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Abstract and Applied Analysis
Volume 2013 (2013), Article ID 827482, 4 pages
-Stability for Differential Systems Relative to Initial Time Difference
1College of Electronic and Information Engineering, Hebei University, Baoding 071002, China
2College of Mathematics and Computer Science, Hebei University, Baoding 071002, China
Received 22 March 2013; Accepted 22 April 2013
Academic Editor: Zhenkun Huang
Copyright © 2013 Peiguang Wang and Xiaowei Liu. 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.
This paper investigates the relationship between an unperturbed differential system and a perturbed differential system that have initial time difference. Notions of -stability for differential systems with initial time difference are introduced, and stability criteria are formulated by using variation of parameter techniques.
It is well known that, in applications, asymptotic stability is more important than stability, because the desirable feature is to know the size of the region of asymptotic stability. However, when we study the asymptotic stability, it is not easy to deal with nonexponential types of stability. Pinto  introduced the notion of -stability with the intention of obtaining results about stability for a weakly stable system (at least, weaker than those given exponential stability and uniform Lipschitz stability) under some perturbations and developed the study of exponential stability to a variety of reasonable systems called -systems. Since then, Choi and Ryu , Choi et al. , and Choi and Koo  investigated -stability problem for the nonlinear differential systems respectively, and Choi et al. [5, 6] characterized the -stability in variation for nonlinear difference systems via -similarity and Lyapunov functions and obtained some relative results. For the detailed results of -stability of impulsive dynamic systems on time scale and others systems can be found in [7–10].
At present, the investigation of differential systems with initial time difference has attracted a lot of attention. This is mainly because of the fact that when considering initial value problems, it is impossible not to make errors in the starting time in dealing with real world phenomena, that is, the solutions of the unperturbed differential system may start at some initial time and the solutions of the perturbed systems may start at a different initial time. When we consider such a change of initial time for each solution, we need to deal with the problem of comparing between any two solutions which start at different times. At present, there are two methods to discuss the stability problem with initial time difference: one is the differential inequalities and comparison principle, and the other is the method of variation of parameters. For the pioneering works in this area we can refer to the papers [11, 12]. After that, there are many stability results for various of differential and difference systems; see [13–20]. However, the above results were obtained by using comparison principle and differential inequalities; there are few stability criteria by using the method of variation of parameters; see [21–24].
In this paper, we attempt to extend the notion of -stability to differential systems with initial time difference, namely, initial time difference -stability () and then establish some stability criteria for such differential systems by using the method of variation of parameters. The remainder of this paper is organized in the following manner. Some preliminaries are presented in Section 2. The notions of -stability for differential systems with initial time difference are given in this section. In Section 3, several stability criteria are established. Finally, an example is added to illustrate the result obtained.
Let and denotes the -dimensional Euclidean space with appropriate norm .
Consider the differential systems: and the perturbed differential system of (2): where are locally Lipschitzian and has continuous partial derivatives on . The above assumptions imply the existence and uniqueness of solutions through and . A special case of (3) is where , is the perturbation term. Let . Furthermore, suppose that is the given solution with respect to which we shall study stability criteria.
Let us begin by defining the following notions.
Definition 1. The solution of the system (2) through is said to be initial time difference -stability () with respect to the solution , where is any solution of the system (1), if and only if there exist and a positive bounded continuous function defined on such that for and .
Similarly, we can define initial time difference -stability () with respect to the solution of the system (3) through .
We are now in a position to give the Alekseev’s formula, which is an important tool in the subsequent discussion.
The following lemma will also be needed in our investigations.
3. Stability Criteria
We shall present, in this section, the stability criteria for differential systems with initial time difference.
Theorem 4. Let and be the solutions of (2) and (1) through and , respectively, . Assume that (i), in which ; (ii) there exists a positive bounded continuously differentiable function on such that where ; (iii) is locally Lipschitzian in time such that where .
Then the solution of the system (2) is with respect to the solution .
Proof. Define for , and then . Also, Using a Taylor approximation for and the conditions (i) and (ii), we arrive at And then, from (10), we have Moreover, using the condition (iii), we obtain Then from (12), we get So by Definition 1 with , the solution of (2) is with respect to the solution . This completes the proof.
Remark 5. Set , and then we can obtain Theorem 3.4 in .
Theorem 6. Let be the solution of (3) through . Assume that (i) the solution of (2) is with respect to the solution for , where is any solution of (1); (ii) there exist , and a positive bounded continuous function defined on such that provided that and .
Then the solution of (3) is with respect to the solution .
Proof. Define and , and then . The condition (i) yields
By Lemma 2, it follows that
Now taking the norms of both sides and using the triangle inequality, we have
From (15), we obtain
Setting and using the triangle inequality, we have
By using Lemma 3 and the condition (ii), we obtain
Then we have
where , and .
By Gronwall's inequality, one gets Moreover, set and , we get From Definition 1, it follows that the solution of (3) is with respect to the solution . This completes the proof.
Now, we shall illustrate Theorem 6 by a simple example. Consider the differential systems and the perturbed differential system of (27): Define ; by direct calculation, we have the solution of (27) given by , which exists for all , and , . , and then the solution of system (27) is with respect to .
This paper is supported by the National Natural Science Foundation of China (11271106) and the Natural Science Foundation of Hebei Province of China (A2013201232).
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