Abstract and Applied Analysis

Volume 2013 (2013), Article ID 731021, 6 pages

http://dx.doi.org/10.1155/2013/731021

Research Article

Existence of Oscillatory Solutions of Second Order Delay Differential Equations with Distributed Deviating Arguments

Youjun Liu,¹ Jianwen Zhang,² and Jurang Yan³

¹College of Mathematics and Computer Sciences, Shanxi Datong University, Datong, Shanxi 037009, China

²Institute of Applied Mechanics and Biomedical Engineering, Taiyuan University of Technology, Taiyuan, Shanxi 030024, China

³School of Mathematical Sciences, Shanxi University, Taiyuan, Shanxi 030006, China

Received 9 June 2013; Revised 25 October 2013; Accepted 13 November 2013

Academic Editor: Elena Litsyn

Copyright © 2013 Youjun Liu et al. This 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.

Abstract.

Under weaker hypothesis, we use the Schauder-Tychonoff theorem to obtain new sufficient condition for the global existence of oscillatory solutions for forced second order nonlinear delay differential equations with distributed deviating arguments.

1. Introduction

In this paper, we study the existence of oscillatory solutions for the nonlinear second order delay differential equations with the perturbed term

Under the following conditions:

(1); (2) is increasing function for all , satisfies the local Lipischitz condition.

During the past three decades, the investigation of oscillatory theory for delay differential equations and delay dynamic equations has attracted attention of numerous researchers due to their significance in theory and applications. We mention here the monographs of Myshkis [1], Norkin [2], Shevelo [3], and Agarwal et al. [4]. The oscillation properties of second order delay differential equations were considered also in Koplatadze et al. [5], Shmul’yan [6], and Skubachevskii [7]. Distances between adjacent zeros of oscillating solutions are estimated in [8, 9] for delay and for neutral second order equation in [10]. Distances between zero of solution and zero of its derivative were estimated in [11]. Based on oscillation properties, asymptotic properties of second order delay differential equations were studied in [12]. For related work, we refer the reader to the references [13–24]. However, to the best of our knowledge, the existence of oscillatory solutions for differential equation with distributed deviating arguments has been scarcely investigated. Thus, the research presents its significance.

As usual, a solution of (1) is a function defined on such that and are continuously differentiable on and and (1) holds. Our attention will be restricted to the solution of (1) which satisfy , for . Such a solution is said to be oscillatory if it has a sequence of zeros tending to infinity. Otherwise, it is said to be nonoscillatory.

The purpose of this paper is to prove a general result for (1) on the existence of oscillatory solutions. Throughout this paper, we will use the following notations. For a constant denote the local lipischitz constants of functions .

2. The Main Results

Lemma 1 (see [13]). Let be a locally convex space, nonempty and convex, , and compact. Given a continuous map , there exists such that .

Theorem 2. Assume that there exist such that ,

moreover, there exist two sequences , with such that

Then (1), has an oscillatory solution defined on with , and .

Proof. The proof is based on an application of the well-known Schauder-Tychonoff fixed point theorem. From (2), for any , we have to choose a large such that for all ,

Let denote the locally convex space of all continuous functions with topology of uniform convergence on compact subsets of . Let . Clearly, is a close convex subset of .

Introduce an operator by

It is easy to see that, for any , is well defined on continuously.

From (4), we obtain

Hence, . Thus, we have and is uniformly bounded on .

Let be any sequence and with . Let be large constant with , for any so that

From the compactness of the domain of , there exists a large and a constant ; let and when ,

where . By virtue of (1)–(8), we have that for any and ,

The continuity of on is proved.

Moreover, for all ,

where ;

where . Thus,

where . This implies that is equicontinuous. Hence, by the Ascoli-Arzela Theorem, the operator is completely continuous on . By Lemma 1, there exists satisfying

On the other hand, from (3), we find

which implies that is a bounded oscillatory solution of (1) and . The proof is complete.

Corollary 3. Assume that (2) of Theorem 2 holds, and specially is the ratio of two positive odd integers, there exist two increasing divergent sequences and such that

Then, (1) has an oscillatory solution defined on with , and .

3. Remark

When , and . Let , (1) becomes

According to the results of this paper, is integrable on , and there exist two sequences , with such that

Equation (16) has oscillatory solution.

In paper [14], it was demonstrated that the inequality implied the existence of positive solutions. Neither of the two conditions can be deduced from each other. Moreover, when is bounded, but cannot exist. Likewise, when and claim, can exist, but cannot be hold.

4. Examples

Example 1. Consider second order delay differential equations

Here, , and .

It is easy to see that .

We have

Let ,

It is easy to see from (19) that there exists a such that for all , and . Thus, by Theorem 2, (18) has an oscillatory solution on and . It is not difficult to check that (18) has the oscillatory solution .

Example 2. Consider second order delay differential equations

Here, , and . It is easy to see that ,

Let ,

It is easy to see from (22) that there exists a such that for all and . Thus, by Theorem 2, (21) has an oscillatory solution on and .

Acknowledgments

The author thanks both referees for a careful reading of the paper and useful suggestions that helped to improve the presentation. This research is supported by Natural Sciences Foundation of China (no. 11172194), Natural Sciences Foundation of Shanxi Province (no. 2010011008), and Scientific Research Project Shanxi Datong University (no. 2011 K3).

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