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

Youjun Liu, Jianwen Zhang, and Jurang Yan 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 Correspondence should be addressed to Jianwen Zhang; zjw9791@126.com 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.

Proof.Let Λ be the set of all continuous and bounded vector functions on [ 0 , ∞) and the sup norm.Set  = { ∈ Λ, and define an operator  on  as follows: It is easy to see that x is continuous, for  ≥  1 , x ∈ ; by using (7), we have and taking (7) into account, we have These show that  ⊂ .Since  is a bounded, close, convex subset of Λ, in order to apply the contraction principle, we have to show that  is a contraction mapping on .For all x 1 , x 2 ∈ , and  ≥  1 , 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) x ∈  of x = x.The proof is complete.
Proof.Let Λ be the set of all continuous and bounded vector functions on [ 0 , ∞) and the sup norm.Set  = { ∈ Λ, and define an operator  on  as follows: It is easy to see that x is continuous, for  ≥  1 , x ∈ ; by using (14), we have and taking (14) into account, we have and define an operator  on  as follows It is easy to see that x is continuous.Since the proof is similar to that of Theorem 1, we omit the remaining part of the proof.The proof is complete.Proof.Let Λ be the set of all continuous and bounded vector functions on [ 0 , ∞) and the sup norm.Set  = { ∈ Λ,  7 ≤ ‖x()‖ ≤  8 ,  ≥  0 }, where  7 and  8 are positive constants such that From (6), one can choose a  1 ≥  0 ,  1 ≥  0 + , sufficiently large  ≥  1 such that and define an operator  on  as follows: It is easy to see that x is continuous.The remaining part of the proof is similar to that of Theorem 2; therefore, it is omitted.The proof is complete.
Proof.Let Λ be the set of all continuous and bounded vector functions on [ 0 , ∞) and the sup norm.Set  = { ∈ Λ,  1 ≤ ‖x()‖ ≤  2 ,  ≥  0 }, where  1 and  2 are positive constants such that and define an operator  on  as follows: It is easy to see that x is continuous.Since the proof is similar to that of Theorem 1, we omit the remaining part of the proof.The proof is complete.
Proof.Let Λ be the set of all continuous and bounded vector functions on [ 0 , ∞) and the sup norm.Set  = { ∈ Λ, and define an operator  on  as follows: It is easy to see that x is continuous, for  ≥ It is easy to see that x is continuous.The remaining part of the proof is similar to that of Theorem 2; therefore, it is omitted.The proof is complete.

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