Nonlinear Variation of Parameters Formula for Impulsive Differential Equations with Initial Time Difference and Application

Abstract

This paper establishes variation of parameters formula for impulsive differential equations with initial time difference. As an application, one of the results is used to investigate stability properties of solutions.

1. Introduction

It is now well recognized that impulsive differential equations are suitable mathematical models for many processes and phenomena in biology, physics, technology, and so forth. That is why in recent years the mathematical theory of such systems has gained increasing significance. We notice that most of the studies about initial value problems of impulsive differential equations are investigated only for perturbation or change of dependent variable keeping the initial time unchanged. However, in dealing with real world phenomena, it is impossible not to make errors in the starting 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, the investigation of differential systems with initial time difference has attracted a lot of attention. There are two methods of comparing the differences of the two solutions. One is the differential inequalities technique and comparison principle; the other is variation of parameters. For the pioneering works in this area we can refer to the papers [1, 2]. Ever since then, many results for various differential and difference systems have been obtained. The results obtained by the former method can be seen in [3–10]; and those done by the latter can be found in [11–15]. However, up till now, to the best of our knowledge, there are few results for impulsive differential equations with initial time difference. To be specific, there are no results on variation of parameters formula for impulsive differential equations relative to initial time difference. The method of variation of parameters is an important and fruitful technique since it is a practical tool in the investigation of the properties of solutions. It has been applied to the study of the relations of unperturbed and perturbed systems with different initial conditions.

In this paper, we will develop variation of parameters formula for impulsive differential equations with initial time changed and investigate Lipschitz stability by using one of the results obtained. The remainder of this paper is organized in the following manner. Some preliminaries are presented in Section 2, and various types of nonlinear variation of parameters formulae are established in Section 3. Finally, as an application, one of the results is applied to impulsive differential equations and the stability properties are obtained.

2. Preliminaries

Let and let denote the -dimensional Euclidean space with appropriate norm .

Consider the following unperturbed impulsive differential equations together with the perturbed ones of (2) where(1), and ;(2),  ;(3);(4),  ,  ;(5);(6);(7);(8),  , for all .

We are concerned in this paper with the variation of parameters formula for impulsive differential equations relative to initial time difference. Before we can proceed, we will introduce the following lemmas [12], which are necessary for completing our main results.

Lemma 1. Let the following conditions be fulfilled:)the function is continuous in , and for every and , there exists a finite limit of as , ;()the function is locally Lipschitzian in on ;()for the mapping , , is a homeomorphism;()the system (1) had a solution defined in , .Then there exist a number and a set such that,(i)for every , there exists a unique solution of the system (1) which is defined on ;(ii)the function is continuous for (iii)for , belonging to the interval of existence of solution of (1), ,

Lemma 2. Let the following conditions be fulfilled:()the function is continuous in , , and is continuous in , ;()for every , , there exist finite limits of functions and as , ;()for the mapping is a diffeomorphism and for Then,(i)there exists such that the solution of (1) has continuous derivatives ,  ,  , in the domain (ii)the derivative is a solution of the initial value problem where is the solution of (1) in ;(iii)the derivative satisfies the relation

3. Nonlinear Variation of Parameters Formula

We will present, in this section, the nonlinear variation of parameters formula for impulsive differential equations relative to initial time difference. It is very useful for investigating the stability properties of solutions.

Theorem 3. Let the system (1) satisfy the conditions of Lemma 2 and let be a solution of (1). Then for any solution of the system (3). The following formula is valid:

Proof. Set , where , . Then for , we have When , we have two cases.
Case 1. Consider Case 2. Consider Integrating (12) from to and using (13) and (14), we have Now let , where , . Then we have Integrating (16) from 0 to 1, we arrive at Combining (15) and (17) yields The proof is complete.

Corollary 4. Suppose that the assumptions of Theorem 3 hold except that being replaced with ; then the following formula is valid:

Theorem 5. Suppose that the assumptions of Theorem 3 hold; then the following formula is valid:

Proof. Set , where , . Then for , we have where .
If , we have three cases.
Case 1. Consider Case 2. Consider Case 3. Consider Integrating (21) from to and using (22) and (24), we have The proof is complete.

4. Application

In this section, we turn to the Lipschitz stability of system (3): where .

Definition 6. The solution of the system (3) is said to be initial time difference Lipschitz stable with respect to the solution for , where is any solution of the system (1), if and only if there exists an such that

Theorem 7. Let the following conditions be fulfilled:the assumptions of Corollary 4 hold;the zero solution of (1) is Lipschitz stable;  for ; and is a constant; and are constants; and are constants;, and .Then the solution of the system (3) is with respect to the solution .

Proof. From Corollary 4, it follows that Taking the norm and using the triangle inequality on both sides, we have From conditions , we obtain Setting , we have Since , as long as , then we have Applying Gronwall’s inequality to (32), we get Setting    , we have From condition , it follows that the solution of the system (3) is with respect to the solution .
The proof is complete.

Conflict of Interests

The authors declare that they have no competing interests.

Authors’ Contribution

All authors completed the paper together. All authors read and approved the final paper.

Acknowledgments

The authors would like to thank the reviewers for their valuable suggestions and comments. This paper is supported by the National Natural Science Foundation of China (11271106) and the Natural Science Foundation of Hebei Province of China (A2013201232).