Discrete Dynamics in Nature and Society

Volume 2013 (2013), Article ID 391973, 8 pages

http://dx.doi.org/10.1155/2013/391973

Research Article

Existence of Nonoscillatory Solutions for System of Higher-Order Neutral Differential Equations with Distributed Deviating Arguments

Youjun Liu,1,2 Jianwen Zhang,¹ and Jurang Yan³

¹Institute of Applied Mechanics and Biomedical Engineering, Taiyuan University of Technology, Taiyuan, Shanxi 030024, China

²College of Mathematics and Computer Sciences, Shanxi Datong University, Datong, Shanxi 037009, China

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

Received 25 July 2013; Accepted 17 October 2013

Academic Editor: Garyfalos Papaschinopoulos

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.

In this paper, we consider the existence of nonoscillatory solutions for system of variable coefficients higher-order neutral differential equations with distributed deviating arguments. We use the Banach contraction principle to obtain new sufficient conditions for the existence of nonoscillatory solutions.

1. Introduction and Preliminary

In this paper, we consider the system of higher-order neutral differential equations with distributed deviating arguments:

where is a positive integer, , , , and ; , , and ;, is continuous matrix on , , and matrix coefficients system of higher order neutral differential equations with distributed deviating arguments:

where is a positive integer, , , , and ; , , and is nonsingular constant matrix; , and is continuous matrix on , .

Recently there have been a lot of activities concerning the existence of nonoscillatory solutions for neutral differential equations with positive and negative coefficients. In 2013, Candan [1] has investigated existence of nonoscillatory solutions for system of higher-order nonlinear neutral differential equations:

and matrix coefficient system of higher order neutral functional differential equation:

In 2012, Candan [2] studies higher-order nonlinear differential equation:

has obtained sufficient conditions for the existence of nonoscillatory solutions. For related work, we refer the reader to the books [3–12].

A solution of system of (1) and (2) is a continuous function defined on , for some , such that and are times continuously differentiable, and and are continuously differentiable, and system of (1) and (2) holds for all . Here, .

2. The Main Results

Theorem 1. Assume that and

Then, (1) has a bounded nonoscillatory solution.

Proof. Let be the set of all continuous and bounded vector functions on and the sup norm. Set , where and are positive constants and is a constant vector, such that . From (6), one can choose a , , sufficiently large such that

and define an operator on as follows:

It is easy to see that is continuous, for , ; by using (7), we have

and taking (7) into account, we have

These show that . Since is a bounded, close, convex subset of , in order to apply the contraction principle, we have to show that is a contraction mapping on . For all , and ,

Using (7),

This implies, with the sup norm, that

which shows that is a contraction mapping on , and therefore there exists a unique solution. Consequently there exists a unique solution of (1) of . The proof is complete.

Theorem 2. Assume that and that (6) holds.

Then, (1) has a bounded nonoscillatory solution.

Proof. Let be the set of all continuous and bounded vector functions on and the sup norm. Set , where and are positive constants such that . From (6), one can choose a , , sufficiently large such that

and define an operator on as follows:

It is easy to see that is continuous, for , ; by using (14), we have

and taking (14) into account, we have

These show that . Since is a bounded, close, convex subset of , in order to apply the contraction principle, we have to show that is a contraction mapping on . For all , and ,

or using (14),

This implies, with the sup norm, that

which shows that is a contraction mapping on , and therefore there exists a unique solution. Consequently there exists a unique solution of (1) of . The proof is complete.

Theorem 3. Assume that and that (6) holds.

Then, (1) has a bounded nonoscillatory solution.

Proof. Let be the set of all continuous and bounded vector functions on and the sup norm. Set , where and are positive constants such that . From (6), one can choose a , , sufficiently large such that

and define an operator on as follows

It is easy to see that is continuous. Since the proof is similar to that of Theorem 1, we omit the remaining part of the proof. The proof is complete.

Theorem 4. Assume that and that (6) holds.

Then, (1) has a bounded nonoscillatory solution.

Proof. Let be the set of all continuous and bounded vector functions on and the sup norm. Set , where and are positive constants such that . From (6), one can choose a , , sufficiently large such that

and define an operator on as follows:

It is easy to see that is continuous. The remaining part of the proof is similar to that of Theorem 2; therefore, it is omitted. The proof is complete.

Theorem 5. Assume that and that (6) holds.

Then, (2) has a bounded nonoscillatory solution.

Proof. Let be the set of all continuous and bounded vector functions on and the sup norm. Set , where and are positive constants such that . From (6), one can choose a , , sufficiently large such that

and define an operator on as follows:

It is easy to see that is continuous. Since the proof is similar to that of Theorem 1, we omit the remaining part of the proof. The proof is complete.

Theorem 6. Assume that and that (6) holds.

Then, (2) has a bounded nonoscillatory solution.

Proof. Let be the set of all continuous and bounded vector functions on and the sup norm. Set , where and are positive constants such that . From (6), one can choose a , , sufficiently large such that

and define an operator on as follows:

It is easy to see that is continuous, for , ; by using (27), we have

and taking (27) into account, we have

It is easy to see that is continuous. The remaining part of the proof is similar to that of Theorem 2; therefore, it is omitted. The proof is complete.

3. Example

Example 1. Consider high-order neutral differential equation with distributed deviating arguments:

Here, , , , , , , , and .

It is easy to see that

thus Theorem 1 holds. In fact, is a nonoscillatory solution of (31).

Example 2. Consider high-order neutral differential equation with distributed deviating arguments:

Here, , , , , , , , and .

It is easy to see that

thus Theorem 5 holds. In fact, is a nonoscillatory solution of (33).

Acknowledgments

This research is supported by the Natural Sciences Foundation of China (no. 11172194), the Natural Sciences Foundation of Shanxi Province (no. 2010011008), and the Scientific Research Project Shanxi Datong University (no. 2011K3).

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