Theorems 16 and 17 wemake the following observation Since
1199101(119909) = 119886 (119909) cos (120593 (119909)) 119910
2(119909) = 119886 (119909) sin (120593 (119909))
(93)
are solutions of (26) we see that 119910(119909) = 11988811199101(119909) + 119888
21199102(119909)
is a fundamental system of all solutions of (26) Assume that1198882
1+ 1198882
2gt 0 and set 119878(119905) = 119888
1sin 119905 + 119888
2cos 119905 Then 119878(119905) clearly
satisfies (5)
10 International Journal of Differential Equations
Proof of Theorem 16 Applying Theorem 8 on 119910(119909) =
11988811199101(119909) + 119888
21199102(119909) we conclude that 119910(119909) is fractal oscillatory
with the fractal dimension 119904
Proof of Theorem 17 Theorem 11 implies that 119910(119909) =
11988811199101(119909) + 119888
21199102(119909) is fractal oscillatory near 119909 = 0 with the
fractal dimension 1
Appendix
Proof of Remark 9 By (17) there exists 1198881gt 0 such that
1198881le [minus120593
1015840(119909)]
2minus119904
int
119909
0
119886 (120585) 119889120585 119909 isin (0 1199050] (A1)
From (15) it follows that
1198881le [minus120593
1015840(119909)]
2minus119904
119886 (119909) int
119909
0
119889120585 = 119909119886 (119909) [minus1205931015840(119909)]
2minus119904
119909 isin (0 1199050]
(A2)
which implies that1
minus1205931015840(119909)
le 119888minus1(2minus119904)
11199091(2minus119904)
[119886 (119909)]1(2minus119904)
119909 isin (0 1199050] (A3)
By (19) there exists 1198882gt 0 such that
119909119886 (119909) [minus1205931015840(119909)]
2minus119904
le 1198882 119909 isin (0 119905
0] (A4)
and hence
minus1205931015840(119909) le 119888
1(2minus119904)
2119909minus1(2minus119904)
[119886 (119909)]minus1(2minus119904)
119909 isin (0 1199050]
(A5)Therefore we see that
[
1
minus1205931015840(119909)
]
119904minus1
int
1199050
119909
119886 (120585) [minus1205931015840(120585)] 119889120585
le 119888minus(119904minus1)(2minus119904)
1119909(119904minus1)(2minus119904)
[119886 (119909)](119904minus1)(2minus119904)
1198881(2minus119904)
2
times int
1199050
119909
120585minus1(2minus119904)
[119886 (120585)]minus(119904minus1)(2minus119904)
119889120585
le 1198883119909(119904minus1)(2minus119904)
[119886 (119909)](119904minus1)(2minus119904)
[119886 (119909)]minus(119904minus1)(2minus119904)
times int
1199050
119909
120585minus1(2minus119904)
119889120585
= 1198883
2 minus 119904
119904 minus 1
(1 minus 119909(119904minus1)(2minus119904)
119905minus(119904minus1)(2minus119904)
0)
le 1198883
2 minus 119904
119904 minus 1
119909 isin (0 1199050]
(A6)
so that (20) holds We conclude that
[119909119886 (119909)](119904minus1)(2minus119904)
int
1199050
119909
119886 (120585) [minus1205931015840(120585)] 119889120585
= (119909119886 (119909) [minus1205931015840(119909)]
2minus119904
)
(119904minus1)(2minus119904)
times ([
1
minus1205931015840(119909)
]
119904minus1
int
1199050
119909
119886 (120585) [minus1205931015840(120585)] 119889120585)
(A7)
By (19) and (20) we conclude that (18) is satisfied The proofof Remark 9 is complete
References
[1] C Tricot Curves and Fractal Dimension Springer New YorkNY USA 1995
[2] M Pasic ldquoFractal oscillations for a class of second order lineardifferential equations of Euler typerdquo Journal of MathematicalAnalysis and Applications vol 341 no 1 pp 211ndash223 2008
[3] M K Kwong M Pasic and J S W Wong ldquoRectifiable oscil-lations in second-order linear differential equationsrdquo Journal ofDifferential Equations vol 245 no 8 pp 2333ndash2351 2008
[4] M Pasic and J S W Wong ldquoRectifiable oscillations in second-order half-linear differential equationsrdquo Annali di MatematicaPura ed Applicata Series 4 vol 188 no 3 pp 517ndash541 2009
[5] M Pasic and S Tanaka ldquoFractal oscillations of self-adjoint anddamped linear differential equations of second-orderrdquo AppliedMathematics and Computation vol 218 no 5 pp 2281ndash22932011
[6] Y Naito M Pasic S Tanaka and D Zubrinic ldquoFractal oscilla-tions near domain boundary of radially symmetric solutions ofp-Laplace equations fractal geometry and dynamical systemsin pure and applied mathematicsrdquo Contemporary MathematicsAmerican Mathematical Society In press
[7] W A Coppel Stability and Asymptotic Behavior of DifferentialEquations D C Heath andCompany BostonMass USA 1965
[8] P Hartman Ordinary Differential Equations BirkhauserBoston Mass USA 2nd edition 1982
[9] F Neuman ldquoA general construction of linear differentialequations with solutions of prescribed propertiesrdquo AppliedMathematics Letters vol 17 no 1 pp 71ndash76 2004
[10] F Neuman ldquoStructure of solution spaces via transformationrdquoApplied Mathematics Letters vol 21 no 5 pp 529ndash533 2008
[11] P Borgnat and P Flandrin ldquoOn the chirp decomposition ofWeierstrass-Mandelbrot functions and their time-frequencyinterpretationrdquo Applied and Computational Harmonic Analysisvol 15 no 2 pp 134ndash146 2003
[12] E J Candes P R Charlton and H Helgason ldquoDetectinghighly oscillatory signals by chirplet path pursuitrdquo Applied andComputational Harmonic Analysis vol 24 no 1 pp 14ndash402008
[13] S Jaffard andYMeyer ldquoWaveletmethods for pointwise regular-ity and local oscillations of functionsrdquoMemoirs of the AmericanMathematical Society vol 123 no 587 pp 1ndash110 1996
[14] Y Meyer and H Xu ldquoWavelet analysis and chirpsrdquo Applied andComputational Harmonic Analysis vol 4 no 4 pp 366ndash3791997
[15] G RenQChen P Cerejeiras andUKaehle ldquoChirp transformsand chirp seriesrdquo Journal of Mathematical Analysis and Applica-tions vol 373 no 2 pp 356ndash369 2011
[16] M Kepesi and L Weruaga ldquoAdaptive chirp-based time-frequency analysis of speech signalsrdquo Speech Communicationvol 48 no 5 pp 474ndash492 2006
[17] L Weruaga and M Kepesi ldquoThe fan-chirp transform for non-stationary harmonic signalsrdquo Signal Processing vol 87 no 6 pp1504ndash1522 2007
[18] E Barlow A J Mulholland A Nordon and A GachaganldquoTheoretical analysis of chirp excitation of contrast agentsrdquoPhysics Procedia vol 3 no 1 pp 743ndash747 2009
International Journal of Differential Equations 11
[19] M H Pedersen T X Misaridis and J A Jensen ldquoClinicalevaluation of chirp-coded excitation in medical ultrasoundrdquoUltrasound in Medicine and Biology vol 29 no 6 pp 895ndash9052003
[20] T Paavle M Min and T Parve ldquoUsing of chirp excitation forbioimpedance estimation theoretical aspects andmodelingrdquo inProceedings of the 11th International Biennial Baltic ElectronicsConference (BECrsquo08) pp 325ndash328 Tallinn Estonia October2008
[21] M Pasic ldquoRectifiable and unrectifiable oscillations for a class ofsecond-order linear differential equations of Euler typerdquo Journalof Mathematical Analysis and Applications vol 335 no 1 pp724ndash738 2007
[22] K Falconer Fractal Geometry Mathematical Foundations andApplications John Wiley amp Sons Hoboken NJ USA 1999
[23] P Mattila Geometry of Sets and Measures in Euclidean SpacesFractals and Rectifiability vol 44 of Cambridge Studies inAdvanced Mathematics Cambridge University Press Cam-bridge UK 1995
[24] K J Falconer ldquoOn the Minkowski measurability of fractalsrdquoProceedings of the American Mathematical Society vol 123 no4 pp 1115ndash1124 1995
[25] C Q He and M L Lapidus Generalized Minkowski ContentSpectrum of Fractal Drums Fractal Strings and the RiemannZeta-Function vol 127 of Memoirs of the American Mathemat-ical Society American Mathematical Society Providence RIUSA 1997
[26] M L Lapidus and M van Frankenhuijsen Fractal geometryComplex Dimensions and Zeta Functions Geometry and Spec-tra of Fractal Strings Springer Monographs in MathematicsSpringer New York NY USA 2006
[27] M Pasic ldquoMinkowski-Bouligand dimension of solutions ofthe one-dimensional 119901-Laplacianrdquo Journal of Differential Equa-tions vol 190 no 1 pp 268ndash305 2003
[28] J S W Wong ldquoOn rectifiable oscillation of Euler type secondorder linear differential equationsrdquo Electronic Journal of Quali-tativeTheory of Differential Equations vol 2007 no 20 pp 1ndash122007
[29] M Pasic ldquoRectifiable and unrectifiable oscillations for a gen-eralization of the Riemann-Weber version of Euler differentialequationrdquo Georgian Mathematical Journal vol 15 no 4 pp759ndash774 2008
[30] M Pasic and S Tanaka ldquoRectifiable oscillations of self-adjointand damped linear differential equations of second-orderrdquoJournal of Mathematical Analysis and Applications vol 381 no1 pp 27ndash42 2011
[31] M Pasic D Zubrinic and V Zupanovic ldquoOscillatory andphase dimensions of solutions of some second-order differentialequationsrdquo Bulletin des Sciences Mathematiques vol 133 no 8pp 859ndash874 2009
[32] L Korkut and M Resman ldquoFractal oscillations of chirp-likefunctionsrdquo Georgian Mathematical Journal vol 19 no 4 pp705ndash720 2012
Submit your manuscripts athttpwwwhindawicom
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Mathematical Problems in Engineering
Hindawi Publishing Corporationhttpwwwhindawicom
Differential EquationsInternational Journal of
Volume 2014
Applied MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Journal of
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Mathematical PhysicsAdvances in
Complex AnalysisJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
OptimizationJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of
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Operations ResearchAdvances in
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Function Spaces
Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of Mathematics and Mathematical Sciences
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Algebra
Discrete Dynamics in Nature and Society
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Decision SciencesAdvances in
Discrete MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom
Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Stochastic AnalysisInternational Journal of
International Journal of Differential Equations 5
where 119878der(119891) denotes the Schwarzian derivative of119891 definedby
119878der (119891) =11989110158401015840
119891
minus
3
2
(
1198911015840
119891
)
2
(28)
Hence from Theorem 16 we obtain the following conse-quence
Corollary 18 Let the functions 119886 isin 1198622((0 1199050]) and 120593 isin
1198623((0 1199050]) satisfy structural conditions (3) (4) and (15) as
well as the conditions (16) (17) and (18) in respect to a givenreal number 119904 isin (1 2) Let 1198861015840119886 = minus(12)120593101584010158401205931015840 Then everynontrivial solution 119910 isin 1198622((0 119905
0]) of (27) is fractal oscillatory
near 119909 = 0 with the fractal dimension 119904
As a consequence of Theorem 16 and Corollary 18 wederive the following examples for linear differential equationsof second order having all the solutions to be fractal oscilla-tory near 119909 = 0
Example 19 The so-called damped chirp equation
11991010158401015840+
120573 minus 2120572 + 1
119909
1199101015840+ (
1205732
1199092120573+2
minus
120572 (120573 minus 120572)
1199092
)119910 = 0
119909 isin (0 1199050]
(29)
is fractal oscillatory near 119909 = 0 with the fractal dimension2 minus (1 + 120572)(1 + 120573) where 120573 gt 120572 ge 0 When 119886(119909) = 119909120572 and120593(119909) = 119909
minus120573 (26) becomes (29) It is easy to see that (3) (4)(15) and (16) are satisfied In the same as in Example 12 wesee that (17) and (18) hold for 119904 = 2 minus (1 + 120572)(1 + 120573) HenceTheorem 16 proves that every nontrivial solution of (29) isfractal oscillatory near 119909 = 0 with the fractal dimension 2 minus(1 + 120572)(1 + 120573)
Nowwe assume that 0 lt 120573 lt 120572ThenTheorem 17 impliesthat (29) is fractal oscillatory near 119909 = 0 with the fractaldimension 1
Example 20 The following equation
11991010158401015840+
1 minus 2120574
119909
1199101015840+
1205742+ 1205882
1199092119910 = 0 119909 isin (0 119905
0] (30)
is fractal oscillatory near 119909 = 0 with the fractal dimension1 where 120574 gt 0 and 120588 gt 0 In the case where 119886(119909) = 119909
120574
and 120593(119909) = minus120588 log119909 (26) becomes (30) We see that 119886 isin119862([0 119905
0]) cap 119862
2((0 1199050]) 120593 isin 1198622((0 119905
0]) lim
119909rarr+0120593(119909) = infin
1198861015840isin 1198711((0 1199050]) and 1205931015840119886 isin 1198711((0 119905
0]) Therefore Theorem 17
implies that every nontrivial solution of (30) is fractaloscillatory near 119909 = 0 with the fractal dimension 1
Question 2 What can we say about the application of Theo-rem 16 on the case of 120593(119909) given in Example 3(iii)
At the end of this section we suggest that the readerstudies some invariant properties of fractal oscillations of thechirp function (1) in respect to the translation and reflexionAnalogously to Definitions 1 and 5 one can define the fractaloscillations near an arbitrary real point 119909 = 119909
0as follows
minus05
minus025 minus02 minus015 minus01
minus01
minus03
05
1
minus005
Figure 2 119910 is oscillatory near 119909 = 0 from the left side
Definition 21 Let 1199090isin R and 120575 gt 0 A function 119910 isin
119862((1199090 1199090+ 120575]) is oscillatory near 119909 = 119909
0 if there is a
decreasing sequence 119887119899isin (1199090 1199090+ 120575] such that 119910(119887
119899) = 0
for 119899 isin N and 119887119899 1199090as 119899 rarr infin Moreover if the graph
Γ(119910) satisfies the condition (12) for some 119904 isin [1 2) then 119910(119909)is said to be fractal oscillatory near 119909 = 119909
0with the fractal
dimension 119904
Definition 22 Let 1199090isin R and 120575 gt 0 It is said that a function
119910 isin 119862([1199090minus 120575 119909
0)) is fractal oscillatory near 119909 = 0 from
the left side with the fractal dimension 119904 isin [1 2) if there is anincreasing sequence 119886
119899isin [1199090minus 120575 119909
0) such that 119910(119886
119899) = 0 for
119899 isin N 119886119899 1199090as 119899 rarr infin and the graph Γ(119910) satisfies the
condition (12) see Figure 2
Definition 23 Let 1199090isin R and let 120575 gt 0 It is said that a
function 119910 isin 119862([1199090minus120575 1199090) cup (1199090 1199090+120575]) is two-sided fractal
oscillatory near 119909 = 0 with the fractal dimension 119904 isin [1 2)if there is an increasing sequence 119886
119899isin [1199090minus 120575 119909
0) and a
decreasing sequence 119887119899isin (1199090 1199090+ 120575] such that 119910(119886
119899) =
119910(119887119899) = 0 for 119899 isin N 119886
119899 1199090and 119887119899 1199090as 119899 rarr infin
and the graph Γ(119910) satisfies the condition (12) see Figure 3
With the help ofTheorem 8 we state the following result
Theorem 24 Let 1199090isin R and let 120575 gt 0 Let the functions
119886 120596 isin 1198621((0 120575]) satisfy structural conditions (3) (4) (5) (15)
and (16) as well as conditions (17) and (18) in respect to a givenreal number 119904 isin (1 2) Then one has
(i) the chirp function 119910(119909) = 119886(119909 minus 1199090)119878(120593(119909 minus 119909
0)) 119909 isin
(1199090 1199090+ 120575] is fractal oscillatory near 119909 = 119909
0with the
fractal dimension 119904
(ii) the chirp function 119910(119909) = 119886(1199090minus 119909)119878(120593(119909
0minus 119909)) 119909 isin
[1199090minus 120575 119909
0) is fractal oscillatory near 119909 = 119909
0from the
left side with the fractal dimension 119904
(iii) the chirp function 119910(119909) = 119886(|119909minus1199090|)119878(120593(|119909minus119909
0|)) 119909 isin
[1199090minus120575 1199090)cup(1199090 1199090+120575] is two-sided fractal oscillatory
near 119909 = 1199090with the fractal dimension 119904
6 International Journal of Differential Equations
minus03 minus02 minus01
minus1
01 02 03
minus05
1
05
Figure 3 119910 is two-sided oscillatory near 119909 = 0
3 Proof for the Fractal Oscillations ofChirp Functions
In this section we give the proofs of the main results dealingwith the fractal oscillations of chirp functions
By Definition 5 it follows that if for a prescribed realnumber 119904 isin (1 2) there are two positive constants 119888
1and 1198882
such that
11988811205762minus119904le1003816100381610038161003816Γ120576(119910)1003816100381610038161003816le 11988821205762minus119904 120576 isin (0 120576
0) (31)
for some 1205760gt 0 then a function 119910 isin 119862((0 119905
0]) is fractal
oscillatory near 119909 = 0 with the fractal dimension 119904 Thefollowing lemma plays the essential role in the proof of (31)
Lemma 25 Let 119910 isin 119862((0 1199050]) be a bounded function on
(0 1199050] and let 119886
119899isin (0 119905
0] be a decreasing sequence of con-
secutive zeros of 119910(119909) such that 119886119899rarr 0 Let 120576
0gt 0 and let
119896 = 119896(120576) be an index function satisfying1003816100381610038161003816119886119899minus 119886119899+1
1003816100381610038161003816le 120576 forall119899 ge 119896 (120576) 120576 isin (0 120576
0) (32)
Then1003816100381610038161003816Γ120576(119910)1003816100381610038161003816ge sum
119899ge119896(120576)
max119909isin[119886119899+1119886119899]
1003816100381610038161003816119910 (119909)
1003816100381610038161003816(119886119899minus 119886119899+1)
forall120576 isin (0 1205760)
(33)
Proof It is exactly the same as [3 pp 2350]
Let us remark that for some 1205760gt 0 we say that a function
119896 = 119896(120576) is an index function on (0 1205760) if 119896 (0 120576
0) rarr N and
lim120576rarr0
119896(120576) = infin
Lemma 26 Let 119910 isin 119862[119886 119887] Then
1003816100381610038161003816Γ120576(119910)1003816100381610038161003816le 4120587120576length (119910) + 41205871205762 for every 120576 gt 0 (34)
Proof Let 120576 gt 0 Set 1199091= 119886 and set
119909119894+1= max 119909 isin [119909
119894 119887] 119889 ((119905 119910 (119905)) (119909
119894 119910 (119909119894)))
le 120576 for 119905 isin [119909119894 119909]
(35)
for 119894 = 1 2 Then there exists 119899 ge 2 such that 119909119899= 119887 Set
119873 = max119899 isin N 119909119899lt 119887 We see that119873 ge 1
119886 = 1199091lt 1199092lt sdot sdot sdot lt 119909
119894lt 119909119894+1lt sdot sdot sdot lt 119909
119873lt 119909119873+1
= 119887 (36)
and if119873 ge 2 then
119889 ((119909119894 119910 (119909119894)) (119909
119894+1 119910 (119909119894+1))) = 120576 119894 = 1 2 119873 minus 1
(37)
We will show that
Γ120576(119910) sub
119873
⋃
119894=1
1198612120576(119909119894 119910 (119909119894)) (38)
where
1198612120576(1199051 1199052) = (120591
1 1205912) isin R2 119889 ((119905
1 1199052) (1205911 1205912)) le 2120576
(39)
Let (1205911 1205912) isin Γ
120576(119910) Then there exists 120585 isin [119886 119887] such that
119889((1205911 1205912) (120585 119910(120585))) le 120576 Because of the definition of 119909
119894+1 we
find that 120585 isin [119909119894 119909119894+1] for some 119894 isin 1 2 119873 so that
119889 ((120585 119910 (120585)) (119909119894 119910 (119909119894))) le 120576 (40)
Hence it follows that
119889 ((1205911 1205912) (119909119894 119910 (119909119894))) le 119889 ((120591
1 1205912) (120585 119910 (120585)))
+ 119889 ((120585 119910 (120585)) (119909119894 119910 (119909119894))) le 2120576
(41)
which means that (1205911 1205912) isin 119861
2120576(119909119894 119910(119909119894)) Therefore we
obtain (38) By (38) we conclude that
1003816100381610038161003816Γ120576(119910)1003816100381610038161003816le
119873
sum
119894=1
10038161003816100381610038161198612120576(119909119894 119910 (119909119894))1003816100381610038161003816= 4119873120587120576
2 (42)
When119873 = 1 from (42) it follows that1003816100381610038161003816Γ120576(119910)1003816100381610038161003816le 4120587120576
2le 4120587120576length (119910) + 41205871205762 (43)
Now we assume that119873 ge 2 We observe that
length (119910) =119873
sum
119894=1
length (119910|[119909119894 119909119894+1]
)
ge
119873
sum
119894=1
119889 ((119909119894 119910 (119909119894)) (119909
119894+1 119910 (119909119894+1)))
ge
119873minus1
sum
119894=1
119889 ((119909119894 119910 (119909119894)) (119909
119894+1 119910 (119909119894+1)))
= (119873 minus 1) 120576
(44)
that is
119873120576 le length (119910) + 120576 (45)
Combining (42) with (45) we obtain1003816100381610038161003816Γ120576(119910)1003816100381610038161003816le 4120587120576length (119910) + 41205871205762 (46)
International Journal of Differential Equations 7
Lemma 27 Let 119910 isin 119862(0 1199050] be bounded on (0 119905
0] and let
120576 gt 0 Then there exists 119862 gt 0 such that for every 120576 gt 0 and119905 isin (0 119905
0]
1003816100381610038161003816Γ120576(119910)1003816100381610038161003816le 119862[119905 sup
119909isin(0119905]
1003816100381610038161003816119910 (119909)
1003816100381610038161003816+ 120576length (119910|
[1199051199050]) + 120576 + 120576
2]
(47)
Proof Let 120576 gt 0 and let 119905 isin (0 1199050] From the geometry point
of view it is clear that1003816100381610038161003816Γ120576(119910|(0119905])1003816100381610038161003816sub 119877 (48)
where 119877 is a rectangle in R2 defined by
119877 = (1199051 1199052) 1199051isin [minus120576 119905 + 120576]
1199052isin [minus120576 minus sup
119909isin(0119905]
1003816100381610038161003816119910 (119909)
1003816100381610038161003816 120576 + sup119909isin(0119905]
1003816100381610038161003816119910 (119909)
1003816100381610038161003816]
(49)
Hence we have
1003816100381610038161003816Γ120576(119910(0119905])1003816100381610038161003816le (2120576 + 119905) (2120576 + 2 sup
119909isin(0119905]
1003816100381610038161003816119910 (119909)
1003816100381610038161003816)
= 41205762+ 2(119905 + 2 sup
119909isin(0119905]
1003816100381610038161003816119910 (119909)
1003816100381610038161003816) 120576
+ 2119905 sup119909isin(0119905]
1003816100381610038161003816119910 (119909)
1003816100381610038161003816
le 41205762+ 1198621120576 + 2119905 sup
119909isin(0119905]
1003816100381610038161003816119910 (119909)
1003816100381610038161003816
(50)
where 1198621= 2(1199050+ 2sup
119909isin(01199050]|119910(119909)|) Lemma 26 implies that
10038161003816100381610038161003816Γ120576(119910|[1199051199050])
10038161003816100381610038161003816le 4120587120576length (119910|
[1199051199050]) + 4120587120576
2 (51)
Consequently we see that1003816100381610038161003816Γ120576(119910)1003816100381610038161003816le1003816100381610038161003816Γ120576(119910|(0119905])1003816100381610038161003816+
10038161003816100381610038161003816Γ120576(119910|[1199051199050])
10038161003816100381610038161003816
le 41205762+ 1198621120576 + 2119905 sup
119909isin(0119905]
1003816100381610038161003816119910 (119909)
1003816100381610038161003816
+ 4120587120576length (119910|[1199051199050]) + 4120587120576
2
le 119862[119905 sup119909isin(0119905]
1003816100381610038161003816119910 (119909)
1003816100381610038161003816+ 120576length (119910|
[1199051199050]) + 120576 + 120576
2]
(52)
where 119862 = max1198621 4(120587 + 1)
Lemma 28 Let 119910 isin 1198621(0 1199050] be bounded on (0 119905
0] Assume
that
int
1199050
0
100381610038161003816100381610038161199101015840(119909)
10038161003816100381610038161003816119889119909 = infin (53)
lim sup119909rarr0
[119909 sup120585isin(0119909]
1003816100381610038161003816119910 (120585)
1003816100381610038161003816]
(119904minus1)(2minus119904)
int
1199050
119909
100381610038161003816100381610038161199101015840(120585)
10038161003816100381610038161003816119889120585 lt infin (54)
for some 119904 isin (1 2) Then there exists 1198882gt 0 such that
1003816100381610038161003816Γ120576(119910)1003816100381610038161003816le 11988821205762minus119904 120576 isin (0 1) (55)
Proof Let 120576 isin (0 1) and let 119904 isin (1 2) be satisfy (54) By (53)there exists 119909
1isin (0 119905
0) which depends on 120576 and 119904 such that
int
1199050
1199091
100381610038161003816100381610038161199101015840(119905)
10038161003816100381610038161003816119889119905 = 120576
1minus119904 (56)
Therefore we have
length (119910|[1199091 1199050]
) = int
1199050
1199091
radic1 +10038161003816100381610038161199101015840(119909)1003816100381610038161003816
2119889119909
le int
1199050
1199091
(1 +
100381610038161003816100381610038161199101015840(119909)
10038161003816100381610038161003816) 119889119909
= 1199050minus 1199091+ 1205761minus119904
le 1199050+ 1205761minus119904
(57)
By (54) there exists 1198621gt 0 such that
[119909 sup120585isin(0119909]
1003816100381610038161003816119910 (120585)
1003816100381610038161003816]
(119904minus1)(2minus119904)
int
1199050
119909
100381610038161003816100381610038161199101015840(120585)
10038161003816100381610038161003816119889120585 le 119862
1 119909 isin (0 119905
0]
(58)
which implies that
[
[
1199091sup120585isin(01199091]
1003816100381610038161003816119910 (120585)
1003816100381610038161003816]
]
(119904minus1)(2minus119904)
int
1199050
1199091
100381610038161003816100381610038161199101015840(120585)
10038161003816100381610038161003816119889120585 le 119862
1 (59)
that is
1199091sup120585isin(01199091]
1003816100381610038161003816119910 (120585)
1003816100381610038161003816le 11986221205762minus119904 (60)
where 1198622= 119862(2minus119904)(119904minus1)
1 From Lemma 27 (57) and (60) it
follows that1003816100381610038161003816Γ120576(119910)1003816100381610038161003816le 119862 [119862
21205762minus119904+ 120576 (1199050+ 1205761minus119904) + 120576 + 120576
2]
= 119862 [(1198622+ 1) + (119905
0+ 1) 120576
119904minus1+ 120576119904] 1205762minus119904
le 119862 (1198622+ 1199050+ 3) 120576
2minus119904 120576 isin (0 1)
(61)
In order to showTheorems 8 and 11 we need the followingtwo geometric lemmas
Lemma 29 (see [1]) If Γ sube R2 is a simple curve (ie itsparameterization is a bijection) and length(Γ) lt infin then
length (Γ) = lim120576rarr0
1003816100381610038161003816Γ120576
1003816100381610038161003816
2120576
(62)
where Γ120576denotes the 120576-neighborhood of the graph Γ
8 International Journal of Differential Equations
Now we are able to proveTheorem 8
Proof of Theorem 8 Let 119904 isin (1 2) and let 119910(119909) be a chirpfunction given by (1) We note here that it is enough to showthat 119910(119909) satisfies (31)
At the first let 119909119899be a sequence defined by 119909
119899= 120593minus1(1205910+
119899120591) for all sufficiently large 119899 isin N From (4) it follows that120593minus1(119905) is decreasing Hence 119909
119899is decreasing as well as 119909
119899rarr
0 as 119899 rarr infin because of lim119909rarr+0
120593(119909) = infin (see (4))We notethat 119910(119909
119899) = 0 and 119910(119909) = 0 on (119909
119899+1 119909119899) for all sufficiently
large 119899 isin N Also minus11205931015840(119909) is an increasing function becauseof (16) The mean value theorem shows that
120591
minus1205931015840(119909119899+1)
le 119909119899minus 119909119899+1le
120591
minus1205931015840(119909119899)
(63)
Now let 1205760isin (0 1) Let 119896(120576) be the smallest natural number
satisfying120591
minus1205931015840(119909119896(120576))
le 120576 forall120576 isin (0 1205760) (64)
Such 119896(120576) exists for every 120576 isin (0 1205760) since 119909
119899rarr 0 as 119899 rarr
infin and lim119909rarr+0
1205931015840(119909) = minusinfin (this equality is true because
1205931015840notin 1198711(0 1199050) since lim
119909rarr+0120593(119909) = infin) Moreover since 119909
119899
is decreasing and minus11205931015840(119909) is increasing we obtain
minus1205931015840(119909119899) ge 120591120576
minus1forall119899 ge 119896 (120576) (65)
Combining (63) and (65) it is easy to deduce that suchdefined 119896(120576) satisfies condition (32)
By (16) there exists 119871 gt 0 such that (1(minus1205931015840(119909)))1015840 le 119871 for119909 isin (0 119905
0] which means that
minus
minus12059310158401015840(119909)
minus1205931015840(119909)
le minus1198711205931015840(119909) 119909 isin (0 119905
0] (66)
Integrating (66) on [120593minus1(119905 + 2120591) 120593minus1(119905)] we have
logminus1205931015840(120593minus1(119905 + 2120591))
minus1205931015840(120593minus1(119905))
le 2119871120591 119905 isin [120593 (1199050) infin) (67)
that is
minus1205931015840(120593minus1(119905)) ge minus119890
minus21198711205911205931015840(120593minus1(119905 + 120591)) 119905 isin [120593 (119905
0) infin)
(68)
By the definition of 119896(120576) and (64) we see that120591
minus1205931015840(119909119896(120576)minus1
)
gt 120576 (69)
Hence from (68) it follows that
120591120576minus1gt minus1205931015840(119909119896(120576)minus1
)
= minus1205931015840(120593minus1(1205910+ (119896 (120576) minus 1) 120591))
ge minus119890minus2119871120591
1205931015840(120593minus1(1205910+ (119896 (120576) minus 1) 120591 + 2120591))
= minus119890minus2119871120591
1205931015840(120593minus1(1205910+ (119896 (120576) + 1) 120591))
= minus119890minus2119871120591
1205931015840(119909119896(120576)+1
)
(70)
which implies that
minus1205931015840(119909119896(120576)+1
) le 119872120576minus1 120576 gt 0 (71)
where119872 = 1205911198902119871120591
Next it is clear that the assumption (5) ensures a realnumber119872
0gt 0 such that
1198720= max119905isinR
|119878 (119905)| = max119905isin[0120591]
|119878 (119905)| (72)
Hence from (15) we have
max119909isin[119909119899+1 119909119899]
1003816100381610038161003816119910 (119909)
1003816100381610038161003816= max119909isin[119909119899+1119909119899]
(119886 (119909)1003816100381610038161003816119878 (120593 (119909))
1003816100381610038161003816)
ge 119886 (119909119899+1) max119909isin[119909119899+1119909119899]
1003816100381610038161003816119878 (120593 (119909))
1003816100381610038161003816
= 1198720119886 (119909119899+1)
(73)
Now from (63) and (73) it follows that
max119909isin[119909119899+1 119909119899]
1003816100381610038161003816119910 (119909)
1003816100381610038161003816(119909119899minus 119909119899+1) ge 119872
0119886 (119909119899+1) (119909119899minus 119909119899+1)
ge 1205911198720
119886 (119909119899+1)
minus1205931015840(119909119899+1)
= 1205911198720
119886 (120593minus1(1205910+(119899+1) 120591))
minus1205931015840(120593minus1(1205910+(119899+1) 120591))
(74)
Since 120593minus1(119905) is decreasing as well as minus11205931015840(119909) is increasingand 119886(119909) is nondecreasing and positive we conclude that thefunction
119886 (120593minus1(119905))
minus1205931015840(120593minus1(119905))
is nonincreasing (75)
Hence we observe that
sum
119899ge119896(120576)
max119909isin[119909119899+1119909119899]
1003816100381610038161003816119910 (119909)
1003816100381610038161003816(119909119899minus 119909119899+1)
ge 1205911198720sum
119899ge119896(120576)
119886 (120593minus1(1205910+ (119899 + 1) 120591))
minus1205931015840(120593minus1(1205910+ (119899 + 1) 120591))
ge 1205911198720int
infin
119896(120576)
119886 (120593minus1(1205910+ (119905 + 1) 120591))
minus1205931015840(120593minus1(1205910+ (119905 + 1) 120591))
119889119905
= 1198720int
120593minus1(1205910+(119896(120576)+1)120591)
0
119886 (120585) 119889120585
= 1198720int
119909119896(120576)+1
0
119886 (120585) 119889120585
(76)
Next from assumption (17) we get some1198721gt 0 such that
int
119909
0
119886 (120585) 119889120585 ge 1198721[minus1205931015840(119909)]
minus(2minus119904)
119909 isin (0 1199050] (77)
International Journal of Differential Equations 9
Now from (76) (77) and (71) we obtain
sum
119899ge119896(120576)
max119909isin[119909119899+1119909119899]
1003816100381610038161003816119910 (119909)
1003816100381610038161003816(119909119899minus 119909119899+1)
ge 11987201198721[minus1205931015840(119909119896(120576)+1
)]
minus(2minus119904)
ge 11987201198721(119872120576minus1)
minus(2minus119904)
= 11987201198721119872minus(2minus119904)
1205762minus119904
(78)
Thus using Lemma 25 we have
1003816100381610038161003816Γ120576(119910)1003816100381610038161003816ge 11988811205762minus119904 120576 isin (0 120576
0) (79)
for some 1198881gt 0
Next we prove that 119910(119909) satisfies (53) Assume to thecontrary that
int
1199050
0
100381610038161003816100381610038161199101015840(119909)
10038161003816100381610038161003816119889119909 lt infin (80)
Then we have
length (119910) = int1199050
0
radic1 +10038161003816100381610038161199101015840(119909)1003816100381610038161003816
2119889119909
le int
1199050
0
(1 +
100381610038161003816100381610038161199101015840(119909)
10038161003816100381610038161003816) 119889119909 lt infin
(81)
From Lemma 29 it follows that
length (119910) = lim120576rarr+0
1003816100381610038161003816Γ120576(119910)1003816100381610038161003816
2120576
(82)
On the other hand by (79) we see that
lim120576rarr+0
1003816100381610038161003816Γ120576(119910)1003816100381610038161003816
2120576
= infin (83)
This is a contaradiction and hence 119910(119909) satisfies (53)Finally we show that 119910(119909) also satisfies inequality (54)
Because of (5) we obtain a constant 119888 gt 0 such that
max119905isinR
|119878 (119905)| = max119905isin[0120591]
|119878 (119905)| le 119888
max119905isinR
100381610038161003816100381610038161198781015840(119905)
10038161003816100381610038161003816= max119905isin[0120591]
100381610038161003816100381610038161198781015840(119905)
10038161003816100381610038161003816le 119888
(84)
By (15) we find that
sup119905isin(0119909]
1003816100381610038161003816119910 (119905)
1003816100381610038161003816= sup119905isin(0119909]
|119886 (119905)|1003816100381610038161003816119878 (120593 (119905))
1003816100381610038161003816le 119888119886 (119909) (85)
Hence from (1) and (15) it follows that100381610038161003816100381610038161199101015840(119909)
10038161003816100381610038161003816le 119888 (119886
1015840(119909) + 119886 (119909) [minus120593
1015840(119909)]) (86)
and therefore
int
1199050
119909
100381610038161003816100381610038161199101015840(120585)
10038161003816100381610038161003816119889120585 le 119888119886 (119905
0) + 119888 int
1199050
119909
119886 (120585) [minus1205931015840(120585)] 119889120585 (87)
Using (85) and (87) we see that
[119909 sup120585isin(0119909]
1003816100381610038161003816119910 (120585)
1003816100381610038161003816]
(119904minus1)(2minus119904)
int
1199050
119909
100381610038161003816100381610038161199101015840(120585)
10038161003816100381610038161003816119889120585
le 119888119886 (1199050) [119888119909119886 (119909)]
(119904minus1)(2minus119904)
+ 119888[119888119909119886 (119909)](119904minus1)(2minus119904)
int
1199050
119909
119886 (120585) [minus1205931015840(120585)] 119889120585
le 119888119886 (1199050) [1198881199050119886 (1199050)](119904minus1)(2minus119904)
+ 1198881(2minus119904)
[119909119886 (119909)](119904minus1)(2minus119904)
int
1199050
119909
119886 (120585) [minus1205931015840(120585)] 119889120585
(88)
By (18) we conclude that (54) holdsBy Lemma 28 there exists 119888
2gt 0 such that
1003816100381610038161003816Γ120576(119910)1003816100381610038161003816le 11988821205762minus119904 120576 isin (0 120576
0) (89)
Thus we have proved that the chirp function119910(119909) given by (1)satisfies the desired inequality (31) This completes the proofof Theorem 8
Proof of Theorem 11 Let 119910(119909) be the chirp function (1) It iseasy to see that 119910(119909) is oscillatory near 119909 = 0 By (5) thereexists119872 gt 0 such that
|119878 (119905)| le 119872
100381610038161003816100381610038161198781015840(119905)
10038161003816100381610038161003816le 119872 119905 isin R (90)
Hence we observe that100381610038161003816100381610038161199101015840(119909)
10038161003816100381610038161003816le
100381610038161003816100381610038161198861015840(119909) 119878 (120593 (119909))
10038161003816100381610038161003816+
10038161003816100381610038161003816119886 (119909) 119878 (120593 (119909)) 120593
1015840(119909)
10038161003816100381610038161003816
le 119872
100381610038161003816100381610038161198861015840(119909)
10038161003816100381610038161003816+ 119872
10038161003816100381610038161003816119886 (119909) 120593
1015840(119909)
10038161003816100381610038161003816
(91)
for 119909 isin (0 1199050] From (81) it follows that length(119910) lt infin
Lemma 29 shows that
length (119910) = lim120576rarr0
1003816100381610038161003816Γ120576(119910)1003816100381610038161003816
2120576
(92)
which implies that 120576length(Γ) le |Γ120576(119910)| le 4120576length(Γ) 120576 isin
(0 1205760) for some 120576
0gt 0 Therefore 119910(119909) is fractal oscillatory
near 119909 = 0 with the fractal dimension 1
4 Proof for the Fractal Oscillations of (26)In this section we give the proofs for the fractal oscillations ofthe linear second-order differential equation (26) consideredas an application of the main results on the fractal oscillationof chirp functions
Before we present the proofs of Theorems 16 and 17 wemake the following observation Since
1199101(119909) = 119886 (119909) cos (120593 (119909)) 119910
2(119909) = 119886 (119909) sin (120593 (119909))
(93)
are solutions of (26) we see that 119910(119909) = 11988811199101(119909) + 119888
21199102(119909)
is a fundamental system of all solutions of (26) Assume that1198882
1+ 1198882
2gt 0 and set 119878(119905) = 119888
1sin 119905 + 119888
2cos 119905 Then 119878(119905) clearly
satisfies (5)
10 International Journal of Differential Equations
Proof of Theorem 16 Applying Theorem 8 on 119910(119909) =
11988811199101(119909) + 119888
21199102(119909) we conclude that 119910(119909) is fractal oscillatory
with the fractal dimension 119904
Proof of Theorem 17 Theorem 11 implies that 119910(119909) =
11988811199101(119909) + 119888
21199102(119909) is fractal oscillatory near 119909 = 0 with the
fractal dimension 1
Appendix
Proof of Remark 9 By (17) there exists 1198881gt 0 such that
1198881le [minus120593
1015840(119909)]
2minus119904
int
119909
0
119886 (120585) 119889120585 119909 isin (0 1199050] (A1)
From (15) it follows that
1198881le [minus120593
1015840(119909)]
2minus119904
119886 (119909) int
119909
0
119889120585 = 119909119886 (119909) [minus1205931015840(119909)]
2minus119904
119909 isin (0 1199050]
(A2)
which implies that1
minus1205931015840(119909)
le 119888minus1(2minus119904)
11199091(2minus119904)
[119886 (119909)]1(2minus119904)
119909 isin (0 1199050] (A3)
By (19) there exists 1198882gt 0 such that
119909119886 (119909) [minus1205931015840(119909)]
2minus119904
le 1198882 119909 isin (0 119905
0] (A4)
and hence
minus1205931015840(119909) le 119888
1(2minus119904)
2119909minus1(2minus119904)
[119886 (119909)]minus1(2minus119904)
119909 isin (0 1199050]
(A5)Therefore we see that
[
1
minus1205931015840(119909)
]
119904minus1
int
1199050
119909
119886 (120585) [minus1205931015840(120585)] 119889120585
le 119888minus(119904minus1)(2minus119904)
1119909(119904minus1)(2minus119904)
[119886 (119909)](119904minus1)(2minus119904)
1198881(2minus119904)
2
times int
1199050
119909
120585minus1(2minus119904)
[119886 (120585)]minus(119904minus1)(2minus119904)
119889120585
le 1198883119909(119904minus1)(2minus119904)
[119886 (119909)](119904minus1)(2minus119904)
[119886 (119909)]minus(119904minus1)(2minus119904)
times int
1199050
119909
120585minus1(2minus119904)
119889120585
= 1198883
2 minus 119904
119904 minus 1
(1 minus 119909(119904minus1)(2minus119904)
119905minus(119904minus1)(2minus119904)
0)
le 1198883
2 minus 119904
119904 minus 1
119909 isin (0 1199050]
(A6)
so that (20) holds We conclude that
[119909119886 (119909)](119904minus1)(2minus119904)
int
1199050
119909
119886 (120585) [minus1205931015840(120585)] 119889120585
= (119909119886 (119909) [minus1205931015840(119909)]
2minus119904
)
(119904minus1)(2minus119904)
times ([
1
minus1205931015840(119909)
]
119904minus1
int
1199050
119909
119886 (120585) [minus1205931015840(120585)] 119889120585)
(A7)
By (19) and (20) we conclude that (18) is satisfied The proofof Remark 9 is complete
References
[1] C Tricot Curves and Fractal Dimension Springer New YorkNY USA 1995
[2] M Pasic ldquoFractal oscillations for a class of second order lineardifferential equations of Euler typerdquo Journal of MathematicalAnalysis and Applications vol 341 no 1 pp 211ndash223 2008
[3] M K Kwong M Pasic and J S W Wong ldquoRectifiable oscil-lations in second-order linear differential equationsrdquo Journal ofDifferential Equations vol 245 no 8 pp 2333ndash2351 2008
[4] M Pasic and J S W Wong ldquoRectifiable oscillations in second-order half-linear differential equationsrdquo Annali di MatematicaPura ed Applicata Series 4 vol 188 no 3 pp 517ndash541 2009
[5] M Pasic and S Tanaka ldquoFractal oscillations of self-adjoint anddamped linear differential equations of second-orderrdquo AppliedMathematics and Computation vol 218 no 5 pp 2281ndash22932011
[6] Y Naito M Pasic S Tanaka and D Zubrinic ldquoFractal oscilla-tions near domain boundary of radially symmetric solutions ofp-Laplace equations fractal geometry and dynamical systemsin pure and applied mathematicsrdquo Contemporary MathematicsAmerican Mathematical Society In press
[7] W A Coppel Stability and Asymptotic Behavior of DifferentialEquations D C Heath andCompany BostonMass USA 1965
[8] P Hartman Ordinary Differential Equations BirkhauserBoston Mass USA 2nd edition 1982
[9] F Neuman ldquoA general construction of linear differentialequations with solutions of prescribed propertiesrdquo AppliedMathematics Letters vol 17 no 1 pp 71ndash76 2004
[10] F Neuman ldquoStructure of solution spaces via transformationrdquoApplied Mathematics Letters vol 21 no 5 pp 529ndash533 2008
[11] P Borgnat and P Flandrin ldquoOn the chirp decomposition ofWeierstrass-Mandelbrot functions and their time-frequencyinterpretationrdquo Applied and Computational Harmonic Analysisvol 15 no 2 pp 134ndash146 2003
[12] E J Candes P R Charlton and H Helgason ldquoDetectinghighly oscillatory signals by chirplet path pursuitrdquo Applied andComputational Harmonic Analysis vol 24 no 1 pp 14ndash402008
[13] S Jaffard andYMeyer ldquoWaveletmethods for pointwise regular-ity and local oscillations of functionsrdquoMemoirs of the AmericanMathematical Society vol 123 no 587 pp 1ndash110 1996
[14] Y Meyer and H Xu ldquoWavelet analysis and chirpsrdquo Applied andComputational Harmonic Analysis vol 4 no 4 pp 366ndash3791997
[15] G RenQChen P Cerejeiras andUKaehle ldquoChirp transformsand chirp seriesrdquo Journal of Mathematical Analysis and Applica-tions vol 373 no 2 pp 356ndash369 2011
[16] M Kepesi and L Weruaga ldquoAdaptive chirp-based time-frequency analysis of speech signalsrdquo Speech Communicationvol 48 no 5 pp 474ndash492 2006
[17] L Weruaga and M Kepesi ldquoThe fan-chirp transform for non-stationary harmonic signalsrdquo Signal Processing vol 87 no 6 pp1504ndash1522 2007
[18] E Barlow A J Mulholland A Nordon and A GachaganldquoTheoretical analysis of chirp excitation of contrast agentsrdquoPhysics Procedia vol 3 no 1 pp 743ndash747 2009
International Journal of Differential Equations 11
[19] M H Pedersen T X Misaridis and J A Jensen ldquoClinicalevaluation of chirp-coded excitation in medical ultrasoundrdquoUltrasound in Medicine and Biology vol 29 no 6 pp 895ndash9052003
[20] T Paavle M Min and T Parve ldquoUsing of chirp excitation forbioimpedance estimation theoretical aspects andmodelingrdquo inProceedings of the 11th International Biennial Baltic ElectronicsConference (BECrsquo08) pp 325ndash328 Tallinn Estonia October2008
[21] M Pasic ldquoRectifiable and unrectifiable oscillations for a class ofsecond-order linear differential equations of Euler typerdquo Journalof Mathematical Analysis and Applications vol 335 no 1 pp724ndash738 2007
[22] K Falconer Fractal Geometry Mathematical Foundations andApplications John Wiley amp Sons Hoboken NJ USA 1999
[23] P Mattila Geometry of Sets and Measures in Euclidean SpacesFractals and Rectifiability vol 44 of Cambridge Studies inAdvanced Mathematics Cambridge University Press Cam-bridge UK 1995
[24] K J Falconer ldquoOn the Minkowski measurability of fractalsrdquoProceedings of the American Mathematical Society vol 123 no4 pp 1115ndash1124 1995
[25] C Q He and M L Lapidus Generalized Minkowski ContentSpectrum of Fractal Drums Fractal Strings and the RiemannZeta-Function vol 127 of Memoirs of the American Mathemat-ical Society American Mathematical Society Providence RIUSA 1997
[26] M L Lapidus and M van Frankenhuijsen Fractal geometryComplex Dimensions and Zeta Functions Geometry and Spec-tra of Fractal Strings Springer Monographs in MathematicsSpringer New York NY USA 2006
[27] M Pasic ldquoMinkowski-Bouligand dimension of solutions ofthe one-dimensional 119901-Laplacianrdquo Journal of Differential Equa-tions vol 190 no 1 pp 268ndash305 2003
[28] J S W Wong ldquoOn rectifiable oscillation of Euler type secondorder linear differential equationsrdquo Electronic Journal of Quali-tativeTheory of Differential Equations vol 2007 no 20 pp 1ndash122007
[29] M Pasic ldquoRectifiable and unrectifiable oscillations for a gen-eralization of the Riemann-Weber version of Euler differentialequationrdquo Georgian Mathematical Journal vol 15 no 4 pp759ndash774 2008
[30] M Pasic and S Tanaka ldquoRectifiable oscillations of self-adjointand damped linear differential equations of second-orderrdquoJournal of Mathematical Analysis and Applications vol 381 no1 pp 27ndash42 2011
[31] M Pasic D Zubrinic and V Zupanovic ldquoOscillatory andphase dimensions of solutions of some second-order differentialequationsrdquo Bulletin des Sciences Mathematiques vol 133 no 8pp 859ndash874 2009
[32] L Korkut and M Resman ldquoFractal oscillations of chirp-likefunctionsrdquo Georgian Mathematical Journal vol 19 no 4 pp705ndash720 2012
Submit your manuscripts athttpwwwhindawicom
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
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Mathematical Problems in Engineering
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Differential EquationsInternational Journal of
Volume 2014
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OptimizationJournal of
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International Journal of
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Function Spaces
Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
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Algebra
Discrete Dynamics in Nature and Society
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
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Decision SciencesAdvances in
Discrete MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom
Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Stochastic AnalysisInternational Journal of
6 International Journal of Differential Equations
minus03 minus02 minus01
minus1
01 02 03
minus05
1
05
Figure 3 119910 is two-sided oscillatory near 119909 = 0
3 Proof for the Fractal Oscillations ofChirp Functions
In this section we give the proofs of the main results dealingwith the fractal oscillations of chirp functions
By Definition 5 it follows that if for a prescribed realnumber 119904 isin (1 2) there are two positive constants 119888
1and 1198882
such that
11988811205762minus119904le1003816100381610038161003816Γ120576(119910)1003816100381610038161003816le 11988821205762minus119904 120576 isin (0 120576
0) (31)
for some 1205760gt 0 then a function 119910 isin 119862((0 119905
0]) is fractal
oscillatory near 119909 = 0 with the fractal dimension 119904 Thefollowing lemma plays the essential role in the proof of (31)
Lemma 25 Let 119910 isin 119862((0 1199050]) be a bounded function on
(0 1199050] and let 119886
119899isin (0 119905
0] be a decreasing sequence of con-
secutive zeros of 119910(119909) such that 119886119899rarr 0 Let 120576
0gt 0 and let
119896 = 119896(120576) be an index function satisfying1003816100381610038161003816119886119899minus 119886119899+1
1003816100381610038161003816le 120576 forall119899 ge 119896 (120576) 120576 isin (0 120576
0) (32)
Then1003816100381610038161003816Γ120576(119910)1003816100381610038161003816ge sum
119899ge119896(120576)
max119909isin[119886119899+1119886119899]
1003816100381610038161003816119910 (119909)
1003816100381610038161003816(119886119899minus 119886119899+1)
forall120576 isin (0 1205760)
(33)
Proof It is exactly the same as [3 pp 2350]
Let us remark that for some 1205760gt 0 we say that a function
119896 = 119896(120576) is an index function on (0 1205760) if 119896 (0 120576
0) rarr N and
lim120576rarr0
119896(120576) = infin
Lemma 26 Let 119910 isin 119862[119886 119887] Then
1003816100381610038161003816Γ120576(119910)1003816100381610038161003816le 4120587120576length (119910) + 41205871205762 for every 120576 gt 0 (34)
Proof Let 120576 gt 0 Set 1199091= 119886 and set
119909119894+1= max 119909 isin [119909
119894 119887] 119889 ((119905 119910 (119905)) (119909
119894 119910 (119909119894)))
le 120576 for 119905 isin [119909119894 119909]
(35)
for 119894 = 1 2 Then there exists 119899 ge 2 such that 119909119899= 119887 Set
119873 = max119899 isin N 119909119899lt 119887 We see that119873 ge 1
119886 = 1199091lt 1199092lt sdot sdot sdot lt 119909
119894lt 119909119894+1lt sdot sdot sdot lt 119909
119873lt 119909119873+1
= 119887 (36)
and if119873 ge 2 then
119889 ((119909119894 119910 (119909119894)) (119909
119894+1 119910 (119909119894+1))) = 120576 119894 = 1 2 119873 minus 1
(37)
We will show that
Γ120576(119910) sub
119873
⋃
119894=1
1198612120576(119909119894 119910 (119909119894)) (38)
where
1198612120576(1199051 1199052) = (120591
1 1205912) isin R2 119889 ((119905
1 1199052) (1205911 1205912)) le 2120576
(39)
Let (1205911 1205912) isin Γ
120576(119910) Then there exists 120585 isin [119886 119887] such that
119889((1205911 1205912) (120585 119910(120585))) le 120576 Because of the definition of 119909
119894+1 we
find that 120585 isin [119909119894 119909119894+1] for some 119894 isin 1 2 119873 so that
119889 ((120585 119910 (120585)) (119909119894 119910 (119909119894))) le 120576 (40)
Hence it follows that
119889 ((1205911 1205912) (119909119894 119910 (119909119894))) le 119889 ((120591
1 1205912) (120585 119910 (120585)))
+ 119889 ((120585 119910 (120585)) (119909119894 119910 (119909119894))) le 2120576
(41)
which means that (1205911 1205912) isin 119861
2120576(119909119894 119910(119909119894)) Therefore we
obtain (38) By (38) we conclude that
1003816100381610038161003816Γ120576(119910)1003816100381610038161003816le
119873
sum
119894=1
10038161003816100381610038161198612120576(119909119894 119910 (119909119894))1003816100381610038161003816= 4119873120587120576
2 (42)
When119873 = 1 from (42) it follows that1003816100381610038161003816Γ120576(119910)1003816100381610038161003816le 4120587120576
2le 4120587120576length (119910) + 41205871205762 (43)
Now we assume that119873 ge 2 We observe that
length (119910) =119873
sum
119894=1
length (119910|[119909119894 119909119894+1]
)
ge
119873
sum
119894=1
119889 ((119909119894 119910 (119909119894)) (119909
119894+1 119910 (119909119894+1)))
ge
119873minus1
sum
119894=1
119889 ((119909119894 119910 (119909119894)) (119909
119894+1 119910 (119909119894+1)))
= (119873 minus 1) 120576
(44)
that is
119873120576 le length (119910) + 120576 (45)
Combining (42) with (45) we obtain1003816100381610038161003816Γ120576(119910)1003816100381610038161003816le 4120587120576length (119910) + 41205871205762 (46)
International Journal of Differential Equations 7
Lemma 27 Let 119910 isin 119862(0 1199050] be bounded on (0 119905
0] and let
120576 gt 0 Then there exists 119862 gt 0 such that for every 120576 gt 0 and119905 isin (0 119905
0]
1003816100381610038161003816Γ120576(119910)1003816100381610038161003816le 119862[119905 sup
119909isin(0119905]
1003816100381610038161003816119910 (119909)
1003816100381610038161003816+ 120576length (119910|
[1199051199050]) + 120576 + 120576
2]
(47)
Proof Let 120576 gt 0 and let 119905 isin (0 1199050] From the geometry point
of view it is clear that1003816100381610038161003816Γ120576(119910|(0119905])1003816100381610038161003816sub 119877 (48)
where 119877 is a rectangle in R2 defined by
119877 = (1199051 1199052) 1199051isin [minus120576 119905 + 120576]
1199052isin [minus120576 minus sup
119909isin(0119905]
1003816100381610038161003816119910 (119909)
1003816100381610038161003816 120576 + sup119909isin(0119905]
1003816100381610038161003816119910 (119909)
1003816100381610038161003816]
(49)
Hence we have
1003816100381610038161003816Γ120576(119910(0119905])1003816100381610038161003816le (2120576 + 119905) (2120576 + 2 sup
119909isin(0119905]
1003816100381610038161003816119910 (119909)
1003816100381610038161003816)
= 41205762+ 2(119905 + 2 sup
119909isin(0119905]
1003816100381610038161003816119910 (119909)
1003816100381610038161003816) 120576
+ 2119905 sup119909isin(0119905]
1003816100381610038161003816119910 (119909)
1003816100381610038161003816
le 41205762+ 1198621120576 + 2119905 sup
119909isin(0119905]
1003816100381610038161003816119910 (119909)
1003816100381610038161003816
(50)
where 1198621= 2(1199050+ 2sup
119909isin(01199050]|119910(119909)|) Lemma 26 implies that
10038161003816100381610038161003816Γ120576(119910|[1199051199050])
10038161003816100381610038161003816le 4120587120576length (119910|
[1199051199050]) + 4120587120576
2 (51)
Consequently we see that1003816100381610038161003816Γ120576(119910)1003816100381610038161003816le1003816100381610038161003816Γ120576(119910|(0119905])1003816100381610038161003816+
10038161003816100381610038161003816Γ120576(119910|[1199051199050])
10038161003816100381610038161003816
le 41205762+ 1198621120576 + 2119905 sup
119909isin(0119905]
1003816100381610038161003816119910 (119909)
1003816100381610038161003816
+ 4120587120576length (119910|[1199051199050]) + 4120587120576
2
le 119862[119905 sup119909isin(0119905]
1003816100381610038161003816119910 (119909)
1003816100381610038161003816+ 120576length (119910|
[1199051199050]) + 120576 + 120576
2]
(52)
where 119862 = max1198621 4(120587 + 1)
Lemma 28 Let 119910 isin 1198621(0 1199050] be bounded on (0 119905
0] Assume
that
int
1199050
0
100381610038161003816100381610038161199101015840(119909)
10038161003816100381610038161003816119889119909 = infin (53)
lim sup119909rarr0
[119909 sup120585isin(0119909]
1003816100381610038161003816119910 (120585)
1003816100381610038161003816]
(119904minus1)(2minus119904)
int
1199050
119909
100381610038161003816100381610038161199101015840(120585)
10038161003816100381610038161003816119889120585 lt infin (54)
for some 119904 isin (1 2) Then there exists 1198882gt 0 such that
1003816100381610038161003816Γ120576(119910)1003816100381610038161003816le 11988821205762minus119904 120576 isin (0 1) (55)
Proof Let 120576 isin (0 1) and let 119904 isin (1 2) be satisfy (54) By (53)there exists 119909
1isin (0 119905
0) which depends on 120576 and 119904 such that
int
1199050
1199091
100381610038161003816100381610038161199101015840(119905)
10038161003816100381610038161003816119889119905 = 120576
1minus119904 (56)
Therefore we have
length (119910|[1199091 1199050]
) = int
1199050
1199091
radic1 +10038161003816100381610038161199101015840(119909)1003816100381610038161003816
2119889119909
le int
1199050
1199091
(1 +
100381610038161003816100381610038161199101015840(119909)
10038161003816100381610038161003816) 119889119909
= 1199050minus 1199091+ 1205761minus119904
le 1199050+ 1205761minus119904
(57)
By (54) there exists 1198621gt 0 such that
[119909 sup120585isin(0119909]
1003816100381610038161003816119910 (120585)
1003816100381610038161003816]
(119904minus1)(2minus119904)
int
1199050
119909
100381610038161003816100381610038161199101015840(120585)
10038161003816100381610038161003816119889120585 le 119862
1 119909 isin (0 119905
0]
(58)
which implies that
[
[
1199091sup120585isin(01199091]
1003816100381610038161003816119910 (120585)
1003816100381610038161003816]
]
(119904minus1)(2minus119904)
int
1199050
1199091
100381610038161003816100381610038161199101015840(120585)
10038161003816100381610038161003816119889120585 le 119862
1 (59)
that is
1199091sup120585isin(01199091]
1003816100381610038161003816119910 (120585)
1003816100381610038161003816le 11986221205762minus119904 (60)
where 1198622= 119862(2minus119904)(119904minus1)
1 From Lemma 27 (57) and (60) it
follows that1003816100381610038161003816Γ120576(119910)1003816100381610038161003816le 119862 [119862
21205762minus119904+ 120576 (1199050+ 1205761minus119904) + 120576 + 120576
2]
= 119862 [(1198622+ 1) + (119905
0+ 1) 120576
119904minus1+ 120576119904] 1205762minus119904
le 119862 (1198622+ 1199050+ 3) 120576
2minus119904 120576 isin (0 1)
(61)
In order to showTheorems 8 and 11 we need the followingtwo geometric lemmas
Lemma 29 (see [1]) If Γ sube R2 is a simple curve (ie itsparameterization is a bijection) and length(Γ) lt infin then
length (Γ) = lim120576rarr0
1003816100381610038161003816Γ120576
1003816100381610038161003816
2120576
(62)
where Γ120576denotes the 120576-neighborhood of the graph Γ
8 International Journal of Differential Equations
Now we are able to proveTheorem 8
Proof of Theorem 8 Let 119904 isin (1 2) and let 119910(119909) be a chirpfunction given by (1) We note here that it is enough to showthat 119910(119909) satisfies (31)
At the first let 119909119899be a sequence defined by 119909
119899= 120593minus1(1205910+
119899120591) for all sufficiently large 119899 isin N From (4) it follows that120593minus1(119905) is decreasing Hence 119909
119899is decreasing as well as 119909
119899rarr
0 as 119899 rarr infin because of lim119909rarr+0
120593(119909) = infin (see (4))We notethat 119910(119909
119899) = 0 and 119910(119909) = 0 on (119909
119899+1 119909119899) for all sufficiently
large 119899 isin N Also minus11205931015840(119909) is an increasing function becauseof (16) The mean value theorem shows that
120591
minus1205931015840(119909119899+1)
le 119909119899minus 119909119899+1le
120591
minus1205931015840(119909119899)
(63)
Now let 1205760isin (0 1) Let 119896(120576) be the smallest natural number
satisfying120591
minus1205931015840(119909119896(120576))
le 120576 forall120576 isin (0 1205760) (64)
Such 119896(120576) exists for every 120576 isin (0 1205760) since 119909
119899rarr 0 as 119899 rarr
infin and lim119909rarr+0
1205931015840(119909) = minusinfin (this equality is true because
1205931015840notin 1198711(0 1199050) since lim
119909rarr+0120593(119909) = infin) Moreover since 119909
119899
is decreasing and minus11205931015840(119909) is increasing we obtain
minus1205931015840(119909119899) ge 120591120576
minus1forall119899 ge 119896 (120576) (65)
Combining (63) and (65) it is easy to deduce that suchdefined 119896(120576) satisfies condition (32)
By (16) there exists 119871 gt 0 such that (1(minus1205931015840(119909)))1015840 le 119871 for119909 isin (0 119905
0] which means that
minus
minus12059310158401015840(119909)
minus1205931015840(119909)
le minus1198711205931015840(119909) 119909 isin (0 119905
0] (66)
Integrating (66) on [120593minus1(119905 + 2120591) 120593minus1(119905)] we have
logminus1205931015840(120593minus1(119905 + 2120591))
minus1205931015840(120593minus1(119905))
le 2119871120591 119905 isin [120593 (1199050) infin) (67)
that is
minus1205931015840(120593minus1(119905)) ge minus119890
minus21198711205911205931015840(120593minus1(119905 + 120591)) 119905 isin [120593 (119905
0) infin)
(68)
By the definition of 119896(120576) and (64) we see that120591
minus1205931015840(119909119896(120576)minus1
)
gt 120576 (69)
Hence from (68) it follows that
120591120576minus1gt minus1205931015840(119909119896(120576)minus1
)
= minus1205931015840(120593minus1(1205910+ (119896 (120576) minus 1) 120591))
ge minus119890minus2119871120591
1205931015840(120593minus1(1205910+ (119896 (120576) minus 1) 120591 + 2120591))
= minus119890minus2119871120591
1205931015840(120593minus1(1205910+ (119896 (120576) + 1) 120591))
= minus119890minus2119871120591
1205931015840(119909119896(120576)+1
)
(70)
which implies that
minus1205931015840(119909119896(120576)+1
) le 119872120576minus1 120576 gt 0 (71)
where119872 = 1205911198902119871120591
Next it is clear that the assumption (5) ensures a realnumber119872
0gt 0 such that
1198720= max119905isinR
|119878 (119905)| = max119905isin[0120591]
|119878 (119905)| (72)
Hence from (15) we have
max119909isin[119909119899+1 119909119899]
1003816100381610038161003816119910 (119909)
1003816100381610038161003816= max119909isin[119909119899+1119909119899]
(119886 (119909)1003816100381610038161003816119878 (120593 (119909))
1003816100381610038161003816)
ge 119886 (119909119899+1) max119909isin[119909119899+1119909119899]
1003816100381610038161003816119878 (120593 (119909))
1003816100381610038161003816
= 1198720119886 (119909119899+1)
(73)
Now from (63) and (73) it follows that
max119909isin[119909119899+1 119909119899]
1003816100381610038161003816119910 (119909)
1003816100381610038161003816(119909119899minus 119909119899+1) ge 119872
0119886 (119909119899+1) (119909119899minus 119909119899+1)
ge 1205911198720
119886 (119909119899+1)
minus1205931015840(119909119899+1)
= 1205911198720
119886 (120593minus1(1205910+(119899+1) 120591))
minus1205931015840(120593minus1(1205910+(119899+1) 120591))
(74)
Since 120593minus1(119905) is decreasing as well as minus11205931015840(119909) is increasingand 119886(119909) is nondecreasing and positive we conclude that thefunction
119886 (120593minus1(119905))
minus1205931015840(120593minus1(119905))
is nonincreasing (75)
Hence we observe that
sum
119899ge119896(120576)
max119909isin[119909119899+1119909119899]
1003816100381610038161003816119910 (119909)
1003816100381610038161003816(119909119899minus 119909119899+1)
ge 1205911198720sum
119899ge119896(120576)
119886 (120593minus1(1205910+ (119899 + 1) 120591))
minus1205931015840(120593minus1(1205910+ (119899 + 1) 120591))
ge 1205911198720int
infin
119896(120576)
119886 (120593minus1(1205910+ (119905 + 1) 120591))
minus1205931015840(120593minus1(1205910+ (119905 + 1) 120591))
119889119905
= 1198720int
120593minus1(1205910+(119896(120576)+1)120591)
0
119886 (120585) 119889120585
= 1198720int
119909119896(120576)+1
0
119886 (120585) 119889120585
(76)
Next from assumption (17) we get some1198721gt 0 such that
int
119909
0
119886 (120585) 119889120585 ge 1198721[minus1205931015840(119909)]
minus(2minus119904)
119909 isin (0 1199050] (77)
International Journal of Differential Equations 9
Now from (76) (77) and (71) we obtain
sum
119899ge119896(120576)
max119909isin[119909119899+1119909119899]
1003816100381610038161003816119910 (119909)
1003816100381610038161003816(119909119899minus 119909119899+1)
ge 11987201198721[minus1205931015840(119909119896(120576)+1
)]
minus(2minus119904)
ge 11987201198721(119872120576minus1)
minus(2minus119904)
= 11987201198721119872minus(2minus119904)
1205762minus119904
(78)
Thus using Lemma 25 we have
1003816100381610038161003816Γ120576(119910)1003816100381610038161003816ge 11988811205762minus119904 120576 isin (0 120576
0) (79)
for some 1198881gt 0
Next we prove that 119910(119909) satisfies (53) Assume to thecontrary that
int
1199050
0
100381610038161003816100381610038161199101015840(119909)
10038161003816100381610038161003816119889119909 lt infin (80)
Then we have
length (119910) = int1199050
0
radic1 +10038161003816100381610038161199101015840(119909)1003816100381610038161003816
2119889119909
le int
1199050
0
(1 +
100381610038161003816100381610038161199101015840(119909)
10038161003816100381610038161003816) 119889119909 lt infin
(81)
From Lemma 29 it follows that
length (119910) = lim120576rarr+0
1003816100381610038161003816Γ120576(119910)1003816100381610038161003816
2120576
(82)
On the other hand by (79) we see that
lim120576rarr+0
1003816100381610038161003816Γ120576(119910)1003816100381610038161003816
2120576
= infin (83)
This is a contaradiction and hence 119910(119909) satisfies (53)Finally we show that 119910(119909) also satisfies inequality (54)
Because of (5) we obtain a constant 119888 gt 0 such that
max119905isinR
|119878 (119905)| = max119905isin[0120591]
|119878 (119905)| le 119888
max119905isinR
100381610038161003816100381610038161198781015840(119905)
10038161003816100381610038161003816= max119905isin[0120591]
100381610038161003816100381610038161198781015840(119905)
10038161003816100381610038161003816le 119888
(84)
By (15) we find that
sup119905isin(0119909]
1003816100381610038161003816119910 (119905)
1003816100381610038161003816= sup119905isin(0119909]
|119886 (119905)|1003816100381610038161003816119878 (120593 (119905))
1003816100381610038161003816le 119888119886 (119909) (85)
Hence from (1) and (15) it follows that100381610038161003816100381610038161199101015840(119909)
10038161003816100381610038161003816le 119888 (119886
1015840(119909) + 119886 (119909) [minus120593
1015840(119909)]) (86)
and therefore
int
1199050
119909
100381610038161003816100381610038161199101015840(120585)
10038161003816100381610038161003816119889120585 le 119888119886 (119905
0) + 119888 int
1199050
119909
119886 (120585) [minus1205931015840(120585)] 119889120585 (87)
Using (85) and (87) we see that
[119909 sup120585isin(0119909]
1003816100381610038161003816119910 (120585)
1003816100381610038161003816]
(119904minus1)(2minus119904)
int
1199050
119909
100381610038161003816100381610038161199101015840(120585)
10038161003816100381610038161003816119889120585
le 119888119886 (1199050) [119888119909119886 (119909)]
(119904minus1)(2minus119904)
+ 119888[119888119909119886 (119909)](119904minus1)(2minus119904)
int
1199050
119909
119886 (120585) [minus1205931015840(120585)] 119889120585
le 119888119886 (1199050) [1198881199050119886 (1199050)](119904minus1)(2minus119904)
+ 1198881(2minus119904)
[119909119886 (119909)](119904minus1)(2minus119904)
int
1199050
119909
119886 (120585) [minus1205931015840(120585)] 119889120585
(88)
By (18) we conclude that (54) holdsBy Lemma 28 there exists 119888
2gt 0 such that
1003816100381610038161003816Γ120576(119910)1003816100381610038161003816le 11988821205762minus119904 120576 isin (0 120576
0) (89)
Thus we have proved that the chirp function119910(119909) given by (1)satisfies the desired inequality (31) This completes the proofof Theorem 8
Proof of Theorem 11 Let 119910(119909) be the chirp function (1) It iseasy to see that 119910(119909) is oscillatory near 119909 = 0 By (5) thereexists119872 gt 0 such that
|119878 (119905)| le 119872
100381610038161003816100381610038161198781015840(119905)
10038161003816100381610038161003816le 119872 119905 isin R (90)
Hence we observe that100381610038161003816100381610038161199101015840(119909)
10038161003816100381610038161003816le
100381610038161003816100381610038161198861015840(119909) 119878 (120593 (119909))
10038161003816100381610038161003816+
10038161003816100381610038161003816119886 (119909) 119878 (120593 (119909)) 120593
1015840(119909)
10038161003816100381610038161003816
le 119872
100381610038161003816100381610038161198861015840(119909)
10038161003816100381610038161003816+ 119872
10038161003816100381610038161003816119886 (119909) 120593
1015840(119909)
10038161003816100381610038161003816
(91)
for 119909 isin (0 1199050] From (81) it follows that length(119910) lt infin
Lemma 29 shows that
length (119910) = lim120576rarr0
1003816100381610038161003816Γ120576(119910)1003816100381610038161003816
2120576
(92)
which implies that 120576length(Γ) le |Γ120576(119910)| le 4120576length(Γ) 120576 isin
(0 1205760) for some 120576
0gt 0 Therefore 119910(119909) is fractal oscillatory
near 119909 = 0 with the fractal dimension 1
4 Proof for the Fractal Oscillations of (26)In this section we give the proofs for the fractal oscillations ofthe linear second-order differential equation (26) consideredas an application of the main results on the fractal oscillationof chirp functions
Before we present the proofs of Theorems 16 and 17 wemake the following observation Since
1199101(119909) = 119886 (119909) cos (120593 (119909)) 119910
2(119909) = 119886 (119909) sin (120593 (119909))
(93)
are solutions of (26) we see that 119910(119909) = 11988811199101(119909) + 119888
21199102(119909)
is a fundamental system of all solutions of (26) Assume that1198882
1+ 1198882
2gt 0 and set 119878(119905) = 119888
1sin 119905 + 119888
2cos 119905 Then 119878(119905) clearly
satisfies (5)
10 International Journal of Differential Equations
Proof of Theorem 16 Applying Theorem 8 on 119910(119909) =
11988811199101(119909) + 119888
21199102(119909) we conclude that 119910(119909) is fractal oscillatory
with the fractal dimension 119904
Proof of Theorem 17 Theorem 11 implies that 119910(119909) =
11988811199101(119909) + 119888
21199102(119909) is fractal oscillatory near 119909 = 0 with the
fractal dimension 1
Appendix
Proof of Remark 9 By (17) there exists 1198881gt 0 such that
1198881le [minus120593
1015840(119909)]
2minus119904
int
119909
0
119886 (120585) 119889120585 119909 isin (0 1199050] (A1)
From (15) it follows that
1198881le [minus120593
1015840(119909)]
2minus119904
119886 (119909) int
119909
0
119889120585 = 119909119886 (119909) [minus1205931015840(119909)]
2minus119904
119909 isin (0 1199050]
(A2)
which implies that1
minus1205931015840(119909)
le 119888minus1(2minus119904)
11199091(2minus119904)
[119886 (119909)]1(2minus119904)
119909 isin (0 1199050] (A3)
By (19) there exists 1198882gt 0 such that
119909119886 (119909) [minus1205931015840(119909)]
2minus119904
le 1198882 119909 isin (0 119905
0] (A4)
and hence
minus1205931015840(119909) le 119888
1(2minus119904)
2119909minus1(2minus119904)
[119886 (119909)]minus1(2minus119904)
119909 isin (0 1199050]
(A5)Therefore we see that
[
1
minus1205931015840(119909)
]
119904minus1
int
1199050
119909
119886 (120585) [minus1205931015840(120585)] 119889120585
le 119888minus(119904minus1)(2minus119904)
1119909(119904minus1)(2minus119904)
[119886 (119909)](119904minus1)(2minus119904)
1198881(2minus119904)
2
times int
1199050
119909
120585minus1(2minus119904)
[119886 (120585)]minus(119904minus1)(2minus119904)
119889120585
le 1198883119909(119904minus1)(2minus119904)
[119886 (119909)](119904minus1)(2minus119904)
[119886 (119909)]minus(119904minus1)(2minus119904)
times int
1199050
119909
120585minus1(2minus119904)
119889120585
= 1198883
2 minus 119904
119904 minus 1
(1 minus 119909(119904minus1)(2minus119904)
119905minus(119904minus1)(2minus119904)
0)
le 1198883
2 minus 119904
119904 minus 1
119909 isin (0 1199050]
(A6)
so that (20) holds We conclude that
[119909119886 (119909)](119904minus1)(2minus119904)
int
1199050
119909
119886 (120585) [minus1205931015840(120585)] 119889120585
= (119909119886 (119909) [minus1205931015840(119909)]
2minus119904
)
(119904minus1)(2minus119904)
times ([
1
minus1205931015840(119909)
]
119904minus1
int
1199050
119909
119886 (120585) [minus1205931015840(120585)] 119889120585)
(A7)
By (19) and (20) we conclude that (18) is satisfied The proofof Remark 9 is complete
References
[1] C Tricot Curves and Fractal Dimension Springer New YorkNY USA 1995
[2] M Pasic ldquoFractal oscillations for a class of second order lineardifferential equations of Euler typerdquo Journal of MathematicalAnalysis and Applications vol 341 no 1 pp 211ndash223 2008
[3] M K Kwong M Pasic and J S W Wong ldquoRectifiable oscil-lations in second-order linear differential equationsrdquo Journal ofDifferential Equations vol 245 no 8 pp 2333ndash2351 2008
[4] M Pasic and J S W Wong ldquoRectifiable oscillations in second-order half-linear differential equationsrdquo Annali di MatematicaPura ed Applicata Series 4 vol 188 no 3 pp 517ndash541 2009
[5] M Pasic and S Tanaka ldquoFractal oscillations of self-adjoint anddamped linear differential equations of second-orderrdquo AppliedMathematics and Computation vol 218 no 5 pp 2281ndash22932011
[6] Y Naito M Pasic S Tanaka and D Zubrinic ldquoFractal oscilla-tions near domain boundary of radially symmetric solutions ofp-Laplace equations fractal geometry and dynamical systemsin pure and applied mathematicsrdquo Contemporary MathematicsAmerican Mathematical Society In press
[7] W A Coppel Stability and Asymptotic Behavior of DifferentialEquations D C Heath andCompany BostonMass USA 1965
[8] P Hartman Ordinary Differential Equations BirkhauserBoston Mass USA 2nd edition 1982
[9] F Neuman ldquoA general construction of linear differentialequations with solutions of prescribed propertiesrdquo AppliedMathematics Letters vol 17 no 1 pp 71ndash76 2004
[10] F Neuman ldquoStructure of solution spaces via transformationrdquoApplied Mathematics Letters vol 21 no 5 pp 529ndash533 2008
[11] P Borgnat and P Flandrin ldquoOn the chirp decomposition ofWeierstrass-Mandelbrot functions and their time-frequencyinterpretationrdquo Applied and Computational Harmonic Analysisvol 15 no 2 pp 134ndash146 2003
[12] E J Candes P R Charlton and H Helgason ldquoDetectinghighly oscillatory signals by chirplet path pursuitrdquo Applied andComputational Harmonic Analysis vol 24 no 1 pp 14ndash402008
[13] S Jaffard andYMeyer ldquoWaveletmethods for pointwise regular-ity and local oscillations of functionsrdquoMemoirs of the AmericanMathematical Society vol 123 no 587 pp 1ndash110 1996
[14] Y Meyer and H Xu ldquoWavelet analysis and chirpsrdquo Applied andComputational Harmonic Analysis vol 4 no 4 pp 366ndash3791997
[15] G RenQChen P Cerejeiras andUKaehle ldquoChirp transformsand chirp seriesrdquo Journal of Mathematical Analysis and Applica-tions vol 373 no 2 pp 356ndash369 2011
[16] M Kepesi and L Weruaga ldquoAdaptive chirp-based time-frequency analysis of speech signalsrdquo Speech Communicationvol 48 no 5 pp 474ndash492 2006
[17] L Weruaga and M Kepesi ldquoThe fan-chirp transform for non-stationary harmonic signalsrdquo Signal Processing vol 87 no 6 pp1504ndash1522 2007
[18] E Barlow A J Mulholland A Nordon and A GachaganldquoTheoretical analysis of chirp excitation of contrast agentsrdquoPhysics Procedia vol 3 no 1 pp 743ndash747 2009
International Journal of Differential Equations 11
[19] M H Pedersen T X Misaridis and J A Jensen ldquoClinicalevaluation of chirp-coded excitation in medical ultrasoundrdquoUltrasound in Medicine and Biology vol 29 no 6 pp 895ndash9052003
[20] T Paavle M Min and T Parve ldquoUsing of chirp excitation forbioimpedance estimation theoretical aspects andmodelingrdquo inProceedings of the 11th International Biennial Baltic ElectronicsConference (BECrsquo08) pp 325ndash328 Tallinn Estonia October2008
[21] M Pasic ldquoRectifiable and unrectifiable oscillations for a class ofsecond-order linear differential equations of Euler typerdquo Journalof Mathematical Analysis and Applications vol 335 no 1 pp724ndash738 2007
[22] K Falconer Fractal Geometry Mathematical Foundations andApplications John Wiley amp Sons Hoboken NJ USA 1999
[23] P Mattila Geometry of Sets and Measures in Euclidean SpacesFractals and Rectifiability vol 44 of Cambridge Studies inAdvanced Mathematics Cambridge University Press Cam-bridge UK 1995
[24] K J Falconer ldquoOn the Minkowski measurability of fractalsrdquoProceedings of the American Mathematical Society vol 123 no4 pp 1115ndash1124 1995
[25] C Q He and M L Lapidus Generalized Minkowski ContentSpectrum of Fractal Drums Fractal Strings and the RiemannZeta-Function vol 127 of Memoirs of the American Mathemat-ical Society American Mathematical Society Providence RIUSA 1997
[26] M L Lapidus and M van Frankenhuijsen Fractal geometryComplex Dimensions and Zeta Functions Geometry and Spec-tra of Fractal Strings Springer Monographs in MathematicsSpringer New York NY USA 2006
[27] M Pasic ldquoMinkowski-Bouligand dimension of solutions ofthe one-dimensional 119901-Laplacianrdquo Journal of Differential Equa-tions vol 190 no 1 pp 268ndash305 2003
[28] J S W Wong ldquoOn rectifiable oscillation of Euler type secondorder linear differential equationsrdquo Electronic Journal of Quali-tativeTheory of Differential Equations vol 2007 no 20 pp 1ndash122007
[29] M Pasic ldquoRectifiable and unrectifiable oscillations for a gen-eralization of the Riemann-Weber version of Euler differentialequationrdquo Georgian Mathematical Journal vol 15 no 4 pp759ndash774 2008
[30] M Pasic and S Tanaka ldquoRectifiable oscillations of self-adjointand damped linear differential equations of second-orderrdquoJournal of Mathematical Analysis and Applications vol 381 no1 pp 27ndash42 2011
[31] M Pasic D Zubrinic and V Zupanovic ldquoOscillatory andphase dimensions of solutions of some second-order differentialequationsrdquo Bulletin des Sciences Mathematiques vol 133 no 8pp 859ndash874 2009
[32] L Korkut and M Resman ldquoFractal oscillations of chirp-likefunctionsrdquo Georgian Mathematical Journal vol 19 no 4 pp705ndash720 2012
Submit your manuscripts athttpwwwhindawicom
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Mathematical Problems in Engineering
Hindawi Publishing Corporationhttpwwwhindawicom
Differential EquationsInternational Journal of
Volume 2014
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Stochastic AnalysisInternational Journal of
International Journal of Differential Equations 7
Lemma 27 Let 119910 isin 119862(0 1199050] be bounded on (0 119905
0] and let
120576 gt 0 Then there exists 119862 gt 0 such that for every 120576 gt 0 and119905 isin (0 119905
0]
1003816100381610038161003816Γ120576(119910)1003816100381610038161003816le 119862[119905 sup
119909isin(0119905]
1003816100381610038161003816119910 (119909)
1003816100381610038161003816+ 120576length (119910|
[1199051199050]) + 120576 + 120576
2]
(47)
Proof Let 120576 gt 0 and let 119905 isin (0 1199050] From the geometry point
of view it is clear that1003816100381610038161003816Γ120576(119910|(0119905])1003816100381610038161003816sub 119877 (48)
where 119877 is a rectangle in R2 defined by
119877 = (1199051 1199052) 1199051isin [minus120576 119905 + 120576]
1199052isin [minus120576 minus sup
119909isin(0119905]
1003816100381610038161003816119910 (119909)
1003816100381610038161003816 120576 + sup119909isin(0119905]
1003816100381610038161003816119910 (119909)
1003816100381610038161003816]
(49)
Hence we have
1003816100381610038161003816Γ120576(119910(0119905])1003816100381610038161003816le (2120576 + 119905) (2120576 + 2 sup
119909isin(0119905]
1003816100381610038161003816119910 (119909)
1003816100381610038161003816)
= 41205762+ 2(119905 + 2 sup
119909isin(0119905]
1003816100381610038161003816119910 (119909)
1003816100381610038161003816) 120576
+ 2119905 sup119909isin(0119905]
1003816100381610038161003816119910 (119909)
1003816100381610038161003816
le 41205762+ 1198621120576 + 2119905 sup
119909isin(0119905]
1003816100381610038161003816119910 (119909)
1003816100381610038161003816
(50)
where 1198621= 2(1199050+ 2sup
119909isin(01199050]|119910(119909)|) Lemma 26 implies that
10038161003816100381610038161003816Γ120576(119910|[1199051199050])
10038161003816100381610038161003816le 4120587120576length (119910|
[1199051199050]) + 4120587120576
2 (51)
Consequently we see that1003816100381610038161003816Γ120576(119910)1003816100381610038161003816le1003816100381610038161003816Γ120576(119910|(0119905])1003816100381610038161003816+
10038161003816100381610038161003816Γ120576(119910|[1199051199050])
10038161003816100381610038161003816
le 41205762+ 1198621120576 + 2119905 sup
119909isin(0119905]
1003816100381610038161003816119910 (119909)
1003816100381610038161003816
+ 4120587120576length (119910|[1199051199050]) + 4120587120576
2
le 119862[119905 sup119909isin(0119905]
1003816100381610038161003816119910 (119909)
1003816100381610038161003816+ 120576length (119910|
[1199051199050]) + 120576 + 120576
2]
(52)
where 119862 = max1198621 4(120587 + 1)
Lemma 28 Let 119910 isin 1198621(0 1199050] be bounded on (0 119905
0] Assume
that
int
1199050
0
100381610038161003816100381610038161199101015840(119909)
10038161003816100381610038161003816119889119909 = infin (53)
lim sup119909rarr0
[119909 sup120585isin(0119909]
1003816100381610038161003816119910 (120585)
1003816100381610038161003816]
(119904minus1)(2minus119904)
int
1199050
119909
100381610038161003816100381610038161199101015840(120585)
10038161003816100381610038161003816119889120585 lt infin (54)
for some 119904 isin (1 2) Then there exists 1198882gt 0 such that
1003816100381610038161003816Γ120576(119910)1003816100381610038161003816le 11988821205762minus119904 120576 isin (0 1) (55)
Proof Let 120576 isin (0 1) and let 119904 isin (1 2) be satisfy (54) By (53)there exists 119909
1isin (0 119905
0) which depends on 120576 and 119904 such that
int
1199050
1199091
100381610038161003816100381610038161199101015840(119905)
10038161003816100381610038161003816119889119905 = 120576
1minus119904 (56)
Therefore we have
length (119910|[1199091 1199050]
) = int
1199050
1199091
radic1 +10038161003816100381610038161199101015840(119909)1003816100381610038161003816
2119889119909
le int
1199050
1199091
(1 +
100381610038161003816100381610038161199101015840(119909)
10038161003816100381610038161003816) 119889119909
= 1199050minus 1199091+ 1205761minus119904
le 1199050+ 1205761minus119904
(57)
By (54) there exists 1198621gt 0 such that
[119909 sup120585isin(0119909]
1003816100381610038161003816119910 (120585)
1003816100381610038161003816]
(119904minus1)(2minus119904)
int
1199050
119909
100381610038161003816100381610038161199101015840(120585)
10038161003816100381610038161003816119889120585 le 119862
1 119909 isin (0 119905
0]
(58)
which implies that
[
[
1199091sup120585isin(01199091]
1003816100381610038161003816119910 (120585)
1003816100381610038161003816]
]
(119904minus1)(2minus119904)
int
1199050
1199091
100381610038161003816100381610038161199101015840(120585)
10038161003816100381610038161003816119889120585 le 119862
1 (59)
that is
1199091sup120585isin(01199091]
1003816100381610038161003816119910 (120585)
1003816100381610038161003816le 11986221205762minus119904 (60)
where 1198622= 119862(2minus119904)(119904minus1)
1 From Lemma 27 (57) and (60) it
follows that1003816100381610038161003816Γ120576(119910)1003816100381610038161003816le 119862 [119862
21205762minus119904+ 120576 (1199050+ 1205761minus119904) + 120576 + 120576
2]
= 119862 [(1198622+ 1) + (119905
0+ 1) 120576
119904minus1+ 120576119904] 1205762minus119904
le 119862 (1198622+ 1199050+ 3) 120576
2minus119904 120576 isin (0 1)
(61)
In order to showTheorems 8 and 11 we need the followingtwo geometric lemmas
Lemma 29 (see [1]) If Γ sube R2 is a simple curve (ie itsparameterization is a bijection) and length(Γ) lt infin then
length (Γ) = lim120576rarr0
1003816100381610038161003816Γ120576
1003816100381610038161003816
2120576
(62)
where Γ120576denotes the 120576-neighborhood of the graph Γ
8 International Journal of Differential Equations
Now we are able to proveTheorem 8
Proof of Theorem 8 Let 119904 isin (1 2) and let 119910(119909) be a chirpfunction given by (1) We note here that it is enough to showthat 119910(119909) satisfies (31)
At the first let 119909119899be a sequence defined by 119909
119899= 120593minus1(1205910+
119899120591) for all sufficiently large 119899 isin N From (4) it follows that120593minus1(119905) is decreasing Hence 119909
119899is decreasing as well as 119909
119899rarr
0 as 119899 rarr infin because of lim119909rarr+0
120593(119909) = infin (see (4))We notethat 119910(119909
119899) = 0 and 119910(119909) = 0 on (119909
119899+1 119909119899) for all sufficiently
large 119899 isin N Also minus11205931015840(119909) is an increasing function becauseof (16) The mean value theorem shows that
120591
minus1205931015840(119909119899+1)
le 119909119899minus 119909119899+1le
120591
minus1205931015840(119909119899)
(63)
Now let 1205760isin (0 1) Let 119896(120576) be the smallest natural number
satisfying120591
minus1205931015840(119909119896(120576))
le 120576 forall120576 isin (0 1205760) (64)
Such 119896(120576) exists for every 120576 isin (0 1205760) since 119909
119899rarr 0 as 119899 rarr
infin and lim119909rarr+0
1205931015840(119909) = minusinfin (this equality is true because
1205931015840notin 1198711(0 1199050) since lim
119909rarr+0120593(119909) = infin) Moreover since 119909
119899
is decreasing and minus11205931015840(119909) is increasing we obtain
minus1205931015840(119909119899) ge 120591120576
minus1forall119899 ge 119896 (120576) (65)
Combining (63) and (65) it is easy to deduce that suchdefined 119896(120576) satisfies condition (32)
By (16) there exists 119871 gt 0 such that (1(minus1205931015840(119909)))1015840 le 119871 for119909 isin (0 119905
0] which means that
minus
minus12059310158401015840(119909)
minus1205931015840(119909)
le minus1198711205931015840(119909) 119909 isin (0 119905
0] (66)
Integrating (66) on [120593minus1(119905 + 2120591) 120593minus1(119905)] we have
logminus1205931015840(120593minus1(119905 + 2120591))
minus1205931015840(120593minus1(119905))
le 2119871120591 119905 isin [120593 (1199050) infin) (67)
that is
minus1205931015840(120593minus1(119905)) ge minus119890
minus21198711205911205931015840(120593minus1(119905 + 120591)) 119905 isin [120593 (119905
0) infin)
(68)
By the definition of 119896(120576) and (64) we see that120591
minus1205931015840(119909119896(120576)minus1
)
gt 120576 (69)
Hence from (68) it follows that
120591120576minus1gt minus1205931015840(119909119896(120576)minus1
)
= minus1205931015840(120593minus1(1205910+ (119896 (120576) minus 1) 120591))
ge minus119890minus2119871120591
1205931015840(120593minus1(1205910+ (119896 (120576) minus 1) 120591 + 2120591))
= minus119890minus2119871120591
1205931015840(120593minus1(1205910+ (119896 (120576) + 1) 120591))
= minus119890minus2119871120591
1205931015840(119909119896(120576)+1
)
(70)
which implies that
minus1205931015840(119909119896(120576)+1
) le 119872120576minus1 120576 gt 0 (71)
where119872 = 1205911198902119871120591
Next it is clear that the assumption (5) ensures a realnumber119872
0gt 0 such that
1198720= max119905isinR
|119878 (119905)| = max119905isin[0120591]
|119878 (119905)| (72)
Hence from (15) we have
max119909isin[119909119899+1 119909119899]
1003816100381610038161003816119910 (119909)
1003816100381610038161003816= max119909isin[119909119899+1119909119899]
(119886 (119909)1003816100381610038161003816119878 (120593 (119909))
1003816100381610038161003816)
ge 119886 (119909119899+1) max119909isin[119909119899+1119909119899]
1003816100381610038161003816119878 (120593 (119909))
1003816100381610038161003816
= 1198720119886 (119909119899+1)
(73)
Now from (63) and (73) it follows that
max119909isin[119909119899+1 119909119899]
1003816100381610038161003816119910 (119909)
1003816100381610038161003816(119909119899minus 119909119899+1) ge 119872
0119886 (119909119899+1) (119909119899minus 119909119899+1)
ge 1205911198720
119886 (119909119899+1)
minus1205931015840(119909119899+1)
= 1205911198720
119886 (120593minus1(1205910+(119899+1) 120591))
minus1205931015840(120593minus1(1205910+(119899+1) 120591))
(74)
Since 120593minus1(119905) is decreasing as well as minus11205931015840(119909) is increasingand 119886(119909) is nondecreasing and positive we conclude that thefunction
119886 (120593minus1(119905))
minus1205931015840(120593minus1(119905))
is nonincreasing (75)
Hence we observe that
sum
119899ge119896(120576)
max119909isin[119909119899+1119909119899]
1003816100381610038161003816119910 (119909)
1003816100381610038161003816(119909119899minus 119909119899+1)
ge 1205911198720sum
119899ge119896(120576)
119886 (120593minus1(1205910+ (119899 + 1) 120591))
minus1205931015840(120593minus1(1205910+ (119899 + 1) 120591))
ge 1205911198720int
infin
119896(120576)
119886 (120593minus1(1205910+ (119905 + 1) 120591))
minus1205931015840(120593minus1(1205910+ (119905 + 1) 120591))
119889119905
= 1198720int
120593minus1(1205910+(119896(120576)+1)120591)
0
119886 (120585) 119889120585
= 1198720int
119909119896(120576)+1
0
119886 (120585) 119889120585
(76)
Next from assumption (17) we get some1198721gt 0 such that
int
119909
0
119886 (120585) 119889120585 ge 1198721[minus1205931015840(119909)]
minus(2minus119904)
119909 isin (0 1199050] (77)
International Journal of Differential Equations 9
Now from (76) (77) and (71) we obtain
sum
119899ge119896(120576)
max119909isin[119909119899+1119909119899]
1003816100381610038161003816119910 (119909)
1003816100381610038161003816(119909119899minus 119909119899+1)
ge 11987201198721[minus1205931015840(119909119896(120576)+1
)]
minus(2minus119904)
ge 11987201198721(119872120576minus1)
minus(2minus119904)
= 11987201198721119872minus(2minus119904)
1205762minus119904
(78)
Thus using Lemma 25 we have
1003816100381610038161003816Γ120576(119910)1003816100381610038161003816ge 11988811205762minus119904 120576 isin (0 120576
0) (79)
for some 1198881gt 0
Next we prove that 119910(119909) satisfies (53) Assume to thecontrary that
int
1199050
0
100381610038161003816100381610038161199101015840(119909)
10038161003816100381610038161003816119889119909 lt infin (80)
Then we have
length (119910) = int1199050
0
radic1 +10038161003816100381610038161199101015840(119909)1003816100381610038161003816
2119889119909
le int
1199050
0
(1 +
100381610038161003816100381610038161199101015840(119909)
10038161003816100381610038161003816) 119889119909 lt infin
(81)
From Lemma 29 it follows that
length (119910) = lim120576rarr+0
1003816100381610038161003816Γ120576(119910)1003816100381610038161003816
2120576
(82)
On the other hand by (79) we see that
lim120576rarr+0
1003816100381610038161003816Γ120576(119910)1003816100381610038161003816
2120576
= infin (83)
This is a contaradiction and hence 119910(119909) satisfies (53)Finally we show that 119910(119909) also satisfies inequality (54)
Because of (5) we obtain a constant 119888 gt 0 such that
max119905isinR
|119878 (119905)| = max119905isin[0120591]
|119878 (119905)| le 119888
max119905isinR
100381610038161003816100381610038161198781015840(119905)
10038161003816100381610038161003816= max119905isin[0120591]
100381610038161003816100381610038161198781015840(119905)
10038161003816100381610038161003816le 119888
(84)
By (15) we find that
sup119905isin(0119909]
1003816100381610038161003816119910 (119905)
1003816100381610038161003816= sup119905isin(0119909]
|119886 (119905)|1003816100381610038161003816119878 (120593 (119905))
1003816100381610038161003816le 119888119886 (119909) (85)
Hence from (1) and (15) it follows that100381610038161003816100381610038161199101015840(119909)
10038161003816100381610038161003816le 119888 (119886
1015840(119909) + 119886 (119909) [minus120593
1015840(119909)]) (86)
and therefore
int
1199050
119909
100381610038161003816100381610038161199101015840(120585)
10038161003816100381610038161003816119889120585 le 119888119886 (119905
0) + 119888 int
1199050
119909
119886 (120585) [minus1205931015840(120585)] 119889120585 (87)
Using (85) and (87) we see that
[119909 sup120585isin(0119909]
1003816100381610038161003816119910 (120585)
1003816100381610038161003816]
(119904minus1)(2minus119904)
int
1199050
119909
100381610038161003816100381610038161199101015840(120585)
10038161003816100381610038161003816119889120585
le 119888119886 (1199050) [119888119909119886 (119909)]
(119904minus1)(2minus119904)
+ 119888[119888119909119886 (119909)](119904minus1)(2minus119904)
int
1199050
119909
119886 (120585) [minus1205931015840(120585)] 119889120585
le 119888119886 (1199050) [1198881199050119886 (1199050)](119904minus1)(2minus119904)
+ 1198881(2minus119904)
[119909119886 (119909)](119904minus1)(2minus119904)
int
1199050
119909
119886 (120585) [minus1205931015840(120585)] 119889120585
(88)
By (18) we conclude that (54) holdsBy Lemma 28 there exists 119888
2gt 0 such that
1003816100381610038161003816Γ120576(119910)1003816100381610038161003816le 11988821205762minus119904 120576 isin (0 120576
0) (89)
Thus we have proved that the chirp function119910(119909) given by (1)satisfies the desired inequality (31) This completes the proofof Theorem 8
Proof of Theorem 11 Let 119910(119909) be the chirp function (1) It iseasy to see that 119910(119909) is oscillatory near 119909 = 0 By (5) thereexists119872 gt 0 such that
|119878 (119905)| le 119872
100381610038161003816100381610038161198781015840(119905)
10038161003816100381610038161003816le 119872 119905 isin R (90)
Hence we observe that100381610038161003816100381610038161199101015840(119909)
10038161003816100381610038161003816le
100381610038161003816100381610038161198861015840(119909) 119878 (120593 (119909))
10038161003816100381610038161003816+
10038161003816100381610038161003816119886 (119909) 119878 (120593 (119909)) 120593
1015840(119909)
10038161003816100381610038161003816
le 119872
100381610038161003816100381610038161198861015840(119909)
10038161003816100381610038161003816+ 119872
10038161003816100381610038161003816119886 (119909) 120593
1015840(119909)
10038161003816100381610038161003816
(91)
for 119909 isin (0 1199050] From (81) it follows that length(119910) lt infin
Lemma 29 shows that
length (119910) = lim120576rarr0
1003816100381610038161003816Γ120576(119910)1003816100381610038161003816
2120576
(92)
which implies that 120576length(Γ) le |Γ120576(119910)| le 4120576length(Γ) 120576 isin
(0 1205760) for some 120576
0gt 0 Therefore 119910(119909) is fractal oscillatory
near 119909 = 0 with the fractal dimension 1
4 Proof for the Fractal Oscillations of (26)In this section we give the proofs for the fractal oscillations ofthe linear second-order differential equation (26) consideredas an application of the main results on the fractal oscillationof chirp functions
Before we present the proofs of Theorems 16 and 17 wemake the following observation Since
1199101(119909) = 119886 (119909) cos (120593 (119909)) 119910
2(119909) = 119886 (119909) sin (120593 (119909))
(93)
are solutions of (26) we see that 119910(119909) = 11988811199101(119909) + 119888
21199102(119909)
is a fundamental system of all solutions of (26) Assume that1198882
1+ 1198882
2gt 0 and set 119878(119905) = 119888
1sin 119905 + 119888
2cos 119905 Then 119878(119905) clearly
satisfies (5)
10 International Journal of Differential Equations
Proof of Theorem 16 Applying Theorem 8 on 119910(119909) =
11988811199101(119909) + 119888
21199102(119909) we conclude that 119910(119909) is fractal oscillatory
with the fractal dimension 119904
Proof of Theorem 17 Theorem 11 implies that 119910(119909) =
11988811199101(119909) + 119888
21199102(119909) is fractal oscillatory near 119909 = 0 with the
fractal dimension 1
Appendix
Proof of Remark 9 By (17) there exists 1198881gt 0 such that
1198881le [minus120593
1015840(119909)]
2minus119904
int
119909
0
119886 (120585) 119889120585 119909 isin (0 1199050] (A1)
From (15) it follows that
1198881le [minus120593
1015840(119909)]
2minus119904
119886 (119909) int
119909
0
119889120585 = 119909119886 (119909) [minus1205931015840(119909)]
2minus119904
119909 isin (0 1199050]
(A2)
which implies that1
minus1205931015840(119909)
le 119888minus1(2minus119904)
11199091(2minus119904)
[119886 (119909)]1(2minus119904)
119909 isin (0 1199050] (A3)
By (19) there exists 1198882gt 0 such that
119909119886 (119909) [minus1205931015840(119909)]
2minus119904
le 1198882 119909 isin (0 119905
0] (A4)
and hence
minus1205931015840(119909) le 119888
1(2minus119904)
2119909minus1(2minus119904)
[119886 (119909)]minus1(2minus119904)
119909 isin (0 1199050]
(A5)Therefore we see that
[
1
minus1205931015840(119909)
]
119904minus1
int
1199050
119909
119886 (120585) [minus1205931015840(120585)] 119889120585
le 119888minus(119904minus1)(2minus119904)
1119909(119904minus1)(2minus119904)
[119886 (119909)](119904minus1)(2minus119904)
1198881(2minus119904)
2
times int
1199050
119909
120585minus1(2minus119904)
[119886 (120585)]minus(119904minus1)(2minus119904)
119889120585
le 1198883119909(119904minus1)(2minus119904)
[119886 (119909)](119904minus1)(2minus119904)
[119886 (119909)]minus(119904minus1)(2minus119904)
times int
1199050
119909
120585minus1(2minus119904)
119889120585
= 1198883
2 minus 119904
119904 minus 1
(1 minus 119909(119904minus1)(2minus119904)
119905minus(119904minus1)(2minus119904)
0)
le 1198883
2 minus 119904
119904 minus 1
119909 isin (0 1199050]
(A6)
so that (20) holds We conclude that
[119909119886 (119909)](119904minus1)(2minus119904)
int
1199050
119909
119886 (120585) [minus1205931015840(120585)] 119889120585
= (119909119886 (119909) [minus1205931015840(119909)]
2minus119904
)
(119904minus1)(2minus119904)
times ([
1
minus1205931015840(119909)
]
119904minus1
int
1199050
119909
119886 (120585) [minus1205931015840(120585)] 119889120585)
(A7)
By (19) and (20) we conclude that (18) is satisfied The proofof Remark 9 is complete
References
[1] C Tricot Curves and Fractal Dimension Springer New YorkNY USA 1995
[2] M Pasic ldquoFractal oscillations for a class of second order lineardifferential equations of Euler typerdquo Journal of MathematicalAnalysis and Applications vol 341 no 1 pp 211ndash223 2008
[3] M K Kwong M Pasic and J S W Wong ldquoRectifiable oscil-lations in second-order linear differential equationsrdquo Journal ofDifferential Equations vol 245 no 8 pp 2333ndash2351 2008
[4] M Pasic and J S W Wong ldquoRectifiable oscillations in second-order half-linear differential equationsrdquo Annali di MatematicaPura ed Applicata Series 4 vol 188 no 3 pp 517ndash541 2009
[5] M Pasic and S Tanaka ldquoFractal oscillations of self-adjoint anddamped linear differential equations of second-orderrdquo AppliedMathematics and Computation vol 218 no 5 pp 2281ndash22932011
[6] Y Naito M Pasic S Tanaka and D Zubrinic ldquoFractal oscilla-tions near domain boundary of radially symmetric solutions ofp-Laplace equations fractal geometry and dynamical systemsin pure and applied mathematicsrdquo Contemporary MathematicsAmerican Mathematical Society In press
[7] W A Coppel Stability and Asymptotic Behavior of DifferentialEquations D C Heath andCompany BostonMass USA 1965
[8] P Hartman Ordinary Differential Equations BirkhauserBoston Mass USA 2nd edition 1982
[9] F Neuman ldquoA general construction of linear differentialequations with solutions of prescribed propertiesrdquo AppliedMathematics Letters vol 17 no 1 pp 71ndash76 2004
[10] F Neuman ldquoStructure of solution spaces via transformationrdquoApplied Mathematics Letters vol 21 no 5 pp 529ndash533 2008
[11] P Borgnat and P Flandrin ldquoOn the chirp decomposition ofWeierstrass-Mandelbrot functions and their time-frequencyinterpretationrdquo Applied and Computational Harmonic Analysisvol 15 no 2 pp 134ndash146 2003
[12] E J Candes P R Charlton and H Helgason ldquoDetectinghighly oscillatory signals by chirplet path pursuitrdquo Applied andComputational Harmonic Analysis vol 24 no 1 pp 14ndash402008
[13] S Jaffard andYMeyer ldquoWaveletmethods for pointwise regular-ity and local oscillations of functionsrdquoMemoirs of the AmericanMathematical Society vol 123 no 587 pp 1ndash110 1996
[14] Y Meyer and H Xu ldquoWavelet analysis and chirpsrdquo Applied andComputational Harmonic Analysis vol 4 no 4 pp 366ndash3791997
[15] G RenQChen P Cerejeiras andUKaehle ldquoChirp transformsand chirp seriesrdquo Journal of Mathematical Analysis and Applica-tions vol 373 no 2 pp 356ndash369 2011
[16] M Kepesi and L Weruaga ldquoAdaptive chirp-based time-frequency analysis of speech signalsrdquo Speech Communicationvol 48 no 5 pp 474ndash492 2006
[17] L Weruaga and M Kepesi ldquoThe fan-chirp transform for non-stationary harmonic signalsrdquo Signal Processing vol 87 no 6 pp1504ndash1522 2007
[18] E Barlow A J Mulholland A Nordon and A GachaganldquoTheoretical analysis of chirp excitation of contrast agentsrdquoPhysics Procedia vol 3 no 1 pp 743ndash747 2009
International Journal of Differential Equations 11
[19] M H Pedersen T X Misaridis and J A Jensen ldquoClinicalevaluation of chirp-coded excitation in medical ultrasoundrdquoUltrasound in Medicine and Biology vol 29 no 6 pp 895ndash9052003
[20] T Paavle M Min and T Parve ldquoUsing of chirp excitation forbioimpedance estimation theoretical aspects andmodelingrdquo inProceedings of the 11th International Biennial Baltic ElectronicsConference (BECrsquo08) pp 325ndash328 Tallinn Estonia October2008
[21] M Pasic ldquoRectifiable and unrectifiable oscillations for a class ofsecond-order linear differential equations of Euler typerdquo Journalof Mathematical Analysis and Applications vol 335 no 1 pp724ndash738 2007
[22] K Falconer Fractal Geometry Mathematical Foundations andApplications John Wiley amp Sons Hoboken NJ USA 1999
[23] P Mattila Geometry of Sets and Measures in Euclidean SpacesFractals and Rectifiability vol 44 of Cambridge Studies inAdvanced Mathematics Cambridge University Press Cam-bridge UK 1995
[24] K J Falconer ldquoOn the Minkowski measurability of fractalsrdquoProceedings of the American Mathematical Society vol 123 no4 pp 1115ndash1124 1995
[25] C Q He and M L Lapidus Generalized Minkowski ContentSpectrum of Fractal Drums Fractal Strings and the RiemannZeta-Function vol 127 of Memoirs of the American Mathemat-ical Society American Mathematical Society Providence RIUSA 1997
[26] M L Lapidus and M van Frankenhuijsen Fractal geometryComplex Dimensions and Zeta Functions Geometry and Spec-tra of Fractal Strings Springer Monographs in MathematicsSpringer New York NY USA 2006
[27] M Pasic ldquoMinkowski-Bouligand dimension of solutions ofthe one-dimensional 119901-Laplacianrdquo Journal of Differential Equa-tions vol 190 no 1 pp 268ndash305 2003
[28] J S W Wong ldquoOn rectifiable oscillation of Euler type secondorder linear differential equationsrdquo Electronic Journal of Quali-tativeTheory of Differential Equations vol 2007 no 20 pp 1ndash122007
[29] M Pasic ldquoRectifiable and unrectifiable oscillations for a gen-eralization of the Riemann-Weber version of Euler differentialequationrdquo Georgian Mathematical Journal vol 15 no 4 pp759ndash774 2008
[30] M Pasic and S Tanaka ldquoRectifiable oscillations of self-adjointand damped linear differential equations of second-orderrdquoJournal of Mathematical Analysis and Applications vol 381 no1 pp 27ndash42 2011
[31] M Pasic D Zubrinic and V Zupanovic ldquoOscillatory andphase dimensions of solutions of some second-order differentialequationsrdquo Bulletin des Sciences Mathematiques vol 133 no 8pp 859ndash874 2009
[32] L Korkut and M Resman ldquoFractal oscillations of chirp-likefunctionsrdquo Georgian Mathematical Journal vol 19 no 4 pp705ndash720 2012
Submit your manuscripts athttpwwwhindawicom
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Mathematical Problems in Engineering
Hindawi Publishing Corporationhttpwwwhindawicom
Differential EquationsInternational Journal of
Volume 2014
Applied MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Mathematical PhysicsAdvances in
Complex AnalysisJournal of
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OptimizationJournal of
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CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Operations ResearchAdvances in
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Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Function Spaces
Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of Mathematics and Mathematical Sciences
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Algebra
Discrete Dynamics in Nature and Society
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
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Decision SciencesAdvances in
Discrete MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom
Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Stochastic AnalysisInternational Journal of
8 International Journal of Differential Equations
Now we are able to proveTheorem 8
Proof of Theorem 8 Let 119904 isin (1 2) and let 119910(119909) be a chirpfunction given by (1) We note here that it is enough to showthat 119910(119909) satisfies (31)
At the first let 119909119899be a sequence defined by 119909
119899= 120593minus1(1205910+
119899120591) for all sufficiently large 119899 isin N From (4) it follows that120593minus1(119905) is decreasing Hence 119909
119899is decreasing as well as 119909
119899rarr
0 as 119899 rarr infin because of lim119909rarr+0
120593(119909) = infin (see (4))We notethat 119910(119909
119899) = 0 and 119910(119909) = 0 on (119909
119899+1 119909119899) for all sufficiently
large 119899 isin N Also minus11205931015840(119909) is an increasing function becauseof (16) The mean value theorem shows that
120591
minus1205931015840(119909119899+1)
le 119909119899minus 119909119899+1le
120591
minus1205931015840(119909119899)
(63)
Now let 1205760isin (0 1) Let 119896(120576) be the smallest natural number
satisfying120591
minus1205931015840(119909119896(120576))
le 120576 forall120576 isin (0 1205760) (64)
Such 119896(120576) exists for every 120576 isin (0 1205760) since 119909
119899rarr 0 as 119899 rarr
infin and lim119909rarr+0
1205931015840(119909) = minusinfin (this equality is true because
1205931015840notin 1198711(0 1199050) since lim
119909rarr+0120593(119909) = infin) Moreover since 119909
119899
is decreasing and minus11205931015840(119909) is increasing we obtain
minus1205931015840(119909119899) ge 120591120576
minus1forall119899 ge 119896 (120576) (65)
Combining (63) and (65) it is easy to deduce that suchdefined 119896(120576) satisfies condition (32)
By (16) there exists 119871 gt 0 such that (1(minus1205931015840(119909)))1015840 le 119871 for119909 isin (0 119905
0] which means that
minus
minus12059310158401015840(119909)
minus1205931015840(119909)
le minus1198711205931015840(119909) 119909 isin (0 119905
0] (66)
Integrating (66) on [120593minus1(119905 + 2120591) 120593minus1(119905)] we have
logminus1205931015840(120593minus1(119905 + 2120591))
minus1205931015840(120593minus1(119905))
le 2119871120591 119905 isin [120593 (1199050) infin) (67)
that is
minus1205931015840(120593minus1(119905)) ge minus119890
minus21198711205911205931015840(120593minus1(119905 + 120591)) 119905 isin [120593 (119905
0) infin)
(68)
By the definition of 119896(120576) and (64) we see that120591
minus1205931015840(119909119896(120576)minus1
)
gt 120576 (69)
Hence from (68) it follows that
120591120576minus1gt minus1205931015840(119909119896(120576)minus1
)
= minus1205931015840(120593minus1(1205910+ (119896 (120576) minus 1) 120591))
ge minus119890minus2119871120591
1205931015840(120593minus1(1205910+ (119896 (120576) minus 1) 120591 + 2120591))
= minus119890minus2119871120591
1205931015840(120593minus1(1205910+ (119896 (120576) + 1) 120591))
= minus119890minus2119871120591
1205931015840(119909119896(120576)+1
)
(70)
which implies that
minus1205931015840(119909119896(120576)+1
) le 119872120576minus1 120576 gt 0 (71)
where119872 = 1205911198902119871120591
Next it is clear that the assumption (5) ensures a realnumber119872
0gt 0 such that
1198720= max119905isinR
|119878 (119905)| = max119905isin[0120591]
|119878 (119905)| (72)
Hence from (15) we have
max119909isin[119909119899+1 119909119899]
1003816100381610038161003816119910 (119909)
1003816100381610038161003816= max119909isin[119909119899+1119909119899]
(119886 (119909)1003816100381610038161003816119878 (120593 (119909))
1003816100381610038161003816)
ge 119886 (119909119899+1) max119909isin[119909119899+1119909119899]
1003816100381610038161003816119878 (120593 (119909))
1003816100381610038161003816
= 1198720119886 (119909119899+1)
(73)
Now from (63) and (73) it follows that
max119909isin[119909119899+1 119909119899]
1003816100381610038161003816119910 (119909)
1003816100381610038161003816(119909119899minus 119909119899+1) ge 119872
0119886 (119909119899+1) (119909119899minus 119909119899+1)
ge 1205911198720
119886 (119909119899+1)
minus1205931015840(119909119899+1)
= 1205911198720
119886 (120593minus1(1205910+(119899+1) 120591))
minus1205931015840(120593minus1(1205910+(119899+1) 120591))
(74)
Since 120593minus1(119905) is decreasing as well as minus11205931015840(119909) is increasingand 119886(119909) is nondecreasing and positive we conclude that thefunction
119886 (120593minus1(119905))
minus1205931015840(120593minus1(119905))
is nonincreasing (75)
Hence we observe that
sum
119899ge119896(120576)
max119909isin[119909119899+1119909119899]
1003816100381610038161003816119910 (119909)
1003816100381610038161003816(119909119899minus 119909119899+1)
ge 1205911198720sum
119899ge119896(120576)
119886 (120593minus1(1205910+ (119899 + 1) 120591))
minus1205931015840(120593minus1(1205910+ (119899 + 1) 120591))
ge 1205911198720int
infin
119896(120576)
119886 (120593minus1(1205910+ (119905 + 1) 120591))
minus1205931015840(120593minus1(1205910+ (119905 + 1) 120591))
119889119905
= 1198720int
120593minus1(1205910+(119896(120576)+1)120591)
0
119886 (120585) 119889120585
= 1198720int
119909119896(120576)+1
0
119886 (120585) 119889120585
(76)
Next from assumption (17) we get some1198721gt 0 such that
int
119909
0
119886 (120585) 119889120585 ge 1198721[minus1205931015840(119909)]
minus(2minus119904)
119909 isin (0 1199050] (77)
International Journal of Differential Equations 9
Now from (76) (77) and (71) we obtain
sum
119899ge119896(120576)
max119909isin[119909119899+1119909119899]
1003816100381610038161003816119910 (119909)
1003816100381610038161003816(119909119899minus 119909119899+1)
ge 11987201198721[minus1205931015840(119909119896(120576)+1
)]
minus(2minus119904)
ge 11987201198721(119872120576minus1)
minus(2minus119904)
= 11987201198721119872minus(2minus119904)
1205762minus119904
(78)
Thus using Lemma 25 we have
1003816100381610038161003816Γ120576(119910)1003816100381610038161003816ge 11988811205762minus119904 120576 isin (0 120576
0) (79)
for some 1198881gt 0
Next we prove that 119910(119909) satisfies (53) Assume to thecontrary that
int
1199050
0
100381610038161003816100381610038161199101015840(119909)
10038161003816100381610038161003816119889119909 lt infin (80)
Then we have
length (119910) = int1199050
0
radic1 +10038161003816100381610038161199101015840(119909)1003816100381610038161003816
2119889119909
le int
1199050
0
(1 +
100381610038161003816100381610038161199101015840(119909)
10038161003816100381610038161003816) 119889119909 lt infin
(81)
From Lemma 29 it follows that
length (119910) = lim120576rarr+0
1003816100381610038161003816Γ120576(119910)1003816100381610038161003816
2120576
(82)
On the other hand by (79) we see that
lim120576rarr+0
1003816100381610038161003816Γ120576(119910)1003816100381610038161003816
2120576
= infin (83)
This is a contaradiction and hence 119910(119909) satisfies (53)Finally we show that 119910(119909) also satisfies inequality (54)
Because of (5) we obtain a constant 119888 gt 0 such that
max119905isinR
|119878 (119905)| = max119905isin[0120591]
|119878 (119905)| le 119888
max119905isinR
100381610038161003816100381610038161198781015840(119905)
10038161003816100381610038161003816= max119905isin[0120591]
100381610038161003816100381610038161198781015840(119905)
10038161003816100381610038161003816le 119888
(84)
By (15) we find that
sup119905isin(0119909]
1003816100381610038161003816119910 (119905)
1003816100381610038161003816= sup119905isin(0119909]
|119886 (119905)|1003816100381610038161003816119878 (120593 (119905))
1003816100381610038161003816le 119888119886 (119909) (85)
Hence from (1) and (15) it follows that100381610038161003816100381610038161199101015840(119909)
10038161003816100381610038161003816le 119888 (119886
1015840(119909) + 119886 (119909) [minus120593
1015840(119909)]) (86)
and therefore
int
1199050
119909
100381610038161003816100381610038161199101015840(120585)
10038161003816100381610038161003816119889120585 le 119888119886 (119905
0) + 119888 int
1199050
119909
119886 (120585) [minus1205931015840(120585)] 119889120585 (87)
Using (85) and (87) we see that
[119909 sup120585isin(0119909]
1003816100381610038161003816119910 (120585)
1003816100381610038161003816]
(119904minus1)(2minus119904)
int
1199050
119909
100381610038161003816100381610038161199101015840(120585)
10038161003816100381610038161003816119889120585
le 119888119886 (1199050) [119888119909119886 (119909)]
(119904minus1)(2minus119904)
+ 119888[119888119909119886 (119909)](119904minus1)(2minus119904)
int
1199050
119909
119886 (120585) [minus1205931015840(120585)] 119889120585
le 119888119886 (1199050) [1198881199050119886 (1199050)](119904minus1)(2minus119904)
+ 1198881(2minus119904)
[119909119886 (119909)](119904minus1)(2minus119904)
int
1199050
119909
119886 (120585) [minus1205931015840(120585)] 119889120585
(88)
By (18) we conclude that (54) holdsBy Lemma 28 there exists 119888
2gt 0 such that
1003816100381610038161003816Γ120576(119910)1003816100381610038161003816le 11988821205762minus119904 120576 isin (0 120576
0) (89)
Thus we have proved that the chirp function119910(119909) given by (1)satisfies the desired inequality (31) This completes the proofof Theorem 8
Proof of Theorem 11 Let 119910(119909) be the chirp function (1) It iseasy to see that 119910(119909) is oscillatory near 119909 = 0 By (5) thereexists119872 gt 0 such that
|119878 (119905)| le 119872
100381610038161003816100381610038161198781015840(119905)
10038161003816100381610038161003816le 119872 119905 isin R (90)
Hence we observe that100381610038161003816100381610038161199101015840(119909)
10038161003816100381610038161003816le
100381610038161003816100381610038161198861015840(119909) 119878 (120593 (119909))
10038161003816100381610038161003816+
10038161003816100381610038161003816119886 (119909) 119878 (120593 (119909)) 120593
1015840(119909)
10038161003816100381610038161003816
le 119872
100381610038161003816100381610038161198861015840(119909)
10038161003816100381610038161003816+ 119872
10038161003816100381610038161003816119886 (119909) 120593
1015840(119909)
10038161003816100381610038161003816
(91)
for 119909 isin (0 1199050] From (81) it follows that length(119910) lt infin
Lemma 29 shows that
length (119910) = lim120576rarr0
1003816100381610038161003816Γ120576(119910)1003816100381610038161003816
2120576
(92)
which implies that 120576length(Γ) le |Γ120576(119910)| le 4120576length(Γ) 120576 isin
(0 1205760) for some 120576
0gt 0 Therefore 119910(119909) is fractal oscillatory
near 119909 = 0 with the fractal dimension 1
4 Proof for the Fractal Oscillations of (26)In this section we give the proofs for the fractal oscillations ofthe linear second-order differential equation (26) consideredas an application of the main results on the fractal oscillationof chirp functions
Before we present the proofs of Theorems 16 and 17 wemake the following observation Since
1199101(119909) = 119886 (119909) cos (120593 (119909)) 119910
2(119909) = 119886 (119909) sin (120593 (119909))
(93)
are solutions of (26) we see that 119910(119909) = 11988811199101(119909) + 119888
21199102(119909)
is a fundamental system of all solutions of (26) Assume that1198882
1+ 1198882
2gt 0 and set 119878(119905) = 119888
1sin 119905 + 119888
2cos 119905 Then 119878(119905) clearly
satisfies (5)
10 International Journal of Differential Equations
Proof of Theorem 16 Applying Theorem 8 on 119910(119909) =
11988811199101(119909) + 119888
21199102(119909) we conclude that 119910(119909) is fractal oscillatory
with the fractal dimension 119904
Proof of Theorem 17 Theorem 11 implies that 119910(119909) =
11988811199101(119909) + 119888
21199102(119909) is fractal oscillatory near 119909 = 0 with the
fractal dimension 1
Appendix
Proof of Remark 9 By (17) there exists 1198881gt 0 such that
1198881le [minus120593
1015840(119909)]
2minus119904
int
119909
0
119886 (120585) 119889120585 119909 isin (0 1199050] (A1)
From (15) it follows that
1198881le [minus120593
1015840(119909)]
2minus119904
119886 (119909) int
119909
0
119889120585 = 119909119886 (119909) [minus1205931015840(119909)]
2minus119904
119909 isin (0 1199050]
(A2)
which implies that1
minus1205931015840(119909)
le 119888minus1(2minus119904)
11199091(2minus119904)
[119886 (119909)]1(2minus119904)
119909 isin (0 1199050] (A3)
By (19) there exists 1198882gt 0 such that
119909119886 (119909) [minus1205931015840(119909)]
2minus119904
le 1198882 119909 isin (0 119905
0] (A4)
and hence
minus1205931015840(119909) le 119888
1(2minus119904)
2119909minus1(2minus119904)
[119886 (119909)]minus1(2minus119904)
119909 isin (0 1199050]
(A5)Therefore we see that
[
1
minus1205931015840(119909)
]
119904minus1
int
1199050
119909
119886 (120585) [minus1205931015840(120585)] 119889120585
le 119888minus(119904minus1)(2minus119904)
1119909(119904minus1)(2minus119904)
[119886 (119909)](119904minus1)(2minus119904)
1198881(2minus119904)
2
times int
1199050
119909
120585minus1(2minus119904)
[119886 (120585)]minus(119904minus1)(2minus119904)
119889120585
le 1198883119909(119904minus1)(2minus119904)
[119886 (119909)](119904minus1)(2minus119904)
[119886 (119909)]minus(119904minus1)(2minus119904)
times int
1199050
119909
120585minus1(2minus119904)
119889120585
= 1198883
2 minus 119904
119904 minus 1
(1 minus 119909(119904minus1)(2minus119904)
119905minus(119904minus1)(2minus119904)
0)
le 1198883
2 minus 119904
119904 minus 1
119909 isin (0 1199050]
(A6)
so that (20) holds We conclude that
[119909119886 (119909)](119904minus1)(2minus119904)
int
1199050
119909
119886 (120585) [minus1205931015840(120585)] 119889120585
= (119909119886 (119909) [minus1205931015840(119909)]
2minus119904
)
(119904minus1)(2minus119904)
times ([
1
minus1205931015840(119909)
]
119904minus1
int
1199050
119909
119886 (120585) [minus1205931015840(120585)] 119889120585)
(A7)
By (19) and (20) we conclude that (18) is satisfied The proofof Remark 9 is complete
References
[1] C Tricot Curves and Fractal Dimension Springer New YorkNY USA 1995
[2] M Pasic ldquoFractal oscillations for a class of second order lineardifferential equations of Euler typerdquo Journal of MathematicalAnalysis and Applications vol 341 no 1 pp 211ndash223 2008
[3] M K Kwong M Pasic and J S W Wong ldquoRectifiable oscil-lations in second-order linear differential equationsrdquo Journal ofDifferential Equations vol 245 no 8 pp 2333ndash2351 2008
[4] M Pasic and J S W Wong ldquoRectifiable oscillations in second-order half-linear differential equationsrdquo Annali di MatematicaPura ed Applicata Series 4 vol 188 no 3 pp 517ndash541 2009
[5] M Pasic and S Tanaka ldquoFractal oscillations of self-adjoint anddamped linear differential equations of second-orderrdquo AppliedMathematics and Computation vol 218 no 5 pp 2281ndash22932011
[6] Y Naito M Pasic S Tanaka and D Zubrinic ldquoFractal oscilla-tions near domain boundary of radially symmetric solutions ofp-Laplace equations fractal geometry and dynamical systemsin pure and applied mathematicsrdquo Contemporary MathematicsAmerican Mathematical Society In press
[7] W A Coppel Stability and Asymptotic Behavior of DifferentialEquations D C Heath andCompany BostonMass USA 1965
[8] P Hartman Ordinary Differential Equations BirkhauserBoston Mass USA 2nd edition 1982
[9] F Neuman ldquoA general construction of linear differentialequations with solutions of prescribed propertiesrdquo AppliedMathematics Letters vol 17 no 1 pp 71ndash76 2004
[10] F Neuman ldquoStructure of solution spaces via transformationrdquoApplied Mathematics Letters vol 21 no 5 pp 529ndash533 2008
[11] P Borgnat and P Flandrin ldquoOn the chirp decomposition ofWeierstrass-Mandelbrot functions and their time-frequencyinterpretationrdquo Applied and Computational Harmonic Analysisvol 15 no 2 pp 134ndash146 2003
[12] E J Candes P R Charlton and H Helgason ldquoDetectinghighly oscillatory signals by chirplet path pursuitrdquo Applied andComputational Harmonic Analysis vol 24 no 1 pp 14ndash402008
[13] S Jaffard andYMeyer ldquoWaveletmethods for pointwise regular-ity and local oscillations of functionsrdquoMemoirs of the AmericanMathematical Society vol 123 no 587 pp 1ndash110 1996
[14] Y Meyer and H Xu ldquoWavelet analysis and chirpsrdquo Applied andComputational Harmonic Analysis vol 4 no 4 pp 366ndash3791997
[15] G RenQChen P Cerejeiras andUKaehle ldquoChirp transformsand chirp seriesrdquo Journal of Mathematical Analysis and Applica-tions vol 373 no 2 pp 356ndash369 2011
[16] M Kepesi and L Weruaga ldquoAdaptive chirp-based time-frequency analysis of speech signalsrdquo Speech Communicationvol 48 no 5 pp 474ndash492 2006
[17] L Weruaga and M Kepesi ldquoThe fan-chirp transform for non-stationary harmonic signalsrdquo Signal Processing vol 87 no 6 pp1504ndash1522 2007
[18] E Barlow A J Mulholland A Nordon and A GachaganldquoTheoretical analysis of chirp excitation of contrast agentsrdquoPhysics Procedia vol 3 no 1 pp 743ndash747 2009
International Journal of Differential Equations 11
[19] M H Pedersen T X Misaridis and J A Jensen ldquoClinicalevaluation of chirp-coded excitation in medical ultrasoundrdquoUltrasound in Medicine and Biology vol 29 no 6 pp 895ndash9052003
[20] T Paavle M Min and T Parve ldquoUsing of chirp excitation forbioimpedance estimation theoretical aspects andmodelingrdquo inProceedings of the 11th International Biennial Baltic ElectronicsConference (BECrsquo08) pp 325ndash328 Tallinn Estonia October2008
[21] M Pasic ldquoRectifiable and unrectifiable oscillations for a class ofsecond-order linear differential equations of Euler typerdquo Journalof Mathematical Analysis and Applications vol 335 no 1 pp724ndash738 2007
[22] K Falconer Fractal Geometry Mathematical Foundations andApplications John Wiley amp Sons Hoboken NJ USA 1999
[23] P Mattila Geometry of Sets and Measures in Euclidean SpacesFractals and Rectifiability vol 44 of Cambridge Studies inAdvanced Mathematics Cambridge University Press Cam-bridge UK 1995
[24] K J Falconer ldquoOn the Minkowski measurability of fractalsrdquoProceedings of the American Mathematical Society vol 123 no4 pp 1115ndash1124 1995
[25] C Q He and M L Lapidus Generalized Minkowski ContentSpectrum of Fractal Drums Fractal Strings and the RiemannZeta-Function vol 127 of Memoirs of the American Mathemat-ical Society American Mathematical Society Providence RIUSA 1997
[26] M L Lapidus and M van Frankenhuijsen Fractal geometryComplex Dimensions and Zeta Functions Geometry and Spec-tra of Fractal Strings Springer Monographs in MathematicsSpringer New York NY USA 2006
[27] M Pasic ldquoMinkowski-Bouligand dimension of solutions ofthe one-dimensional 119901-Laplacianrdquo Journal of Differential Equa-tions vol 190 no 1 pp 268ndash305 2003
[28] J S W Wong ldquoOn rectifiable oscillation of Euler type secondorder linear differential equationsrdquo Electronic Journal of Quali-tativeTheory of Differential Equations vol 2007 no 20 pp 1ndash122007
[29] M Pasic ldquoRectifiable and unrectifiable oscillations for a gen-eralization of the Riemann-Weber version of Euler differentialequationrdquo Georgian Mathematical Journal vol 15 no 4 pp759ndash774 2008
[30] M Pasic and S Tanaka ldquoRectifiable oscillations of self-adjointand damped linear differential equations of second-orderrdquoJournal of Mathematical Analysis and Applications vol 381 no1 pp 27ndash42 2011
[31] M Pasic D Zubrinic and V Zupanovic ldquoOscillatory andphase dimensions of solutions of some second-order differentialequationsrdquo Bulletin des Sciences Mathematiques vol 133 no 8pp 859ndash874 2009
[32] L Korkut and M Resman ldquoFractal oscillations of chirp-likefunctionsrdquo Georgian Mathematical Journal vol 19 no 4 pp705ndash720 2012
Submit your manuscripts athttpwwwhindawicom
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Mathematical Problems in Engineering
Hindawi Publishing Corporationhttpwwwhindawicom
Differential EquationsInternational Journal of
Volume 2014
Applied MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Mathematical PhysicsAdvances in
Complex AnalysisJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
OptimizationJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Operations ResearchAdvances in
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Function Spaces
Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of Mathematics and Mathematical Sciences
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Algebra
Discrete Dynamics in Nature and Society
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Decision SciencesAdvances in
Discrete MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom
Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Stochastic AnalysisInternational Journal of
International Journal of Differential Equations 9
Now from (76) (77) and (71) we obtain
sum
119899ge119896(120576)
max119909isin[119909119899+1119909119899]
1003816100381610038161003816119910 (119909)
1003816100381610038161003816(119909119899minus 119909119899+1)
ge 11987201198721[minus1205931015840(119909119896(120576)+1
)]
minus(2minus119904)
ge 11987201198721(119872120576minus1)
minus(2minus119904)
= 11987201198721119872minus(2minus119904)
1205762minus119904
(78)
Thus using Lemma 25 we have
1003816100381610038161003816Γ120576(119910)1003816100381610038161003816ge 11988811205762minus119904 120576 isin (0 120576
0) (79)
for some 1198881gt 0
Next we prove that 119910(119909) satisfies (53) Assume to thecontrary that
int
1199050
0
100381610038161003816100381610038161199101015840(119909)
10038161003816100381610038161003816119889119909 lt infin (80)
Then we have
length (119910) = int1199050
0
radic1 +10038161003816100381610038161199101015840(119909)1003816100381610038161003816
2119889119909
le int
1199050
0
(1 +
100381610038161003816100381610038161199101015840(119909)
10038161003816100381610038161003816) 119889119909 lt infin
(81)
From Lemma 29 it follows that
length (119910) = lim120576rarr+0
1003816100381610038161003816Γ120576(119910)1003816100381610038161003816
2120576
(82)
On the other hand by (79) we see that
lim120576rarr+0
1003816100381610038161003816Γ120576(119910)1003816100381610038161003816
2120576
= infin (83)
This is a contaradiction and hence 119910(119909) satisfies (53)Finally we show that 119910(119909) also satisfies inequality (54)
Because of (5) we obtain a constant 119888 gt 0 such that
max119905isinR
|119878 (119905)| = max119905isin[0120591]
|119878 (119905)| le 119888
max119905isinR
100381610038161003816100381610038161198781015840(119905)
10038161003816100381610038161003816= max119905isin[0120591]
100381610038161003816100381610038161198781015840(119905)
10038161003816100381610038161003816le 119888
(84)
By (15) we find that
sup119905isin(0119909]
1003816100381610038161003816119910 (119905)
1003816100381610038161003816= sup119905isin(0119909]
|119886 (119905)|1003816100381610038161003816119878 (120593 (119905))
1003816100381610038161003816le 119888119886 (119909) (85)
Hence from (1) and (15) it follows that100381610038161003816100381610038161199101015840(119909)
10038161003816100381610038161003816le 119888 (119886
1015840(119909) + 119886 (119909) [minus120593
1015840(119909)]) (86)
and therefore
int
1199050
119909
100381610038161003816100381610038161199101015840(120585)
10038161003816100381610038161003816119889120585 le 119888119886 (119905
0) + 119888 int
1199050
119909
119886 (120585) [minus1205931015840(120585)] 119889120585 (87)
Using (85) and (87) we see that
[119909 sup120585isin(0119909]
1003816100381610038161003816119910 (120585)
1003816100381610038161003816]
(119904minus1)(2minus119904)
int
1199050
119909
100381610038161003816100381610038161199101015840(120585)
10038161003816100381610038161003816119889120585
le 119888119886 (1199050) [119888119909119886 (119909)]
(119904minus1)(2minus119904)
+ 119888[119888119909119886 (119909)](119904minus1)(2minus119904)
int
1199050
119909
119886 (120585) [minus1205931015840(120585)] 119889120585
le 119888119886 (1199050) [1198881199050119886 (1199050)](119904minus1)(2minus119904)
+ 1198881(2minus119904)
[119909119886 (119909)](119904minus1)(2minus119904)
int
1199050
119909
119886 (120585) [minus1205931015840(120585)] 119889120585
(88)
By (18) we conclude that (54) holdsBy Lemma 28 there exists 119888
2gt 0 such that
1003816100381610038161003816Γ120576(119910)1003816100381610038161003816le 11988821205762minus119904 120576 isin (0 120576
0) (89)
Thus we have proved that the chirp function119910(119909) given by (1)satisfies the desired inequality (31) This completes the proofof Theorem 8
Proof of Theorem 11 Let 119910(119909) be the chirp function (1) It iseasy to see that 119910(119909) is oscillatory near 119909 = 0 By (5) thereexists119872 gt 0 such that
|119878 (119905)| le 119872
100381610038161003816100381610038161198781015840(119905)
10038161003816100381610038161003816le 119872 119905 isin R (90)
Hence we observe that100381610038161003816100381610038161199101015840(119909)
10038161003816100381610038161003816le
100381610038161003816100381610038161198861015840(119909) 119878 (120593 (119909))
10038161003816100381610038161003816+
10038161003816100381610038161003816119886 (119909) 119878 (120593 (119909)) 120593
1015840(119909)
10038161003816100381610038161003816
le 119872
100381610038161003816100381610038161198861015840(119909)
10038161003816100381610038161003816+ 119872
10038161003816100381610038161003816119886 (119909) 120593
1015840(119909)
10038161003816100381610038161003816
(91)
for 119909 isin (0 1199050] From (81) it follows that length(119910) lt infin
Lemma 29 shows that
length (119910) = lim120576rarr0
1003816100381610038161003816Γ120576(119910)1003816100381610038161003816
2120576
(92)
which implies that 120576length(Γ) le |Γ120576(119910)| le 4120576length(Γ) 120576 isin
(0 1205760) for some 120576
0gt 0 Therefore 119910(119909) is fractal oscillatory
near 119909 = 0 with the fractal dimension 1
4 Proof for the Fractal Oscillations of (26)In this section we give the proofs for the fractal oscillations ofthe linear second-order differential equation (26) consideredas an application of the main results on the fractal oscillationof chirp functions
Before we present the proofs of Theorems 16 and 17 wemake the following observation Since
1199101(119909) = 119886 (119909) cos (120593 (119909)) 119910
2(119909) = 119886 (119909) sin (120593 (119909))
(93)
are solutions of (26) we see that 119910(119909) = 11988811199101(119909) + 119888
21199102(119909)
is a fundamental system of all solutions of (26) Assume that1198882
1+ 1198882
2gt 0 and set 119878(119905) = 119888
1sin 119905 + 119888
2cos 119905 Then 119878(119905) clearly
satisfies (5)
10 International Journal of Differential Equations
Proof of Theorem 16 Applying Theorem 8 on 119910(119909) =
11988811199101(119909) + 119888
21199102(119909) we conclude that 119910(119909) is fractal oscillatory
with the fractal dimension 119904
Proof of Theorem 17 Theorem 11 implies that 119910(119909) =
11988811199101(119909) + 119888
21199102(119909) is fractal oscillatory near 119909 = 0 with the
fractal dimension 1
Appendix
Proof of Remark 9 By (17) there exists 1198881gt 0 such that
1198881le [minus120593
1015840(119909)]
2minus119904
int
119909
0
119886 (120585) 119889120585 119909 isin (0 1199050] (A1)
From (15) it follows that
1198881le [minus120593
1015840(119909)]
2minus119904
119886 (119909) int
119909
0
119889120585 = 119909119886 (119909) [minus1205931015840(119909)]
2minus119904
119909 isin (0 1199050]
(A2)
which implies that1
minus1205931015840(119909)
le 119888minus1(2minus119904)
11199091(2minus119904)
[119886 (119909)]1(2minus119904)
119909 isin (0 1199050] (A3)
By (19) there exists 1198882gt 0 such that
119909119886 (119909) [minus1205931015840(119909)]
2minus119904
le 1198882 119909 isin (0 119905
0] (A4)
and hence
minus1205931015840(119909) le 119888
1(2minus119904)
2119909minus1(2minus119904)
[119886 (119909)]minus1(2minus119904)
119909 isin (0 1199050]
(A5)Therefore we see that
[
1
minus1205931015840(119909)
]
119904minus1
int
1199050
119909
119886 (120585) [minus1205931015840(120585)] 119889120585
le 119888minus(119904minus1)(2minus119904)
1119909(119904minus1)(2minus119904)
[119886 (119909)](119904minus1)(2minus119904)
1198881(2minus119904)
2
times int
1199050
119909
120585minus1(2minus119904)
[119886 (120585)]minus(119904minus1)(2minus119904)
119889120585
le 1198883119909(119904minus1)(2minus119904)
[119886 (119909)](119904minus1)(2minus119904)
[119886 (119909)]minus(119904minus1)(2minus119904)
times int
1199050
119909
120585minus1(2minus119904)
119889120585
= 1198883
2 minus 119904
119904 minus 1
(1 minus 119909(119904minus1)(2minus119904)
119905minus(119904minus1)(2minus119904)
0)
le 1198883
2 minus 119904
119904 minus 1
119909 isin (0 1199050]
(A6)
so that (20) holds We conclude that
[119909119886 (119909)](119904minus1)(2minus119904)
int
1199050
119909
119886 (120585) [minus1205931015840(120585)] 119889120585
= (119909119886 (119909) [minus1205931015840(119909)]
2minus119904
)
(119904minus1)(2minus119904)
times ([
1
minus1205931015840(119909)
]
119904minus1
int
1199050
119909
119886 (120585) [minus1205931015840(120585)] 119889120585)
(A7)
By (19) and (20) we conclude that (18) is satisfied The proofof Remark 9 is complete
References
[1] C Tricot Curves and Fractal Dimension Springer New YorkNY USA 1995
[2] M Pasic ldquoFractal oscillations for a class of second order lineardifferential equations of Euler typerdquo Journal of MathematicalAnalysis and Applications vol 341 no 1 pp 211ndash223 2008
[3] M K Kwong M Pasic and J S W Wong ldquoRectifiable oscil-lations in second-order linear differential equationsrdquo Journal ofDifferential Equations vol 245 no 8 pp 2333ndash2351 2008
[4] M Pasic and J S W Wong ldquoRectifiable oscillations in second-order half-linear differential equationsrdquo Annali di MatematicaPura ed Applicata Series 4 vol 188 no 3 pp 517ndash541 2009
[5] M Pasic and S Tanaka ldquoFractal oscillations of self-adjoint anddamped linear differential equations of second-orderrdquo AppliedMathematics and Computation vol 218 no 5 pp 2281ndash22932011
[6] Y Naito M Pasic S Tanaka and D Zubrinic ldquoFractal oscilla-tions near domain boundary of radially symmetric solutions ofp-Laplace equations fractal geometry and dynamical systemsin pure and applied mathematicsrdquo Contemporary MathematicsAmerican Mathematical Society In press
[7] W A Coppel Stability and Asymptotic Behavior of DifferentialEquations D C Heath andCompany BostonMass USA 1965
[8] P Hartman Ordinary Differential Equations BirkhauserBoston Mass USA 2nd edition 1982
[9] F Neuman ldquoA general construction of linear differentialequations with solutions of prescribed propertiesrdquo AppliedMathematics Letters vol 17 no 1 pp 71ndash76 2004
[10] F Neuman ldquoStructure of solution spaces via transformationrdquoApplied Mathematics Letters vol 21 no 5 pp 529ndash533 2008
[11] P Borgnat and P Flandrin ldquoOn the chirp decomposition ofWeierstrass-Mandelbrot functions and their time-frequencyinterpretationrdquo Applied and Computational Harmonic Analysisvol 15 no 2 pp 134ndash146 2003
[12] E J Candes P R Charlton and H Helgason ldquoDetectinghighly oscillatory signals by chirplet path pursuitrdquo Applied andComputational Harmonic Analysis vol 24 no 1 pp 14ndash402008
[13] S Jaffard andYMeyer ldquoWaveletmethods for pointwise regular-ity and local oscillations of functionsrdquoMemoirs of the AmericanMathematical Society vol 123 no 587 pp 1ndash110 1996
[14] Y Meyer and H Xu ldquoWavelet analysis and chirpsrdquo Applied andComputational Harmonic Analysis vol 4 no 4 pp 366ndash3791997
[15] G RenQChen P Cerejeiras andUKaehle ldquoChirp transformsand chirp seriesrdquo Journal of Mathematical Analysis and Applica-tions vol 373 no 2 pp 356ndash369 2011
[16] M Kepesi and L Weruaga ldquoAdaptive chirp-based time-frequency analysis of speech signalsrdquo Speech Communicationvol 48 no 5 pp 474ndash492 2006
[17] L Weruaga and M Kepesi ldquoThe fan-chirp transform for non-stationary harmonic signalsrdquo Signal Processing vol 87 no 6 pp1504ndash1522 2007
[18] E Barlow A J Mulholland A Nordon and A GachaganldquoTheoretical analysis of chirp excitation of contrast agentsrdquoPhysics Procedia vol 3 no 1 pp 743ndash747 2009
International Journal of Differential Equations 11
[19] M H Pedersen T X Misaridis and J A Jensen ldquoClinicalevaluation of chirp-coded excitation in medical ultrasoundrdquoUltrasound in Medicine and Biology vol 29 no 6 pp 895ndash9052003
[20] T Paavle M Min and T Parve ldquoUsing of chirp excitation forbioimpedance estimation theoretical aspects andmodelingrdquo inProceedings of the 11th International Biennial Baltic ElectronicsConference (BECrsquo08) pp 325ndash328 Tallinn Estonia October2008
[21] M Pasic ldquoRectifiable and unrectifiable oscillations for a class ofsecond-order linear differential equations of Euler typerdquo Journalof Mathematical Analysis and Applications vol 335 no 1 pp724ndash738 2007
[22] K Falconer Fractal Geometry Mathematical Foundations andApplications John Wiley amp Sons Hoboken NJ USA 1999
[23] P Mattila Geometry of Sets and Measures in Euclidean SpacesFractals and Rectifiability vol 44 of Cambridge Studies inAdvanced Mathematics Cambridge University Press Cam-bridge UK 1995
[24] K J Falconer ldquoOn the Minkowski measurability of fractalsrdquoProceedings of the American Mathematical Society vol 123 no4 pp 1115ndash1124 1995
[25] C Q He and M L Lapidus Generalized Minkowski ContentSpectrum of Fractal Drums Fractal Strings and the RiemannZeta-Function vol 127 of Memoirs of the American Mathemat-ical Society American Mathematical Society Providence RIUSA 1997
[26] M L Lapidus and M van Frankenhuijsen Fractal geometryComplex Dimensions and Zeta Functions Geometry and Spec-tra of Fractal Strings Springer Monographs in MathematicsSpringer New York NY USA 2006
[27] M Pasic ldquoMinkowski-Bouligand dimension of solutions ofthe one-dimensional 119901-Laplacianrdquo Journal of Differential Equa-tions vol 190 no 1 pp 268ndash305 2003
[28] J S W Wong ldquoOn rectifiable oscillation of Euler type secondorder linear differential equationsrdquo Electronic Journal of Quali-tativeTheory of Differential Equations vol 2007 no 20 pp 1ndash122007
[29] M Pasic ldquoRectifiable and unrectifiable oscillations for a gen-eralization of the Riemann-Weber version of Euler differentialequationrdquo Georgian Mathematical Journal vol 15 no 4 pp759ndash774 2008
[30] M Pasic and S Tanaka ldquoRectifiable oscillations of self-adjointand damped linear differential equations of second-orderrdquoJournal of Mathematical Analysis and Applications vol 381 no1 pp 27ndash42 2011
[31] M Pasic D Zubrinic and V Zupanovic ldquoOscillatory andphase dimensions of solutions of some second-order differentialequationsrdquo Bulletin des Sciences Mathematiques vol 133 no 8pp 859ndash874 2009
[32] L Korkut and M Resman ldquoFractal oscillations of chirp-likefunctionsrdquo Georgian Mathematical Journal vol 19 no 4 pp705ndash720 2012
Submit your manuscripts athttpwwwhindawicom
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Mathematical Problems in Engineering
Hindawi Publishing Corporationhttpwwwhindawicom
Differential EquationsInternational Journal of
Volume 2014
Applied MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Mathematical PhysicsAdvances in
Complex AnalysisJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
OptimizationJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Operations ResearchAdvances in
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Function Spaces
Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of Mathematics and Mathematical Sciences
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Algebra
Discrete Dynamics in Nature and Society
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Decision SciencesAdvances in
Discrete MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom
Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Stochastic AnalysisInternational Journal of
10 International Journal of Differential Equations
Proof of Theorem 16 Applying Theorem 8 on 119910(119909) =
11988811199101(119909) + 119888
21199102(119909) we conclude that 119910(119909) is fractal oscillatory
with the fractal dimension 119904
Proof of Theorem 17 Theorem 11 implies that 119910(119909) =
11988811199101(119909) + 119888
21199102(119909) is fractal oscillatory near 119909 = 0 with the
fractal dimension 1
Appendix
Proof of Remark 9 By (17) there exists 1198881gt 0 such that
1198881le [minus120593
1015840(119909)]
2minus119904
int
119909
0
119886 (120585) 119889120585 119909 isin (0 1199050] (A1)
From (15) it follows that
1198881le [minus120593
1015840(119909)]
2minus119904
119886 (119909) int
119909
0
119889120585 = 119909119886 (119909) [minus1205931015840(119909)]
2minus119904
119909 isin (0 1199050]
(A2)
which implies that1
minus1205931015840(119909)
le 119888minus1(2minus119904)
11199091(2minus119904)
[119886 (119909)]1(2minus119904)
119909 isin (0 1199050] (A3)
By (19) there exists 1198882gt 0 such that
119909119886 (119909) [minus1205931015840(119909)]
2minus119904
le 1198882 119909 isin (0 119905
0] (A4)
and hence
minus1205931015840(119909) le 119888
1(2minus119904)
2119909minus1(2minus119904)
[119886 (119909)]minus1(2minus119904)
119909 isin (0 1199050]
(A5)Therefore we see that
[
1
minus1205931015840(119909)
]
119904minus1
int
1199050
119909
119886 (120585) [minus1205931015840(120585)] 119889120585
le 119888minus(119904minus1)(2minus119904)
1119909(119904minus1)(2minus119904)
[119886 (119909)](119904minus1)(2minus119904)
1198881(2minus119904)
2
times int
1199050
119909
120585minus1(2minus119904)
[119886 (120585)]minus(119904minus1)(2minus119904)
119889120585
le 1198883119909(119904minus1)(2minus119904)
[119886 (119909)](119904minus1)(2minus119904)
[119886 (119909)]minus(119904minus1)(2minus119904)
times int
1199050
119909
120585minus1(2minus119904)
119889120585
= 1198883
2 minus 119904
119904 minus 1
(1 minus 119909(119904minus1)(2minus119904)
119905minus(119904minus1)(2minus119904)
0)
le 1198883
2 minus 119904
119904 minus 1
119909 isin (0 1199050]
(A6)
so that (20) holds We conclude that
[119909119886 (119909)](119904minus1)(2minus119904)
int
1199050
119909
119886 (120585) [minus1205931015840(120585)] 119889120585
= (119909119886 (119909) [minus1205931015840(119909)]
2minus119904
)
(119904minus1)(2minus119904)
times ([
1
minus1205931015840(119909)
]
119904minus1
int
1199050
119909
119886 (120585) [minus1205931015840(120585)] 119889120585)
(A7)
By (19) and (20) we conclude that (18) is satisfied The proofof Remark 9 is complete
References
[1] C Tricot Curves and Fractal Dimension Springer New YorkNY USA 1995
[2] M Pasic ldquoFractal oscillations for a class of second order lineardifferential equations of Euler typerdquo Journal of MathematicalAnalysis and Applications vol 341 no 1 pp 211ndash223 2008
[3] M K Kwong M Pasic and J S W Wong ldquoRectifiable oscil-lations in second-order linear differential equationsrdquo Journal ofDifferential Equations vol 245 no 8 pp 2333ndash2351 2008
[4] M Pasic and J S W Wong ldquoRectifiable oscillations in second-order half-linear differential equationsrdquo Annali di MatematicaPura ed Applicata Series 4 vol 188 no 3 pp 517ndash541 2009
[5] M Pasic and S Tanaka ldquoFractal oscillations of self-adjoint anddamped linear differential equations of second-orderrdquo AppliedMathematics and Computation vol 218 no 5 pp 2281ndash22932011
[6] Y Naito M Pasic S Tanaka and D Zubrinic ldquoFractal oscilla-tions near domain boundary of radially symmetric solutions ofp-Laplace equations fractal geometry and dynamical systemsin pure and applied mathematicsrdquo Contemporary MathematicsAmerican Mathematical Society In press
[7] W A Coppel Stability and Asymptotic Behavior of DifferentialEquations D C Heath andCompany BostonMass USA 1965
[8] P Hartman Ordinary Differential Equations BirkhauserBoston Mass USA 2nd edition 1982
[9] F Neuman ldquoA general construction of linear differentialequations with solutions of prescribed propertiesrdquo AppliedMathematics Letters vol 17 no 1 pp 71ndash76 2004
[10] F Neuman ldquoStructure of solution spaces via transformationrdquoApplied Mathematics Letters vol 21 no 5 pp 529ndash533 2008
[11] P Borgnat and P Flandrin ldquoOn the chirp decomposition ofWeierstrass-Mandelbrot functions and their time-frequencyinterpretationrdquo Applied and Computational Harmonic Analysisvol 15 no 2 pp 134ndash146 2003
[12] E J Candes P R Charlton and H Helgason ldquoDetectinghighly oscillatory signals by chirplet path pursuitrdquo Applied andComputational Harmonic Analysis vol 24 no 1 pp 14ndash402008
[13] S Jaffard andYMeyer ldquoWaveletmethods for pointwise regular-ity and local oscillations of functionsrdquoMemoirs of the AmericanMathematical Society vol 123 no 587 pp 1ndash110 1996
[14] Y Meyer and H Xu ldquoWavelet analysis and chirpsrdquo Applied andComputational Harmonic Analysis vol 4 no 4 pp 366ndash3791997
[15] G RenQChen P Cerejeiras andUKaehle ldquoChirp transformsand chirp seriesrdquo Journal of Mathematical Analysis and Applica-tions vol 373 no 2 pp 356ndash369 2011
[16] M Kepesi and L Weruaga ldquoAdaptive chirp-based time-frequency analysis of speech signalsrdquo Speech Communicationvol 48 no 5 pp 474ndash492 2006
[17] L Weruaga and M Kepesi ldquoThe fan-chirp transform for non-stationary harmonic signalsrdquo Signal Processing vol 87 no 6 pp1504ndash1522 2007
[18] E Barlow A J Mulholland A Nordon and A GachaganldquoTheoretical analysis of chirp excitation of contrast agentsrdquoPhysics Procedia vol 3 no 1 pp 743ndash747 2009
International Journal of Differential Equations 11
[19] M H Pedersen T X Misaridis and J A Jensen ldquoClinicalevaluation of chirp-coded excitation in medical ultrasoundrdquoUltrasound in Medicine and Biology vol 29 no 6 pp 895ndash9052003
[20] T Paavle M Min and T Parve ldquoUsing of chirp excitation forbioimpedance estimation theoretical aspects andmodelingrdquo inProceedings of the 11th International Biennial Baltic ElectronicsConference (BECrsquo08) pp 325ndash328 Tallinn Estonia October2008
[21] M Pasic ldquoRectifiable and unrectifiable oscillations for a class ofsecond-order linear differential equations of Euler typerdquo Journalof Mathematical Analysis and Applications vol 335 no 1 pp724ndash738 2007
[22] K Falconer Fractal Geometry Mathematical Foundations andApplications John Wiley amp Sons Hoboken NJ USA 1999
[23] P Mattila Geometry of Sets and Measures in Euclidean SpacesFractals and Rectifiability vol 44 of Cambridge Studies inAdvanced Mathematics Cambridge University Press Cam-bridge UK 1995
[24] K J Falconer ldquoOn the Minkowski measurability of fractalsrdquoProceedings of the American Mathematical Society vol 123 no4 pp 1115ndash1124 1995
[25] C Q He and M L Lapidus Generalized Minkowski ContentSpectrum of Fractal Drums Fractal Strings and the RiemannZeta-Function vol 127 of Memoirs of the American Mathemat-ical Society American Mathematical Society Providence RIUSA 1997
[26] M L Lapidus and M van Frankenhuijsen Fractal geometryComplex Dimensions and Zeta Functions Geometry and Spec-tra of Fractal Strings Springer Monographs in MathematicsSpringer New York NY USA 2006
[27] M Pasic ldquoMinkowski-Bouligand dimension of solutions ofthe one-dimensional 119901-Laplacianrdquo Journal of Differential Equa-tions vol 190 no 1 pp 268ndash305 2003
[28] J S W Wong ldquoOn rectifiable oscillation of Euler type secondorder linear differential equationsrdquo Electronic Journal of Quali-tativeTheory of Differential Equations vol 2007 no 20 pp 1ndash122007
[29] M Pasic ldquoRectifiable and unrectifiable oscillations for a gen-eralization of the Riemann-Weber version of Euler differentialequationrdquo Georgian Mathematical Journal vol 15 no 4 pp759ndash774 2008
[30] M Pasic and S Tanaka ldquoRectifiable oscillations of self-adjointand damped linear differential equations of second-orderrdquoJournal of Mathematical Analysis and Applications vol 381 no1 pp 27ndash42 2011
[31] M Pasic D Zubrinic and V Zupanovic ldquoOscillatory andphase dimensions of solutions of some second-order differentialequationsrdquo Bulletin des Sciences Mathematiques vol 133 no 8pp 859ndash874 2009
[32] L Korkut and M Resman ldquoFractal oscillations of chirp-likefunctionsrdquo Georgian Mathematical Journal vol 19 no 4 pp705ndash720 2012
Submit your manuscripts athttpwwwhindawicom
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Mathematical Problems in Engineering
Hindawi Publishing Corporationhttpwwwhindawicom
Differential EquationsInternational Journal of
Volume 2014
Applied MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Mathematical PhysicsAdvances in
Complex AnalysisJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
OptimizationJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Operations ResearchAdvances in
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Function Spaces
Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of Mathematics and Mathematical Sciences
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Algebra
Discrete Dynamics in Nature and Society
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Decision SciencesAdvances in
Discrete MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom
Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Stochastic AnalysisInternational Journal of
International Journal of Differential Equations 11
[19] M H Pedersen T X Misaridis and J A Jensen ldquoClinicalevaluation of chirp-coded excitation in medical ultrasoundrdquoUltrasound in Medicine and Biology vol 29 no 6 pp 895ndash9052003
[20] T Paavle M Min and T Parve ldquoUsing of chirp excitation forbioimpedance estimation theoretical aspects andmodelingrdquo inProceedings of the 11th International Biennial Baltic ElectronicsConference (BECrsquo08) pp 325ndash328 Tallinn Estonia October2008
[21] M Pasic ldquoRectifiable and unrectifiable oscillations for a class ofsecond-order linear differential equations of Euler typerdquo Journalof Mathematical Analysis and Applications vol 335 no 1 pp724ndash738 2007
[22] K Falconer Fractal Geometry Mathematical Foundations andApplications John Wiley amp Sons Hoboken NJ USA 1999
[23] P Mattila Geometry of Sets and Measures in Euclidean SpacesFractals and Rectifiability vol 44 of Cambridge Studies inAdvanced Mathematics Cambridge University Press Cam-bridge UK 1995
[24] K J Falconer ldquoOn the Minkowski measurability of fractalsrdquoProceedings of the American Mathematical Society vol 123 no4 pp 1115ndash1124 1995
[25] C Q He and M L Lapidus Generalized Minkowski ContentSpectrum of Fractal Drums Fractal Strings and the RiemannZeta-Function vol 127 of Memoirs of the American Mathemat-ical Society American Mathematical Society Providence RIUSA 1997
[26] M L Lapidus and M van Frankenhuijsen Fractal geometryComplex Dimensions and Zeta Functions Geometry and Spec-tra of Fractal Strings Springer Monographs in MathematicsSpringer New York NY USA 2006
[27] M Pasic ldquoMinkowski-Bouligand dimension of solutions ofthe one-dimensional 119901-Laplacianrdquo Journal of Differential Equa-tions vol 190 no 1 pp 268ndash305 2003
[28] J S W Wong ldquoOn rectifiable oscillation of Euler type secondorder linear differential equationsrdquo Electronic Journal of Quali-tativeTheory of Differential Equations vol 2007 no 20 pp 1ndash122007
[29] M Pasic ldquoRectifiable and unrectifiable oscillations for a gen-eralization of the Riemann-Weber version of Euler differentialequationrdquo Georgian Mathematical Journal vol 15 no 4 pp759ndash774 2008
[30] M Pasic and S Tanaka ldquoRectifiable oscillations of self-adjointand damped linear differential equations of second-orderrdquoJournal of Mathematical Analysis and Applications vol 381 no1 pp 27ndash42 2011
[31] M Pasic D Zubrinic and V Zupanovic ldquoOscillatory andphase dimensions of solutions of some second-order differentialequationsrdquo Bulletin des Sciences Mathematiques vol 133 no 8pp 859ndash874 2009
[32] L Korkut and M Resman ldquoFractal oscillations of chirp-likefunctionsrdquo Georgian Mathematical Journal vol 19 no 4 pp705ndash720 2012
Submit your manuscripts athttpwwwhindawicom
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Mathematical Problems in Engineering
Hindawi Publishing Corporationhttpwwwhindawicom
Differential EquationsInternational Journal of
Volume 2014
Applied MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Mathematical PhysicsAdvances in
Complex AnalysisJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
OptimizationJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Operations ResearchAdvances in
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Function Spaces
Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of Mathematics and Mathematical Sciences
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Algebra
Discrete Dynamics in Nature and Society
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Decision SciencesAdvances in
Discrete MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom
Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Stochastic AnalysisInternational Journal of
Submit your manuscripts athttpwwwhindawicom
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Mathematical Problems in Engineering
Hindawi Publishing Corporationhttpwwwhindawicom
Differential EquationsInternational Journal of
Volume 2014
Applied MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Mathematical PhysicsAdvances in
Complex AnalysisJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
OptimizationJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Operations ResearchAdvances in
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Function Spaces
Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of Mathematics and Mathematical Sciences
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Algebra
Discrete Dynamics in Nature and Society
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Decision SciencesAdvances in
Discrete MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom
Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Stochastic AnalysisInternational Journal of
