Oscillation of impulsive functional differential equations with oscillatory potentials and Riemann-Stieltjes integrals
© Liu and Sun; licensee Springer 2012
Received: 16 July 2012
Accepted: 20 September 2012
Published: 5 October 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.
Keywordsoscillation impulse delay Riemann-Stieltjes integral
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].
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 .
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.
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].
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.
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.
This completes the proof of Lemma 2.1. □
where is defined as in Lemma 2.1.
which implies (7). The proof of Lemma 2.2 is complete. □
It is easy to see that is a piecewise continuous function on for given .
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 .
where we use the convention that and .
3 Main results
and is defined as (9). Then Eq. (1) is oscillatory.
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) , .
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 .
- (i)when () is an odd number,(18)
- (ii)when () is an even number,(19)
Then Eq. (1) is oscillatory.
, the number of impulse moments in the interval , is odd;
is an even number.
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).
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).
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