Research Article Interval Oscillation Criteria for a Class of Fractional Differential ... · 2019....
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Hindawi Publishing Corporation Mathematical Problems in Engineering Volume 2013, Article ID 301085, 8 pages http://dx.doi.org/10.1155/2013/301085 Research Article Interval Oscillation Criteria for a Class of Fractional Differential Equations with Damping Term Chunxia Qi and Junmo Cheng School of Business, Shandong University of Technology, Zibo, Shandong 255049, China Correspondence should be addressed to Chunxia Qi; [email protected] Received 20 January 2013; Accepted 10 March 2013 Academic Editor: Sotiris Ntouyas Copyright © 2013 C. Qi and J. Cheng. 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. Some new interval oscillation criteria are established based on the certain Riccati transformation and inequality technique for a class of fractional differential equations with damping term. For illustrating the validity of the established results, we also present some applications for them. 1. Introduction Fractional differential equations are generalizations of clas- sical differential equations of integer order and can find their applications in many fields of science and engineering. In the last few decades, research on various aspects of fractional differential equations, for example, the existence, uniqueness, and stability of solutions of fractional differential equations, the numerical methods for fractional differential equations, and so on, has been paid much attention by many authors (e.g., we refer the reader to see [1–8] and the references therein). In these investigations, we notice that very little attention is paid to oscillation of fractional differential equations. Recent results in this direction only include Chen’s work [9], in which some new oscillation criteria are established for the following fractional differential equation: [ () ( ()) ] − () (∫ ∞ (V − ) − (V)V) = 0, > 0, (1) where , are positive-valued functions and is the quotient of two odd positive numbers. In this paper, we are concerned with oscillation of solutions of fractional differential equations of the following form: ( () [ () ()] ) + () [ () ()] − () ∫ ∞ ( − ) − () = 0, ∈ [ 0 , ∞) , (2) where ∈ 1 ([ 0 , ∞), R + ), ∈ 2 ([ 0 , ∞), R + ), , ∈ ([ 0 , ∞), R + ), ∈ (0, 1), () denotes the Liouville right-sided fractional derivative of order of , and () = −1/(Γ(1 − ))(/) ∫ ∞ ( − ) − (). A solution of (2) is said to be oscillatory if it is neither eventually positive nor eventually negative; otherwise it is nonoscillatory. Equation (2) is said to be oscillatory in case all its solutions are oscillatory. e organization of the rest of this paper is as follows. In Section 2, we establish some new interval oscillation criteria for (2) by a generalized Riccati transformation and inequality technique and present some applications for our results in Section 3. roughout this paper, R denotes the set of real numbers, and R + = (0, ∞). For more details about the theory of fractional differential equations, we refer the reader to [10– 12]. 2. Main Results For the sake of convenience, in the rest of this paper, we set () = ∫ ∞ ( − ) − (), () = ∫ 0 (()/()), 1 (, ) =∫ (1/ () ()), and 2 (, ) = ∫ ( 1 (, )/()).
Transcript of Research Article Interval Oscillation Criteria for a Class of Fractional Differential ... · 2019....
Hindawi Publishing CorporationMathematical Problems in EngineeringVolume 2013 Article ID 301085 8 pageshttpdxdoiorg1011552013301085
Research ArticleInterval Oscillation Criteria for a Class of FractionalDifferential Equations with Damping Term
Chunxia Qi and Junmo Cheng
School of Business Shandong University of Technology Zibo Shandong 255049 China
Correspondence should be addressed to Chunxia Qi qichunxiasdut126com
Received 20 January 2013 Accepted 10 March 2013
Academic Editor Sotiris Ntouyas
Copyright copy 2013 C Qi and J Cheng This is an open access article distributed under the Creative Commons Attribution Licensewhich permits unrestricted use distribution and reproduction in any medium provided the original work is properly cited
Some new interval oscillation criteria are established based on the certain Riccati transformation and inequality technique for a classof fractional differential equations with damping term For illustrating the validity of the established results we also present someapplications for them
1 Introduction
Fractional differential equations are generalizations of clas-sical differential equations of integer order and can findtheir applications in many fields of science and engineeringIn the last few decades research on various aspects offractional differential equations for example the existenceuniqueness and stability of solutions of fractional differentialequations the numerical methods for fractional differentialequations and so on has been paid much attention bymany authors (eg we refer the reader to see [1ndash8] andthe references therein) In these investigations we noticethat very little attention is paid to oscillation of fractionaldifferential equations Recent results in this direction onlyinclude Chenrsquos work [9] in which some new oscillationcriteria are established for the following fractional differentialequation
[119903 (119905) (119863120572119910 (119905))120578]1015840
minus 119902 (119905) 119891(intinfin
119905
(V minus 119905)minus120572119910 (V) 119889V) = 0
119905 gt 0(1)
where 119903 119902 are positive-valued functions and 120578 is the quotientof two odd positive numbers
In this paper we are concerned with oscillation ofsolutions of fractional differential equations of the followingform
(119886 (119905) [119903 (119905) 119863120572119909 (119905)]1015840)1015840
+ 119901 (119905) [119903 (119905) 119863120572119909 (119905)]1015840
minus 119902 (119905) intinfin
119905
(120585 minus 119905)minus120572119909 (120585) 119889120585 = 0 119905 isin [1199050infin)
(2)
where 119886 isin 1198621([1199050infin)R
+) 119903 isin 1198622([119905
0infin)R
+) 119901 119902 isin
119862([1199050infin)R
+) 120572 isin (0 1) 119863120572119909(119905) denotes the Liouville
right-sided fractional derivative of order 120572 of 119909 and119863120572119909(119905) =minus1(Γ(1 minus 120572))(119889119889119905) intinfin
119905
(120585 minus 119905)minus120572119909(120585)119889120585A solution of (2) is said to be oscillatory if it is neither
eventually positive nor eventually negative otherwise it isnonoscillatory Equation (2) is said to be oscillatory in caseall its solutions are oscillatory
The organization of the rest of this paper is as follows InSection 2 we establish some new interval oscillation criteriafor (2) by a generalized Riccati transformation and inequalitytechnique and present some applications for our results inSection 3 Throughout this paper R denotes the set of realnumbers andR
+= (0infin) Formore details about the theory
of fractional differential equations we refer the reader to [10ndash12]
2 Main Results
For the sake of convenience in the rest of this paper we set119883(119905) = intinfin
119905
(120585 minus 119905)minus120572119909(120585)119889120585 119860(119905) = int1199051199050
(119901(119904)119886(119904))119889119904 1205751(119905 119886)
= int119905119886
(1119890119860(119904)119886(119904))119889119904 and 1205752(119905 119886) = int119905
119886
(1205751(119904 119886)119903(119904))119889119904
2 Mathematical Problems in Engineering
Lemma 1 Assume 119909 is a solution of (2) Then1198831015840(119905) = minusΓ(1 minus120572)119863120572119909(119905)
Lemma 2 Assume 119909 is an eventually positive solution of (2)and
intinfin
1199050
1119890119860(119904)119886 (119904)
119889119904 = infin (3)
intinfin
1199050
1119903 (119904)
119889119904 = infin (4)
intinfin
1199050
1119903 (120585)
intinfin
120585
1119890119860(120591)119886 (120591)
intinfin
120591
119890119860(119904)119902 (119904) 119889119904 119889120591 119889120585 = infin (5)
Then there exists a sufficiently large 119879 such that
[119903 (119905)119863120572119909 (119905)]1015840 lt 0 119900119899 [119879infin) (6)
and either119863120572119909(119905) lt 0 on [119879infin) or lim119905rarrinfin
119883(119905) = 0
Proof Since 119909 is an eventually positive solution of (2) thereexists 119905
1such that 119909(119905) gt 0 on [119905
1infin) So119883(119905) gt 0 on [119905
1infin)
and we have
(119890119860(119905)119886 (119905) [119903 (119905)119863120572119909 (119905)]1015840)1015840
= 119890119860(119905)(119886 (119905) [119903 (119905)119863120572119909 (119905)]1015840)1015840
+ 119890119860(119905)119901 (119905) [119903 (119905) 119863120572119909 (119905)]1015840
= 119890119860(119905) (119886 (119905) [119903 (119905) 119863120572119909 (119905)]1015840)1015840
+119901 (119905) [119903 (119905) 119863120572119909 (119905)]1015840
= 119890119860(119905)119902 (119905)119883 (119905) gt 0
(7)
Then 119890119860(119905)119886(119905)[119903(119905)119863120572119909(119905)]1015840 is strictly increasing on [1199051infin)
and thus [119903(119905)119863120572119909(119905)]1015840 is eventually of one sign We claim[119903(119905)119863120572119909(119905)]1015840 lt 0 on [119905
2infin) where 119905
2gt 1199051is sufficiently
large Otherwise assume there exists a sufficiently large 1199053gt
1199052such that [119903(119905)119863120572119909(119905)]1015840 gt 0 on [119905
3infin)Then for 119905 isin [119905
3infin)
we have
119903 (119905)119863120572119909 (119905) minus 119903 (1199053)119863120572119909 (119905
3)
= int119905
1199053
119890119860(119904)119886 (119904) [119903 (119904)119863120572119909 (119904)]1015840
119890119860(119904)119886 (119904)119889119904
ge 119890119860(1199053)119886 (1199053) [119903 (1199053)119863120572119909 (119905
3)]1015840 int119905
1199053
1119890119860(119904)119886 (119904)
119889119904
(8)
By (3) we have
lim119905997888rarrinfin
119903 (119905)119863120572119909 (119905) = infin (9)
which implies for some sufficiently large 1199054gt 1199053 119863120572119909(119905) gt
0 and 119905 isin [1199054infin) By Lemma 1 we have
119883 (119905) minus 119883 (1199054) = int119905
1199054
1198831015840 (119904) 119889119904
= minusΓ (1 minus 120572)int119905
1199054
119863120572119909 (119904) 119889119904
= minusΓ (1 minus 120572)int119905
1199054
119903 (119904)119863120572119909 (119904)119903 (119904)
119889119904
le minusΓ (1 minus 120572) 119903 (1199054)119863120572119909 (119905
4) int119905
1199054
1119903 (119904)
119889119904
(10)
By (4) we obtain lim119905rarrinfin
119883(119905) = minusinfin which contradicts119883(119905) gt 0 on [119905
1infin) So [119903(119905)119863120572119909(119905)]1015840 lt 0 on [119905
2infin) Thus
119863120572119909(119905) is eventually of one sign Now we assume 119863120572119909(119905) gt0 119905 isin [119905
5infin) for some sufficiently 119905
5gt 1199054 Then by Lemma 1
1198831015840(119905) lt 0 for 119905 isin [1199055infin) Since 119883(119905) gt 0 furthermore we
have lim119905rarrinfin
119883(119905) = 120573 ge 0 We claim 120573 = 0 Otherwiseassume 120573 gt 0 Then119883(119905) ge 120573 on [119905
5infin) and for 119905 isin [119905
5infin)
by (7) we have
(119890119860(119905)119886 (119905) [119903 (119905)119863120572119909 (119905)]1015840)1015840
ge 119890119860(119905)119902 (119905)119883 (119905) ge 120573119890119860(119905)119902 (119905) (11)
Substituting 119905with 119904 in (11) an integration for (11) with respectto 119904 from 119905 toinfin yields
minus 119890119860(119905)119886 (119905) [119903 (119905) 119863120572119909 (119905)]1015840
ge minus lim119905997888rarrinfin
119890119860(119905)119886 (119905) [119903 (119905)119863120572119909 (119905)]1015840 + 120573intinfin
119905
119890119860(119904)119902 (119904) 119889119904
gt 120573int119905
1199055
119890119860(119904)119902 (119904) 119889119904
(12)
which means
(119903 (119905)119863120572119909 (119905))1015840 lt minus120573
119890119860(119905)119886 (119905)intinfin
119905
119890119860(119904)119902 (119904) 119889119904 (13)
Substituting 119905 with 120591 in (13) an integration for (13) withrespect to 120591 from 119905 toinfin yields
minus119903 (119905)119863120572119909 (119905) lt minus lim119905997888rarrinfin
119903 (119905)119863120572119909 (119905)
minus 120573intinfin
119905
1119890119860(120591)119886 (120591)
intinfin
120591
119890119860(119904)119902 (119904) 119889119904 119889120591
lt minus120573intinfin
119905
1119890119860(120591)119886 (120591)
intinfin
120591
119890119860(119904)119902 (119904) 119889119904 119889120591
(14)
That is
1198831015840 (119905) lt minusΓ (1 minus 120572) 120573119903 (119905)
intinfin
119905
1119890119860(120591)119886 (120591)
intinfin
120591
119890119860(119904)119902 (119904) 119889119904 119889120591
(15)
Mathematical Problems in Engineering 3
Substituting 119905 with 120585 in (15) an integration for (15) withrespect to 120585 from 119905
5to 119905 yields
119883 (119905) minus 119883 (1199055)
lt minusΓ (1 minus 120572) 120573int119905
1199055
1119903 (120585)
intinfin
120585
1119890119860(120591)119886 (120591)
times intinfin
120591
119890119860(119904)119902 (119904) 119889119904 119889120591 119889120585
(16)
By (5) one can see lim119905rarrinfin
119883(119905) = minusinfin which is a contradic-tion So the proof is complete
Lemma 3 Assume that 119909 is an eventually positive solution of(2) such that
[119903 (119905)119863120572119909 (119905)]1015840 lt 0 119863120572119909 (119905) lt 0 119900119899 [1199051infin) (17)
where 1199051ge 1199050is sufficiently large Then we have
1198831015840 (119905) ge minusΓ (1 minus 120572) 120575
1(119905 1199051) 119890119860(119905)119886 (119905) (119903 (119905)119863120572119909 (119905))1015840
119903 (119905) (18)
119883(119905) ge minusΓ (1 minus 120572) 1205752(119905 1199051) 119890119860(119905)119886 (119905) [119903 (119905)119863120572119909 (119905)]1015840 (19)
Proof By Lemma 2 we have 119890119860(119905)119886(119905)[119903(119905)119863120572119909(119905)]1015840 is strictlyincreasing on [119905
1infin) So
119903 (119905)119863120572119909 (119905) le 119903 (119905)119863120572119909 (119905) minus 119903 (1199051)119863120572119909 (119905)
= int119905
1199051
119890119860(119904)119886 (119904) [119903 (119904)119863120572119909 (119904)]1015840
119890119860(119904)119886 (119904)119889119904
le 119890119860(119905)119886 (119905) [119903 (119905) 119863120572119909 (119905)]1015840 int119905
1199051
1119890119860(119904)119886 (119904)
119889119904
= 1205751(119905 1199051) 119890119860(119905)119886 (119905) [119903 (119905)119863120572119909 (119905)]1015840
(20)
Using Lemma 1 we obtain that
1198831015840 (119905) ge minusΓ (1 minus 120572) 120575
1(119905 1199051) 119890119860(119905)119886 (119905) [119903 (119905)119863120572119909 (119905)]1015840
119903 (119905) (21)
Then
119883 (119905) ge 119883 (119905) minus 119883 (1199051)
ge minusint119905
1199051
Γ (1 minus 120572) 1205751(119904 1199051) 119890119860(119904)119886 (119904) [119903 (119904)119863120572119909 (119904)]1015840
119903 (119904)119889119904
ge minusΓ (1 minus 120572) 119890119860(119905)119886 (119905) [119903 (119905) 119863120572119909 (119905)]1015840 int119905
1199051
1205751(119904 1199051)
119903 (119904)119889119904
= minusΓ (1 minus 120572) 1205752(119905 1199051) 119890119860(119905)119886 (119905) [119903 (119905)119863120572119909 (119905)]1015840
(22)
Theorem 4 Assume (3)ndash(5) hold and there exist two func-tions 120601 isin 1198621([119905
0infin)R
+) and 120593 isin 1198621([119905
0infin) [0infin)) such
that
intinfin
119879
120601 (119904) 119902 (119904) 119890119860(119904) minus 120601 (119904) 1205931015840 (119904)
+120601 (119904) Γ (1 minus 120572) 120575
1(119904 119879) 1205932 (119904)
119903 (119904)
minus [2120593 (119904) 120601 (119904) Γ (1 minus 120572) 1205751(119904 119879) + 119903 (119904) 1206011015840 (119904)]
2
times(4Γ (1 minus 120572) 120601 (119904) 1205751(119904 119879) 119903 (119904))minus1 119889119904 = infin
(23)
for all sufficiently large 119879 Then every solution of (2) is oscilla-tory or satisfies lim
119905rarrinfin119883(119905) = 0
Proof Assume (2) has a nonoscillatory solution 119909 on [1199050infin)
Without loss of generality we may assume 119909(119905) gt 0 on[1199051infin) where 119905
1is sufficiently large By Lemma 2 we have
(119903(119905)119863120572119909(119905))1015840 lt 0 119905 isin [1199052infin) where 119905
2gt 1199051is sufficiently
large and either 119863120572119909(119905) lt 0 on [1199052infin) or lim
119905rarrinfin119883(119905) = 0
Define the generalized Riccati function
120596 (119905) = 120601 (119905) minus119890119860(119905)119886 (119905) [119903 (119905) 119863120572119909 (119905)]1015840
119883(119905)+ 120593 (119905) (24)
Then for 119905 isin [1199052infin) we have
1205961015840 (119905) = minus1206011015840 (119905)119890119860(119905)119886 (119905) [119903 (119905)119863120572119909 (119905)]1015840
119883 (119905)
+ 120601 (119905) minus119890119860(119905)119886 (119905) [119903 (119905)119863120572119909 (119905)]1015840
119883(119905)1015840
+ 1206011015840 (119905) 120593 (119905) + 120601 (119905) 1205931015840 (119905)
= minus120601 (119905) (119883 (119905) (119890119860(119905)119886 (119905) [119903 (119905)119863120572119909 (119905)]1015840)1015840
minus1198831015840 (119905) 119890119860(119905)119886 (119905) [119903 (119905)119863120572119909 (119905)]1015840)
times (1198832 (119905))minus1
+1206011015840 (119905)120601 (119905)
120596 (119905) + 120601 (119905) 1205931015840 (119905)
= minus120601 (119905) (119883 (119905) 119890119860(119905)(119886 (119905) [119903 (119905) 119863120572119909 (119905)]1015840)1015840
+ 119890119860(119905)119901 (119905) [119903 (119905)119863120572119909 (119905)]1015840
minus1198831015840 (119905) 119890119860(119905)119886 (119905) [119903 (119905) 119863120572119909 (119905)]1015840)
times (1198832 (119905))minus1
+1206011015840 (119905)120601 (119905)
120596 (119905) + 120601 (119905) 1205931015840 (119905)
4 Mathematical Problems in Engineering
= minus120601 (119905) 119902 (119905) 119890119860(119905)
+120601 (119905)1198831015840 (119905) 119886 (119905) 119890119860(119905)[119903 (119905)119863120572119909 (119905)]1015840
1198832 (119905)
+1206011015840 (119905)120601 (119905)
120596 (119905) + 120601 (119905) 1205931015840 (119905)
(25)
By Lemma 3 and the definition of 119891 we get that
1205961015840 (119905) le minus120601 (119905) 119902 (119905) 119890119860(119905)
minus (120601 (119905) Γ (1 minus 120572) 1205751(119905 1199052) 119890119860(119905)119886 (119905)
times (119903 (119905)119863120572119909 (119905))1015840119886 (119905) 119890119860(119905)(119903 (119905)119863120572119909 (119905))1015840)
times (119903 (119905)1198832 (119905))minus1
+1206011015840 (119905)120601 (119905)
120596 (119905) + 120601 (119905) 1205931015840 (119905)
= minus120601 (119905) 119902 (119905) 119890119860(119905) minus120601 (119905) Γ (1 minus 120572) 120575
1(119905 1199052)
119903 (119905)
times [120596 (119905)120601 (119905)
minus 120593 (119905)]2
+1206011015840 (119905)120601 (119905)
120596 (119905) + 120601 (119905) 1205931015840 (119905)
= minus120601 (119905) 119902 (119905) 119890119860(119905) + 120601 (119905) 1205931015840 (119905)
minus120601 (119905) Γ (1 minus 120572) 120575
1(119905 1199052) 1205932 (119905)
119903 (119905)
minus120601 (119905) Γ (1 minus 120572) 120575
1(119905 1199052)
119903 (119905)1205962 (119905)1206012 (119905)
+2120593 (119905) 120601 (119905) Γ (1 minus 120572) 120575
1(119905 1199052) + 119903 (119905) 1206011015840 (119905)
119903 (119905) 120601 (119905)120596 (119905)
le minus120601 (119905) 119902 (119905) 119890119860(119905) + 120601 (119905) 1205931015840 (119905)
minus120601 (119905) Γ (1 minus 120572) 120575
1(119905 1199052) 1205932 (119905)
119903 (119905)
+[2120593 (119905) 120601 (119905) Γ (1 minus 120572) 120575
1(119905 1199052) + 119903 (119905) 1206011015840 (119905)]
2
4Γ (1 minus 120572) 120601 (119905) 1205751(119905 1199052) 119903 (119905)
(26)
Substituting 119905 with 119904 in (26) an integration for (26) withrespect to 119904 from 119905
2to 119905 yields
int119905
1199052
120601 (119904) 119902 (119904) 119890119860(119904) minus 120601 (119904) 1205931015840 (119904)
+120601 (119904) Γ (1 minus 120572) 120575
1(119904 1199052) 1205932 (119904)
119903 (119904)
minus [2120593 (119904) 120601 (119904) Γ (1 minus 120572) 1205751(119904 1199052) + 119903 (119904) 1206011015840 (119904)]
2
times(4Γ (1 minus 120572) 120601 (119904) 1205751(119904 1199052) 119903 (119904))minus1 119889119904
le 120596 (1199052) minus 120596 (119905) le 120596 (119905
2) lt infin
(27)which contradicts (23) So the proof is complete
Theorem 5 Define D = (119905 119904) | 119905 ge 119904 ge 1199050 Assume (3)ndash(5)
hold and there exists a function119867 isin 1198621(DR) such that119867(119905 119905) = 0 119891119900119903 119905 ge 119905
0 119867 (119905 119904) gt 0 119891119900119903 119905 gt 119904 ge 119905
0
(28)
and119867 has a nonpositive continuous partial derivative1198671015840119904(119905 119904)
and
lim119905997888rarrinfin
sup 1119867 (119905 119905
0)
times int119905
1199050
119867(119905 119904) 120601 (119904) 119902 (119904) 119890119860(119904) minus 120601 (119904) 1205931015840 (119904)
+120601 (119904) Γ (1 minus 120572) 120575
1(119904 119879) 1205932 (119904)
119903 (119904)
minus [2120593 (119904) 120601 (119904) Γ (1 minus 120572)
times 1205751(119904 119879) +119903 (119904) 1206011015840 (119904)]
2
times (4Γ (1 minus 120572) 120601 (119904) 1205751
times (119904 119879) 119903 (119904) )minus1 119889119904 = infin
(29)
for all sufficiently large 119879 where 120601 120593 are defined as inTheorem 4 Then every solution of (2) is oscillatory or satisfieslim119905rarrinfin
119883(119905) = 0
Proof Assume (2) has a nonoscillatory solution 119909 on [1199050infin)
Without loss of generality we may assume 119909(119905) gt 0 on[1199051infin) where 119905
1is sufficiently large By Lemma 2 we have
119863120572119909(119905) lt 0 on [1199052infin) for some sufficiently large 119905
2gt 1199051 Let
120596(119905) be defined as in Theorem 4 By (26) we have
120601 (119905) 119902 (119905) 119890119860(119905) + 120601 (119905) 1205931015840 (119905) minus120601 (119905) Γ (1 minus 120572) 120575
1(119905 1199052) 1205932 (119905)
119903 (119905)
+[2120593 (119905) 120601 (119905) Γ (1 minus 120572) 120575
1(119905 1199052) + 119903 (119905) 1206011015840 (119905)]
2
4Γ (1 minus 120572) 120601 (119905) 1205751(119905 1199052) 119903 (119905)
le minus1205961015840 (119905)
(30)Substituting 119905with 119904 in (30) multiplying both sides by119867(119905 119904)and then integrating it with respect to 119904 from 119905
2to 119905 yield
int119905
1199052
119867(119905 119904)
times120601 (119904) 119902 (119904) 119890119860(119904) minus 120601 (119904) 1205931015840 (119904)
+120601 (119904) Γ (1 minus 120572) 120575
1(119904 1199052) 1205932 (119904)
119903 (119904)
Mathematical Problems in Engineering 5
minus[2120593 (119904) 120601 (119904) Γ (1 minus 120572) 120575
1(119904 1199052) + 119903 (119904) 1206011015840 (119904)]
2
4Γ (1 minus 120572) 120601 (119904) 1205751(119904 1199052) 119903 (119904)
119889119904
le minusint119905
1199052
119867(119905 119904) 1205961015840 (119904) 119889119904
= 119867 (119905 1199052) 120596 (1199052) + int119905
1199052
1198671015840119904(119905 119904) 120596 (119904) Δ119904
le 119867 (119905 1199052) 120596 (1199052) le 119867 (119905 119905
0) 120596 (1199052)
(31)
Then
int119905
1199050
119867(119905 119904)
times 120601 (119904) 119902 (119904) 119890119860(119904) minus 120601 (119904) 1205931015840 (119904)
+120601 (119904) Γ (1 minus 120572) 120575
1(119904 1199052) 1205932 (119904)
119903 (119904)
minus [2120593 (119904) 120601 (119904) Γ (1 minus 120572)
times1205751(119904 1199052) + 119903 (119904) 1206011015840 (119904)]
2
times(4Γ (1 minus 120572) 120601 (119904) 1205751(119904 1199052) 119903 (119904))minus1 119889119904
= int1199052
1199050
119867(119905 119904)
times120601 (119904) 119902 (119904) 119890119860(119904) minus 120601 (119904) 1205931015840 (119904)
+120601 (119904) Γ (1 minus 120572) 120575
1(119904 1199052) 1205932 (119904)
119903 (119904)
minus[2120593 (119904) 120601 (119904) Γ (1 minus 120572) 120575
1(119904 1199052) + 119903 (119904) 1206011015840 (119904)]
2
4Γ (1 minus 120572) 120601 (119904) 1205751(119904 1199052) 119903 (119904)
119889119904
+ int119905
1199052
119867(119905 119904)
times120601 (119904) 119902 (119904) 119890119860(119904) minus 120601 (119904) 1205931015840 (119904)
+120601 (119904) Γ (1 minus 120572) 120575
1(119904 1199052) 1205932 (119904)
119903 (119904)
minus[2120593 (119904) 120601 (119904) Γ (1 minus 120572) 120575
1(119904 1199052) + 119903 (119904) 1206011015840 (119904)]
2
4Γ (1 minus 120572) 120601 (119904) 1205751(119904 1199052) 119903 (119904)
times 119889119904
le 119867 (119905 1199050) 120596 (1199052) + 119867 (119905 119905
0)
times int1199052
1199050
10038161003816100381610038161003816120601 (119904) 119902 (119904) 119890119860(119904) minus 120601 (119904) 1205931015840 (119904)
+ (120601 (119904) Γ (1 minus 120572) 1205751(119904 1199052) 1205932 (119904))
times (119903 (119904))minus1
minus [2120593 (119904) 120601 (119904) Γ (1 minus 120572) 1205751
times (119904 1199052) +119903 (119904) 1206011015840 (119904)]
2
times (4Γ (1 minus 120572) 120601 (119904) 1205751(119904 1199052) 119903 (119904))minus110038161003816100381610038161003816 119889119904
(32)
So
lim119905997888rarrinfin
sup 1119867 (119905 119905
0)
times int119905
1199050
119867(119905 119904) 120601 (119904) 119902 (119904) 119890119860(119904) minus 120601 (119904) 1205931015840 (119904)
+ (120601 (119904) Γ (1 minus 120572) 1205751(119904 1199052) 1205932 (119904))
times (119903 (119904))minus1 minus [2120593 (119904) 120601 (119904) Γ (1 minus 120572)
times 1205751(119904 1199052) + 119903 (119904) 1206011015840 (119904)]
2
times (4Γ (1 minus 120572) 120601 (119904) 1205751(119904 1199052) 119903 (119904))minus1 119889119904
le 120596 (1199052)
+ int1199052
1199050
10038161003816100381610038161003816120601 (119904) 119902 (119904) 119890119860(119904) minus 120601 (119904) 1205931015840 (119904)
+120601 (119904) Γ (1 minus 120572) 120575
1(119904 1199052) 1205932 (119904)
119903 (119904)
minus [2120593 (119904) 120601 (119904) Γ (1 minus 120572) 1205751(119904 1199052) + 119903 (119904) 1206011015840 (119904)]
2
times(4Γ (1 minus 120572) 120601 (119904) 1205751(119904 1199052) 119903 (119904))minus110038161003816100381610038161003816 119889119904 lt infin
(33)
which contradicts (29) So the proof is complete
In Theorems 5 if we take 119867(119905 119904) for some special func-tions such as (119905 minus 119904)119898 or ln (119905119904) then we can obtain somecorollaries as follows
Corollary 6 Assume (3)ndash(5) hold and
lim119905997888rarrinfin
sup 1(119905 minus 1199050)119898
times int119905
1199050
(119905 minus 119904)119898 120601 (119904) 119902 (119904) 119890119860(119904) minus 120601 (119904) 1205931015840 (119904)
+ (120601 (119904) Γ (1 minus 120572) 1205751(119904 119879) 1205932 (119904))
6 Mathematical Problems in Engineering
times (119903 (119904))minus1
minus [2120593 (119904) 120601 (119904) Γ (1 minus 120572) 1205751(119904 119879)
+119903 (119904) 1206011015840 (119904)]2
times (4Γ (1 minus 120572) 120601 (119904) 1205751
times (119904 119879) 119903 (119904) )minus1 119889119904 = infin
(34)
for all sufficiently large 119879 Then every solution of (2) is oscilla-tory or satisfies lim
119905rarrinfin119883(119905) = 0
Corollary 7 Assume (3)ndash(5) hold and
lim119905997888rarrinfin
sup 1(ln 119905 minus ln 119905
0)
times int119905
1199050
(ln 119905 minus ln 119904) 120601 (119904) 119902 (119904) 119890119860(119904) minus 120601 (119904) 1205931015840 (119904)
+120601 (119904) Γ (1 minus 120572) 120575
1(119904 119879) 1205932 (119904)
119903 (119904)
minus [2120593 (119904) 120601 (119904) Γ (1 minus 120572) 1205751(119904 119879)
+119903 (119904) 1206011015840 (119904)]2
times (4Γ (1 minus 120572) 120601 (119904)
times1205751(119904 119879) 119903 (119904))minus1 119889119904 = infin
(35)
for all sufficiently large 119879 Then every solution of (2) is oscilla-tory or satisfies lim
119905rarrinfin119883(119905) = 0
3 Applications
In this section we will present some applications for theabove established results
Example 8 Consider
(119905(119863120572119909 (119905))1015840)1015840
+ 119905minus2(119863120572119909 (119905))1015840
minus119872119905minus2 intinfin
119905
(120585 minus 119905)minus120572119909 (120585) 119889120585 = 0 119905 isin [2infin) (36)
where119872 gt 0 is a quotient of two odd positive integers
We have in (2) 119886(119905) = 119905 119901(119905) = 119905minus2 119902(119905) = 119872119905minus2 119903(119905) =1 and 119905
0= 2 Then
1 le 119890119860(119905) = 119890int119905
2(119901(119904)119886(119904))119889119904 = 119890int
119905
2119904minus3119889119904 = 119890minus(12)[119905
minus2minus2minus2] le 11989018
(37)
Moreover we have
intinfin
1199050
1119890119860(119904)119886 (119904)
119889119904 = intinfin
2
1119890119860(119904)119886 (119904)
119889119904 ge 119890minus18 intinfin
2
1119904119889119904 = infin
intinfin
1199050
1119903 (119904)
119889119904 = infin
(38)
Furthermore
intinfin
1199050
1119903 (120585)
intinfin
120585
1119890119860(120591)119886 (120591)
intinfin
120591
119890119860(119904)119902 (119904) 119889119904 119889120591 119889120585
= 119872intinfin
2
intinfin
120585
1119890119860(120591)120591
intinfin
120591
119890119860(119904)
1199042119889119904 119889120591 119889120585
ge 119872119890minus18 intinfin
2
intinfin
120585
1120591intinfin
120591
11199042119889119904 119889120591 119889120585
= 119872119890minus18 intinfin
2
intinfin
120585
11205912119889120591 119889120585
= 119872119890minus18 intinfin
2
1120585119889120585 = infin
(39)
On the other hand for a sufficiently large 119879 we have
1205751(119905 119879) = int
119905
119879
1119890119860(119904)119886 (119904)
119889119904
= int119905
119879
1119890119860(119904)119904
119889119904 ge 119890minus18 int119905
119879
1119904119889119904 997888rarr infin
(40)
So we can take 119879lowast gt 119879 such that 1205751(119905 119879) gt 1 for 119905 isin [119879lowastinfin)
Taking 120601(119905) = 119905 120593(119905) = 0 in (23) we get that
intinfin
119879
120601 (119904) 119902 (119904) 119890119860(119904) minus
[119903 (119904) 1206011015840 (119904)]2
4 [Γ (1 minus 120572) 120601 (119904) 1205751(119904 119879)] 119903 (119904)
119889119904
ge intinfin
119879
[119872 minus 14Γ (1 minus 120572) 120575
1(119904 119879)
] 1119904119889119904
= int119879lowast
119879
[119872 minus 14Γ (1 minus 120572) 120575
1(119904 119879)
] 1119904119889119904
+ intinfin
119879lowast
[119872 minus 14Γ (1 minus 120572) 120575
1(119904 119879)
] 1119904119889119904
ge int119879lowast
119879
[119872 minus 14Γ (1 minus 120572) 120575
1(119904 119879)
] 1119904119889119904
+ intinfin
119879lowast
[119872 minus 14Γ (1 minus 120572)
] 1119904119889119904 = infin
(41)
provided that119872 gt 1(4Γ(1minus120572)) So (3)ndash(5) and (23) all holdand by Theorem 4 we deduce that every solution of (36) isoscillatory or satisfies lim
119905rarrinfin119883(119905) = 0 under condition119872 gt
14Γ(1 minus 120572)
Mathematical Problems in Engineering 7
Example 9 Consider
(radic119905(119863120572119909 (119905))1015840)1015840
+ 119890minus119905(119863120572119909 (119905))1015840
minus (119872119905minus52 + ln 119905) intinfin
119905
(120585 minus 119905)minus120572119909 (120585) 119889120585 = 0 119905 isin [2infin)
(42)
where 120572 isin (0 1) and119872 gt 0 is a constant
We have in (2) 119886(119905) = radic119905 119901(119905) = 119890minus119905 119902(119905) = 119872119905minus52 +ln 119905 119903(119905) = 1 and 119905
0= 2 Then
1 le 119890119860(119905) = 119890int119905
2(119901(119904)119886(119904))119889119904 = 119890int
119905
2(1119890119904radic119904)119889119904
le 119890int119905
2119904minus(32)119889119904
= 119890minus2[119905minus12minus2minus12] le 119890radic2
(43)
So we have
intinfin
1199050
1119890119860(119904)119886 (119904)
119889119904 = intinfin
2
1119890119860(119904)radic119904
119889119904 ge 119890minusradic2 intinfin
2
1radic119904119889119904 = infin
intinfin
1199050
1119903 (119904)
119889119904 = infin
(44)
Furthermore
intinfin
1199050
1119903 (120585)
intinfin
120585
1119890119860(120591)119886 (120591)
intinfin
120591
119890119860(119904)119902 (119904) 119889119904 119889120591 119889120585
= intinfin
2
intinfin
120585
1119890119860(120591)radic120591
intinfin
120591
119890119860(119904) (119872119904minus52 + ln 119904) 119889119904 119889120591 119889120585
ge 119872119890minusradic2 intinfin
2
intinfin
120585
1radic120591
intinfin
120591
119904minus52119889119904 119889120591 119889120585
= 2119872119890minusradic2
3intinfin
2
[intinfin
120585
11205912119889120591] 119889120585
= 2119872119890minusradic2
3intinfin
2
1120585119889120585 = infin
(45)
On the other hand Taking 120601(119905) = 1199052 120593(119905) = 0 and 119867(119905 119904) =119905 minus 119904 in (29) we get that
lim119905997888rarrinfin
sup 1119905 minus 1199050
int119905
1199050
(119905 minus 119904)
times120601 (119904) 119902 (119904) 119890119860(119904)
minus[119903 (119904) 1206011015840 (119904)]
2
4Γ (1 minus 120572) 120601 (119904) 1205751(119904 119879) 119903 (119904)
119889119904
ge lim119905997888rarrinfin
sup 1119905 minus 2
times int119905
2
(119905 minus 119904) (119872119904minus52 + ln 119904) 1199042
minus 1Γ (1 minus 120572) 120575
1(119904 119879)
119889119904
ge lim119905997888rarrinfin
sup 1119905 minus 2
int119905
2
(119905 minus 119904)119872119904minus12119889119904 = infin
(46)
So (3)ndash(5) and (29) all hold and by Corollary 6 with 119898 = 1we deduce that every solution of (42) is oscillatory or satisfieslim119905rarrinfin
119883(119905) = 0
Acknowledgments
This work is partially supported by Planning Fund projectof the Ministry of Education of China (10YJA630019) Theauthors would thank the reviewers very much for theirvaluable suggestions on this paper
References
[1] A Saadatmandi and M Dehghan ldquoA new operational matrixfor solving fractional-order differential equationsrdquo Computersamp Mathematics with Applications vol 59 no 3 pp 1326ndash13362010
[2] Y Zhou F Jiao and J Li ldquoExistence and uniqueness for p-typefractional neutral differential equationsrdquo Nonlinear AnalysisTheory Methods amp Applications vol 71 no 7-8 pp 2724ndash27332009
[3] L Galeone and R Garrappa ldquoExplicit methods for fractionaldifferential equations and their stability propertiesrdquo Journal ofComputational and Applied Mathematics vol 228 no 2 pp548ndash560 2009
[4] J C Trigeassou N Maamri J Sabatier and A OustaloupldquoA Lyapunov approach to the stability of fractional differentialequationsrdquo Signal Process vol 91 pp 437ndash445 2011
[5] W Deng ldquoSmoothness and stability of the solutions fornonlinear fractional differential equationsrdquo Nonlinear AnalysisTheory Methods amp Applications vol 72 no 3-4 pp 1768ndash17772010
[6] F Ghoreishi and S Yazdani ldquoAn extension of the spectralTau method for numerical solution of multi-order fractionaldifferential equations with convergence analysisrdquo Computers ampMathematics with Applications vol 61 no 1 pp 30ndash43 2011
[7] J T Edwards N J Ford and A C Simpson ldquoThe numericalsolution of linear multi-term fractional differential equationssystems of equationsrdquo Journal of Computational and AppliedMathematics vol 148 no 2 pp 401ndash418 2002
[8] M Muslim ldquoExistence and approximation of solutions tofractional differential equationsrdquo Mathematical and ComputerModelling vol 49 no 5-6 pp 1164ndash1172 2009
[9] D-X Chen ldquoOscillation criteria of fractional differential equa-tionsrdquo Advances in Difference Equations vol 2012 article 332012
[10] I Podlubny Fractional Differential Equations vol 198 Aca-demic Press San Diego Calif USA 1999
8 Mathematical Problems in Engineering
[11] A A Kilbas H M Srivastava and J J Trujillo Theoryand Applications of Fractional Differential Equations vol 204Elsevier Science BV Amsterdam The Netherlands 2006
[12] S Das Functional Fractional Calculus Springer Berlin Ger-many 2nd edition 2011
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Mathematical Problems in Engineering
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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
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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
2 Mathematical Problems in Engineering
Lemma 1 Assume 119909 is a solution of (2) Then1198831015840(119905) = minusΓ(1 minus120572)119863120572119909(119905)
Lemma 2 Assume 119909 is an eventually positive solution of (2)and
intinfin
1199050
1119890119860(119904)119886 (119904)
119889119904 = infin (3)
intinfin
1199050
1119903 (119904)
119889119904 = infin (4)
intinfin
1199050
1119903 (120585)
intinfin
120585
1119890119860(120591)119886 (120591)
intinfin
120591
119890119860(119904)119902 (119904) 119889119904 119889120591 119889120585 = infin (5)
Then there exists a sufficiently large 119879 such that
[119903 (119905)119863120572119909 (119905)]1015840 lt 0 119900119899 [119879infin) (6)
and either119863120572119909(119905) lt 0 on [119879infin) or lim119905rarrinfin
119883(119905) = 0
Proof Since 119909 is an eventually positive solution of (2) thereexists 119905
1such that 119909(119905) gt 0 on [119905
1infin) So119883(119905) gt 0 on [119905
1infin)
and we have
(119890119860(119905)119886 (119905) [119903 (119905)119863120572119909 (119905)]1015840)1015840
= 119890119860(119905)(119886 (119905) [119903 (119905)119863120572119909 (119905)]1015840)1015840
+ 119890119860(119905)119901 (119905) [119903 (119905) 119863120572119909 (119905)]1015840
= 119890119860(119905) (119886 (119905) [119903 (119905) 119863120572119909 (119905)]1015840)1015840
+119901 (119905) [119903 (119905) 119863120572119909 (119905)]1015840
= 119890119860(119905)119902 (119905)119883 (119905) gt 0
(7)
Then 119890119860(119905)119886(119905)[119903(119905)119863120572119909(119905)]1015840 is strictly increasing on [1199051infin)
and thus [119903(119905)119863120572119909(119905)]1015840 is eventually of one sign We claim[119903(119905)119863120572119909(119905)]1015840 lt 0 on [119905
2infin) where 119905
2gt 1199051is sufficiently
large Otherwise assume there exists a sufficiently large 1199053gt
1199052such that [119903(119905)119863120572119909(119905)]1015840 gt 0 on [119905
3infin)Then for 119905 isin [119905
3infin)
we have
119903 (119905)119863120572119909 (119905) minus 119903 (1199053)119863120572119909 (119905
3)
= int119905
1199053
119890119860(119904)119886 (119904) [119903 (119904)119863120572119909 (119904)]1015840
119890119860(119904)119886 (119904)119889119904
ge 119890119860(1199053)119886 (1199053) [119903 (1199053)119863120572119909 (119905
3)]1015840 int119905
1199053
1119890119860(119904)119886 (119904)
119889119904
(8)
By (3) we have
lim119905997888rarrinfin
119903 (119905)119863120572119909 (119905) = infin (9)
which implies for some sufficiently large 1199054gt 1199053 119863120572119909(119905) gt
0 and 119905 isin [1199054infin) By Lemma 1 we have
119883 (119905) minus 119883 (1199054) = int119905
1199054
1198831015840 (119904) 119889119904
= minusΓ (1 minus 120572)int119905
1199054
119863120572119909 (119904) 119889119904
= minusΓ (1 minus 120572)int119905
1199054
119903 (119904)119863120572119909 (119904)119903 (119904)
119889119904
le minusΓ (1 minus 120572) 119903 (1199054)119863120572119909 (119905
4) int119905
1199054
1119903 (119904)
119889119904
(10)
By (4) we obtain lim119905rarrinfin
119883(119905) = minusinfin which contradicts119883(119905) gt 0 on [119905
1infin) So [119903(119905)119863120572119909(119905)]1015840 lt 0 on [119905
2infin) Thus
119863120572119909(119905) is eventually of one sign Now we assume 119863120572119909(119905) gt0 119905 isin [119905
5infin) for some sufficiently 119905
5gt 1199054 Then by Lemma 1
1198831015840(119905) lt 0 for 119905 isin [1199055infin) Since 119883(119905) gt 0 furthermore we
have lim119905rarrinfin
119883(119905) = 120573 ge 0 We claim 120573 = 0 Otherwiseassume 120573 gt 0 Then119883(119905) ge 120573 on [119905
5infin) and for 119905 isin [119905
5infin)
by (7) we have
(119890119860(119905)119886 (119905) [119903 (119905)119863120572119909 (119905)]1015840)1015840
ge 119890119860(119905)119902 (119905)119883 (119905) ge 120573119890119860(119905)119902 (119905) (11)
Substituting 119905with 119904 in (11) an integration for (11) with respectto 119904 from 119905 toinfin yields
minus 119890119860(119905)119886 (119905) [119903 (119905) 119863120572119909 (119905)]1015840
ge minus lim119905997888rarrinfin
119890119860(119905)119886 (119905) [119903 (119905)119863120572119909 (119905)]1015840 + 120573intinfin
119905
119890119860(119904)119902 (119904) 119889119904
gt 120573int119905
1199055
119890119860(119904)119902 (119904) 119889119904
(12)
which means
(119903 (119905)119863120572119909 (119905))1015840 lt minus120573
119890119860(119905)119886 (119905)intinfin
119905
119890119860(119904)119902 (119904) 119889119904 (13)
Substituting 119905 with 120591 in (13) an integration for (13) withrespect to 120591 from 119905 toinfin yields
minus119903 (119905)119863120572119909 (119905) lt minus lim119905997888rarrinfin
119903 (119905)119863120572119909 (119905)
minus 120573intinfin
119905
1119890119860(120591)119886 (120591)
intinfin
120591
119890119860(119904)119902 (119904) 119889119904 119889120591
lt minus120573intinfin
119905
1119890119860(120591)119886 (120591)
intinfin
120591
119890119860(119904)119902 (119904) 119889119904 119889120591
(14)
That is
1198831015840 (119905) lt minusΓ (1 minus 120572) 120573119903 (119905)
intinfin
119905
1119890119860(120591)119886 (120591)
intinfin
120591
119890119860(119904)119902 (119904) 119889119904 119889120591
(15)
Mathematical Problems in Engineering 3
Substituting 119905 with 120585 in (15) an integration for (15) withrespect to 120585 from 119905
5to 119905 yields
119883 (119905) minus 119883 (1199055)
lt minusΓ (1 minus 120572) 120573int119905
1199055
1119903 (120585)
intinfin
120585
1119890119860(120591)119886 (120591)
times intinfin
120591
119890119860(119904)119902 (119904) 119889119904 119889120591 119889120585
(16)
By (5) one can see lim119905rarrinfin
119883(119905) = minusinfin which is a contradic-tion So the proof is complete
Lemma 3 Assume that 119909 is an eventually positive solution of(2) such that
[119903 (119905)119863120572119909 (119905)]1015840 lt 0 119863120572119909 (119905) lt 0 119900119899 [1199051infin) (17)
where 1199051ge 1199050is sufficiently large Then we have
1198831015840 (119905) ge minusΓ (1 minus 120572) 120575
1(119905 1199051) 119890119860(119905)119886 (119905) (119903 (119905)119863120572119909 (119905))1015840
119903 (119905) (18)
119883(119905) ge minusΓ (1 minus 120572) 1205752(119905 1199051) 119890119860(119905)119886 (119905) [119903 (119905)119863120572119909 (119905)]1015840 (19)
Proof By Lemma 2 we have 119890119860(119905)119886(119905)[119903(119905)119863120572119909(119905)]1015840 is strictlyincreasing on [119905
1infin) So
119903 (119905)119863120572119909 (119905) le 119903 (119905)119863120572119909 (119905) minus 119903 (1199051)119863120572119909 (119905)
= int119905
1199051
119890119860(119904)119886 (119904) [119903 (119904)119863120572119909 (119904)]1015840
119890119860(119904)119886 (119904)119889119904
le 119890119860(119905)119886 (119905) [119903 (119905) 119863120572119909 (119905)]1015840 int119905
1199051
1119890119860(119904)119886 (119904)
119889119904
= 1205751(119905 1199051) 119890119860(119905)119886 (119905) [119903 (119905)119863120572119909 (119905)]1015840
(20)
Using Lemma 1 we obtain that
1198831015840 (119905) ge minusΓ (1 minus 120572) 120575
1(119905 1199051) 119890119860(119905)119886 (119905) [119903 (119905)119863120572119909 (119905)]1015840
119903 (119905) (21)
Then
119883 (119905) ge 119883 (119905) minus 119883 (1199051)
ge minusint119905
1199051
Γ (1 minus 120572) 1205751(119904 1199051) 119890119860(119904)119886 (119904) [119903 (119904)119863120572119909 (119904)]1015840
119903 (119904)119889119904
ge minusΓ (1 minus 120572) 119890119860(119905)119886 (119905) [119903 (119905) 119863120572119909 (119905)]1015840 int119905
1199051
1205751(119904 1199051)
119903 (119904)119889119904
= minusΓ (1 minus 120572) 1205752(119905 1199051) 119890119860(119905)119886 (119905) [119903 (119905)119863120572119909 (119905)]1015840
(22)
Theorem 4 Assume (3)ndash(5) hold and there exist two func-tions 120601 isin 1198621([119905
0infin)R
+) and 120593 isin 1198621([119905
0infin) [0infin)) such
that
intinfin
119879
120601 (119904) 119902 (119904) 119890119860(119904) minus 120601 (119904) 1205931015840 (119904)
+120601 (119904) Γ (1 minus 120572) 120575
1(119904 119879) 1205932 (119904)
119903 (119904)
minus [2120593 (119904) 120601 (119904) Γ (1 minus 120572) 1205751(119904 119879) + 119903 (119904) 1206011015840 (119904)]
2
times(4Γ (1 minus 120572) 120601 (119904) 1205751(119904 119879) 119903 (119904))minus1 119889119904 = infin
(23)
for all sufficiently large 119879 Then every solution of (2) is oscilla-tory or satisfies lim
119905rarrinfin119883(119905) = 0
Proof Assume (2) has a nonoscillatory solution 119909 on [1199050infin)
Without loss of generality we may assume 119909(119905) gt 0 on[1199051infin) where 119905
1is sufficiently large By Lemma 2 we have
(119903(119905)119863120572119909(119905))1015840 lt 0 119905 isin [1199052infin) where 119905
2gt 1199051is sufficiently
large and either 119863120572119909(119905) lt 0 on [1199052infin) or lim
119905rarrinfin119883(119905) = 0
Define the generalized Riccati function
120596 (119905) = 120601 (119905) minus119890119860(119905)119886 (119905) [119903 (119905) 119863120572119909 (119905)]1015840
119883(119905)+ 120593 (119905) (24)
Then for 119905 isin [1199052infin) we have
1205961015840 (119905) = minus1206011015840 (119905)119890119860(119905)119886 (119905) [119903 (119905)119863120572119909 (119905)]1015840
119883 (119905)
+ 120601 (119905) minus119890119860(119905)119886 (119905) [119903 (119905)119863120572119909 (119905)]1015840
119883(119905)1015840
+ 1206011015840 (119905) 120593 (119905) + 120601 (119905) 1205931015840 (119905)
= minus120601 (119905) (119883 (119905) (119890119860(119905)119886 (119905) [119903 (119905)119863120572119909 (119905)]1015840)1015840
minus1198831015840 (119905) 119890119860(119905)119886 (119905) [119903 (119905)119863120572119909 (119905)]1015840)
times (1198832 (119905))minus1
+1206011015840 (119905)120601 (119905)
120596 (119905) + 120601 (119905) 1205931015840 (119905)
= minus120601 (119905) (119883 (119905) 119890119860(119905)(119886 (119905) [119903 (119905) 119863120572119909 (119905)]1015840)1015840
+ 119890119860(119905)119901 (119905) [119903 (119905)119863120572119909 (119905)]1015840
minus1198831015840 (119905) 119890119860(119905)119886 (119905) [119903 (119905) 119863120572119909 (119905)]1015840)
times (1198832 (119905))minus1
+1206011015840 (119905)120601 (119905)
120596 (119905) + 120601 (119905) 1205931015840 (119905)
4 Mathematical Problems in Engineering
= minus120601 (119905) 119902 (119905) 119890119860(119905)
+120601 (119905)1198831015840 (119905) 119886 (119905) 119890119860(119905)[119903 (119905)119863120572119909 (119905)]1015840
1198832 (119905)
+1206011015840 (119905)120601 (119905)
120596 (119905) + 120601 (119905) 1205931015840 (119905)
(25)
By Lemma 3 and the definition of 119891 we get that
1205961015840 (119905) le minus120601 (119905) 119902 (119905) 119890119860(119905)
minus (120601 (119905) Γ (1 minus 120572) 1205751(119905 1199052) 119890119860(119905)119886 (119905)
times (119903 (119905)119863120572119909 (119905))1015840119886 (119905) 119890119860(119905)(119903 (119905)119863120572119909 (119905))1015840)
times (119903 (119905)1198832 (119905))minus1
+1206011015840 (119905)120601 (119905)
120596 (119905) + 120601 (119905) 1205931015840 (119905)
= minus120601 (119905) 119902 (119905) 119890119860(119905) minus120601 (119905) Γ (1 minus 120572) 120575
1(119905 1199052)
119903 (119905)
times [120596 (119905)120601 (119905)
minus 120593 (119905)]2
+1206011015840 (119905)120601 (119905)
120596 (119905) + 120601 (119905) 1205931015840 (119905)
= minus120601 (119905) 119902 (119905) 119890119860(119905) + 120601 (119905) 1205931015840 (119905)
minus120601 (119905) Γ (1 minus 120572) 120575
1(119905 1199052) 1205932 (119905)
119903 (119905)
minus120601 (119905) Γ (1 minus 120572) 120575
1(119905 1199052)
119903 (119905)1205962 (119905)1206012 (119905)
+2120593 (119905) 120601 (119905) Γ (1 minus 120572) 120575
1(119905 1199052) + 119903 (119905) 1206011015840 (119905)
119903 (119905) 120601 (119905)120596 (119905)
le minus120601 (119905) 119902 (119905) 119890119860(119905) + 120601 (119905) 1205931015840 (119905)
minus120601 (119905) Γ (1 minus 120572) 120575
1(119905 1199052) 1205932 (119905)
119903 (119905)
+[2120593 (119905) 120601 (119905) Γ (1 minus 120572) 120575
1(119905 1199052) + 119903 (119905) 1206011015840 (119905)]
2
4Γ (1 minus 120572) 120601 (119905) 1205751(119905 1199052) 119903 (119905)
(26)
Substituting 119905 with 119904 in (26) an integration for (26) withrespect to 119904 from 119905
2to 119905 yields
int119905
1199052
120601 (119904) 119902 (119904) 119890119860(119904) minus 120601 (119904) 1205931015840 (119904)
+120601 (119904) Γ (1 minus 120572) 120575
1(119904 1199052) 1205932 (119904)
119903 (119904)
minus [2120593 (119904) 120601 (119904) Γ (1 minus 120572) 1205751(119904 1199052) + 119903 (119904) 1206011015840 (119904)]
2
times(4Γ (1 minus 120572) 120601 (119904) 1205751(119904 1199052) 119903 (119904))minus1 119889119904
le 120596 (1199052) minus 120596 (119905) le 120596 (119905
2) lt infin
(27)which contradicts (23) So the proof is complete
Theorem 5 Define D = (119905 119904) | 119905 ge 119904 ge 1199050 Assume (3)ndash(5)
hold and there exists a function119867 isin 1198621(DR) such that119867(119905 119905) = 0 119891119900119903 119905 ge 119905
0 119867 (119905 119904) gt 0 119891119900119903 119905 gt 119904 ge 119905
0
(28)
and119867 has a nonpositive continuous partial derivative1198671015840119904(119905 119904)
and
lim119905997888rarrinfin
sup 1119867 (119905 119905
0)
times int119905
1199050
119867(119905 119904) 120601 (119904) 119902 (119904) 119890119860(119904) minus 120601 (119904) 1205931015840 (119904)
+120601 (119904) Γ (1 minus 120572) 120575
1(119904 119879) 1205932 (119904)
119903 (119904)
minus [2120593 (119904) 120601 (119904) Γ (1 minus 120572)
times 1205751(119904 119879) +119903 (119904) 1206011015840 (119904)]
2
times (4Γ (1 minus 120572) 120601 (119904) 1205751
times (119904 119879) 119903 (119904) )minus1 119889119904 = infin
(29)
for all sufficiently large 119879 where 120601 120593 are defined as inTheorem 4 Then every solution of (2) is oscillatory or satisfieslim119905rarrinfin
119883(119905) = 0
Proof Assume (2) has a nonoscillatory solution 119909 on [1199050infin)
Without loss of generality we may assume 119909(119905) gt 0 on[1199051infin) where 119905
1is sufficiently large By Lemma 2 we have
119863120572119909(119905) lt 0 on [1199052infin) for some sufficiently large 119905
2gt 1199051 Let
120596(119905) be defined as in Theorem 4 By (26) we have
120601 (119905) 119902 (119905) 119890119860(119905) + 120601 (119905) 1205931015840 (119905) minus120601 (119905) Γ (1 minus 120572) 120575
1(119905 1199052) 1205932 (119905)
119903 (119905)
+[2120593 (119905) 120601 (119905) Γ (1 minus 120572) 120575
1(119905 1199052) + 119903 (119905) 1206011015840 (119905)]
2
4Γ (1 minus 120572) 120601 (119905) 1205751(119905 1199052) 119903 (119905)
le minus1205961015840 (119905)
(30)Substituting 119905with 119904 in (30) multiplying both sides by119867(119905 119904)and then integrating it with respect to 119904 from 119905
2to 119905 yield
int119905
1199052
119867(119905 119904)
times120601 (119904) 119902 (119904) 119890119860(119904) minus 120601 (119904) 1205931015840 (119904)
+120601 (119904) Γ (1 minus 120572) 120575
1(119904 1199052) 1205932 (119904)
119903 (119904)
Mathematical Problems in Engineering 5
minus[2120593 (119904) 120601 (119904) Γ (1 minus 120572) 120575
1(119904 1199052) + 119903 (119904) 1206011015840 (119904)]
2
4Γ (1 minus 120572) 120601 (119904) 1205751(119904 1199052) 119903 (119904)
119889119904
le minusint119905
1199052
119867(119905 119904) 1205961015840 (119904) 119889119904
= 119867 (119905 1199052) 120596 (1199052) + int119905
1199052
1198671015840119904(119905 119904) 120596 (119904) Δ119904
le 119867 (119905 1199052) 120596 (1199052) le 119867 (119905 119905
0) 120596 (1199052)
(31)
Then
int119905
1199050
119867(119905 119904)
times 120601 (119904) 119902 (119904) 119890119860(119904) minus 120601 (119904) 1205931015840 (119904)
+120601 (119904) Γ (1 minus 120572) 120575
1(119904 1199052) 1205932 (119904)
119903 (119904)
minus [2120593 (119904) 120601 (119904) Γ (1 minus 120572)
times1205751(119904 1199052) + 119903 (119904) 1206011015840 (119904)]
2
times(4Γ (1 minus 120572) 120601 (119904) 1205751(119904 1199052) 119903 (119904))minus1 119889119904
= int1199052
1199050
119867(119905 119904)
times120601 (119904) 119902 (119904) 119890119860(119904) minus 120601 (119904) 1205931015840 (119904)
+120601 (119904) Γ (1 minus 120572) 120575
1(119904 1199052) 1205932 (119904)
119903 (119904)
minus[2120593 (119904) 120601 (119904) Γ (1 minus 120572) 120575
1(119904 1199052) + 119903 (119904) 1206011015840 (119904)]
2
4Γ (1 minus 120572) 120601 (119904) 1205751(119904 1199052) 119903 (119904)
119889119904
+ int119905
1199052
119867(119905 119904)
times120601 (119904) 119902 (119904) 119890119860(119904) minus 120601 (119904) 1205931015840 (119904)
+120601 (119904) Γ (1 minus 120572) 120575
1(119904 1199052) 1205932 (119904)
119903 (119904)
minus[2120593 (119904) 120601 (119904) Γ (1 minus 120572) 120575
1(119904 1199052) + 119903 (119904) 1206011015840 (119904)]
2
4Γ (1 minus 120572) 120601 (119904) 1205751(119904 1199052) 119903 (119904)
times 119889119904
le 119867 (119905 1199050) 120596 (1199052) + 119867 (119905 119905
0)
times int1199052
1199050
10038161003816100381610038161003816120601 (119904) 119902 (119904) 119890119860(119904) minus 120601 (119904) 1205931015840 (119904)
+ (120601 (119904) Γ (1 minus 120572) 1205751(119904 1199052) 1205932 (119904))
times (119903 (119904))minus1
minus [2120593 (119904) 120601 (119904) Γ (1 minus 120572) 1205751
times (119904 1199052) +119903 (119904) 1206011015840 (119904)]
2
times (4Γ (1 minus 120572) 120601 (119904) 1205751(119904 1199052) 119903 (119904))minus110038161003816100381610038161003816 119889119904
(32)
So
lim119905997888rarrinfin
sup 1119867 (119905 119905
0)
times int119905
1199050
119867(119905 119904) 120601 (119904) 119902 (119904) 119890119860(119904) minus 120601 (119904) 1205931015840 (119904)
+ (120601 (119904) Γ (1 minus 120572) 1205751(119904 1199052) 1205932 (119904))
times (119903 (119904))minus1 minus [2120593 (119904) 120601 (119904) Γ (1 minus 120572)
times 1205751(119904 1199052) + 119903 (119904) 1206011015840 (119904)]
2
times (4Γ (1 minus 120572) 120601 (119904) 1205751(119904 1199052) 119903 (119904))minus1 119889119904
le 120596 (1199052)
+ int1199052
1199050
10038161003816100381610038161003816120601 (119904) 119902 (119904) 119890119860(119904) minus 120601 (119904) 1205931015840 (119904)
+120601 (119904) Γ (1 minus 120572) 120575
1(119904 1199052) 1205932 (119904)
119903 (119904)
minus [2120593 (119904) 120601 (119904) Γ (1 minus 120572) 1205751(119904 1199052) + 119903 (119904) 1206011015840 (119904)]
2
times(4Γ (1 minus 120572) 120601 (119904) 1205751(119904 1199052) 119903 (119904))minus110038161003816100381610038161003816 119889119904 lt infin
(33)
which contradicts (29) So the proof is complete
In Theorems 5 if we take 119867(119905 119904) for some special func-tions such as (119905 minus 119904)119898 or ln (119905119904) then we can obtain somecorollaries as follows
Corollary 6 Assume (3)ndash(5) hold and
lim119905997888rarrinfin
sup 1(119905 minus 1199050)119898
times int119905
1199050
(119905 minus 119904)119898 120601 (119904) 119902 (119904) 119890119860(119904) minus 120601 (119904) 1205931015840 (119904)
+ (120601 (119904) Γ (1 minus 120572) 1205751(119904 119879) 1205932 (119904))
6 Mathematical Problems in Engineering
times (119903 (119904))minus1
minus [2120593 (119904) 120601 (119904) Γ (1 minus 120572) 1205751(119904 119879)
+119903 (119904) 1206011015840 (119904)]2
times (4Γ (1 minus 120572) 120601 (119904) 1205751
times (119904 119879) 119903 (119904) )minus1 119889119904 = infin
(34)
for all sufficiently large 119879 Then every solution of (2) is oscilla-tory or satisfies lim
119905rarrinfin119883(119905) = 0
Corollary 7 Assume (3)ndash(5) hold and
lim119905997888rarrinfin
sup 1(ln 119905 minus ln 119905
0)
times int119905
1199050
(ln 119905 minus ln 119904) 120601 (119904) 119902 (119904) 119890119860(119904) minus 120601 (119904) 1205931015840 (119904)
+120601 (119904) Γ (1 minus 120572) 120575
1(119904 119879) 1205932 (119904)
119903 (119904)
minus [2120593 (119904) 120601 (119904) Γ (1 minus 120572) 1205751(119904 119879)
+119903 (119904) 1206011015840 (119904)]2
times (4Γ (1 minus 120572) 120601 (119904)
times1205751(119904 119879) 119903 (119904))minus1 119889119904 = infin
(35)
for all sufficiently large 119879 Then every solution of (2) is oscilla-tory or satisfies lim
119905rarrinfin119883(119905) = 0
3 Applications
In this section we will present some applications for theabove established results
Example 8 Consider
(119905(119863120572119909 (119905))1015840)1015840
+ 119905minus2(119863120572119909 (119905))1015840
minus119872119905minus2 intinfin
119905
(120585 minus 119905)minus120572119909 (120585) 119889120585 = 0 119905 isin [2infin) (36)
where119872 gt 0 is a quotient of two odd positive integers
We have in (2) 119886(119905) = 119905 119901(119905) = 119905minus2 119902(119905) = 119872119905minus2 119903(119905) =1 and 119905
0= 2 Then
1 le 119890119860(119905) = 119890int119905
2(119901(119904)119886(119904))119889119904 = 119890int
119905
2119904minus3119889119904 = 119890minus(12)[119905
minus2minus2minus2] le 11989018
(37)
Moreover we have
intinfin
1199050
1119890119860(119904)119886 (119904)
119889119904 = intinfin
2
1119890119860(119904)119886 (119904)
119889119904 ge 119890minus18 intinfin
2
1119904119889119904 = infin
intinfin
1199050
1119903 (119904)
119889119904 = infin
(38)
Furthermore
intinfin
1199050
1119903 (120585)
intinfin
120585
1119890119860(120591)119886 (120591)
intinfin
120591
119890119860(119904)119902 (119904) 119889119904 119889120591 119889120585
= 119872intinfin
2
intinfin
120585
1119890119860(120591)120591
intinfin
120591
119890119860(119904)
1199042119889119904 119889120591 119889120585
ge 119872119890minus18 intinfin
2
intinfin
120585
1120591intinfin
120591
11199042119889119904 119889120591 119889120585
= 119872119890minus18 intinfin
2
intinfin
120585
11205912119889120591 119889120585
= 119872119890minus18 intinfin
2
1120585119889120585 = infin
(39)
On the other hand for a sufficiently large 119879 we have
1205751(119905 119879) = int
119905
119879
1119890119860(119904)119886 (119904)
119889119904
= int119905
119879
1119890119860(119904)119904
119889119904 ge 119890minus18 int119905
119879
1119904119889119904 997888rarr infin
(40)
So we can take 119879lowast gt 119879 such that 1205751(119905 119879) gt 1 for 119905 isin [119879lowastinfin)
Taking 120601(119905) = 119905 120593(119905) = 0 in (23) we get that
intinfin
119879
120601 (119904) 119902 (119904) 119890119860(119904) minus
[119903 (119904) 1206011015840 (119904)]2
4 [Γ (1 minus 120572) 120601 (119904) 1205751(119904 119879)] 119903 (119904)
119889119904
ge intinfin
119879
[119872 minus 14Γ (1 minus 120572) 120575
1(119904 119879)
] 1119904119889119904
= int119879lowast
119879
[119872 minus 14Γ (1 minus 120572) 120575
1(119904 119879)
] 1119904119889119904
+ intinfin
119879lowast
[119872 minus 14Γ (1 minus 120572) 120575
1(119904 119879)
] 1119904119889119904
ge int119879lowast
119879
[119872 minus 14Γ (1 minus 120572) 120575
1(119904 119879)
] 1119904119889119904
+ intinfin
119879lowast
[119872 minus 14Γ (1 minus 120572)
] 1119904119889119904 = infin
(41)
provided that119872 gt 1(4Γ(1minus120572)) So (3)ndash(5) and (23) all holdand by Theorem 4 we deduce that every solution of (36) isoscillatory or satisfies lim
119905rarrinfin119883(119905) = 0 under condition119872 gt
14Γ(1 minus 120572)
Mathematical Problems in Engineering 7
Example 9 Consider
(radic119905(119863120572119909 (119905))1015840)1015840
+ 119890minus119905(119863120572119909 (119905))1015840
minus (119872119905minus52 + ln 119905) intinfin
119905
(120585 minus 119905)minus120572119909 (120585) 119889120585 = 0 119905 isin [2infin)
(42)
where 120572 isin (0 1) and119872 gt 0 is a constant
We have in (2) 119886(119905) = radic119905 119901(119905) = 119890minus119905 119902(119905) = 119872119905minus52 +ln 119905 119903(119905) = 1 and 119905
0= 2 Then
1 le 119890119860(119905) = 119890int119905
2(119901(119904)119886(119904))119889119904 = 119890int
119905
2(1119890119904radic119904)119889119904
le 119890int119905
2119904minus(32)119889119904
= 119890minus2[119905minus12minus2minus12] le 119890radic2
(43)
So we have
intinfin
1199050
1119890119860(119904)119886 (119904)
119889119904 = intinfin
2
1119890119860(119904)radic119904
119889119904 ge 119890minusradic2 intinfin
2
1radic119904119889119904 = infin
intinfin
1199050
1119903 (119904)
119889119904 = infin
(44)
Furthermore
intinfin
1199050
1119903 (120585)
intinfin
120585
1119890119860(120591)119886 (120591)
intinfin
120591
119890119860(119904)119902 (119904) 119889119904 119889120591 119889120585
= intinfin
2
intinfin
120585
1119890119860(120591)radic120591
intinfin
120591
119890119860(119904) (119872119904minus52 + ln 119904) 119889119904 119889120591 119889120585
ge 119872119890minusradic2 intinfin
2
intinfin
120585
1radic120591
intinfin
120591
119904minus52119889119904 119889120591 119889120585
= 2119872119890minusradic2
3intinfin
2
[intinfin
120585
11205912119889120591] 119889120585
= 2119872119890minusradic2
3intinfin
2
1120585119889120585 = infin
(45)
On the other hand Taking 120601(119905) = 1199052 120593(119905) = 0 and 119867(119905 119904) =119905 minus 119904 in (29) we get that
lim119905997888rarrinfin
sup 1119905 minus 1199050
int119905
1199050
(119905 minus 119904)
times120601 (119904) 119902 (119904) 119890119860(119904)
minus[119903 (119904) 1206011015840 (119904)]
2
4Γ (1 minus 120572) 120601 (119904) 1205751(119904 119879) 119903 (119904)
119889119904
ge lim119905997888rarrinfin
sup 1119905 minus 2
times int119905
2
(119905 minus 119904) (119872119904minus52 + ln 119904) 1199042
minus 1Γ (1 minus 120572) 120575
1(119904 119879)
119889119904
ge lim119905997888rarrinfin
sup 1119905 minus 2
int119905
2
(119905 minus 119904)119872119904minus12119889119904 = infin
(46)
So (3)ndash(5) and (29) all hold and by Corollary 6 with 119898 = 1we deduce that every solution of (42) is oscillatory or satisfieslim119905rarrinfin
119883(119905) = 0
Acknowledgments
This work is partially supported by Planning Fund projectof the Ministry of Education of China (10YJA630019) Theauthors would thank the reviewers very much for theirvaluable suggestions on this paper
References
[1] A Saadatmandi and M Dehghan ldquoA new operational matrixfor solving fractional-order differential equationsrdquo Computersamp Mathematics with Applications vol 59 no 3 pp 1326ndash13362010
[2] Y Zhou F Jiao and J Li ldquoExistence and uniqueness for p-typefractional neutral differential equationsrdquo Nonlinear AnalysisTheory Methods amp Applications vol 71 no 7-8 pp 2724ndash27332009
[3] L Galeone and R Garrappa ldquoExplicit methods for fractionaldifferential equations and their stability propertiesrdquo Journal ofComputational and Applied Mathematics vol 228 no 2 pp548ndash560 2009
[4] J C Trigeassou N Maamri J Sabatier and A OustaloupldquoA Lyapunov approach to the stability of fractional differentialequationsrdquo Signal Process vol 91 pp 437ndash445 2011
[5] W Deng ldquoSmoothness and stability of the solutions fornonlinear fractional differential equationsrdquo Nonlinear AnalysisTheory Methods amp Applications vol 72 no 3-4 pp 1768ndash17772010
[6] F Ghoreishi and S Yazdani ldquoAn extension of the spectralTau method for numerical solution of multi-order fractionaldifferential equations with convergence analysisrdquo Computers ampMathematics with Applications vol 61 no 1 pp 30ndash43 2011
[7] J T Edwards N J Ford and A C Simpson ldquoThe numericalsolution of linear multi-term fractional differential equationssystems of equationsrdquo Journal of Computational and AppliedMathematics vol 148 no 2 pp 401ndash418 2002
[8] M Muslim ldquoExistence and approximation of solutions tofractional differential equationsrdquo Mathematical and ComputerModelling vol 49 no 5-6 pp 1164ndash1172 2009
[9] D-X Chen ldquoOscillation criteria of fractional differential equa-tionsrdquo Advances in Difference Equations vol 2012 article 332012
[10] I Podlubny Fractional Differential Equations vol 198 Aca-demic Press San Diego Calif USA 1999
8 Mathematical Problems in Engineering
[11] A A Kilbas H M Srivastava and J J Trujillo Theoryand Applications of Fractional Differential Equations vol 204Elsevier Science BV Amsterdam The Netherlands 2006
[12] S Das Functional Fractional Calculus Springer Berlin Ger-many 2nd edition 2011
Submit your manuscripts athttpwwwhindawicom
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
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Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Mathematical Problems in Engineering
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Differential EquationsInternational Journal of
Volume 2014
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Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Journal of
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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
International Journal of Mathematics and Mathematical Sciences
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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
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Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Stochastic AnalysisInternational Journal of
Mathematical Problems in Engineering 3
Substituting 119905 with 120585 in (15) an integration for (15) withrespect to 120585 from 119905
5to 119905 yields
119883 (119905) minus 119883 (1199055)
lt minusΓ (1 minus 120572) 120573int119905
1199055
1119903 (120585)
intinfin
120585
1119890119860(120591)119886 (120591)
times intinfin
120591
119890119860(119904)119902 (119904) 119889119904 119889120591 119889120585
(16)
By (5) one can see lim119905rarrinfin
119883(119905) = minusinfin which is a contradic-tion So the proof is complete
Lemma 3 Assume that 119909 is an eventually positive solution of(2) such that
[119903 (119905)119863120572119909 (119905)]1015840 lt 0 119863120572119909 (119905) lt 0 119900119899 [1199051infin) (17)
where 1199051ge 1199050is sufficiently large Then we have
1198831015840 (119905) ge minusΓ (1 minus 120572) 120575
1(119905 1199051) 119890119860(119905)119886 (119905) (119903 (119905)119863120572119909 (119905))1015840
119903 (119905) (18)
119883(119905) ge minusΓ (1 minus 120572) 1205752(119905 1199051) 119890119860(119905)119886 (119905) [119903 (119905)119863120572119909 (119905)]1015840 (19)
Proof By Lemma 2 we have 119890119860(119905)119886(119905)[119903(119905)119863120572119909(119905)]1015840 is strictlyincreasing on [119905
1infin) So
119903 (119905)119863120572119909 (119905) le 119903 (119905)119863120572119909 (119905) minus 119903 (1199051)119863120572119909 (119905)
= int119905
1199051
119890119860(119904)119886 (119904) [119903 (119904)119863120572119909 (119904)]1015840
119890119860(119904)119886 (119904)119889119904
le 119890119860(119905)119886 (119905) [119903 (119905) 119863120572119909 (119905)]1015840 int119905
1199051
1119890119860(119904)119886 (119904)
119889119904
= 1205751(119905 1199051) 119890119860(119905)119886 (119905) [119903 (119905)119863120572119909 (119905)]1015840
(20)
Using Lemma 1 we obtain that
1198831015840 (119905) ge minusΓ (1 minus 120572) 120575
1(119905 1199051) 119890119860(119905)119886 (119905) [119903 (119905)119863120572119909 (119905)]1015840
119903 (119905) (21)
Then
119883 (119905) ge 119883 (119905) minus 119883 (1199051)
ge minusint119905
1199051
Γ (1 minus 120572) 1205751(119904 1199051) 119890119860(119904)119886 (119904) [119903 (119904)119863120572119909 (119904)]1015840
119903 (119904)119889119904
ge minusΓ (1 minus 120572) 119890119860(119905)119886 (119905) [119903 (119905) 119863120572119909 (119905)]1015840 int119905
1199051
1205751(119904 1199051)
119903 (119904)119889119904
= minusΓ (1 minus 120572) 1205752(119905 1199051) 119890119860(119905)119886 (119905) [119903 (119905)119863120572119909 (119905)]1015840
(22)
Theorem 4 Assume (3)ndash(5) hold and there exist two func-tions 120601 isin 1198621([119905
0infin)R
+) and 120593 isin 1198621([119905
0infin) [0infin)) such
that
intinfin
119879
120601 (119904) 119902 (119904) 119890119860(119904) minus 120601 (119904) 1205931015840 (119904)
+120601 (119904) Γ (1 minus 120572) 120575
1(119904 119879) 1205932 (119904)
119903 (119904)
minus [2120593 (119904) 120601 (119904) Γ (1 minus 120572) 1205751(119904 119879) + 119903 (119904) 1206011015840 (119904)]
2
times(4Γ (1 minus 120572) 120601 (119904) 1205751(119904 119879) 119903 (119904))minus1 119889119904 = infin
(23)
for all sufficiently large 119879 Then every solution of (2) is oscilla-tory or satisfies lim
119905rarrinfin119883(119905) = 0
Proof Assume (2) has a nonoscillatory solution 119909 on [1199050infin)
Without loss of generality we may assume 119909(119905) gt 0 on[1199051infin) where 119905
1is sufficiently large By Lemma 2 we have
(119903(119905)119863120572119909(119905))1015840 lt 0 119905 isin [1199052infin) where 119905
2gt 1199051is sufficiently
large and either 119863120572119909(119905) lt 0 on [1199052infin) or lim
119905rarrinfin119883(119905) = 0
Define the generalized Riccati function
120596 (119905) = 120601 (119905) minus119890119860(119905)119886 (119905) [119903 (119905) 119863120572119909 (119905)]1015840
119883(119905)+ 120593 (119905) (24)
Then for 119905 isin [1199052infin) we have
1205961015840 (119905) = minus1206011015840 (119905)119890119860(119905)119886 (119905) [119903 (119905)119863120572119909 (119905)]1015840
119883 (119905)
+ 120601 (119905) minus119890119860(119905)119886 (119905) [119903 (119905)119863120572119909 (119905)]1015840
119883(119905)1015840
+ 1206011015840 (119905) 120593 (119905) + 120601 (119905) 1205931015840 (119905)
= minus120601 (119905) (119883 (119905) (119890119860(119905)119886 (119905) [119903 (119905)119863120572119909 (119905)]1015840)1015840
minus1198831015840 (119905) 119890119860(119905)119886 (119905) [119903 (119905)119863120572119909 (119905)]1015840)
times (1198832 (119905))minus1
+1206011015840 (119905)120601 (119905)
120596 (119905) + 120601 (119905) 1205931015840 (119905)
= minus120601 (119905) (119883 (119905) 119890119860(119905)(119886 (119905) [119903 (119905) 119863120572119909 (119905)]1015840)1015840
+ 119890119860(119905)119901 (119905) [119903 (119905)119863120572119909 (119905)]1015840
minus1198831015840 (119905) 119890119860(119905)119886 (119905) [119903 (119905) 119863120572119909 (119905)]1015840)
times (1198832 (119905))minus1
+1206011015840 (119905)120601 (119905)
120596 (119905) + 120601 (119905) 1205931015840 (119905)
4 Mathematical Problems in Engineering
= minus120601 (119905) 119902 (119905) 119890119860(119905)
+120601 (119905)1198831015840 (119905) 119886 (119905) 119890119860(119905)[119903 (119905)119863120572119909 (119905)]1015840
1198832 (119905)
+1206011015840 (119905)120601 (119905)
120596 (119905) + 120601 (119905) 1205931015840 (119905)
(25)
By Lemma 3 and the definition of 119891 we get that
1205961015840 (119905) le minus120601 (119905) 119902 (119905) 119890119860(119905)
minus (120601 (119905) Γ (1 minus 120572) 1205751(119905 1199052) 119890119860(119905)119886 (119905)
times (119903 (119905)119863120572119909 (119905))1015840119886 (119905) 119890119860(119905)(119903 (119905)119863120572119909 (119905))1015840)
times (119903 (119905)1198832 (119905))minus1
+1206011015840 (119905)120601 (119905)
120596 (119905) + 120601 (119905) 1205931015840 (119905)
= minus120601 (119905) 119902 (119905) 119890119860(119905) minus120601 (119905) Γ (1 minus 120572) 120575
1(119905 1199052)
119903 (119905)
times [120596 (119905)120601 (119905)
minus 120593 (119905)]2
+1206011015840 (119905)120601 (119905)
120596 (119905) + 120601 (119905) 1205931015840 (119905)
= minus120601 (119905) 119902 (119905) 119890119860(119905) + 120601 (119905) 1205931015840 (119905)
minus120601 (119905) Γ (1 minus 120572) 120575
1(119905 1199052) 1205932 (119905)
119903 (119905)
minus120601 (119905) Γ (1 minus 120572) 120575
1(119905 1199052)
119903 (119905)1205962 (119905)1206012 (119905)
+2120593 (119905) 120601 (119905) Γ (1 minus 120572) 120575
1(119905 1199052) + 119903 (119905) 1206011015840 (119905)
119903 (119905) 120601 (119905)120596 (119905)
le minus120601 (119905) 119902 (119905) 119890119860(119905) + 120601 (119905) 1205931015840 (119905)
minus120601 (119905) Γ (1 minus 120572) 120575
1(119905 1199052) 1205932 (119905)
119903 (119905)
+[2120593 (119905) 120601 (119905) Γ (1 minus 120572) 120575
1(119905 1199052) + 119903 (119905) 1206011015840 (119905)]
2
4Γ (1 minus 120572) 120601 (119905) 1205751(119905 1199052) 119903 (119905)
(26)
Substituting 119905 with 119904 in (26) an integration for (26) withrespect to 119904 from 119905
2to 119905 yields
int119905
1199052
120601 (119904) 119902 (119904) 119890119860(119904) minus 120601 (119904) 1205931015840 (119904)
+120601 (119904) Γ (1 minus 120572) 120575
1(119904 1199052) 1205932 (119904)
119903 (119904)
minus [2120593 (119904) 120601 (119904) Γ (1 minus 120572) 1205751(119904 1199052) + 119903 (119904) 1206011015840 (119904)]
2
times(4Γ (1 minus 120572) 120601 (119904) 1205751(119904 1199052) 119903 (119904))minus1 119889119904
le 120596 (1199052) minus 120596 (119905) le 120596 (119905
2) lt infin
(27)which contradicts (23) So the proof is complete
Theorem 5 Define D = (119905 119904) | 119905 ge 119904 ge 1199050 Assume (3)ndash(5)
hold and there exists a function119867 isin 1198621(DR) such that119867(119905 119905) = 0 119891119900119903 119905 ge 119905
0 119867 (119905 119904) gt 0 119891119900119903 119905 gt 119904 ge 119905
0
(28)
and119867 has a nonpositive continuous partial derivative1198671015840119904(119905 119904)
and
lim119905997888rarrinfin
sup 1119867 (119905 119905
0)
times int119905
1199050
119867(119905 119904) 120601 (119904) 119902 (119904) 119890119860(119904) minus 120601 (119904) 1205931015840 (119904)
+120601 (119904) Γ (1 minus 120572) 120575
1(119904 119879) 1205932 (119904)
119903 (119904)
minus [2120593 (119904) 120601 (119904) Γ (1 minus 120572)
times 1205751(119904 119879) +119903 (119904) 1206011015840 (119904)]
2
times (4Γ (1 minus 120572) 120601 (119904) 1205751
times (119904 119879) 119903 (119904) )minus1 119889119904 = infin
(29)
for all sufficiently large 119879 where 120601 120593 are defined as inTheorem 4 Then every solution of (2) is oscillatory or satisfieslim119905rarrinfin
119883(119905) = 0
Proof Assume (2) has a nonoscillatory solution 119909 on [1199050infin)
Without loss of generality we may assume 119909(119905) gt 0 on[1199051infin) where 119905
1is sufficiently large By Lemma 2 we have
119863120572119909(119905) lt 0 on [1199052infin) for some sufficiently large 119905
2gt 1199051 Let
120596(119905) be defined as in Theorem 4 By (26) we have
120601 (119905) 119902 (119905) 119890119860(119905) + 120601 (119905) 1205931015840 (119905) minus120601 (119905) Γ (1 minus 120572) 120575
1(119905 1199052) 1205932 (119905)
119903 (119905)
+[2120593 (119905) 120601 (119905) Γ (1 minus 120572) 120575
1(119905 1199052) + 119903 (119905) 1206011015840 (119905)]
2
4Γ (1 minus 120572) 120601 (119905) 1205751(119905 1199052) 119903 (119905)
le minus1205961015840 (119905)
(30)Substituting 119905with 119904 in (30) multiplying both sides by119867(119905 119904)and then integrating it with respect to 119904 from 119905
2to 119905 yield
int119905
1199052
119867(119905 119904)
times120601 (119904) 119902 (119904) 119890119860(119904) minus 120601 (119904) 1205931015840 (119904)
+120601 (119904) Γ (1 minus 120572) 120575
1(119904 1199052) 1205932 (119904)
119903 (119904)
Mathematical Problems in Engineering 5
minus[2120593 (119904) 120601 (119904) Γ (1 minus 120572) 120575
1(119904 1199052) + 119903 (119904) 1206011015840 (119904)]
2
4Γ (1 minus 120572) 120601 (119904) 1205751(119904 1199052) 119903 (119904)
119889119904
le minusint119905
1199052
119867(119905 119904) 1205961015840 (119904) 119889119904
= 119867 (119905 1199052) 120596 (1199052) + int119905
1199052
1198671015840119904(119905 119904) 120596 (119904) Δ119904
le 119867 (119905 1199052) 120596 (1199052) le 119867 (119905 119905
0) 120596 (1199052)
(31)
Then
int119905
1199050
119867(119905 119904)
times 120601 (119904) 119902 (119904) 119890119860(119904) minus 120601 (119904) 1205931015840 (119904)
+120601 (119904) Γ (1 minus 120572) 120575
1(119904 1199052) 1205932 (119904)
119903 (119904)
minus [2120593 (119904) 120601 (119904) Γ (1 minus 120572)
times1205751(119904 1199052) + 119903 (119904) 1206011015840 (119904)]
2
times(4Γ (1 minus 120572) 120601 (119904) 1205751(119904 1199052) 119903 (119904))minus1 119889119904
= int1199052
1199050
119867(119905 119904)
times120601 (119904) 119902 (119904) 119890119860(119904) minus 120601 (119904) 1205931015840 (119904)
+120601 (119904) Γ (1 minus 120572) 120575
1(119904 1199052) 1205932 (119904)
119903 (119904)
minus[2120593 (119904) 120601 (119904) Γ (1 minus 120572) 120575
1(119904 1199052) + 119903 (119904) 1206011015840 (119904)]
2
4Γ (1 minus 120572) 120601 (119904) 1205751(119904 1199052) 119903 (119904)
119889119904
+ int119905
1199052
119867(119905 119904)
times120601 (119904) 119902 (119904) 119890119860(119904) minus 120601 (119904) 1205931015840 (119904)
+120601 (119904) Γ (1 minus 120572) 120575
1(119904 1199052) 1205932 (119904)
119903 (119904)
minus[2120593 (119904) 120601 (119904) Γ (1 minus 120572) 120575
1(119904 1199052) + 119903 (119904) 1206011015840 (119904)]
2
4Γ (1 minus 120572) 120601 (119904) 1205751(119904 1199052) 119903 (119904)
times 119889119904
le 119867 (119905 1199050) 120596 (1199052) + 119867 (119905 119905
0)
times int1199052
1199050
10038161003816100381610038161003816120601 (119904) 119902 (119904) 119890119860(119904) minus 120601 (119904) 1205931015840 (119904)
+ (120601 (119904) Γ (1 minus 120572) 1205751(119904 1199052) 1205932 (119904))
times (119903 (119904))minus1
minus [2120593 (119904) 120601 (119904) Γ (1 minus 120572) 1205751
times (119904 1199052) +119903 (119904) 1206011015840 (119904)]
2
times (4Γ (1 minus 120572) 120601 (119904) 1205751(119904 1199052) 119903 (119904))minus110038161003816100381610038161003816 119889119904
(32)
So
lim119905997888rarrinfin
sup 1119867 (119905 119905
0)
times int119905
1199050
119867(119905 119904) 120601 (119904) 119902 (119904) 119890119860(119904) minus 120601 (119904) 1205931015840 (119904)
+ (120601 (119904) Γ (1 minus 120572) 1205751(119904 1199052) 1205932 (119904))
times (119903 (119904))minus1 minus [2120593 (119904) 120601 (119904) Γ (1 minus 120572)
times 1205751(119904 1199052) + 119903 (119904) 1206011015840 (119904)]
2
times (4Γ (1 minus 120572) 120601 (119904) 1205751(119904 1199052) 119903 (119904))minus1 119889119904
le 120596 (1199052)
+ int1199052
1199050
10038161003816100381610038161003816120601 (119904) 119902 (119904) 119890119860(119904) minus 120601 (119904) 1205931015840 (119904)
+120601 (119904) Γ (1 minus 120572) 120575
1(119904 1199052) 1205932 (119904)
119903 (119904)
minus [2120593 (119904) 120601 (119904) Γ (1 minus 120572) 1205751(119904 1199052) + 119903 (119904) 1206011015840 (119904)]
2
times(4Γ (1 minus 120572) 120601 (119904) 1205751(119904 1199052) 119903 (119904))minus110038161003816100381610038161003816 119889119904 lt infin
(33)
which contradicts (29) So the proof is complete
In Theorems 5 if we take 119867(119905 119904) for some special func-tions such as (119905 minus 119904)119898 or ln (119905119904) then we can obtain somecorollaries as follows
Corollary 6 Assume (3)ndash(5) hold and
lim119905997888rarrinfin
sup 1(119905 minus 1199050)119898
times int119905
1199050
(119905 minus 119904)119898 120601 (119904) 119902 (119904) 119890119860(119904) minus 120601 (119904) 1205931015840 (119904)
+ (120601 (119904) Γ (1 minus 120572) 1205751(119904 119879) 1205932 (119904))
6 Mathematical Problems in Engineering
times (119903 (119904))minus1
minus [2120593 (119904) 120601 (119904) Γ (1 minus 120572) 1205751(119904 119879)
+119903 (119904) 1206011015840 (119904)]2
times (4Γ (1 minus 120572) 120601 (119904) 1205751
times (119904 119879) 119903 (119904) )minus1 119889119904 = infin
(34)
for all sufficiently large 119879 Then every solution of (2) is oscilla-tory or satisfies lim
119905rarrinfin119883(119905) = 0
Corollary 7 Assume (3)ndash(5) hold and
lim119905997888rarrinfin
sup 1(ln 119905 minus ln 119905
0)
times int119905
1199050
(ln 119905 minus ln 119904) 120601 (119904) 119902 (119904) 119890119860(119904) minus 120601 (119904) 1205931015840 (119904)
+120601 (119904) Γ (1 minus 120572) 120575
1(119904 119879) 1205932 (119904)
119903 (119904)
minus [2120593 (119904) 120601 (119904) Γ (1 minus 120572) 1205751(119904 119879)
+119903 (119904) 1206011015840 (119904)]2
times (4Γ (1 minus 120572) 120601 (119904)
times1205751(119904 119879) 119903 (119904))minus1 119889119904 = infin
(35)
for all sufficiently large 119879 Then every solution of (2) is oscilla-tory or satisfies lim
119905rarrinfin119883(119905) = 0
3 Applications
In this section we will present some applications for theabove established results
Example 8 Consider
(119905(119863120572119909 (119905))1015840)1015840
+ 119905minus2(119863120572119909 (119905))1015840
minus119872119905minus2 intinfin
119905
(120585 minus 119905)minus120572119909 (120585) 119889120585 = 0 119905 isin [2infin) (36)
where119872 gt 0 is a quotient of two odd positive integers
We have in (2) 119886(119905) = 119905 119901(119905) = 119905minus2 119902(119905) = 119872119905minus2 119903(119905) =1 and 119905
0= 2 Then
1 le 119890119860(119905) = 119890int119905
2(119901(119904)119886(119904))119889119904 = 119890int
119905
2119904minus3119889119904 = 119890minus(12)[119905
minus2minus2minus2] le 11989018
(37)
Moreover we have
intinfin
1199050
1119890119860(119904)119886 (119904)
119889119904 = intinfin
2
1119890119860(119904)119886 (119904)
119889119904 ge 119890minus18 intinfin
2
1119904119889119904 = infin
intinfin
1199050
1119903 (119904)
119889119904 = infin
(38)
Furthermore
intinfin
1199050
1119903 (120585)
intinfin
120585
1119890119860(120591)119886 (120591)
intinfin
120591
119890119860(119904)119902 (119904) 119889119904 119889120591 119889120585
= 119872intinfin
2
intinfin
120585
1119890119860(120591)120591
intinfin
120591
119890119860(119904)
1199042119889119904 119889120591 119889120585
ge 119872119890minus18 intinfin
2
intinfin
120585
1120591intinfin
120591
11199042119889119904 119889120591 119889120585
= 119872119890minus18 intinfin
2
intinfin
120585
11205912119889120591 119889120585
= 119872119890minus18 intinfin
2
1120585119889120585 = infin
(39)
On the other hand for a sufficiently large 119879 we have
1205751(119905 119879) = int
119905
119879
1119890119860(119904)119886 (119904)
119889119904
= int119905
119879
1119890119860(119904)119904
119889119904 ge 119890minus18 int119905
119879
1119904119889119904 997888rarr infin
(40)
So we can take 119879lowast gt 119879 such that 1205751(119905 119879) gt 1 for 119905 isin [119879lowastinfin)
Taking 120601(119905) = 119905 120593(119905) = 0 in (23) we get that
intinfin
119879
120601 (119904) 119902 (119904) 119890119860(119904) minus
[119903 (119904) 1206011015840 (119904)]2
4 [Γ (1 minus 120572) 120601 (119904) 1205751(119904 119879)] 119903 (119904)
119889119904
ge intinfin
119879
[119872 minus 14Γ (1 minus 120572) 120575
1(119904 119879)
] 1119904119889119904
= int119879lowast
119879
[119872 minus 14Γ (1 minus 120572) 120575
1(119904 119879)
] 1119904119889119904
+ intinfin
119879lowast
[119872 minus 14Γ (1 minus 120572) 120575
1(119904 119879)
] 1119904119889119904
ge int119879lowast
119879
[119872 minus 14Γ (1 minus 120572) 120575
1(119904 119879)
] 1119904119889119904
+ intinfin
119879lowast
[119872 minus 14Γ (1 minus 120572)
] 1119904119889119904 = infin
(41)
provided that119872 gt 1(4Γ(1minus120572)) So (3)ndash(5) and (23) all holdand by Theorem 4 we deduce that every solution of (36) isoscillatory or satisfies lim
119905rarrinfin119883(119905) = 0 under condition119872 gt
14Γ(1 minus 120572)
Mathematical Problems in Engineering 7
Example 9 Consider
(radic119905(119863120572119909 (119905))1015840)1015840
+ 119890minus119905(119863120572119909 (119905))1015840
minus (119872119905minus52 + ln 119905) intinfin
119905
(120585 minus 119905)minus120572119909 (120585) 119889120585 = 0 119905 isin [2infin)
(42)
where 120572 isin (0 1) and119872 gt 0 is a constant
We have in (2) 119886(119905) = radic119905 119901(119905) = 119890minus119905 119902(119905) = 119872119905minus52 +ln 119905 119903(119905) = 1 and 119905
0= 2 Then
1 le 119890119860(119905) = 119890int119905
2(119901(119904)119886(119904))119889119904 = 119890int
119905
2(1119890119904radic119904)119889119904
le 119890int119905
2119904minus(32)119889119904
= 119890minus2[119905minus12minus2minus12] le 119890radic2
(43)
So we have
intinfin
1199050
1119890119860(119904)119886 (119904)
119889119904 = intinfin
2
1119890119860(119904)radic119904
119889119904 ge 119890minusradic2 intinfin
2
1radic119904119889119904 = infin
intinfin
1199050
1119903 (119904)
119889119904 = infin
(44)
Furthermore
intinfin
1199050
1119903 (120585)
intinfin
120585
1119890119860(120591)119886 (120591)
intinfin
120591
119890119860(119904)119902 (119904) 119889119904 119889120591 119889120585
= intinfin
2
intinfin
120585
1119890119860(120591)radic120591
intinfin
120591
119890119860(119904) (119872119904minus52 + ln 119904) 119889119904 119889120591 119889120585
ge 119872119890minusradic2 intinfin
2
intinfin
120585
1radic120591
intinfin
120591
119904minus52119889119904 119889120591 119889120585
= 2119872119890minusradic2
3intinfin
2
[intinfin
120585
11205912119889120591] 119889120585
= 2119872119890minusradic2
3intinfin
2
1120585119889120585 = infin
(45)
On the other hand Taking 120601(119905) = 1199052 120593(119905) = 0 and 119867(119905 119904) =119905 minus 119904 in (29) we get that
lim119905997888rarrinfin
sup 1119905 minus 1199050
int119905
1199050
(119905 minus 119904)
times120601 (119904) 119902 (119904) 119890119860(119904)
minus[119903 (119904) 1206011015840 (119904)]
2
4Γ (1 minus 120572) 120601 (119904) 1205751(119904 119879) 119903 (119904)
119889119904
ge lim119905997888rarrinfin
sup 1119905 minus 2
times int119905
2
(119905 minus 119904) (119872119904minus52 + ln 119904) 1199042
minus 1Γ (1 minus 120572) 120575
1(119904 119879)
119889119904
ge lim119905997888rarrinfin
sup 1119905 minus 2
int119905
2
(119905 minus 119904)119872119904minus12119889119904 = infin
(46)
So (3)ndash(5) and (29) all hold and by Corollary 6 with 119898 = 1we deduce that every solution of (42) is oscillatory or satisfieslim119905rarrinfin
119883(119905) = 0
Acknowledgments
This work is partially supported by Planning Fund projectof the Ministry of Education of China (10YJA630019) Theauthors would thank the reviewers very much for theirvaluable suggestions on this paper
References
[1] A Saadatmandi and M Dehghan ldquoA new operational matrixfor solving fractional-order differential equationsrdquo Computersamp Mathematics with Applications vol 59 no 3 pp 1326ndash13362010
[2] Y Zhou F Jiao and J Li ldquoExistence and uniqueness for p-typefractional neutral differential equationsrdquo Nonlinear AnalysisTheory Methods amp Applications vol 71 no 7-8 pp 2724ndash27332009
[3] L Galeone and R Garrappa ldquoExplicit methods for fractionaldifferential equations and their stability propertiesrdquo Journal ofComputational and Applied Mathematics vol 228 no 2 pp548ndash560 2009
[4] J C Trigeassou N Maamri J Sabatier and A OustaloupldquoA Lyapunov approach to the stability of fractional differentialequationsrdquo Signal Process vol 91 pp 437ndash445 2011
[5] W Deng ldquoSmoothness and stability of the solutions fornonlinear fractional differential equationsrdquo Nonlinear AnalysisTheory Methods amp Applications vol 72 no 3-4 pp 1768ndash17772010
[6] F Ghoreishi and S Yazdani ldquoAn extension of the spectralTau method for numerical solution of multi-order fractionaldifferential equations with convergence analysisrdquo Computers ampMathematics with Applications vol 61 no 1 pp 30ndash43 2011
[7] J T Edwards N J Ford and A C Simpson ldquoThe numericalsolution of linear multi-term fractional differential equationssystems of equationsrdquo Journal of Computational and AppliedMathematics vol 148 no 2 pp 401ndash418 2002
[8] M Muslim ldquoExistence and approximation of solutions tofractional differential equationsrdquo Mathematical and ComputerModelling vol 49 no 5-6 pp 1164ndash1172 2009
[9] D-X Chen ldquoOscillation criteria of fractional differential equa-tionsrdquo Advances in Difference Equations vol 2012 article 332012
[10] I Podlubny Fractional Differential Equations vol 198 Aca-demic Press San Diego Calif USA 1999
8 Mathematical Problems in Engineering
[11] A A Kilbas H M Srivastava and J J Trujillo Theoryand Applications of Fractional Differential Equations vol 204Elsevier Science BV Amsterdam The Netherlands 2006
[12] S Das Functional Fractional Calculus Springer Berlin Ger-many 2nd edition 2011
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Mathematical Problems in Engineering
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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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Decision SciencesAdvances in
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Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Stochastic AnalysisInternational Journal of
4 Mathematical Problems in Engineering
= minus120601 (119905) 119902 (119905) 119890119860(119905)
+120601 (119905)1198831015840 (119905) 119886 (119905) 119890119860(119905)[119903 (119905)119863120572119909 (119905)]1015840
1198832 (119905)
+1206011015840 (119905)120601 (119905)
120596 (119905) + 120601 (119905) 1205931015840 (119905)
(25)
By Lemma 3 and the definition of 119891 we get that
1205961015840 (119905) le minus120601 (119905) 119902 (119905) 119890119860(119905)
minus (120601 (119905) Γ (1 minus 120572) 1205751(119905 1199052) 119890119860(119905)119886 (119905)
times (119903 (119905)119863120572119909 (119905))1015840119886 (119905) 119890119860(119905)(119903 (119905)119863120572119909 (119905))1015840)
times (119903 (119905)1198832 (119905))minus1
+1206011015840 (119905)120601 (119905)
120596 (119905) + 120601 (119905) 1205931015840 (119905)
= minus120601 (119905) 119902 (119905) 119890119860(119905) minus120601 (119905) Γ (1 minus 120572) 120575
1(119905 1199052)
119903 (119905)
times [120596 (119905)120601 (119905)
minus 120593 (119905)]2
+1206011015840 (119905)120601 (119905)
120596 (119905) + 120601 (119905) 1205931015840 (119905)
= minus120601 (119905) 119902 (119905) 119890119860(119905) + 120601 (119905) 1205931015840 (119905)
minus120601 (119905) Γ (1 minus 120572) 120575
1(119905 1199052) 1205932 (119905)
119903 (119905)
minus120601 (119905) Γ (1 minus 120572) 120575
1(119905 1199052)
119903 (119905)1205962 (119905)1206012 (119905)
+2120593 (119905) 120601 (119905) Γ (1 minus 120572) 120575
1(119905 1199052) + 119903 (119905) 1206011015840 (119905)
119903 (119905) 120601 (119905)120596 (119905)
le minus120601 (119905) 119902 (119905) 119890119860(119905) + 120601 (119905) 1205931015840 (119905)
minus120601 (119905) Γ (1 minus 120572) 120575
1(119905 1199052) 1205932 (119905)
119903 (119905)
+[2120593 (119905) 120601 (119905) Γ (1 minus 120572) 120575
1(119905 1199052) + 119903 (119905) 1206011015840 (119905)]
2
4Γ (1 minus 120572) 120601 (119905) 1205751(119905 1199052) 119903 (119905)
(26)
Substituting 119905 with 119904 in (26) an integration for (26) withrespect to 119904 from 119905
2to 119905 yields
int119905
1199052
120601 (119904) 119902 (119904) 119890119860(119904) minus 120601 (119904) 1205931015840 (119904)
+120601 (119904) Γ (1 minus 120572) 120575
1(119904 1199052) 1205932 (119904)
119903 (119904)
minus [2120593 (119904) 120601 (119904) Γ (1 minus 120572) 1205751(119904 1199052) + 119903 (119904) 1206011015840 (119904)]
2
times(4Γ (1 minus 120572) 120601 (119904) 1205751(119904 1199052) 119903 (119904))minus1 119889119904
le 120596 (1199052) minus 120596 (119905) le 120596 (119905
2) lt infin
(27)which contradicts (23) So the proof is complete
Theorem 5 Define D = (119905 119904) | 119905 ge 119904 ge 1199050 Assume (3)ndash(5)
hold and there exists a function119867 isin 1198621(DR) such that119867(119905 119905) = 0 119891119900119903 119905 ge 119905
0 119867 (119905 119904) gt 0 119891119900119903 119905 gt 119904 ge 119905
0
(28)
and119867 has a nonpositive continuous partial derivative1198671015840119904(119905 119904)
and
lim119905997888rarrinfin
sup 1119867 (119905 119905
0)
times int119905
1199050
119867(119905 119904) 120601 (119904) 119902 (119904) 119890119860(119904) minus 120601 (119904) 1205931015840 (119904)
+120601 (119904) Γ (1 minus 120572) 120575
1(119904 119879) 1205932 (119904)
119903 (119904)
minus [2120593 (119904) 120601 (119904) Γ (1 minus 120572)
times 1205751(119904 119879) +119903 (119904) 1206011015840 (119904)]
2
times (4Γ (1 minus 120572) 120601 (119904) 1205751
times (119904 119879) 119903 (119904) )minus1 119889119904 = infin
(29)
for all sufficiently large 119879 where 120601 120593 are defined as inTheorem 4 Then every solution of (2) is oscillatory or satisfieslim119905rarrinfin
119883(119905) = 0
Proof Assume (2) has a nonoscillatory solution 119909 on [1199050infin)
Without loss of generality we may assume 119909(119905) gt 0 on[1199051infin) where 119905
1is sufficiently large By Lemma 2 we have
119863120572119909(119905) lt 0 on [1199052infin) for some sufficiently large 119905
2gt 1199051 Let
120596(119905) be defined as in Theorem 4 By (26) we have
120601 (119905) 119902 (119905) 119890119860(119905) + 120601 (119905) 1205931015840 (119905) minus120601 (119905) Γ (1 minus 120572) 120575
1(119905 1199052) 1205932 (119905)
119903 (119905)
+[2120593 (119905) 120601 (119905) Γ (1 minus 120572) 120575
1(119905 1199052) + 119903 (119905) 1206011015840 (119905)]
2
4Γ (1 minus 120572) 120601 (119905) 1205751(119905 1199052) 119903 (119905)
le minus1205961015840 (119905)
(30)Substituting 119905with 119904 in (30) multiplying both sides by119867(119905 119904)and then integrating it with respect to 119904 from 119905
2to 119905 yield
int119905
1199052
119867(119905 119904)
times120601 (119904) 119902 (119904) 119890119860(119904) minus 120601 (119904) 1205931015840 (119904)
+120601 (119904) Γ (1 minus 120572) 120575
1(119904 1199052) 1205932 (119904)
119903 (119904)
Mathematical Problems in Engineering 5
minus[2120593 (119904) 120601 (119904) Γ (1 minus 120572) 120575
1(119904 1199052) + 119903 (119904) 1206011015840 (119904)]
2
4Γ (1 minus 120572) 120601 (119904) 1205751(119904 1199052) 119903 (119904)
119889119904
le minusint119905
1199052
119867(119905 119904) 1205961015840 (119904) 119889119904
= 119867 (119905 1199052) 120596 (1199052) + int119905
1199052
1198671015840119904(119905 119904) 120596 (119904) Δ119904
le 119867 (119905 1199052) 120596 (1199052) le 119867 (119905 119905
0) 120596 (1199052)
(31)
Then
int119905
1199050
119867(119905 119904)
times 120601 (119904) 119902 (119904) 119890119860(119904) minus 120601 (119904) 1205931015840 (119904)
+120601 (119904) Γ (1 minus 120572) 120575
1(119904 1199052) 1205932 (119904)
119903 (119904)
minus [2120593 (119904) 120601 (119904) Γ (1 minus 120572)
times1205751(119904 1199052) + 119903 (119904) 1206011015840 (119904)]
2
times(4Γ (1 minus 120572) 120601 (119904) 1205751(119904 1199052) 119903 (119904))minus1 119889119904
= int1199052
1199050
119867(119905 119904)
times120601 (119904) 119902 (119904) 119890119860(119904) minus 120601 (119904) 1205931015840 (119904)
+120601 (119904) Γ (1 minus 120572) 120575
1(119904 1199052) 1205932 (119904)
119903 (119904)
minus[2120593 (119904) 120601 (119904) Γ (1 minus 120572) 120575
1(119904 1199052) + 119903 (119904) 1206011015840 (119904)]
2
4Γ (1 minus 120572) 120601 (119904) 1205751(119904 1199052) 119903 (119904)
119889119904
+ int119905
1199052
119867(119905 119904)
times120601 (119904) 119902 (119904) 119890119860(119904) minus 120601 (119904) 1205931015840 (119904)
+120601 (119904) Γ (1 minus 120572) 120575
1(119904 1199052) 1205932 (119904)
119903 (119904)
minus[2120593 (119904) 120601 (119904) Γ (1 minus 120572) 120575
1(119904 1199052) + 119903 (119904) 1206011015840 (119904)]
2
4Γ (1 minus 120572) 120601 (119904) 1205751(119904 1199052) 119903 (119904)
times 119889119904
le 119867 (119905 1199050) 120596 (1199052) + 119867 (119905 119905
0)
times int1199052
1199050
10038161003816100381610038161003816120601 (119904) 119902 (119904) 119890119860(119904) minus 120601 (119904) 1205931015840 (119904)
+ (120601 (119904) Γ (1 minus 120572) 1205751(119904 1199052) 1205932 (119904))
times (119903 (119904))minus1
minus [2120593 (119904) 120601 (119904) Γ (1 minus 120572) 1205751
times (119904 1199052) +119903 (119904) 1206011015840 (119904)]
2
times (4Γ (1 minus 120572) 120601 (119904) 1205751(119904 1199052) 119903 (119904))minus110038161003816100381610038161003816 119889119904
(32)
So
lim119905997888rarrinfin
sup 1119867 (119905 119905
0)
times int119905
1199050
119867(119905 119904) 120601 (119904) 119902 (119904) 119890119860(119904) minus 120601 (119904) 1205931015840 (119904)
+ (120601 (119904) Γ (1 minus 120572) 1205751(119904 1199052) 1205932 (119904))
times (119903 (119904))minus1 minus [2120593 (119904) 120601 (119904) Γ (1 minus 120572)
times 1205751(119904 1199052) + 119903 (119904) 1206011015840 (119904)]
2
times (4Γ (1 minus 120572) 120601 (119904) 1205751(119904 1199052) 119903 (119904))minus1 119889119904
le 120596 (1199052)
+ int1199052
1199050
10038161003816100381610038161003816120601 (119904) 119902 (119904) 119890119860(119904) minus 120601 (119904) 1205931015840 (119904)
+120601 (119904) Γ (1 minus 120572) 120575
1(119904 1199052) 1205932 (119904)
119903 (119904)
minus [2120593 (119904) 120601 (119904) Γ (1 minus 120572) 1205751(119904 1199052) + 119903 (119904) 1206011015840 (119904)]
2
times(4Γ (1 minus 120572) 120601 (119904) 1205751(119904 1199052) 119903 (119904))minus110038161003816100381610038161003816 119889119904 lt infin
(33)
which contradicts (29) So the proof is complete
In Theorems 5 if we take 119867(119905 119904) for some special func-tions such as (119905 minus 119904)119898 or ln (119905119904) then we can obtain somecorollaries as follows
Corollary 6 Assume (3)ndash(5) hold and
lim119905997888rarrinfin
sup 1(119905 minus 1199050)119898
times int119905
1199050
(119905 minus 119904)119898 120601 (119904) 119902 (119904) 119890119860(119904) minus 120601 (119904) 1205931015840 (119904)
+ (120601 (119904) Γ (1 minus 120572) 1205751(119904 119879) 1205932 (119904))
6 Mathematical Problems in Engineering
times (119903 (119904))minus1
minus [2120593 (119904) 120601 (119904) Γ (1 minus 120572) 1205751(119904 119879)
+119903 (119904) 1206011015840 (119904)]2
times (4Γ (1 minus 120572) 120601 (119904) 1205751
times (119904 119879) 119903 (119904) )minus1 119889119904 = infin
(34)
for all sufficiently large 119879 Then every solution of (2) is oscilla-tory or satisfies lim
119905rarrinfin119883(119905) = 0
Corollary 7 Assume (3)ndash(5) hold and
lim119905997888rarrinfin
sup 1(ln 119905 minus ln 119905
0)
times int119905
1199050
(ln 119905 minus ln 119904) 120601 (119904) 119902 (119904) 119890119860(119904) minus 120601 (119904) 1205931015840 (119904)
+120601 (119904) Γ (1 minus 120572) 120575
1(119904 119879) 1205932 (119904)
119903 (119904)
minus [2120593 (119904) 120601 (119904) Γ (1 minus 120572) 1205751(119904 119879)
+119903 (119904) 1206011015840 (119904)]2
times (4Γ (1 minus 120572) 120601 (119904)
times1205751(119904 119879) 119903 (119904))minus1 119889119904 = infin
(35)
for all sufficiently large 119879 Then every solution of (2) is oscilla-tory or satisfies lim
119905rarrinfin119883(119905) = 0
3 Applications
In this section we will present some applications for theabove established results
Example 8 Consider
(119905(119863120572119909 (119905))1015840)1015840
+ 119905minus2(119863120572119909 (119905))1015840
minus119872119905minus2 intinfin
119905
(120585 minus 119905)minus120572119909 (120585) 119889120585 = 0 119905 isin [2infin) (36)
where119872 gt 0 is a quotient of two odd positive integers
We have in (2) 119886(119905) = 119905 119901(119905) = 119905minus2 119902(119905) = 119872119905minus2 119903(119905) =1 and 119905
0= 2 Then
1 le 119890119860(119905) = 119890int119905
2(119901(119904)119886(119904))119889119904 = 119890int
119905
2119904minus3119889119904 = 119890minus(12)[119905
minus2minus2minus2] le 11989018
(37)
Moreover we have
intinfin
1199050
1119890119860(119904)119886 (119904)
119889119904 = intinfin
2
1119890119860(119904)119886 (119904)
119889119904 ge 119890minus18 intinfin
2
1119904119889119904 = infin
intinfin
1199050
1119903 (119904)
119889119904 = infin
(38)
Furthermore
intinfin
1199050
1119903 (120585)
intinfin
120585
1119890119860(120591)119886 (120591)
intinfin
120591
119890119860(119904)119902 (119904) 119889119904 119889120591 119889120585
= 119872intinfin
2
intinfin
120585
1119890119860(120591)120591
intinfin
120591
119890119860(119904)
1199042119889119904 119889120591 119889120585
ge 119872119890minus18 intinfin
2
intinfin
120585
1120591intinfin
120591
11199042119889119904 119889120591 119889120585
= 119872119890minus18 intinfin
2
intinfin
120585
11205912119889120591 119889120585
= 119872119890minus18 intinfin
2
1120585119889120585 = infin
(39)
On the other hand for a sufficiently large 119879 we have
1205751(119905 119879) = int
119905
119879
1119890119860(119904)119886 (119904)
119889119904
= int119905
119879
1119890119860(119904)119904
119889119904 ge 119890minus18 int119905
119879
1119904119889119904 997888rarr infin
(40)
So we can take 119879lowast gt 119879 such that 1205751(119905 119879) gt 1 for 119905 isin [119879lowastinfin)
Taking 120601(119905) = 119905 120593(119905) = 0 in (23) we get that
intinfin
119879
120601 (119904) 119902 (119904) 119890119860(119904) minus
[119903 (119904) 1206011015840 (119904)]2
4 [Γ (1 minus 120572) 120601 (119904) 1205751(119904 119879)] 119903 (119904)
119889119904
ge intinfin
119879
[119872 minus 14Γ (1 minus 120572) 120575
1(119904 119879)
] 1119904119889119904
= int119879lowast
119879
[119872 minus 14Γ (1 minus 120572) 120575
1(119904 119879)
] 1119904119889119904
+ intinfin
119879lowast
[119872 minus 14Γ (1 minus 120572) 120575
1(119904 119879)
] 1119904119889119904
ge int119879lowast
119879
[119872 minus 14Γ (1 minus 120572) 120575
1(119904 119879)
] 1119904119889119904
+ intinfin
119879lowast
[119872 minus 14Γ (1 minus 120572)
] 1119904119889119904 = infin
(41)
provided that119872 gt 1(4Γ(1minus120572)) So (3)ndash(5) and (23) all holdand by Theorem 4 we deduce that every solution of (36) isoscillatory or satisfies lim
119905rarrinfin119883(119905) = 0 under condition119872 gt
14Γ(1 minus 120572)
Mathematical Problems in Engineering 7
Example 9 Consider
(radic119905(119863120572119909 (119905))1015840)1015840
+ 119890minus119905(119863120572119909 (119905))1015840
minus (119872119905minus52 + ln 119905) intinfin
119905
(120585 minus 119905)minus120572119909 (120585) 119889120585 = 0 119905 isin [2infin)
(42)
where 120572 isin (0 1) and119872 gt 0 is a constant
We have in (2) 119886(119905) = radic119905 119901(119905) = 119890minus119905 119902(119905) = 119872119905minus52 +ln 119905 119903(119905) = 1 and 119905
0= 2 Then
1 le 119890119860(119905) = 119890int119905
2(119901(119904)119886(119904))119889119904 = 119890int
119905
2(1119890119904radic119904)119889119904
le 119890int119905
2119904minus(32)119889119904
= 119890minus2[119905minus12minus2minus12] le 119890radic2
(43)
So we have
intinfin
1199050
1119890119860(119904)119886 (119904)
119889119904 = intinfin
2
1119890119860(119904)radic119904
119889119904 ge 119890minusradic2 intinfin
2
1radic119904119889119904 = infin
intinfin
1199050
1119903 (119904)
119889119904 = infin
(44)
Furthermore
intinfin
1199050
1119903 (120585)
intinfin
120585
1119890119860(120591)119886 (120591)
intinfin
120591
119890119860(119904)119902 (119904) 119889119904 119889120591 119889120585
= intinfin
2
intinfin
120585
1119890119860(120591)radic120591
intinfin
120591
119890119860(119904) (119872119904minus52 + ln 119904) 119889119904 119889120591 119889120585
ge 119872119890minusradic2 intinfin
2
intinfin
120585
1radic120591
intinfin
120591
119904minus52119889119904 119889120591 119889120585
= 2119872119890minusradic2
3intinfin
2
[intinfin
120585
11205912119889120591] 119889120585
= 2119872119890minusradic2
3intinfin
2
1120585119889120585 = infin
(45)
On the other hand Taking 120601(119905) = 1199052 120593(119905) = 0 and 119867(119905 119904) =119905 minus 119904 in (29) we get that
lim119905997888rarrinfin
sup 1119905 minus 1199050
int119905
1199050
(119905 minus 119904)
times120601 (119904) 119902 (119904) 119890119860(119904)
minus[119903 (119904) 1206011015840 (119904)]
2
4Γ (1 minus 120572) 120601 (119904) 1205751(119904 119879) 119903 (119904)
119889119904
ge lim119905997888rarrinfin
sup 1119905 minus 2
times int119905
2
(119905 minus 119904) (119872119904minus52 + ln 119904) 1199042
minus 1Γ (1 minus 120572) 120575
1(119904 119879)
119889119904
ge lim119905997888rarrinfin
sup 1119905 minus 2
int119905
2
(119905 minus 119904)119872119904minus12119889119904 = infin
(46)
So (3)ndash(5) and (29) all hold and by Corollary 6 with 119898 = 1we deduce that every solution of (42) is oscillatory or satisfieslim119905rarrinfin
119883(119905) = 0
Acknowledgments
This work is partially supported by Planning Fund projectof the Ministry of Education of China (10YJA630019) Theauthors would thank the reviewers very much for theirvaluable suggestions on this paper
References
[1] A Saadatmandi and M Dehghan ldquoA new operational matrixfor solving fractional-order differential equationsrdquo Computersamp Mathematics with Applications vol 59 no 3 pp 1326ndash13362010
[2] Y Zhou F Jiao and J Li ldquoExistence and uniqueness for p-typefractional neutral differential equationsrdquo Nonlinear AnalysisTheory Methods amp Applications vol 71 no 7-8 pp 2724ndash27332009
[3] L Galeone and R Garrappa ldquoExplicit methods for fractionaldifferential equations and their stability propertiesrdquo Journal ofComputational and Applied Mathematics vol 228 no 2 pp548ndash560 2009
[4] J C Trigeassou N Maamri J Sabatier and A OustaloupldquoA Lyapunov approach to the stability of fractional differentialequationsrdquo Signal Process vol 91 pp 437ndash445 2011
[5] W Deng ldquoSmoothness and stability of the solutions fornonlinear fractional differential equationsrdquo Nonlinear AnalysisTheory Methods amp Applications vol 72 no 3-4 pp 1768ndash17772010
[6] F Ghoreishi and S Yazdani ldquoAn extension of the spectralTau method for numerical solution of multi-order fractionaldifferential equations with convergence analysisrdquo Computers ampMathematics with Applications vol 61 no 1 pp 30ndash43 2011
[7] J T Edwards N J Ford and A C Simpson ldquoThe numericalsolution of linear multi-term fractional differential equationssystems of equationsrdquo Journal of Computational and AppliedMathematics vol 148 no 2 pp 401ndash418 2002
[8] M Muslim ldquoExistence and approximation of solutions tofractional differential equationsrdquo Mathematical and ComputerModelling vol 49 no 5-6 pp 1164ndash1172 2009
[9] D-X Chen ldquoOscillation criteria of fractional differential equa-tionsrdquo Advances in Difference Equations vol 2012 article 332012
[10] I Podlubny Fractional Differential Equations vol 198 Aca-demic Press San Diego Calif USA 1999
8 Mathematical Problems in Engineering
[11] A A Kilbas H M Srivastava and J J Trujillo Theoryand Applications of Fractional Differential Equations vol 204Elsevier Science BV Amsterdam The Netherlands 2006
[12] S Das Functional Fractional Calculus Springer Berlin Ger-many 2nd edition 2011
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
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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
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Stochastic AnalysisInternational Journal of
Mathematical Problems in Engineering 5
minus[2120593 (119904) 120601 (119904) Γ (1 minus 120572) 120575
1(119904 1199052) + 119903 (119904) 1206011015840 (119904)]
2
4Γ (1 minus 120572) 120601 (119904) 1205751(119904 1199052) 119903 (119904)
119889119904
le minusint119905
1199052
119867(119905 119904) 1205961015840 (119904) 119889119904
= 119867 (119905 1199052) 120596 (1199052) + int119905
1199052
1198671015840119904(119905 119904) 120596 (119904) Δ119904
le 119867 (119905 1199052) 120596 (1199052) le 119867 (119905 119905
0) 120596 (1199052)
(31)
Then
int119905
1199050
119867(119905 119904)
times 120601 (119904) 119902 (119904) 119890119860(119904) minus 120601 (119904) 1205931015840 (119904)
+120601 (119904) Γ (1 minus 120572) 120575
1(119904 1199052) 1205932 (119904)
119903 (119904)
minus [2120593 (119904) 120601 (119904) Γ (1 minus 120572)
times1205751(119904 1199052) + 119903 (119904) 1206011015840 (119904)]
2
times(4Γ (1 minus 120572) 120601 (119904) 1205751(119904 1199052) 119903 (119904))minus1 119889119904
= int1199052
1199050
119867(119905 119904)
times120601 (119904) 119902 (119904) 119890119860(119904) minus 120601 (119904) 1205931015840 (119904)
+120601 (119904) Γ (1 minus 120572) 120575
1(119904 1199052) 1205932 (119904)
119903 (119904)
minus[2120593 (119904) 120601 (119904) Γ (1 minus 120572) 120575
1(119904 1199052) + 119903 (119904) 1206011015840 (119904)]
2
4Γ (1 minus 120572) 120601 (119904) 1205751(119904 1199052) 119903 (119904)
119889119904
+ int119905
1199052
119867(119905 119904)
times120601 (119904) 119902 (119904) 119890119860(119904) minus 120601 (119904) 1205931015840 (119904)
+120601 (119904) Γ (1 minus 120572) 120575
1(119904 1199052) 1205932 (119904)
119903 (119904)
minus[2120593 (119904) 120601 (119904) Γ (1 minus 120572) 120575
1(119904 1199052) + 119903 (119904) 1206011015840 (119904)]
2
4Γ (1 minus 120572) 120601 (119904) 1205751(119904 1199052) 119903 (119904)
times 119889119904
le 119867 (119905 1199050) 120596 (1199052) + 119867 (119905 119905
0)
times int1199052
1199050
10038161003816100381610038161003816120601 (119904) 119902 (119904) 119890119860(119904) minus 120601 (119904) 1205931015840 (119904)
+ (120601 (119904) Γ (1 minus 120572) 1205751(119904 1199052) 1205932 (119904))
times (119903 (119904))minus1
minus [2120593 (119904) 120601 (119904) Γ (1 minus 120572) 1205751
times (119904 1199052) +119903 (119904) 1206011015840 (119904)]
2
times (4Γ (1 minus 120572) 120601 (119904) 1205751(119904 1199052) 119903 (119904))minus110038161003816100381610038161003816 119889119904
(32)
So
lim119905997888rarrinfin
sup 1119867 (119905 119905
0)
times int119905
1199050
119867(119905 119904) 120601 (119904) 119902 (119904) 119890119860(119904) minus 120601 (119904) 1205931015840 (119904)
+ (120601 (119904) Γ (1 minus 120572) 1205751(119904 1199052) 1205932 (119904))
times (119903 (119904))minus1 minus [2120593 (119904) 120601 (119904) Γ (1 minus 120572)
times 1205751(119904 1199052) + 119903 (119904) 1206011015840 (119904)]
2
times (4Γ (1 minus 120572) 120601 (119904) 1205751(119904 1199052) 119903 (119904))minus1 119889119904
le 120596 (1199052)
+ int1199052
1199050
10038161003816100381610038161003816120601 (119904) 119902 (119904) 119890119860(119904) minus 120601 (119904) 1205931015840 (119904)
+120601 (119904) Γ (1 minus 120572) 120575
1(119904 1199052) 1205932 (119904)
119903 (119904)
minus [2120593 (119904) 120601 (119904) Γ (1 minus 120572) 1205751(119904 1199052) + 119903 (119904) 1206011015840 (119904)]
2
times(4Γ (1 minus 120572) 120601 (119904) 1205751(119904 1199052) 119903 (119904))minus110038161003816100381610038161003816 119889119904 lt infin
(33)
which contradicts (29) So the proof is complete
In Theorems 5 if we take 119867(119905 119904) for some special func-tions such as (119905 minus 119904)119898 or ln (119905119904) then we can obtain somecorollaries as follows
Corollary 6 Assume (3)ndash(5) hold and
lim119905997888rarrinfin
sup 1(119905 minus 1199050)119898
times int119905
1199050
(119905 minus 119904)119898 120601 (119904) 119902 (119904) 119890119860(119904) minus 120601 (119904) 1205931015840 (119904)
+ (120601 (119904) Γ (1 minus 120572) 1205751(119904 119879) 1205932 (119904))
6 Mathematical Problems in Engineering
times (119903 (119904))minus1
minus [2120593 (119904) 120601 (119904) Γ (1 minus 120572) 1205751(119904 119879)
+119903 (119904) 1206011015840 (119904)]2
times (4Γ (1 minus 120572) 120601 (119904) 1205751
times (119904 119879) 119903 (119904) )minus1 119889119904 = infin
(34)
for all sufficiently large 119879 Then every solution of (2) is oscilla-tory or satisfies lim
119905rarrinfin119883(119905) = 0
Corollary 7 Assume (3)ndash(5) hold and
lim119905997888rarrinfin
sup 1(ln 119905 minus ln 119905
0)
times int119905
1199050
(ln 119905 minus ln 119904) 120601 (119904) 119902 (119904) 119890119860(119904) minus 120601 (119904) 1205931015840 (119904)
+120601 (119904) Γ (1 minus 120572) 120575
1(119904 119879) 1205932 (119904)
119903 (119904)
minus [2120593 (119904) 120601 (119904) Γ (1 minus 120572) 1205751(119904 119879)
+119903 (119904) 1206011015840 (119904)]2
times (4Γ (1 minus 120572) 120601 (119904)
times1205751(119904 119879) 119903 (119904))minus1 119889119904 = infin
(35)
for all sufficiently large 119879 Then every solution of (2) is oscilla-tory or satisfies lim
119905rarrinfin119883(119905) = 0
3 Applications
In this section we will present some applications for theabove established results
Example 8 Consider
(119905(119863120572119909 (119905))1015840)1015840
+ 119905minus2(119863120572119909 (119905))1015840
minus119872119905minus2 intinfin
119905
(120585 minus 119905)minus120572119909 (120585) 119889120585 = 0 119905 isin [2infin) (36)
where119872 gt 0 is a quotient of two odd positive integers
We have in (2) 119886(119905) = 119905 119901(119905) = 119905minus2 119902(119905) = 119872119905minus2 119903(119905) =1 and 119905
0= 2 Then
1 le 119890119860(119905) = 119890int119905
2(119901(119904)119886(119904))119889119904 = 119890int
119905
2119904minus3119889119904 = 119890minus(12)[119905
minus2minus2minus2] le 11989018
(37)
Moreover we have
intinfin
1199050
1119890119860(119904)119886 (119904)
119889119904 = intinfin
2
1119890119860(119904)119886 (119904)
119889119904 ge 119890minus18 intinfin
2
1119904119889119904 = infin
intinfin
1199050
1119903 (119904)
119889119904 = infin
(38)
Furthermore
intinfin
1199050
1119903 (120585)
intinfin
120585
1119890119860(120591)119886 (120591)
intinfin
120591
119890119860(119904)119902 (119904) 119889119904 119889120591 119889120585
= 119872intinfin
2
intinfin
120585
1119890119860(120591)120591
intinfin
120591
119890119860(119904)
1199042119889119904 119889120591 119889120585
ge 119872119890minus18 intinfin
2
intinfin
120585
1120591intinfin
120591
11199042119889119904 119889120591 119889120585
= 119872119890minus18 intinfin
2
intinfin
120585
11205912119889120591 119889120585
= 119872119890minus18 intinfin
2
1120585119889120585 = infin
(39)
On the other hand for a sufficiently large 119879 we have
1205751(119905 119879) = int
119905
119879
1119890119860(119904)119886 (119904)
119889119904
= int119905
119879
1119890119860(119904)119904
119889119904 ge 119890minus18 int119905
119879
1119904119889119904 997888rarr infin
(40)
So we can take 119879lowast gt 119879 such that 1205751(119905 119879) gt 1 for 119905 isin [119879lowastinfin)
Taking 120601(119905) = 119905 120593(119905) = 0 in (23) we get that
intinfin
119879
120601 (119904) 119902 (119904) 119890119860(119904) minus
[119903 (119904) 1206011015840 (119904)]2
4 [Γ (1 minus 120572) 120601 (119904) 1205751(119904 119879)] 119903 (119904)
119889119904
ge intinfin
119879
[119872 minus 14Γ (1 minus 120572) 120575
1(119904 119879)
] 1119904119889119904
= int119879lowast
119879
[119872 minus 14Γ (1 minus 120572) 120575
1(119904 119879)
] 1119904119889119904
+ intinfin
119879lowast
[119872 minus 14Γ (1 minus 120572) 120575
1(119904 119879)
] 1119904119889119904
ge int119879lowast
119879
[119872 minus 14Γ (1 minus 120572) 120575
1(119904 119879)
] 1119904119889119904
+ intinfin
119879lowast
[119872 minus 14Γ (1 minus 120572)
] 1119904119889119904 = infin
(41)
provided that119872 gt 1(4Γ(1minus120572)) So (3)ndash(5) and (23) all holdand by Theorem 4 we deduce that every solution of (36) isoscillatory or satisfies lim
119905rarrinfin119883(119905) = 0 under condition119872 gt
14Γ(1 minus 120572)
Mathematical Problems in Engineering 7
Example 9 Consider
(radic119905(119863120572119909 (119905))1015840)1015840
+ 119890minus119905(119863120572119909 (119905))1015840
minus (119872119905minus52 + ln 119905) intinfin
119905
(120585 minus 119905)minus120572119909 (120585) 119889120585 = 0 119905 isin [2infin)
(42)
where 120572 isin (0 1) and119872 gt 0 is a constant
We have in (2) 119886(119905) = radic119905 119901(119905) = 119890minus119905 119902(119905) = 119872119905minus52 +ln 119905 119903(119905) = 1 and 119905
0= 2 Then
1 le 119890119860(119905) = 119890int119905
2(119901(119904)119886(119904))119889119904 = 119890int
119905
2(1119890119904radic119904)119889119904
le 119890int119905
2119904minus(32)119889119904
= 119890minus2[119905minus12minus2minus12] le 119890radic2
(43)
So we have
intinfin
1199050
1119890119860(119904)119886 (119904)
119889119904 = intinfin
2
1119890119860(119904)radic119904
119889119904 ge 119890minusradic2 intinfin
2
1radic119904119889119904 = infin
intinfin
1199050
1119903 (119904)
119889119904 = infin
(44)
Furthermore
intinfin
1199050
1119903 (120585)
intinfin
120585
1119890119860(120591)119886 (120591)
intinfin
120591
119890119860(119904)119902 (119904) 119889119904 119889120591 119889120585
= intinfin
2
intinfin
120585
1119890119860(120591)radic120591
intinfin
120591
119890119860(119904) (119872119904minus52 + ln 119904) 119889119904 119889120591 119889120585
ge 119872119890minusradic2 intinfin
2
intinfin
120585
1radic120591
intinfin
120591
119904minus52119889119904 119889120591 119889120585
= 2119872119890minusradic2
3intinfin
2
[intinfin
120585
11205912119889120591] 119889120585
= 2119872119890minusradic2
3intinfin
2
1120585119889120585 = infin
(45)
On the other hand Taking 120601(119905) = 1199052 120593(119905) = 0 and 119867(119905 119904) =119905 minus 119904 in (29) we get that
lim119905997888rarrinfin
sup 1119905 minus 1199050
int119905
1199050
(119905 minus 119904)
times120601 (119904) 119902 (119904) 119890119860(119904)
minus[119903 (119904) 1206011015840 (119904)]
2
4Γ (1 minus 120572) 120601 (119904) 1205751(119904 119879) 119903 (119904)
119889119904
ge lim119905997888rarrinfin
sup 1119905 minus 2
times int119905
2
(119905 minus 119904) (119872119904minus52 + ln 119904) 1199042
minus 1Γ (1 minus 120572) 120575
1(119904 119879)
119889119904
ge lim119905997888rarrinfin
sup 1119905 minus 2
int119905
2
(119905 minus 119904)119872119904minus12119889119904 = infin
(46)
So (3)ndash(5) and (29) all hold and by Corollary 6 with 119898 = 1we deduce that every solution of (42) is oscillatory or satisfieslim119905rarrinfin
119883(119905) = 0
Acknowledgments
This work is partially supported by Planning Fund projectof the Ministry of Education of China (10YJA630019) Theauthors would thank the reviewers very much for theirvaluable suggestions on this paper
References
[1] A Saadatmandi and M Dehghan ldquoA new operational matrixfor solving fractional-order differential equationsrdquo Computersamp Mathematics with Applications vol 59 no 3 pp 1326ndash13362010
[2] Y Zhou F Jiao and J Li ldquoExistence and uniqueness for p-typefractional neutral differential equationsrdquo Nonlinear AnalysisTheory Methods amp Applications vol 71 no 7-8 pp 2724ndash27332009
[3] L Galeone and R Garrappa ldquoExplicit methods for fractionaldifferential equations and their stability propertiesrdquo Journal ofComputational and Applied Mathematics vol 228 no 2 pp548ndash560 2009
[4] J C Trigeassou N Maamri J Sabatier and A OustaloupldquoA Lyapunov approach to the stability of fractional differentialequationsrdquo Signal Process vol 91 pp 437ndash445 2011
[5] W Deng ldquoSmoothness and stability of the solutions fornonlinear fractional differential equationsrdquo Nonlinear AnalysisTheory Methods amp Applications vol 72 no 3-4 pp 1768ndash17772010
[6] F Ghoreishi and S Yazdani ldquoAn extension of the spectralTau method for numerical solution of multi-order fractionaldifferential equations with convergence analysisrdquo Computers ampMathematics with Applications vol 61 no 1 pp 30ndash43 2011
[7] J T Edwards N J Ford and A C Simpson ldquoThe numericalsolution of linear multi-term fractional differential equationssystems of equationsrdquo Journal of Computational and AppliedMathematics vol 148 no 2 pp 401ndash418 2002
[8] M Muslim ldquoExistence and approximation of solutions tofractional differential equationsrdquo Mathematical and ComputerModelling vol 49 no 5-6 pp 1164ndash1172 2009
[9] D-X Chen ldquoOscillation criteria of fractional differential equa-tionsrdquo Advances in Difference Equations vol 2012 article 332012
[10] I Podlubny Fractional Differential Equations vol 198 Aca-demic Press San Diego Calif USA 1999
8 Mathematical Problems in Engineering
[11] A A Kilbas H M Srivastava and J J Trujillo Theoryand Applications of Fractional Differential Equations vol 204Elsevier Science BV Amsterdam The Netherlands 2006
[12] S Das Functional Fractional Calculus Springer Berlin Ger-many 2nd edition 2011
Submit your manuscripts athttpwwwhindawicom
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
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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
6 Mathematical Problems in Engineering
times (119903 (119904))minus1
minus [2120593 (119904) 120601 (119904) Γ (1 minus 120572) 1205751(119904 119879)
+119903 (119904) 1206011015840 (119904)]2
times (4Γ (1 minus 120572) 120601 (119904) 1205751
times (119904 119879) 119903 (119904) )minus1 119889119904 = infin
(34)
for all sufficiently large 119879 Then every solution of (2) is oscilla-tory or satisfies lim
119905rarrinfin119883(119905) = 0
Corollary 7 Assume (3)ndash(5) hold and
lim119905997888rarrinfin
sup 1(ln 119905 minus ln 119905
0)
times int119905
1199050
(ln 119905 minus ln 119904) 120601 (119904) 119902 (119904) 119890119860(119904) minus 120601 (119904) 1205931015840 (119904)
+120601 (119904) Γ (1 minus 120572) 120575
1(119904 119879) 1205932 (119904)
119903 (119904)
minus [2120593 (119904) 120601 (119904) Γ (1 minus 120572) 1205751(119904 119879)
+119903 (119904) 1206011015840 (119904)]2
times (4Γ (1 minus 120572) 120601 (119904)
times1205751(119904 119879) 119903 (119904))minus1 119889119904 = infin
(35)
for all sufficiently large 119879 Then every solution of (2) is oscilla-tory or satisfies lim
119905rarrinfin119883(119905) = 0
3 Applications
In this section we will present some applications for theabove established results
Example 8 Consider
(119905(119863120572119909 (119905))1015840)1015840
+ 119905minus2(119863120572119909 (119905))1015840
minus119872119905minus2 intinfin
119905
(120585 minus 119905)minus120572119909 (120585) 119889120585 = 0 119905 isin [2infin) (36)
where119872 gt 0 is a quotient of two odd positive integers
We have in (2) 119886(119905) = 119905 119901(119905) = 119905minus2 119902(119905) = 119872119905minus2 119903(119905) =1 and 119905
0= 2 Then
1 le 119890119860(119905) = 119890int119905
2(119901(119904)119886(119904))119889119904 = 119890int
119905
2119904minus3119889119904 = 119890minus(12)[119905
minus2minus2minus2] le 11989018
(37)
Moreover we have
intinfin
1199050
1119890119860(119904)119886 (119904)
119889119904 = intinfin
2
1119890119860(119904)119886 (119904)
119889119904 ge 119890minus18 intinfin
2
1119904119889119904 = infin
intinfin
1199050
1119903 (119904)
119889119904 = infin
(38)
Furthermore
intinfin
1199050
1119903 (120585)
intinfin
120585
1119890119860(120591)119886 (120591)
intinfin
120591
119890119860(119904)119902 (119904) 119889119904 119889120591 119889120585
= 119872intinfin
2
intinfin
120585
1119890119860(120591)120591
intinfin
120591
119890119860(119904)
1199042119889119904 119889120591 119889120585
ge 119872119890minus18 intinfin
2
intinfin
120585
1120591intinfin
120591
11199042119889119904 119889120591 119889120585
= 119872119890minus18 intinfin
2
intinfin
120585
11205912119889120591 119889120585
= 119872119890minus18 intinfin
2
1120585119889120585 = infin
(39)
On the other hand for a sufficiently large 119879 we have
1205751(119905 119879) = int
119905
119879
1119890119860(119904)119886 (119904)
119889119904
= int119905
119879
1119890119860(119904)119904
119889119904 ge 119890minus18 int119905
119879
1119904119889119904 997888rarr infin
(40)
So we can take 119879lowast gt 119879 such that 1205751(119905 119879) gt 1 for 119905 isin [119879lowastinfin)
Taking 120601(119905) = 119905 120593(119905) = 0 in (23) we get that
intinfin
119879
120601 (119904) 119902 (119904) 119890119860(119904) minus
[119903 (119904) 1206011015840 (119904)]2
4 [Γ (1 minus 120572) 120601 (119904) 1205751(119904 119879)] 119903 (119904)
119889119904
ge intinfin
119879
[119872 minus 14Γ (1 minus 120572) 120575
1(119904 119879)
] 1119904119889119904
= int119879lowast
119879
[119872 minus 14Γ (1 minus 120572) 120575
1(119904 119879)
] 1119904119889119904
+ intinfin
119879lowast
[119872 minus 14Γ (1 minus 120572) 120575
1(119904 119879)
] 1119904119889119904
ge int119879lowast
119879
[119872 minus 14Γ (1 minus 120572) 120575
1(119904 119879)
] 1119904119889119904
+ intinfin
119879lowast
[119872 minus 14Γ (1 minus 120572)
] 1119904119889119904 = infin
(41)
provided that119872 gt 1(4Γ(1minus120572)) So (3)ndash(5) and (23) all holdand by Theorem 4 we deduce that every solution of (36) isoscillatory or satisfies lim
119905rarrinfin119883(119905) = 0 under condition119872 gt
14Γ(1 minus 120572)
Mathematical Problems in Engineering 7
Example 9 Consider
(radic119905(119863120572119909 (119905))1015840)1015840
+ 119890minus119905(119863120572119909 (119905))1015840
minus (119872119905minus52 + ln 119905) intinfin
119905
(120585 minus 119905)minus120572119909 (120585) 119889120585 = 0 119905 isin [2infin)
(42)
where 120572 isin (0 1) and119872 gt 0 is a constant
We have in (2) 119886(119905) = radic119905 119901(119905) = 119890minus119905 119902(119905) = 119872119905minus52 +ln 119905 119903(119905) = 1 and 119905
0= 2 Then
1 le 119890119860(119905) = 119890int119905
2(119901(119904)119886(119904))119889119904 = 119890int
119905
2(1119890119904radic119904)119889119904
le 119890int119905
2119904minus(32)119889119904
= 119890minus2[119905minus12minus2minus12] le 119890radic2
(43)
So we have
intinfin
1199050
1119890119860(119904)119886 (119904)
119889119904 = intinfin
2
1119890119860(119904)radic119904
119889119904 ge 119890minusradic2 intinfin
2
1radic119904119889119904 = infin
intinfin
1199050
1119903 (119904)
119889119904 = infin
(44)
Furthermore
intinfin
1199050
1119903 (120585)
intinfin
120585
1119890119860(120591)119886 (120591)
intinfin
120591
119890119860(119904)119902 (119904) 119889119904 119889120591 119889120585
= intinfin
2
intinfin
120585
1119890119860(120591)radic120591
intinfin
120591
119890119860(119904) (119872119904minus52 + ln 119904) 119889119904 119889120591 119889120585
ge 119872119890minusradic2 intinfin
2
intinfin
120585
1radic120591
intinfin
120591
119904minus52119889119904 119889120591 119889120585
= 2119872119890minusradic2
3intinfin
2
[intinfin
120585
11205912119889120591] 119889120585
= 2119872119890minusradic2
3intinfin
2
1120585119889120585 = infin
(45)
On the other hand Taking 120601(119905) = 1199052 120593(119905) = 0 and 119867(119905 119904) =119905 minus 119904 in (29) we get that
lim119905997888rarrinfin
sup 1119905 minus 1199050
int119905
1199050
(119905 minus 119904)
times120601 (119904) 119902 (119904) 119890119860(119904)
minus[119903 (119904) 1206011015840 (119904)]
2
4Γ (1 minus 120572) 120601 (119904) 1205751(119904 119879) 119903 (119904)
119889119904
ge lim119905997888rarrinfin
sup 1119905 minus 2
times int119905
2
(119905 minus 119904) (119872119904minus52 + ln 119904) 1199042
minus 1Γ (1 minus 120572) 120575
1(119904 119879)
119889119904
ge lim119905997888rarrinfin
sup 1119905 minus 2
int119905
2
(119905 minus 119904)119872119904minus12119889119904 = infin
(46)
So (3)ndash(5) and (29) all hold and by Corollary 6 with 119898 = 1we deduce that every solution of (42) is oscillatory or satisfieslim119905rarrinfin
119883(119905) = 0
Acknowledgments
This work is partially supported by Planning Fund projectof the Ministry of Education of China (10YJA630019) Theauthors would thank the reviewers very much for theirvaluable suggestions on this paper
References
[1] A Saadatmandi and M Dehghan ldquoA new operational matrixfor solving fractional-order differential equationsrdquo Computersamp Mathematics with Applications vol 59 no 3 pp 1326ndash13362010
[2] Y Zhou F Jiao and J Li ldquoExistence and uniqueness for p-typefractional neutral differential equationsrdquo Nonlinear AnalysisTheory Methods amp Applications vol 71 no 7-8 pp 2724ndash27332009
[3] L Galeone and R Garrappa ldquoExplicit methods for fractionaldifferential equations and their stability propertiesrdquo Journal ofComputational and Applied Mathematics vol 228 no 2 pp548ndash560 2009
[4] J C Trigeassou N Maamri J Sabatier and A OustaloupldquoA Lyapunov approach to the stability of fractional differentialequationsrdquo Signal Process vol 91 pp 437ndash445 2011
[5] W Deng ldquoSmoothness and stability of the solutions fornonlinear fractional differential equationsrdquo Nonlinear AnalysisTheory Methods amp Applications vol 72 no 3-4 pp 1768ndash17772010
[6] F Ghoreishi and S Yazdani ldquoAn extension of the spectralTau method for numerical solution of multi-order fractionaldifferential equations with convergence analysisrdquo Computers ampMathematics with Applications vol 61 no 1 pp 30ndash43 2011
[7] J T Edwards N J Ford and A C Simpson ldquoThe numericalsolution of linear multi-term fractional differential equationssystems of equationsrdquo Journal of Computational and AppliedMathematics vol 148 no 2 pp 401ndash418 2002
[8] M Muslim ldquoExistence and approximation of solutions tofractional differential equationsrdquo Mathematical and ComputerModelling vol 49 no 5-6 pp 1164ndash1172 2009
[9] D-X Chen ldquoOscillation criteria of fractional differential equa-tionsrdquo Advances in Difference Equations vol 2012 article 332012
[10] I Podlubny Fractional Differential Equations vol 198 Aca-demic Press San Diego Calif USA 1999
8 Mathematical Problems in Engineering
[11] A A Kilbas H M Srivastava and J J Trujillo Theoryand Applications of Fractional Differential Equations vol 204Elsevier Science BV Amsterdam The Netherlands 2006
[12] S Das Functional Fractional Calculus Springer Berlin Ger-many 2nd edition 2011
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
Mathematical Problems in Engineering 7
Example 9 Consider
(radic119905(119863120572119909 (119905))1015840)1015840
+ 119890minus119905(119863120572119909 (119905))1015840
minus (119872119905minus52 + ln 119905) intinfin
119905
(120585 minus 119905)minus120572119909 (120585) 119889120585 = 0 119905 isin [2infin)
(42)
where 120572 isin (0 1) and119872 gt 0 is a constant
We have in (2) 119886(119905) = radic119905 119901(119905) = 119890minus119905 119902(119905) = 119872119905minus52 +ln 119905 119903(119905) = 1 and 119905
0= 2 Then
1 le 119890119860(119905) = 119890int119905
2(119901(119904)119886(119904))119889119904 = 119890int
119905
2(1119890119904radic119904)119889119904
le 119890int119905
2119904minus(32)119889119904
= 119890minus2[119905minus12minus2minus12] le 119890radic2
(43)
So we have
intinfin
1199050
1119890119860(119904)119886 (119904)
119889119904 = intinfin
2
1119890119860(119904)radic119904
119889119904 ge 119890minusradic2 intinfin
2
1radic119904119889119904 = infin
intinfin
1199050
1119903 (119904)
119889119904 = infin
(44)
Furthermore
intinfin
1199050
1119903 (120585)
intinfin
120585
1119890119860(120591)119886 (120591)
intinfin
120591
119890119860(119904)119902 (119904) 119889119904 119889120591 119889120585
= intinfin
2
intinfin
120585
1119890119860(120591)radic120591
intinfin
120591
119890119860(119904) (119872119904minus52 + ln 119904) 119889119904 119889120591 119889120585
ge 119872119890minusradic2 intinfin
2
intinfin
120585
1radic120591
intinfin
120591
119904minus52119889119904 119889120591 119889120585
= 2119872119890minusradic2
3intinfin
2
[intinfin
120585
11205912119889120591] 119889120585
= 2119872119890minusradic2
3intinfin
2
1120585119889120585 = infin
(45)
On the other hand Taking 120601(119905) = 1199052 120593(119905) = 0 and 119867(119905 119904) =119905 minus 119904 in (29) we get that
lim119905997888rarrinfin
sup 1119905 minus 1199050
int119905
1199050
(119905 minus 119904)
times120601 (119904) 119902 (119904) 119890119860(119904)
minus[119903 (119904) 1206011015840 (119904)]
2
4Γ (1 minus 120572) 120601 (119904) 1205751(119904 119879) 119903 (119904)
119889119904
ge lim119905997888rarrinfin
sup 1119905 minus 2
times int119905
2
(119905 minus 119904) (119872119904minus52 + ln 119904) 1199042
minus 1Γ (1 minus 120572) 120575
1(119904 119879)
119889119904
ge lim119905997888rarrinfin
sup 1119905 minus 2
int119905
2
(119905 minus 119904)119872119904minus12119889119904 = infin
(46)
So (3)ndash(5) and (29) all hold and by Corollary 6 with 119898 = 1we deduce that every solution of (42) is oscillatory or satisfieslim119905rarrinfin
119883(119905) = 0
Acknowledgments
This work is partially supported by Planning Fund projectof the Ministry of Education of China (10YJA630019) Theauthors would thank the reviewers very much for theirvaluable suggestions on this paper
References
[1] A Saadatmandi and M Dehghan ldquoA new operational matrixfor solving fractional-order differential equationsrdquo Computersamp Mathematics with Applications vol 59 no 3 pp 1326ndash13362010
[2] Y Zhou F Jiao and J Li ldquoExistence and uniqueness for p-typefractional neutral differential equationsrdquo Nonlinear AnalysisTheory Methods amp Applications vol 71 no 7-8 pp 2724ndash27332009
[3] L Galeone and R Garrappa ldquoExplicit methods for fractionaldifferential equations and their stability propertiesrdquo Journal ofComputational and Applied Mathematics vol 228 no 2 pp548ndash560 2009
[4] J C Trigeassou N Maamri J Sabatier and A OustaloupldquoA Lyapunov approach to the stability of fractional differentialequationsrdquo Signal Process vol 91 pp 437ndash445 2011
[5] W Deng ldquoSmoothness and stability of the solutions fornonlinear fractional differential equationsrdquo Nonlinear AnalysisTheory Methods amp Applications vol 72 no 3-4 pp 1768ndash17772010
[6] F Ghoreishi and S Yazdani ldquoAn extension of the spectralTau method for numerical solution of multi-order fractionaldifferential equations with convergence analysisrdquo Computers ampMathematics with Applications vol 61 no 1 pp 30ndash43 2011
[7] J T Edwards N J Ford and A C Simpson ldquoThe numericalsolution of linear multi-term fractional differential equationssystems of equationsrdquo Journal of Computational and AppliedMathematics vol 148 no 2 pp 401ndash418 2002
[8] M Muslim ldquoExistence and approximation of solutions tofractional differential equationsrdquo Mathematical and ComputerModelling vol 49 no 5-6 pp 1164ndash1172 2009
[9] D-X Chen ldquoOscillation criteria of fractional differential equa-tionsrdquo Advances in Difference Equations vol 2012 article 332012
[10] I Podlubny Fractional Differential Equations vol 198 Aca-demic Press San Diego Calif USA 1999
8 Mathematical Problems in Engineering
[11] A A Kilbas H M Srivastava and J J Trujillo Theoryand Applications of Fractional Differential Equations vol 204Elsevier Science BV Amsterdam The Netherlands 2006
[12] S Das Functional Fractional Calculus Springer Berlin Ger-many 2nd edition 2011
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
8 Mathematical Problems in Engineering
[11] A A Kilbas H M Srivastava and J J Trujillo Theoryand Applications of Fractional Differential Equations vol 204Elsevier Science BV Amsterdam The Netherlands 2006
[12] S Das Functional Fractional Calculus Springer Berlin Ger-many 2nd edition 2011
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
