Abstract and Applied Analysis

Abstract and Applied Analysis / 2014 / Article

Research Article | Open Access

Volume 2014 |Article ID 632109 | 11 pages | https://doi.org/10.1155/2014/632109

Linearization of Impulsive Differential Equations with Ordinary Dichotomy

Academic Editor: Yongli Song
Received27 Nov 2013
Accepted03 Jan 2014
Published02 Mar 2014

Abstract

This paper presents a linearization theorem for the impulsive differential equations when the linear system has ordinary dichotomy. We prove that when the linear impulsive system has ordinary dichotomy, the nonlinear system , , , , is topologically conjugated to , , , , where , , represents the jump of the solution at . Finally, two examples are given to show the feasibility of our results.

1. Introduction

A basic linearization theorem is the famous Hartman-Grobman theorem (see [1, 2]). Then Palmer successfully generalized the standard Hartman-Grobman theorem to nonautonomous differential equations (see [3]). Then Fenner and Pinto [4] generalized Hartman-Grobman theorem to impulsive differential equations. Since they did not discuss the Hölder regularity of the topologically equivalent function , for this reason, recently, Xia et al. [5] gave a rigorous proof of the Hölder regularity. Xia et al. [6, 7] proved a version of generalized Hartman-Grobman theorem for dynamic systems on time scales. It should be noted that the abovementioned works are based on the linear differential equations with uniform exponential dichotomy. Therefore, motivated by [8], in this paper, we have a version of generalized Hartman-Grobman theorem for the impulsive differential equations when the linear system has ordinary dichotomy.

Our main objective in this paper is to prove that, when the impulsive linear system has an ordinary dichotomy, the nonlinear system is topologically conjugated to its linear part where , , represents the jump of the solution at . Finally, two examples are given to show the feasibility of our results.

2. Definitions

Consider the linear nonautonomous system with impulses at times : where , , represents the jump of the solution at , , and and are matrixes.

Definition 1. System (3) is said to be an ordinary dichotomy, if there exists a projection and a constant such that where is a fundamental matrix of linear system (3) and is given by where is a fundamental matrix of the system , provided that is invertible, for all . In what follows, we will assume that is invertible for all .

Definition 2. In Definition 1, if as , then system (3) is said to possess an ordinary dichotomy with a positive asymptotically stable manifold;if as , then system (3) is said to possess an ordinary dichotomy with a negatively asymptotically stable manifold;if both of them hold, then system (3) is said to possess an ordinary dichotomy with asymptotically stable manifolds.

3. Main Result and Proof

Consider the following nonautonomous impulse systems: where , , represents the jump of the solution at , , and and are matrixes.

Definition 3. Suppose that there exists a function such that (i)for each fixed , is a homeomorphism of into ;(ii) uniformly bounded with respect to ;(iii)assume that has property (ii) also;(iv)if is a solution of system (7), then is a solution of system (6).
If such a map exists, then system (7) is topologically conjugated to (6). is an equivalent function.

Theorem 4. Suppose that the impulsive linear system (6) has an ordinary dichotomy and for any and one has(),(),(),(),(),(), ,where , are integrable functions and , are summable functions in , and and are positive constants. Then system (7) is topologically conjugated to system (6).

Remark 5 (pure continuous case). If impulsive jump operators are absent, then system (6) and (7) reduces to pure continuous systems. That is, Then Theorem 4 reduces to main results in [8].

Remark 6 (Pure discrete cases). A difference system, or a pure discrete-time system, is a special case of systems with impulses. Thus, instead of (6), we have only or for the perturbed case (7), Now, means , so that we may write the linear system in the canonical way as or similarly for the perturbed system, where . Then we have the following.

Corollary 7. Suppose that has an ordinary dichotomy, and for any . If satisfies then system (10) is topologically equivalent to system (9).

Remark 8. We point out that the conditions in Theorem 4 can be approached. For example, taking , if we assume that the interval contains finite number of sequences , then In particular, if , , then

Before the proof of Theorem 4, let us make some discussions about ordinary dichotomy and introduce some lemmas. Note first that if system (6) has an ordinary dichotomy then a fundamental matrix can be chosen such that the projection in (4). In fact, for any projection , there exists an invertible matrix such that . If (4) holds for and ; then (4) holds for and , this implies that is the required projection if is chosen as a fundamental matrix. Furthermore, in (4), we can assume that with being unbounded on for and bounded on for and with , with . Then is bounded on , is bounded on  (and unbounded on ), and is bounded on  (and unbounded on ).

In what follows, we assume that the assumptions in Theorem 4 always hold. Let be a solution of (7) satisfying the initial condition and a solution of (6) satisfying the initial condition .

Lemma 9. For each , the system has a unique bounded solution with .

Proof. For each , the solution of system (16) satisfying is Noting that On the other hand, It follows from (17) that We can assert that and hold. Otherwise, will be unbounded since it concludes unbounded part or/and . Thus Moreover, if satisfies the initial condition , then Now, we prove that is bounded. Due to the boundedness of and , we assume that , , where , are some positive constants. Together with (4) and (), it follows that, if , then we have if , then we have Similarly, if , then we have if , then we have On the other hand, it follows from (4) and () that, if , then we have If , then we have Similarly, if , then we have If , then we have Therefore, we get Similarly, we get It follows from (4), (), (22), and (31) that So is a bounded solution, and the bounded solution is unique with the initial condition . The proof of Lemma 9 is complete.

Lemma 10. For each , the system has a unique bounded solution with .

Proof. For a bounded continuous function of whose norm , we define a map as follows: It follows from (4), (), and (31) that we can also obtain that So, we have . Therefore, is a self-map of a sphere with radius .
Moreover, it follows from (4), (), (), (), and (32) that Let be a positive constant such that ; then . By the contraction principle map has a unique fixed point ; that is, satisfies By direct differentiation, we can verify that is a solution of (34). Furthermore, the solution is bounded with and Now, we are going to prove that the bounded solution with initial condition (39) is unique. For this purpose, we assume that is another solution of (34). Following steps similar to (17)–(22), it is not hard to show that any bounded solution of (34) with initial value condition (39) can be written as follows: Calculate , by (4), (), (), (), and (32) that Hence , and , so we have . This implies that the bounded solution of (34) with initial condition (39) is unique. The proof of Lemma 10 is complete.

Lemma 11. Let be any solution of the system (7); then the system has a unique bounded solution with .

Proof. Obviously, is a bounded solution of system (42) with the initial condition . Now we show that the bounded solution is unique. If not, there is another bounded solution , by Lemma 10, which can be written as follows: Then it follows from (4), (), (), (), and (32) that