Discrete Dynamics in Nature and Society

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

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

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

^{1}Institute of Applied Mechanics and Biomedical Engineering, Taiyuan University of Technology, Taiyuan, Shanxi 030024, China^{2}College of Mathematics and Computer Sciences, Shanxi Datong University, Datong, Shanxi 037009, China^{3}School 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 thatand 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 followsIt 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 haveIt 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).

#### References

- T. Candan, “Existence of nonoscillatory solutions for system of higher order neutral differential equations,”
*Mathematical and Computer Modelling*, vol. 57, no. 3-4, pp. 375–381, 2013. View at Publisher · View at Google Scholar · View at MathSciNet - T. Candan, “The existence of nonoscillatory solutions of higher order nonlinear neutral equations,”
*Applied Mathematics Letters*, vol. 25, no. 3, pp. 412–416, 2012. View at Publisher · View at Google Scholar · View at MathSciNet - T. Candan and R. S. Dahiya, “Existence of nonoscillatory solutions of first and second order neutral differential equations with distributed deviating arguments,”
*Journal of the Franklin Institute*, vol. 347, no. 7, pp. 1309–1316, 2010. View at Publisher · View at Google Scholar · View at MathSciNet - H. El-Metwally, M. R. S. Kulenović, and S. Hadžiomerspahić, “Nonoscillatory solutions for system of neutral delay equation,”
*Nonlinear Analysis: Theory, Methods & Applications*, vol. 54, no. 1, pp. 63–81, 2003. View at Publisher · View at Google Scholar · View at MathSciNet - W. Zhang, W. Feng, J. Yan, and J. Song, “Existence of nonoscillatory solutions of first-order linear neutral delay differential equations,”
*Computers & Mathematics with Applications*, vol. 49, no. 7-8, pp. 1021–1027, 2005. View at Publisher · View at Google Scholar · View at MathSciNet - Y. Zhou and B. G. Zhang, “Existence of nonoscillatory solutions of higher-order neutral differential equations with positive and negative coefficients,”
*Applied Mathematics Letters*, vol. 15, no. 7, pp. 867–874, 2002. View at Publisher · View at Google Scholar · View at MathSciNet - Y. Yu and H. Wang, “Nonoscillatory solutions of second-order nonlinear neutral delay equations,”
*Journal of Mathematical Analysis and Applications*, vol. 311, no. 2, pp. 445–456, 2005. View at Publisher · View at Google Scholar · View at MathSciNet - W.-T. Li, “Positive solutions of an odd-order neutral delay nonlinear differential equations,”
*Nonlinear Analysis: Theory, Methods & Applications*, vol. 36, no. 7, pp. 899–913, 1999. View at Publisher · View at Google Scholar · View at MathSciNet - W.-T. Li, “Positive solutions of second order nonlinear differential equations,”
*Journal of Mathematical Analysis and Applications*, vol. 221, no. 1, pp. 326–337, 1998. View at Publisher · View at Google Scholar · View at MathSciNet - Ö. Öcalan, “Oscillation of neutral differential equation with positive and negative coefficients,”
*Journal of Mathematical Analysis and Applications*, vol. 331, no. 1, pp. 644–654, 2007. View at Publisher · View at Google Scholar · View at MathSciNet - I. Györi and G. Ladas,
*Oscillation Theory of Delay Differential Equations with Applications*, Oxford Mathematical Monographs, The Clarendon Press, Oxford, UK, 1991. View at MathSciNet - J. H. Shen, I. P. Stavroulakis, and X. H. Tang, “Hille type oscillation and nonoscillation criteria for neutral equations with positive and negative coefficients,”
*Studies of the University of Žilina. Mathematical Series*, vol. 14, no. 1, pp. 45–59, 2001. View at Google Scholar · View at MathSciNet