Functional Differential and Difference Equations with Applications 2013View this Special Issue
Asymptotic Behavior of Solutions to a Linear Volterra Integrodifferential System
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.
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.
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  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  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
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).
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.
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.
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.View at: Publisher Site | Google Scholar | Zentralblatt MATH | MathSciNet