Abstract and Applied Analysis

Volume 2013 (2013), Article ID 150684, 5 pages

http://dx.doi.org/10.1155/2013/150684

Research Article

-Stability of Nonlinear Volterra Integro-Differential Systems with Time Delay

Lianzhong Li,1,2 Maoan Han,¹  Xin Xue,² and Yuanyuan Liu¹

¹Department of Mathematics, Shanghai Normal University, Shanghai 200234, China

²School of Mathematics and Statistics, Taishan University, Tai'an, Shandong 271021, China

Received 2 March 2013; Revised 8 April 2013; Accepted 11 April 2013

Academic Editor: Marcia Federson

Copyright © 2013 Lianzhong Li 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.

We give some sufficient conditions for -uniform stability of the trivial solutions of a nonlinear differential system and of nonlinear Volterra integro-differential systems with time delay.

1. Introduction

Akinyele [1] introduced the notion of -stability of the degree with respect to a function , increasing and differentiable on and such that for and , . Constantin [2] introduced the notions of degree of stability and degree of boundedness of solutions of an ordinary differential equation, with respect to a continuous positive and nondecreasing function ; some criteria for these notions are proved there too.

Morchało [3] introduced the notions of -stability, -uniform stability, and -asymptotic stability of trivial solution of the nonlinear system . Several new and sufficient conditions for the mentioned types of stability are proved for the linear system ; in this paper is a scalar continuous function. In [4, 5], Diamandescu gives some sufficient conditions for asymptotic stability and -(uniform) stability of the nonlinear Volterra integro-differential system ; in these papers is a matrix function. Furthermore, in [6], sufficient conditions are given for the uniform Lipschitz stability of the system .

In paper [7], for the nonlinear system

and the nonlinear Volterra integro-differential system

by using the knowledge of fundamental matrix and nonlinear variation of constants, we give some sufficient conditions for -(uniform) stability of trivial solution for the system. The purpose of this paper is to provide sufficient conditions for -uniform stability of trivial solutions for the nonlinear delayed system

and the nonlinear delayed Volterra integro-differential systems

where , for , and with on . The systems studied in [7] do not include time delay, whereas all the systems studied in this paper have time delay.

In this paper, we investigate conditions on the functions under which the trivial solutions of systems (3), (4), and (5) are -stability on ; the main tool used is the integral inequalities and the integral technique. Here is a matrix function whose introduction allows us to obtain a mixed behavior for the components of solutions.

Let denote the Euclidean -space. For , let be the norm of . For an matrix , we define the norm . It is well known that

Let , , be continuous functions and .

Now we give the definitions of -(uniform) stability that we will need in the sequel.

Definition 1 (see [4, 8]). The trivial solution of (3) ((4) or (5)) is said to be -stable on if for every and any , there exists such that any solution of (3) ((4) or (5)), which satisfies the inequality , exists and satisfies the inequality for all .

Definition 2 (see [4, 8]). The trivial solution of (3) ((4) or (5)) is said to be -uniformly stable on if it is -stable on and the previous is independent of .

2. -Stability of the Systems

To prove our theorems, we need the following lemmas.

Lemma 3. Let with , , , . Assume, in addition, that and are nondecreasing functions and for . If satisfies

for , and , then

where , .

Proof. Let be fixed and denote

then ,  and  is nondecreasing on . For , by calculations we get the following:

Suppose that (if , carry out the following arguments with instead of , where is an arbitrary small constant, and subsequently pass to the limit as to complete the proof), then we get

Let

then, we have

Multiplying the above inequality by , we get

Consider now the integral on the interval to obtain

so

for . Let , since , then we have

Since was arbitrarily chosen, considering , we get (8).

Lemma 4. Let be as in Lemma 3. If satisfies

for , and , then

where , .

The proof is similar to the proof of Lemma 3, we omit the details.

Theorem 5. If there exist functions with , such that

for and for all . Moreover,

and , where , are nonnegative constants. If is an increasing diffeomorphism of . Then, the trivial solution of system (3) is -uniformly stable on .

Proof. Suppose that is the unique solution of system (3) which satisfies , since

after performing the change of variables in the second integral, and is the inverse of the diffeomorphism then, it follows that

this implies by Lemma 3 that

so for every , choose , then

for and for all . Hence, the conclusion of the theorem follows.

Theorem 6. Let all the conditions in Theorem 5 hold. Suppose further that there exist functions with , such that

for and for all , moreover,

where is a nonnegative constant. Then, the trivial solutions of systems (4) and (5) are -uniformly stable on .

Proof. For that system (4), suppose is the unique solution of system (4) which satisfies , since

it follows that

after performing the change of variables   (or ) at some intermediate step, and is the inverse of the diffeomorphism . Denote

This implies by Lemma 3 that

for and . So, for every and , let be a constant and choose , then

for and for all . This proves that the trivial solution of system (4) is -uniformly stable on .

Using Lemma 4, the proof of system (5) is similar to that of system (4) and the details are left to the readers.

Remark 7. For , , we obtain the theorems of classical stability and uniform stability.

3. Examples

Example 8. Consider the nonlinear differential system

In (33), , . Let , then for , it is easy to verify that , , and all the assumptions in Theorem 5 satisfied, so the trivial solution of system (33) is -uniformly stable on .

Example 9. Consider the nonlinear Volterra integro-differential system as follows:

In (34), , , , . Choose the same matrix function , then , , for , it is easy to verify that , , and all the assumptions in Theorem 6 are satisfied, so the trivial solution of system (34) is -uniformly stable on .

Acknowledgments

The authors are very grateful to the referees for their valuable comments and suggestions, which helped to shape the paper’s original form. This research was supported by the NNSF of China (10971139), NSF of Shandong Province (ZR2012AL03) and the Shandong Education Fund for College Scientific Research (J11LA51).

References

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