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Oscillation of impulsive functional differential equations with oscillatory potentials and Riemann-Stieltjes integrals
Advances in Difference Equations volume 2012, Article number: 175 (2012)
This paper addresses the oscillation problem of a class of impulsive differential equations with delays and Riemann-Stieltjes integrals that cover many equations in the literature. In the case of oscillatory potentials, both El-Sayed type and Kamenev type oscillation criteria are established by overcoming the difficulty caused by impulses and oscillatory potentials in the estimation of the delayed argument. The main results not only generalize some existing results but also drop a restrictive condition imposed on impulse constants. Finally, two examples are presented to illustrate the theoretical results.
Recent years have witnessed a rapid progress in the theory of impulsive differential equations which provide a natural description of the motion of several real world processes subject to short time perturbations. Due to many applications in physics, chemistry, population dynamics, ecology, biological systems, control theory, etc. [1–6], the theory of impulsive differential equations has been extensively studied in [7–9].
We are here concerned with the oscillation problem of impulsive functional differential equations. Compared to equations without impulses, the oscillation of impulsive differential equations receives less attention [10–20]. In this paper, we investigate the oscillation of the following impulsive differential equation with delay and Riemann-Stieltjes integral:
where , ; denotes the Riemann-Stieltjes integral of the function f on with respect to ξ, and is nondecreasing; is a strictly increasing continuous function on satisfying ; denotes the sequence of impulse moments satisfying and ; , are positive impulse constants, and ; with , , and ; the time delay with is continuous, and for .
With the choice of , and , we see that Eq. (1) reduces to many particular forms considered in the literature. Throughout this paper, we denote
Let be an interval. Define . By a solution of Eq. (1), we mean a function such that , and satisfies Eq. (1) for . A solution of Eq. (1) is said to be oscillatory if it is defined on some ray with , and has arbitrarily large zeros. Otherwise, it is called non-oscillatory. Eq. (1) is said to be oscillatory if all of its nonconstant solutions defined for all large are oscillatory.
Recently, there have been many papers devoted to the oscillation problem for some particular cases of Eq. (1). When there are no impulses and , Sun and Kong  established some interval oscillation criteria for the following equation:
For the case of impulse, Özbekler and Zafer [11, 12] studied forced oscillation of a class of super-half-linear differential equations. Without considering the influence of forced term, some oscillation criteria were given in . Liu and Xu  studied the oscillation of a forced mixed type Emden-Fowler equation (a particular case of Eq. (2))
where is the sequence of impulse moments, , r, q, , e are real valued continuous functions on and . We note that a restrictive condition is imposed on impulse constants that in [15–17]. Some oscillation results for the second-order forced mixed nonlinear impulsive differential equation were also established in [18, 19].
Zafer  investigated a class of second-order sublinear delay impulses differential equation
where is the sequence of impulse moments and . The author established oscillation criteria in two cases of and . We see that is additionally assumed to be continuous in .
Generally speaking, some ideas to oscillation of differential equations without impulses can also be applied to impulsive differential equations. For example, the idea to interval oscillation in [25–29] and the idea of dealing with mixed nonlinearities in . However, when the potentials q, p and e are allowed to change signs, it is difficult to deal with the delayed argument for Eq. (1) as that for differential equations without impulses in . In this paper, we will overcome difficulties caused by oscillatory potentials, delayed argument and impulses, and establish both El-Sayed type and Kamenev type interval oscillation criteria for Eq. (1).
The main contribution of this paper is threefold. First, in the case of oscillatory potentials, we present an estimation on in a bounded interval, which plays a key role in the proof of the main results. Second, the redundant restriction on impulse constants and that is removed by introducing particular El-Sayed type functions in  and using Kong’s technique in  many times based on the number of impulse moments in a bounded interval. Finally, both impulse, delay and Riemann-Stieltjes integral are taken into consideration in this paper. Therefore, most of mixed type Emden-Fowler equations considered in the literature are included as special cases.
The remainder of this paper is organized as follows. In Section 2, some important lemmas are given. Interval oscillation criteria of the El-Sayed type and the Kong type are established in Section 3. Finally, two examples are given in Section 4.
Throughout this paper, we suppose that there are limited impulse moments in any bounded time interval. For the sake of convenience, we introduce the following notations. Denote
where are constants. It is easy to see that are all impulse moments in the interval .
The following two lemmas are crucial in the proof of our main results.
Lemma 2.1 For given constants , assume that , (<0) and (≥0) for . Then we have
Proof It is sufficient to prove one of the cases when on and for . The other case can be proved similarly. By Eq. (1) and , we have that for ,
Since for and , and , we get from (6) that
This completes the proof of Lemma 2.1. □
Remark 1 We see that is a piecewise continuous function on . When there is no impulse moment on , . When there is only one impulse moment on , we have
Lemma 2.2 For given constants , assume that , (<0) and (≥0) for , where . Then we have
where is defined as in Lemma 2.1.
Proof It is sufficient to prove the case when and for . By Lemma 2.1, we obtain
and are all impulse moments, we have that
Integrating (8) from to t with , we get
which implies (7). The proof of Lemma 2.2 is complete. □
Remark 2 For the sake of convenience, we denote
It is easy to see that is a piecewise continuous function on for given .
We denote by the set of Riemann-Stieltjes integrable functions on with respect to ξ. Let such that . We further assume that such that
We see that the condition is satisfied if either or ‘slowly’ as , or is constant in a right neighborhood of 0.
The following two lemmas are given in .
Lemma 2.3 Let
Then for any , there exists such that on ,
Lemma 2.4 Let and satisfying , on and . Then
where we use the convention that and .
3 Main results
Theorem 3.1 If for any , there exist constants for such that , and
For each , let be defined as in Lemma 2.3. Assume further that there exist and such that
and is defined as (9). Then Eq. (1) is oscillatory.
Proof Assume, for the sake of contradiction, that there exists a solution of Eq. (1) which does not have zero in . Without loss of generality, we may assume that for . When for , the proof follows the same argument by using the interval instead of . Put
By Lemma 2.2, we have that for
Similar to the analysis in the proof of Theorem 2.1 in , we can get from Lemmas 2.3 and 2.4 that
By the definition of , multiplying both sides of (16) by , integrating over and using integration by parts, we get
Noting that the second term of the right-hand side of (17) is nonnegative, we get
This contradicts the assumption. □
Next, we will establish a Kamenev type interval oscillation criterion for Eq. (1). First, we introduce a class of functions ℋ which will be used in the sequel. Denote and . A function H is said to belong to the class ℋ if there exist satisfying the following conditions:
(A1) , for ;
(A2) , .
For two constants θ, λ (), we define two operators , by
where is defined as in Theorem 3.1.
Noticing that are all impulse moments in the interval for , we denote the number of impulse moments between and by for . We also mean if .
Theorem 3.2 Suppose that for any , there exist nontrivial subintervals and of , satisfying the conditions of Theorem 3.1. Further assume that for , there exist constants and a function such that
when () is an odd number,(18)
when () is an even number,(19)
Then Eq. (1) is oscillatory.
Proof Otherwise, we may assume that for . Proceeding as in the proof of Theorem 3.1, we have that (16) holds for and . Next, we consider the following two cases:
, the number of impulse moments in the interval , is odd;
is an even number.
For the case (i), we first consider the subinterval . Multiplying both sides of (16) by , then integrating it from to , and using integration by parts, we obtain
It implies that
On the other hand, multiplying both sides of (16) by , integrating it from to , and similar to the above analysis, we can get
Dividing (20) and (21) by and , respectively, and adding them, we have
For the remaining intervals, similar to the analysis in Theorem 2.2 in , we have that for ,
It can be concluded similarly for the case (ii) that
We see that (22) and (23) contradict (18) and (19), respectively. The proof is complete. □
In this section, we give two examples to illustrate our main results. To simplify the computation, we focus our attention on the simple case for Eq. (1).
Example 4.1 Consider the following impulsive differential equation:
where β is a positive constant, , are constants, , for , , , , and . For any , we choose k large enough such that and let , and . Then we have for . Assume that is any function satisfying on (). Let . It is easy to verify that (10) and (11) are valid for ,
Similarly, for , i.e., , we have , and
Choose and . By Theorem 3.1, we know that Eq. (24) is oscillatory if
Example 4.2 Consider the following equation:
where , are constants, , (), , , and . For any , we choose k large enough such that and let , and . Then we have for . Assume that is any function satisfying on (). For any , set
It is easy to verify that (10) and (11) are valid. Let and . We have that . For , we have
For , we obtain
Similarly, for , we have that , and
We choose , so we have
From Theorem 3.2 we know that Eq. (25) is oscillatory if
Ballinger G, Liu X: Permanence of population growth models with impulsive effects. Math. Comput. Model. 1997, 26: 59–72.
Lu Z, Chi X, Chen L: Impulsive control strategies in biological control of pesticide. Theor. Popul. Biol. 2003, 64: 39–47. 10.1016/S0040-5809(03)00048-0
Sun J, Qiao F, Wu Q: Impulsive control of a financial model. Phys. Lett. A 2005, 335: 282–288. 10.1016/j.physleta.2004.12.030
Tang S, Chen L: Global attractivity in a food-limited population model with impulsive effect. J. Math. Anal. Appl. 2004, 292: 211–221. 10.1016/j.jmaa.2003.11.061
Tang S, Xiao Y, Clancy D: New modelling approach concerning integrated disease control and cost-effectivity. Nonlinear Anal. 2005, 63: 439–471. 10.1016/j.na.2005.05.029
Zhang Y, Xiu Z, Chen L: Dynamics complexity of a two-prey one-predator system with impulsive effect. Chaos Solitons Fractals 2005, 26: 131–139. 10.1016/j.chaos.2004.12.037
Lakshmikantham V, Bainov DD, Simeonov PS Series in Modern Applied Mathematics 6. In Theory of Impulsive Differential Equations. World Scientific, Singapore; 1989.
Benchohra M, Henderson J, Ntouyas S Contemporary Mathematics and Its Applications 2. In Impulsive Differential Equations and Inclusions. Hindawi Publishing Corporation, New York; 2006.
Bainov D, Covachev V: Impulsive Differential Equations with a Small Parameter. World Scientific, Singapore; 1995.
Agarwa RP, Karakoç F: Survey on oscillation of impulsive delay differential equations. Comput. Math. Appl. 2010, 60: 1648–1685. 10.1016/j.camwa.2010.06.047
Özbekler A, Zafer A: Forced oscillation of super-half-linear impulsive differential equations. Comput. Math. Appl. 2007, 54: 785–792. 10.1016/j.camwa.2007.03.003
Özbekler A, Zafer A: Interval criteria for the forced oscillation of super-half-linear differential equations under impulse effects. Math. Comput. Model. 2009, 50: 59–65. 10.1016/j.mcm.2008.10.020
Luo Z, Shen J: Oscillation of second order linear differential equations with impulses. Appl. Math. Lett. 2007, 20: 75–81. 10.1016/j.aml.2006.01.019
Chen YS, Feng WZ: Oscillation of second order nonlinear ODE with impulses. J. Math. Anal. Appl. 1997, 210: 150–169. 10.1006/jmaa.1997.5378
Liu XX, Xu ZT: Oscillation of a forced super-linear second order differential equation with impulses. Comput. Math. Appl. 2007, 53: 1740–1749. 10.1016/j.camwa.2006.08.040
Liu XX, Xu ZT: Oscillation criteria for a forced mixed typed Emden-Fowler equation with impulses. Comput. Math. Appl. 2009, 215: 283–291. 10.1016/j.amc.2009.04.072
Muthulakshmi V, Thandapani E: Interval criteria for oscillation of second-order impulsive differential equation with mixed nonlinearities. Electron. J. Differ. Equ. 2011, 2011(40):1–14.
Özbekler A, Zafer A: Oscillation of solutions of second order mixed nonlinear differential equations under impulsive perturbations. Comput. Math. Appl. 2011, 61: 933–940. 10.1016/j.camwa.2010.12.041
Guo Z, Zhou X, Ge W: Interval oscillation criteria for second-order forced impulsive differential equations with mixed nonlinearities. J. Math. Anal. Appl. 2011, 381: 187–201. 10.1016/j.jmaa.2011.02.073
Zafer A: Oscillation of second-order sublinear impulsive differential equations. Abstr. Appl. Anal. 2011., 2011: Article ID 458275
Sun YG, Kong QK: Interval criteria for forced oscillation with nonlinearities given by Riemann-Stieltjes integrals. Comput. Math. Appl. 2011, 62: 243–252. 10.1016/j.camwa.2011.04.072
Sun YG, Wong JSW: Oscillation criteria for second order forced ordinary differential equations with mixed nonlinearities. J. Math. Anal. Appl. 2007, 334: 549–560. 10.1016/j.jmaa.2006.07.109
Sun YG, Meng FW: Oscillation of second-order delay differential equations with mixed nonlinearities. Comput. Math. Appl. 2009, 207: 135–139. 10.1016/j.amc.2008.10.016
Liu Z, Sun YG: Interval criteria for oscillation of a forced impulsive differential equation with Riemann-Stieltjes integral. Comput. Math. Appl. 2012, 63: 1577–1586. 10.1016/j.camwa.2012.03.010
Kong Q: Interval criteria for oscillation of second order linear ordinary differential equations. J. Math. Anal. Appl. 1999, 229: 258–270. 10.1006/jmaa.1998.6159
El-Sayed MA: An oscillation criterion for a forced second order linear differential equation. Proc. Am. Math. Soc. 1993, 118: 813–817.
Yang X: Oscillation criteria for nonlinear differential equations with damping. Comput. Math. Appl. 2003, 136: 549–557. 10.1016/S0096-3003(02)00079-6
Hassan TS, Kong Q: Interval criteria for forced oscillation of differential equations with p -Laplacian and nonlinearities given by Riemann-Stieltjes integrals. J. Korean Math. Soc. 2012, 49: 1017–1030. 10.4134/JKMS.2012.49.5.1017
Erbe L, Hassan TS, Peterson A: Oscillation criteria for nonlinear damped dynamic equations on time scales. Comput. Math. Appl. 2008, 203: 343–357. 10.1016/j.amc.2008.04.038
This work was supported by the National Natural Science Foundation of China (Grant No. 61174217) and the Natural Science Foundation of Shandong Province (Grant Nos. ZR2010AL002 and JQ201119).
The authors declare that they have no competing interests.
All authors carried out the proof. All authors conceived of the study and participated in its design and coordination. All authors read and approved the final manuscript.
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Liu, Z., Sun, Y. Oscillation of impulsive functional differential equations with oscillatory potentials and Riemann-Stieltjes integrals. Adv Differ Equ 2012, 175 (2012). https://doi.org/10.1186/1687-1847-2012-175
- Riemann-Stieltjes integral