Abstract and Applied Analysis

Volume 2013 (2013), Article ID 245905, 5 pages

http://dx.doi.org/10.1155/2013/245905

Research Article

Asymptotic Behavior of Solutions to a Linear Volterra Integrodifferential System

Yue-Wen Cheng and Hui-Sheng Ding

College of Mathematics and Information Science, Jiangxi Normal University, Nanchang, Jiangxi 330022, China

Received 24 July 2013; Accepted 18 September 2013

Academic Editor: Josef Diblík

Copyright © 2013 Yue-Wen Cheng and Hui-Sheng Ding. 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.

We investigate the asymptotic behavior of solutions to a linear Volterra integrodifferential system , We show that under some suitable conditions, there exists a solution for the above integrodifferential system, which is asymptotically equivalent to some given functions. Two examples are given to illustrate our theorem.

1. Introduction

Throughout this paper, we denote by the set of positive integers, by the set of all real numbers, by the set of all nonnegative real numbers, and by the set of all -dimensional real vectors. Moreover, denotes the Banach space of all bounded and continuous functions with the norm

where for .

The aim of this paper is to study some asymptotic behavior of solutions to the following linear Volterra integrodifferential system:

where and , are all continuous functions.

Definition 1. We call a solution of system (2) if is continuously differentiable and satisfies (2).

The asymptotic behavior of solutions has been an important and interesting topic in the qualitative theory of differential and difference equations. Especially, recently, many authors have made interesting and important contributions on the asymptotic behavior of solutions for Volterra type difference equations (e.g., we refer the reader to [1–10] and references therein).

Very recently, Diblík and Schmeidel [6] obtained a very interesting result concerning the asymptotic behavior of solutions for the following linear Volterra difference equation:

More specifically, they proved that for every admissible constant , there exists a solution of (3) such that

where . However, to the best of our knowledge, it seems that there is no literature concerning such asymptotic behavior of solutions for Volterra type differential equations. That is the main motivation of this paper. In this paper, we will adopt the idea in the proof of [6] to investigate some asymptotic behaviors of solutions for Volterra differential system (2).

2. Main Result

Before establishing our main result, we first give an “Arzela-Ascoli” type theorem for the subsets of .

Lemma 2. Let , satisfying (i) is uniformly bounded; (ii) is equiuniformly continuous on every compact subset of ; (iii) for every , there exist and such that for all and . Then is precompact in .

Proof. By the condition (iii), for every , there exist such that

for all and .

Let be a sequence in . By (i) and (ii), it follows from Arzela-Ascoli theorem that for every , there exists a subsequence such that is uniformly convergent on . Then, by choosing the diagonal sequence, we can get a subsequence such that, for every , is uniformly convergent on .

It remains to show that is uniformly convergent on . For every , choose with . Since is uniformly convergent on , for the above , there exists such that

for all ; Combining this with (5), we conclude that

for all , that is, is uniformly convergent on . This completes the proof.

Throughout the rest of this paper, for every , we assume that

where

Theorem 3. Assume that

Let with for all and . Then, there exists a solution of system (2) such that

provided that .

Proof. We define that

for all and . Moreover, we define an operator on

by

for and . It is easy to see that is a nonempty, closed, and convex set in . Next, we divide the remaining proof into two steps.

Step  1. , is continuous, and is compact.

Let . We have

In addition, since , we have

Then, it follows that

for all and . Thus, we conclude that .

For every , there exists a constant such that for all with , we have

which means that is continuous.

Next, we show that is precompact. Firstly, for every , we have

which means that is uniformly bounded. Secondly, for every , and , we have

which yields that is equiuniformly continuous on every compact subsets of . Thirdly, by the definition of , for every , there exists such that for all and , we have

which yields that

and thus . Then, by Lemma 2, we know that is precompact.

Step  2. By Step  1 and Schauder's fixed-point theorem, has a fixed point in ; that is, there exists such that

for all and . Noting that

we have

for all and . Then, it is easy to see that

Combining this with

we have

that is,

Now, define a function by

It follows from (23) that

which yields that

Then, we get

which means that is a solution to system (2). In addition, combining (28) with the assumption

we get

which yields (11).

Example 4. Let , and for all ,

Then, for all , we have ,

In addition, it is easy to see that

Thus, by Theorem 3, we conclude that for every , there exists a solution for (2) such that

Remark 5. It is needed to note that in the above example, is not a solution to (2).

Example 6. Consider the following system:

where

for all , and . By a direct calculation, we get

Moreover, we have

Then, by Theorem 3, for every with and , there exists a solution of system (40) such that

Conflict of Interests

The authors declare that there is no conflict of interests regarding the publication of this paper.

Acknowledgments

The authors are grateful to Professor Diblík and two anonymous referees for their valuable suggestions and comments. In addition, Hui-Sheng Ding acknowledges support from the NSF of China (11101192), the Program for Cultivating Young Scientist of Jiangxi Province (20133BCB23009), and the NSF of Jiangxi Province; Yue-Wen Cheng acknowledges support from the Graduate Innovation Fund of Jiangxi Province.

References

J. A. D. Applelby, I. Győri, and D. W. Reynolds, “On exact convergence rates for solutions of linear systems of Volterra difference equations,” Journal of Difference Equations and Applications, vol. 12, no. 12, pp. 1257–1275, 2006.

J. Diblík, E. Schmeidel, and M. Růžičková, “Existence of asymptotically periodic solutions of system of Volterra difference equations,” Journal of Difference Equations and Applications, vol. 15, no. 11-12, pp. 1165–1177, 2009.

J. Diblík, M. Růžičková, and E. Schmeidel, “Existence of asymptotically periodic solutions of scalar Volterra difference equations,” Tatra Mountains Mathematical Publications, vol. 43, pp. 51–61, 2009.

J. Diblík, E. Schmeidel, and M. Růžičková, “Asymptotically periodic solutions of Volterra system of difference equations,” Computers & Mathematics with Applications, vol. 59, no. 8, pp. 2854–2867, 2010.

J. Diblík, E. Schmeidel, M. Růžičková, and M. Zbaszyniak, “Weighted asymptotically periodic solutions of linear Volterra difference equations,” Abstract and Applied Analysis, vol. 2011, Article ID 370982, 14 pages, 2011.

J. Diblík and E. Schmeidel, “On the existence of solutions of linear Volterra difference equations asymptotically equivalent to a given sequence,” Applied Mathematics and Computation, vol. 218, no. 18, pp. 9310–9320, 2012.

I. Győri and D. W. Reynolds, “On asymptotically periodic solutions of linear discrete Volterra equations,” Fasciculi Mathematici, no. 44, pp. 53–67, 2010.

W. Long and W.-H. Pan, “Asymptotically almost periodic solution to a class of Volterra difference equations,” Advances in Difference Equations, vol. 2012, article 199, 2012.

R. Medina, “Asymptotic behavior of solutions of Volterra difference equations with finite linear part,” Nonlinear Studies, vol. 8, no. 1, pp. 87–95, 2001.

R. Medina, “Asymptotic behavior of Volterra difference equations,” Computers & Mathematics with Applications, vol. 41, no. 5-6, pp. 679–687, 2001.