Advances in Difference Equations Volume 2012 Issue 1 2012 [Doi 10.1186_1687-1847-2012-175] Zhi Liu,...

Liu and Sun Advances in Dierence Equations 2012, 2012:175http://www.advancesindifferenceequations.com/content/2012/1/175

RESEARCH Open Access

Oscillation of impulsive functional differentialequations with oscillatory potentials andRiemann-Stieltjes integralsZhi Liu and Yuangong Sun*

*Correspondence:[email protected] of Mathematical Sciences,University of Jinan, Jinan, Shandong250022, P.R. China

AbstractThis paper addresses the oscillation problem of a class of impulsive dierentialequations with delays and Riemann-Stieltjes integrals that cover many equations inthe literature. In the case of oscillatory potentials, both El-Sayed type and Kamenevtype oscillation criteria are established by overcoming the diculty caused byimpulses and oscillatory potentials in the estimation of the delayed argument. Themain results not only generalize some existing results but also drop a restrictivecondition imposed on impulse constants. Finally, two examples are presented toillustrate the theoretical results.MSC: 34K11

Keywords: oscillation; impulse; delay; Riemann-Stieltjes integral

1 IntroductionRecent years have witnessed a rapid progress in the theory of impulsive dierential equa-tions which provide a natural description of the motion of several real world processessubject to short time perturbations. Due to many applications in physics, chemistry, pop-ulation dynamics, ecology, biological systems, control theory, etc. [], the theory of im-pulsive dierential equations has been extensively studied in [].We are here concerned with the oscillation problem of impulsive functional dierential

equations. Compared to equations without impulses, the oscillation of impulsive dieren-tial equations receives less attention []. In this paper, we investigate the oscillationof the following impulsive dierential equation with delay and Riemann-Stieltjes integral:

(r(t)x

(t)) + q(t)x(t) + h p(t, s)|x( (t, s))|(s) sgnx( (t, s))d (s) = e(t), t = tk ,

x(t+k ) = ckx(tk ), x(t+k ) = dkx(tk ),()

where t t, < h , q, e C[t,),and p C([t,) [,h]); the time delay (t, s) : [t,) [,h] [ ,) with t iscontinuous, (t, s) t and limt+ (t, s) = + for s [,h]. 2012 Liu and Sun; licensee Springer. This is an Open Access article distributed under the terms of the Creative Commons Attribu-tion License (http://creativecommons.org/licenses/by/2.0), which permits unrestricted use, distribution, and reproduction in anymedium, provided the original work is properly cited.

Liu and Sun Advances in Dierence Equations 2012, 2012:175 Page 2 of 14http://www.advancesindifferenceequations.com/content/2012/1/175

With the choice of (s), (s) and (t, s), we see that Eq. () reduces to many particularforms considered in the literature. Throughout this paper, we denote

x(tk)= lim

ttkx(t), x

(tk)= lim

tx(tk +t) x(tk)

t .

Let J R be an interval. Dene PC(J ,R) = {z : J R : z(t) is continuous everywhereexcept some tk at which z(t+k ) and z(tk ) exist and z(tk ) = z(tk)}. By a solution of Eq. (), wemean a function x(t) PC([t,),R) such that x(t),x(t) PC([t,),R), and x(t) satis-es Eq. () for t [t,). A solution of Eq. () is said to be oscillatory if it is dened onsome ray [T ,) with T t, and has arbitrarily large zeros. Otherwise, it is called non-oscillatory. Eq. () is said to be oscillatory if all of its nonconstant solutions dened for alllarge T > are oscillatory.Recently, there have been many papers devoted to the oscillation problem for some par-

ticular cases of Eq. (). When there are no impulses and (t, s) (t), Sun and Kong []established some interval oscillation criteria for the following equation:

(r(t)x(t)

) + q(t)x(t) + h

p(t, s)x( (t))(s) sgnx( (t))d (s) = e(t), ()

which generalize and improve some results for second-order dierential equations withmixed nonlinearities in [, ].For the case of impulse, zbekler and Zafer [, ] studied forced oscillation of a class

of super-half-linear dierential equations. Without considering the inuence of forcedterm, some oscillation criteria were given in []. Liu and Xu [] studied the oscillationof a forced mixed type Emden-Fowler equation (a particular case of Eq. ())

(r(t)x

(t)) + q(t)x(t) +n

i= pi(t)|x(t)|ix(t) = e(t), t = k ,x(+k ) = ckx(k ), x(+k ) = dkx(k ),

()

where {k}k= is the sequence of impulsemoments, > > m > > m+ > > n > , r,q, pi, e are real valued continuous functions on [t,) and r(t) > . We note that a restric-tive condition is imposed on impulse constants that dk ck in []. Some oscillationresults for the second-order forced mixed nonlinear impulsive dierential equation werealso established in [, ].Zafer [] investigated a class of second-order sublinear delay impulses dierential

equation

x(t) + p(t)x(t) + p(t)|x( (t))|x( (t)) = , t = k ,x(t)|t=k + qk|x( (k))|x( (k)) = ,x(t)|t=k = ,

()

where {k}k= is the sequence of impulse moments and < < . The author establishedoscillation criteria in two cases of (t) t and (t) = t. We see that x(t) is additionallyassumed to be continuous in [].


In our recent paper [], we further extend the main results in [] to Eq. () with (t, s) = t. However, the oscillation problem of Eq. () remains untouched in [] for thecase of oscillatory potentials.Generally speaking, some ideas to oscillation of dierential equations without impulses

can also be applied to impulsive dierential equations. For example, the idea to intervaloscillation in [] and the idea of dealing with mixed nonlinearities in []. However,when the potentials q, p and e are allowed to change signs, it is dicult to deal with thedelayed argument x( (t, s)) for Eq. () as that for dierential equations without impulses in[]. In this paper, we will overcome diculties caused by oscillatory potentials, delayedargument and impulses, and establish both El-Sayed type and Kamenev type interval os-cillation criteria for Eq. ().The main contribution of this paper is threefold. First, in the case of oscillatory poten-

tials, we present an estimation on x( (t, s))/x(t) in a bounded interval, which plays a keyrole in the proof of the main results. Second, the redundant restriction on impulse con-stants ck and dk that dk ck is removed by introducing particular El-Sayed type functionsin [] and using Kongs technique in [] many times based on the number of impulsemoments in a bounded interval. Finally, both impulse, delay andRiemann-Stieltjes integralare taken into consideration in this paper. Therefore, most of mixed type Emden-Fowlerequations considered in the literature are included as special cases.The remainder of this paper is organized as follows. In Section , some important lem-

mas are given. Interval oscillation criteria of the El-Sayed type and the Kong type are es-tablished in Section . Finally, two examples are given in Section .

2 PreliminariesThroughout this paper, we suppose that there are limited impulse moments in anybounded time interval. For the sake of convenience, we introduce the following notations.Denote

(a) =min{k|tk > a, tk > t}, (b) =max{k|t < tk < b},

where b > a > t are constants. It is easy to see that t(a), t(a)+, t(a)+, . . . , t(b) are all im-pulse moments in the interval (a,b).The following two lemmas are crucial in the proof of our main results.

Lemma . For given constants b > a t, assume that x(t),x(t),x(t) PC((a,b),R),x(t) > (< ) and (r(t)x(t)) ( ) for t (a,b). Then we have

x(t)x(t)

r(t)R(a, t) , t (a,b), ()

where

R(a, t) =[ t

t(t)

dsr(s) +

d(t)

t(t)t(t)

dsr(s) + +

((t)

k=(a)

dk

) t(a)a

dsr(s)

].

Proof It is sucient to prove one of the cases when x(t) > on (a,b) and (r(t)x(t)) for t (a,b). The other case can be proved similarly. By Eq. () and ck , we have that for


t (a,b),

x(t) x(t) x(t+(t)) + x(t+(t)) x(t(t)) + x(t(t)) x(t+(t)

)+ + x(t(a)) x(a)

= tt(t)

x(s)ds + (c(t) )x(t(t)) + t(t)t(t)

x(s)ds + (c(t) )x(t(t))

+ + t(a)+t(a)

x(s)ds + (c(a) )x(t(a)) + t(a)a

x(s)ds

tt(t)

x(s)ds + t(t)t(t)

x(s)ds + + t(a)+t(a)

x(s)ds + t(a)a

x(s)ds. ()

Since (r(t)x(t)) for t (a,b) and t = tk , and x(tk) = x(tk ) = dk x(t+k ), we get from ()that

x(t) tt(t)

r(s) r(s)x

(s)ds + t(t)t(t)

r(s) r(s)x

(s)ds + + t(a)a

r(s) r(s)x

(s)ds

r(t)x(t) tt(t)

r(s) ds + r(t(t))x

(t(t)) t(t)t(t)

r(s) ds

+ + r(t(a))x(t(a)

) t(a)a

r(s) ds

r(t)x(t) tt(t)

r(s) ds +

r(t)x(t)d(t)

t(t)t(t)

r(s) ds

+ + r(t)x(t)

d(t)d(t) d(a) t(a)a

r(s) ds

= r(t)x(t)R(a, t),

i.e.,

x(t)x(t)

r(t)R(a, t) , t (a,b).

This completes the proof of Lemma ..

Remark We see that R(a, t) is a piecewise continuous function on (a,b). When there isno impulse moment on (a, t), R(a, t) =

ta

r(s) ds. When there is only one impulse moment

tk on (a,b), we have

R(a, t) =

ta

dsr(s) , t (a, tk], t

tkdsr(s) +

dk

tka

dsr(s) , t (tk ,b).

Lemma . For given constants b > a t, assume that x(t),x(t),x(t) PC((a,b),R),x(t) > (< ) and (r(t)x(t)) ( ) for t (ab,b), where ab = min{ (t, s) : (t, s)


(a,b) [,h]}. Then we have

x( (t, s))x(t)

[exp

( t (t,s)

r(u)R(ab,u)

du)] (t)

k=( (t,s))

ck,

(t, s) (ab,b) [,h], ()

where R(ab, t) is dened as in Lemma ..

Proof It is sucient to prove the case when x(t) > and (r(t)x(t)) for t (ab,b). ByLemma ., we obtain

x(t)x(t)

r(t)R(ab, t)

, t (ab,b). ()

Noting that

lnx(t+k) lnx

(tk)= ln x(t

+k )

x(tk )= ln ck ,

and t( (t,s)), t( (t,s))+, . . . , t(t) ( (t, s), t) are all impulse moments, we have that

lnx(t)

x( (t, s)) =(lnx(t) lnx

(t+(t)

))+(lnx

(t+(t)

) lnx

(t(t)

))+(lnx

(t(t)

) lnx

(t+(t)

))+ + (lnx(t+( (t,s))) lnx(t( (t,s))))

+(lnx

(t( (t,s))

) lnx

( (t, s)

))

= tt(t)

x(s)x(s) ds +

t(t)t(t)

x(s)x(s) ds + +

t( (t,s)) (t,s)

x(s)x(s) ds +

(t)k=( (t,s))

ln ck ,

i.e.,

lnx(t)

x( (t, s)) = t (t,s)

x(s)x(s) ds +

(t)k=( (t,s))

ln ck .

Integrating () from (t, s) to t with (t, s) (ab,b) [,h], we get

lnx(t)

x( (t, s)) t (t,s)

r(u)R(ab,u)

du +(t)

k=( (t,s))ln ck .

Therefore,

x(t)x( (t, s)) exp

( t (t,s)

r(u)R(ab,u)

du) (t)

k=( (t,s))ck ,

which implies (). The proof of Lemma . is complete.


Remark For the sake of convenience, we denote

m(t, s;a,b)[exp

( t (t,s;a,b)

r(u)R(ab,u)

du)] (t)

k=( (t,s))

ck. ()

It is easy to see that m(t, s;a,b) is a piecewise continuous function on [a,b] for given s [,h].

We denote by L (,h) the set of Riemann-Stieltjes integrable functions on [,h] withrespect to . Let c (,h) such that (c) = . We further assume that L (,h) suchthat

cd (s) > ,

hc

d (s) > .

We see that the condition L (,h) is satised if either () > or (s) slowlyas s +, or (s) is constant in a right neighborhood of .The following two lemmas are given in [].

Lemma . Let

l =( h

c(s)d (s)

)( hc

d (s))

,

and

l =( c

(s)d (s)

)( cd (s)

).

Then for any (l, l), there exists L (,b) such that (s) > on [,h],

h(s)(s)d (s) = , ()

and

h(s)d (s) = . ()

Lemma . Let C[,h] and L (,h) satisfying , > on [,h] and h (s)d (s) = . Then

h(s)(s)d (s) exp

( h(s) ln

[(s)

]d (s)

), ()

where we use the convention that ln = and e = .


3 Main resultsLet

(aj,bj) ={ C[aj,bj],(t) ,(aj) = (t(aj)) = = (t(bj)) = (bj) =

},

for j = , .

Theorem . If for any T t, there exist constants aj,bj t for j = , such that T a < b a < b, and

p(t, s) , (t, s) (a,b) [,h] (a,b) [,h];e(t) , t (a,b); e(t) , t (a,b).

For each (l, l), let L (,h) be dened as in Lemma .. Assume further that thereexist (aj,bj) and C((a,b) (a,b), (,)) such that

bjaj

[(t)(t)

(q(t) + (t, s;aj,bj)

)

r(t)(t)((t) +

(t)(t) (t)

)]dt > , j = , ,

where

(t, s;aj,bj) =[ |e(t)|

]exp

( h(s) ln p(t, s)[m(t, s;aj,bj)]

(s)

(s) d (s)), ()

and m(t, s;aj,bj) is dened as (). Then Eq. () is oscillatory.

Proof Assume, for the sake of contradiction, that there exists a solution x(t) of Eq. ()which does not have zero in (a,b) (a,b). Without loss of generality, we may assumethat x(t) > for t (a,b). When x(t) < for t (a,b), the proof follows the same argu-ment by using the interval (a,b) instead of (a,b). Put

u(t) = (t) r(t)x(t)

x(t) , t (a,b). ()

We have

u(t) = (t) r(t)x(t)

x(t) (t)(r(t)x(t))

x(t) + (t)r(t)(x(t))

x(t)

= (t)(t) u(t) + (t)q(t) +

(t)x(t)

( h

p(t, s)[x( (t, s)

)](s) d (s) e(t))

+ u(t)

(t)r(t) . ()


By Lemma ., we have that for t (a,b)

(t)x(t)

( h

p(t, s)[x( (t, s)

)](s) d (s) e(t))

= (t) h

p(t, s)[x( (t, s))

x(t)

](s)x(t)(s) d (s) (t)e(t)x(t)

(t) h

p(t, s)[m(t, s;a,b)

](s)x(t)(s) d (s) (t)e(t)x(t) .Similar to the analysis in the proof of Theorem . in [], we can get from Lemmas .and . that

u(t) (t)(t) u(t) + (t)q(t) + (t) (t, s;a,b) +

u(t)(t)r(t) , t (a,b), t = tk . ()

By the denition of (t), multiplying both sides of () by (t), integrating over (a,b)and using integration by parts, we get

ba

(t)(t)(q(t) + (t, s;a,b)

) r(t)(t)

((t) +

(t)(t) (t)

)dt

+ ba

r(t)(t)

[u(t)(t) + r(t)(t)

((t) +

(t)(t) (t)

)]dt. ()

Noting that the second term of the right-hand side of () is nonnegative, we get

ba

[(t)(t)

(q(t) + (t, s;a,b)

) r(t)(t)

((t) +

(t)(t) (t)

)]dt.

This contradicts the assumption.

Next, we will establish a Kamenev type interval oscillation criterion for Eq. (). First, weintroduce a class of functionsH which will be used in the sequel. Denote D = {(t, t*)|t t* t} andH C(D,R). A functionH is said to belong to the classH if there exist h,h Lloc(D,R) satisfying the following conditions:(A) H(t, t) = , H(t, t*) > for t > t*;(A)

t H(t, t*) = h(t, t*)H(t, t*),

t*H(t, t*) = h(t, t*)H(t, t*).

For two constants , ( < ), we dene two operators P, P by

P( ,) =

H(t, )[(t)

(q(t) + (t, s; ,)

) r(t)(t)

( (t)(t) + h(t, )

)]dt,

P( ,) =

H(, t)[(t)

(q(t) + (t, s; ,)

) r(t)(t)

( (t)(t) + h(, t)

)]dt,

where (t, s; ,) is dened as in Theorem ..Noticing that t(aj), t(aj)+, t(aj)+, . . . , t(bj) are all impulsemoments in the interval (aj,bj)

for j = , , we denote the number of impulsemoments between aj and bj by nj := nj(aj,bj) =(bj) (aj) + for j = , . We also mean

mn=l = ifm < l.


Theorem . Suppose that for any T , there exist nontrivial subintervals (a,b) and(a,b) of [T ,), satisfying the conditions of Theorem .. Further assume that for j = , ,there exist constants j ((bj),bj) and a function H H such that(i) when nj =(bj) (aj) + (j = , ) is an odd number,

are constants, t , tk = k + for k =, , , . . . , r(t) = q(t) = , p(t, s) = sin t, (t, s) = t , (s) =

s and h = . For any T ,

we choose k large enough such that k T and let a = k , a = b = k + andb = k + . Then we have p(t, s) for t (a,b) (a,b). Assume that e(t) C[,)is any function satisfying ()je(t) on [aj,bj] (j = , ). Let (s) = . It is easy to verifythat () and () are valid for = ,

R(ab , t) = tt

r(s) ds =

, t (k , k +

],

and

R(ab , t) = tk+

dsr(s) +

dk

k+ t

dsr(s)

= t k +dk

(k + t

)

= (dk )(t k ) + dk, t

(k + , k +

).

Since

m(t, s;a,b) =[exp

( tt

R(ab ,u)

du)] (t)

k=(t )

ck,

we have

m(t, s;a,b) =

e , t (k , k + ];ck [

(dk)(tk )+(dk)(tk )+ ]

dkdk , t (k + , k + ),

and

(t, s;a,b) = exp(

ln( sin t

[m(t, s;a,b)

] s)ds).


Similarly, for j = , i.e., t (k + , k + ), we havem(t, s;a,b) = e , and

(t, s;a,b) = exp(

ln( sin t

[m(t, s;a,b)

] s)ds)

= sin te .

Choose (t) = and (t) = sint. By Theorem ., we know that Eq. () is oscillatory if

bjaj

(t) (t, s;aj,bj)(t) r(t)(t)

((t) +

(t)(t) (t)

)dt

= bjaj

(t, s;aj,bj) sin t dt > , j = , .

Example . Consider the following equation:x

(t) + x(t) + |x(t )|s sgnx(t )ds = e(t), t = tk ,

x(t+k ) = ckx(tk ), x(t+k ) = dkx(tk ),()

where ck , dk > are constants, t , tk = k (k = , , . . .), r(t) = q(t) = p(t, s) =, (t, s) = t , and (s) = s. For any T > , we choose k large enough such that(k ) T and let a = (k ), a = b = k and b = k. Then we have p(t, s) fort (a,b) (a,b). Assume that e(t) C[,) is any function satisfying ()je(t) on[aj,bj] (j = , ). For any (/, ], set

(s) = s()/().

It is easy to verify that () and () are valid. Let (t) = and H(t, s) = (t s). We havethat h(t, s) = h(t, s) . For j = , we have

R(ab , t) =

, t ((k ), k ];(dk)(tk+)+

dk , t (k ,k ),

m(t, s;a,b) =

e , t ((k ), k ];ck [

(dk)(tk+)+(dk)(tk+)+ ]

dkdk , t (k ,k ),

and

(t, s;a,b) =[ |e(t)|

]exp

( (s) ln [m(t, s;a,b)]

s

(s) ds).

For t ((k ), k ), we obtain

H(k ,(k ))P((k ), k

)

= k(k)

H(t, (k )

)[(t)

(q(t) + (t, s;a,b)

)


r(t)(t)( (t)(t) + h

(t, (k )

))]dt

= k(k)

[t (k )

][ + (t, s;a,b)

]dt,

and

ckdk

H(k ,k )P(k ,k )

= ckdk

kk

H(k , t)[(t)

(q(t) + (t, s;a,b)

)

r(t)(t)( (t)(t) + h(k , t)

)]dt

= ckdk

kk

(k t)[ + (t, s;a,b)

]dt.

Similarly, for t (k ,k), we have thatm(t, s;a,b) = e , and

(t, s;a,b) =[ |e(t)|

]exp

( (s) ln [m(t, s;a,b)]

s

(s) ds)

=[ |e(t)|

]exp

( (s)

(s + ln(s)

)ds).

We choose = k , so we have

H( , k )P(k , ) =

kk

[t (k )

][ + (t, s;a,b) ]dt,

and

H(k, )P( , k) =

kk

(k t)[ + (t, s;a,b)

]dt.

From Theorem . we know that Eq. () is oscillatory if

k(k)

[t (k )

][ + (t, s;a,b)

]dt

+ ckdk

kk

(k t)[ + (t, s;a,b)

]dt > ,

and kk

[t (k )

][ + (t, s;a,b)

]dt +

kk

(k t)[ + (t, s;a,b)

]dt > .

Competing interestsThe authors declare that they have no competing interests.


Authors contributionsAll authors carried out the proof. All authors conceived of the study and participated in its design and coordination. Allauthors read and approved the nal manuscript.

AcknowledgementsThis work was supported by the National Natural Science Foundation of China (Grant No. 61174217) and the NaturalScience Foundation of Shandong Province (Grant Nos. ZR2010AL002 and JQ201119).

Received: 16 July 2012 Accepted: 20 September 2012 Published: 5 October 2012

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doi:10.1186/1687-1847-2012-175Cite this article as: Liu and Sun: Oscillation of impulsive functional differential equations with oscillatory potentialsand Riemann-Stieltjes integrals. Advances in Dierence Equations 2012 2012:175.

Oscillation of impulsive functional differential equations with oscillatory potentials and Riemann-Stieltjes integralsAbstractMSCKeywords

IntroductionPreliminariesMain resultsExamplesCompeting interestsAuthors' contributionsAcknowledgementsReferences

Advances in Difference Equations Volume 2012 Issue 1 2012 [Doi 10.1186_1687-1847-2012-175] Zhi Liu,...

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