Open Access

Strict Stability Criteria for Impulsive Functional Differential Systems

Journal of Inequalities and Applications20072008:243863

DOI: 10.1155/2008/243863

Received: 4 September 2007

Accepted: 18 November 2007

Published: 15 December 2007

Abstract

By using Lyapunov functions and Razumikhin techniques, the strict stability of impulsive functional differential systems is investigated. Some comparison theorems are given by virtue of differential inequalities. The corresponding theorems in the literature can be deduced from our results.

1. Introduction

Since time-delay systems are frequently encountered in engineering, biology, economy, and other disciplines, it is significant to study these systems [1]. On the other hand, because many evolution processes in nature are characterized by the fact that at certain moments of time they experience an abrupt change of state, the study of dynamic systems with impulse effects has been assuming greater importance [24]. It is natural to expect that the hybrid systems which are called impulsive functional differential systems can represent a truer framework for mathematical modeling of many real world phenomena. Recently, several papers dealing with stability problem for impulsive functional differential systems have been published [510].

The usual stability concepts do not give any information about the rate of decay of the solutions, and hence are not strict concepts. Consequently, strict-stability concepts have been defined and criteria for such notions to hold are discussed in [11]. Till now, to the best of our knowledge, only the following very little work has been done in this direction [1215].

In this paper, we investigate strict stability for impulsive functional differential systems. The paper is organized as follows. In Section 2, we introduce some basic definitions and notations. In Section 3, we first give two comparison lemmas on differential inequalities. Then, by these lemmas, a comparison theorem is obtained and several direct results are deduced from it. An example is also given to illustrate the advantages of our results.

2. Preliminaries

We consider the following impulsive functional differential system:
https://static-content.springer.com/image/art%3A10.1155%2F2008%2F243863/MediaObjects/13660_2007_Article_1780_Equ1_HTML.gif
(2.1)

where https://static-content.springer.com/image/art%3A10.1155%2F2008%2F243863/MediaObjects/13660_2007_Article_1780_IEq1_HTML.gif is the set of all positive integers, https://static-content.springer.com/image/art%3A10.1155%2F2008%2F243863/MediaObjects/13660_2007_Article_1780_IEq2_HTML.gif , https://static-content.springer.com/image/art%3A10.1155%2F2008%2F243863/MediaObjects/13660_2007_Article_1780_IEq3_HTML.gif is an open set in https://static-content.springer.com/image/art%3A10.1155%2F2008%2F243863/MediaObjects/13660_2007_Article_1780_IEq4_HTML.gif https://static-content.springer.com/image/art%3A10.1155%2F2008%2F243863/MediaObjects/13660_2007_Article_1780_IEq5_HTML.gif https://static-content.springer.com/image/art%3A10.1155%2F2008%2F243863/MediaObjects/13660_2007_Article_1780_IEq6_HTML.gif , here https://static-content.springer.com/image/art%3A10.1155%2F2008%2F243863/MediaObjects/13660_2007_Article_1780_IEq7_HTML.gif , and https://static-content.springer.com/image/art%3A10.1155%2F2008%2F243863/MediaObjects/13660_2007_Article_1780_IEq8_HTML.gif is continuous everywhere except for a finite number of points https://static-content.springer.com/image/art%3A10.1155%2F2008%2F243863/MediaObjects/13660_2007_Article_1780_IEq9_HTML.gif at which https://static-content.springer.com/image/art%3A10.1155%2F2008%2F243863/MediaObjects/13660_2007_Article_1780_IEq10_HTML.gif and https://static-content.springer.com/image/art%3A10.1155%2F2008%2F243863/MediaObjects/13660_2007_Article_1780_IEq11_HTML.gif exist and https://static-content.springer.com/image/art%3A10.1155%2F2008%2F243863/MediaObjects/13660_2007_Article_1780_IEq12_HTML.gif . https://static-content.springer.com/image/art%3A10.1155%2F2008%2F243863/MediaObjects/13660_2007_Article_1780_IEq13_HTML.gif for each https://static-content.springer.com/image/art%3A10.1155%2F2008%2F243863/MediaObjects/13660_2007_Article_1780_IEq14_HTML.gif , where https://static-content.springer.com/image/art%3A10.1155%2F2008%2F243863/MediaObjects/13660_2007_Article_1780_IEq15_HTML.gif denotes the norm of vector in https://static-content.springer.com/image/art%3A10.1155%2F2008%2F243863/MediaObjects/13660_2007_Article_1780_IEq16_HTML.gif , https://static-content.springer.com/image/art%3A10.1155%2F2008%2F243863/MediaObjects/13660_2007_Article_1780_IEq17_HTML.gif with https://static-content.springer.com/image/art%3A10.1155%2F2008%2F243863/MediaObjects/13660_2007_Article_1780_IEq18_HTML.gif as https://static-content.springer.com/image/art%3A10.1155%2F2008%2F243863/MediaObjects/13660_2007_Article_1780_IEq19_HTML.gif and https://static-content.springer.com/image/art%3A10.1155%2F2008%2F243863/MediaObjects/13660_2007_Article_1780_IEq20_HTML.gif denotes the right-hand derivative of https://static-content.springer.com/image/art%3A10.1155%2F2008%2F243863/MediaObjects/13660_2007_Article_1780_IEq21_HTML.gif . For each https://static-content.springer.com/image/art%3A10.1155%2F2008%2F243863/MediaObjects/13660_2007_Article_1780_IEq22_HTML.gif , https://static-content.springer.com/image/art%3A10.1155%2F2008%2F243863/MediaObjects/13660_2007_Article_1780_IEq23_HTML.gif is defined by https://static-content.springer.com/image/art%3A10.1155%2F2008%2F243863/MediaObjects/13660_2007_Article_1780_IEq24_HTML.gif , https://static-content.springer.com/image/art%3A10.1155%2F2008%2F243863/MediaObjects/13660_2007_Article_1780_IEq25_HTML.gif . For https://static-content.springer.com/image/art%3A10.1155%2F2008%2F243863/MediaObjects/13660_2007_Article_1780_IEq26_HTML.gif , https://static-content.springer.com/image/art%3A10.1155%2F2008%2F243863/MediaObjects/13660_2007_Article_1780_IEq27_HTML.gif https://static-content.springer.com/image/art%3A10.1155%2F2008%2F243863/MediaObjects/13660_2007_Article_1780_IEq28_HTML.gif , https://static-content.springer.com/image/art%3A10.1155%2F2008%2F243863/MediaObjects/13660_2007_Article_1780_IEq29_HTML.gif . We assume that https://static-content.springer.com/image/art%3A10.1155%2F2008%2F243863/MediaObjects/13660_2007_Article_1780_IEq30_HTML.gif and https://static-content.springer.com/image/art%3A10.1155%2F2008%2F243863/MediaObjects/13660_2007_Article_1780_IEq31_HTML.gif , so that https://static-content.springer.com/image/art%3A10.1155%2F2008%2F243863/MediaObjects/13660_2007_Article_1780_IEq32_HTML.gif is a solution of (2.1), which we call the zero solution.

Let https://static-content.springer.com/image/art%3A10.1155%2F2008%2F243863/MediaObjects/13660_2007_Article_1780_IEq33_HTML.gif for some https://static-content.springer.com/image/art%3A10.1155%2F2008%2F243863/MediaObjects/13660_2007_Article_1780_IEq34_HTML.gif and https://static-content.springer.com/image/art%3A10.1155%2F2008%2F243863/MediaObjects/13660_2007_Article_1780_IEq35_HTML.gif , a function https://static-content.springer.com/image/art%3A10.1155%2F2008%2F243863/MediaObjects/13660_2007_Article_1780_IEq36_HTML.gif https://static-content.springer.com/image/art%3A10.1155%2F2008%2F243863/MediaObjects/13660_2007_Article_1780_IEq37_HTML.gif is said to be a solution of (2.1) with the initial condition
https://static-content.springer.com/image/art%3A10.1155%2F2008%2F243863/MediaObjects/13660_2007_Article_1780_Equ2_HTML.gif
(2.2)

if it is continuous and satisfies the differential equation https://static-content.springer.com/image/art%3A10.1155%2F2008%2F243863/MediaObjects/13660_2007_Article_1780_IEq38_HTML.gif in each https://static-content.springer.com/image/art%3A10.1155%2F2008%2F243863/MediaObjects/13660_2007_Article_1780_IEq39_HTML.gif https://static-content.springer.com/image/art%3A10.1155%2F2008%2F243863/MediaObjects/13660_2007_Article_1780_IEq40_HTML.gif , and at https://static-content.springer.com/image/art%3A10.1155%2F2008%2F243863/MediaObjects/13660_2007_Article_1780_IEq41_HTML.gif it satisfies https://static-content.springer.com/image/art%3A10.1155%2F2008%2F243863/MediaObjects/13660_2007_Article_1780_IEq42_HTML.gif .

Throughout this paper, we always assume the following conditions hold to ensure the global existence and uniqueness of solution of (2.1) through https://static-content.springer.com/image/art%3A10.1155%2F2008%2F243863/MediaObjects/13660_2007_Article_1780_IEq43_HTML.gif .

(H1) https://static-content.springer.com/image/art%3A10.1155%2F2008%2F243863/MediaObjects/13660_2007_Article_1780_IEq44_HTML.gif is continuous on https://static-content.springer.com/image/art%3A10.1155%2F2008%2F243863/MediaObjects/13660_2007_Article_1780_IEq45_HTML.gif for each https://static-content.springer.com/image/art%3A10.1155%2F2008%2F243863/MediaObjects/13660_2007_Article_1780_IEq46_HTML.gif and for all https://static-content.springer.com/image/art%3A10.1155%2F2008%2F243863/MediaObjects/13660_2007_Article_1780_IEq47_HTML.gif and https://static-content.springer.com/image/art%3A10.1155%2F2008%2F243863/MediaObjects/13660_2007_Article_1780_IEq48_HTML.gif , the limits https://static-content.springer.com/image/art%3A10.1155%2F2008%2F243863/MediaObjects/13660_2007_Article_1780_IEq49_HTML.gif exist.

(H2) https://static-content.springer.com/image/art%3A10.1155%2F2008%2F243863/MediaObjects/13660_2007_Article_1780_IEq50_HTML.gif is Lipschitzian in https://static-content.springer.com/image/art%3A10.1155%2F2008%2F243863/MediaObjects/13660_2007_Article_1780_IEq51_HTML.gif in each compact set in https://static-content.springer.com/image/art%3A10.1155%2F2008%2F243863/MediaObjects/13660_2007_Article_1780_IEq52_HTML.gif .

(H3) https://static-content.springer.com/image/art%3A10.1155%2F2008%2F243863/MediaObjects/13660_2007_Article_1780_IEq53_HTML.gif for all https://static-content.springer.com/image/art%3A10.1155%2F2008%2F243863/MediaObjects/13660_2007_Article_1780_IEq54_HTML.gif and there exists https://static-content.springer.com/image/art%3A10.1155%2F2008%2F243863/MediaObjects/13660_2007_Article_1780_IEq55_HTML.gif such that https://static-content.springer.com/image/art%3A10.1155%2F2008%2F243863/MediaObjects/13660_2007_Article_1780_IEq56_HTML.gif implies that https://static-content.springer.com/image/art%3A10.1155%2F2008%2F243863/MediaObjects/13660_2007_Article_1780_IEq57_HTML.gif for all https://static-content.springer.com/image/art%3A10.1155%2F2008%2F243863/MediaObjects/13660_2007_Article_1780_IEq58_HTML.gif .

The function https://static-content.springer.com/image/art%3A10.1155%2F2008%2F243863/MediaObjects/13660_2007_Article_1780_IEq59_HTML.gif belongs to class https://static-content.springer.com/image/art%3A10.1155%2F2008%2F243863/MediaObjects/13660_2007_Article_1780_IEq60_HTML.gif if the following hold.

(A1) https://static-content.springer.com/image/art%3A10.1155%2F2008%2F243863/MediaObjects/13660_2007_Article_1780_IEq61_HTML.gif is continuous on each of the sets https://static-content.springer.com/image/art%3A10.1155%2F2008%2F243863/MediaObjects/13660_2007_Article_1780_IEq62_HTML.gif and for each https://static-content.springer.com/image/art%3A10.1155%2F2008%2F243863/MediaObjects/13660_2007_Article_1780_IEq63_HTML.gif and https://static-content.springer.com/image/art%3A10.1155%2F2008%2F243863/MediaObjects/13660_2007_Article_1780_IEq64_HTML.gif , https://static-content.springer.com/image/art%3A10.1155%2F2008%2F243863/MediaObjects/13660_2007_Article_1780_IEq65_HTML.gif exists.

(A2) https://static-content.springer.com/image/art%3A10.1155%2F2008%2F243863/MediaObjects/13660_2007_Article_1780_IEq66_HTML.gif is locally Lipschitzian in https://static-content.springer.com/image/art%3A10.1155%2F2008%2F243863/MediaObjects/13660_2007_Article_1780_IEq67_HTML.gif and for https://static-content.springer.com/image/art%3A10.1155%2F2008%2F243863/MediaObjects/13660_2007_Article_1780_IEq68_HTML.gif Let https://static-content.springer.com/image/art%3A10.1155%2F2008%2F243863/MediaObjects/13660_2007_Article_1780_IEq69_HTML.gif , https://static-content.springer.com/image/art%3A10.1155%2F2008%2F243863/MediaObjects/13660_2007_Article_1780_IEq70_HTML.gif along the solution https://static-content.springer.com/image/art%3A10.1155%2F2008%2F243863/MediaObjects/13660_2007_Article_1780_IEq71_HTML.gif of (2.1) is defined as

https://static-content.springer.com/image/art%3A10.1155%2F2008%2F243863/MediaObjects/13660_2007_Article_1780_Equ3_HTML.gif
(2.3)

Let us introduce the following notations for further use:

(i) https://static-content.springer.com/image/art%3A10.1155%2F2008%2F243863/MediaObjects/13660_2007_Article_1780_IEq72_HTML.gif

(ii) https://static-content.springer.com/image/art%3A10.1155%2F2008%2F243863/MediaObjects/13660_2007_Article_1780_IEq73_HTML.gif https://static-content.springer.com/image/art%3A10.1155%2F2008%2F243863/MediaObjects/13660_2007_Article_1780_IEq74_HTML.gif

(iii) https://static-content.springer.com/image/art%3A10.1155%2F2008%2F243863/MediaObjects/13660_2007_Article_1780_IEq75_HTML.gif

(iv) https://static-content.springer.com/image/art%3A10.1155%2F2008%2F243863/MediaObjects/13660_2007_Article_1780_IEq76_HTML.gif ;

(v) https://static-content.springer.com/image/art%3A10.1155%2F2008%2F243863/MediaObjects/13660_2007_Article_1780_IEq77_HTML.gif ;

(vi) https://static-content.springer.com/image/art%3A10.1155%2F2008%2F243863/MediaObjects/13660_2007_Article_1780_IEq78_HTML.gif .

Definition 2.1.

The zero solution of (2.1) is said to be strictly stable (SS), if for any https://static-content.springer.com/image/art%3A10.1155%2F2008%2F243863/MediaObjects/13660_2007_Article_1780_IEq79_HTML.gif and https://static-content.springer.com/image/art%3A10.1155%2F2008%2F243863/MediaObjects/13660_2007_Article_1780_IEq80_HTML.gif , there exists a https://static-content.springer.com/image/art%3A10.1155%2F2008%2F243863/MediaObjects/13660_2007_Article_1780_IEq81_HTML.gif such that https://static-content.springer.com/image/art%3A10.1155%2F2008%2F243863/MediaObjects/13660_2007_Article_1780_IEq82_HTML.gif implies https://static-content.springer.com/image/art%3A10.1155%2F2008%2F243863/MediaObjects/13660_2007_Article_1780_IEq83_HTML.gif for https://static-content.springer.com/image/art%3A10.1155%2F2008%2F243863/MediaObjects/13660_2007_Article_1780_IEq84_HTML.gif and for every https://static-content.springer.com/image/art%3A10.1155%2F2008%2F243863/MediaObjects/13660_2007_Article_1780_IEq85_HTML.gif , there exists an https://static-content.springer.com/image/art%3A10.1155%2F2008%2F243863/MediaObjects/13660_2007_Article_1780_IEq86_HTML.gif such that
https://static-content.springer.com/image/art%3A10.1155%2F2008%2F243863/MediaObjects/13660_2007_Article_1780_Equ4_HTML.gif
(2.4)

Definition 2.2.

The zero solution of (2.1) is said to be strictly uniformly stable (SUS), if https://static-content.springer.com/image/art%3A10.1155%2F2008%2F243863/MediaObjects/13660_2007_Article_1780_IEq87_HTML.gif , and https://static-content.springer.com/image/art%3A10.1155%2F2008%2F243863/MediaObjects/13660_2007_Article_1780_IEq88_HTML.gif in (SS) are independent of https://static-content.springer.com/image/art%3A10.1155%2F2008%2F243863/MediaObjects/13660_2007_Article_1780_IEq89_HTML.gif .

Remark 2.3.

If in (SS) or (SUS), https://static-content.springer.com/image/art%3A10.1155%2F2008%2F243863/MediaObjects/13660_2007_Article_1780_IEq90_HTML.gif , we obtain nonstrict stabilities, that is, the usual stability or uniform stability, respectively. Moreover, strict stability immediately implies that the zero solution is not asymptotically stable.

The preceding notions imply that the motion remains in the tube like domains. To obtain sufficient conditions for such stability concepts to hold, it is necessary to simultaneously obtain both lower and upper bounds of the derivative of Lyapunov function. Thus, we need to consider the following two auxiliary systems:
https://static-content.springer.com/image/art%3A10.1155%2F2008%2F243863/MediaObjects/13660_2007_Article_1780_Equ5_HTML.gif
(2.5)
and
https://static-content.springer.com/image/art%3A10.1155%2F2008%2F243863/MediaObjects/13660_2007_Article_1780_Equ6_HTML.gif
(2.6)

where https://static-content.springer.com/image/art%3A10.1155%2F2008%2F243863/MediaObjects/13660_2007_Article_1780_IEq91_HTML.gif , https://static-content.springer.com/image/art%3A10.1155%2F2008%2F243863/MediaObjects/13660_2007_Article_1780_IEq92_HTML.gif , https://static-content.springer.com/image/art%3A10.1155%2F2008%2F243863/MediaObjects/13660_2007_Article_1780_IEq93_HTML.gif https://static-content.springer.com/image/art%3A10.1155%2F2008%2F243863/MediaObjects/13660_2007_Article_1780_IEq94_HTML.gif https://static-content.springer.com/image/art%3A10.1155%2F2008%2F243863/MediaObjects/13660_2007_Article_1780_IEq95_HTML.gif https://static-content.springer.com/image/art%3A10.1155%2F2008%2F243863/MediaObjects/13660_2007_Article_1780_IEq96_HTML.gif for each https://static-content.springer.com/image/art%3A10.1155%2F2008%2F243863/MediaObjects/13660_2007_Article_1780_IEq97_HTML.gif .

From the theory of impulsive differential systems [2], we obtain that
https://static-content.springer.com/image/art%3A10.1155%2F2008%2F243863/MediaObjects/13660_2007_Article_1780_Equ7_HTML.gif
(2.7)

where https://static-content.springer.com/image/art%3A10.1155%2F2008%2F243863/MediaObjects/13660_2007_Article_1780_IEq98_HTML.gif and https://static-content.springer.com/image/art%3A10.1155%2F2008%2F243863/MediaObjects/13660_2007_Article_1780_IEq99_HTML.gif are the minimal and maximal solutions of (2.5), (2.6), respectively.

The corresponding definitions of strict stability of the auxiliary systems (2.5), (2.6) are as follows.

Definition 2.4.

The zero solutions of comparison systems (2.5), (2.6), as a system, are said to be strictly stable (SS*), if for any https://static-content.springer.com/image/art%3A10.1155%2F2008%2F243863/MediaObjects/13660_2007_Article_1780_IEq100_HTML.gif and https://static-content.springer.com/image/art%3A10.1155%2F2008%2F243863/MediaObjects/13660_2007_Article_1780_IEq101_HTML.gif , there exist a https://static-content.springer.com/image/art%3A10.1155%2F2008%2F243863/MediaObjects/13660_2007_Article_1780_IEq102_HTML.gif and https://static-content.springer.com/image/art%3A10.1155%2F2008%2F243863/MediaObjects/13660_2007_Article_1780_IEq103_HTML.gif satisfying https://static-content.springer.com/image/art%3A10.1155%2F2008%2F243863/MediaObjects/13660_2007_Article_1780_IEq104_HTML.gif such that
https://static-content.springer.com/image/art%3A10.1155%2F2008%2F243863/MediaObjects/13660_2007_Article_1780_Equ8_HTML.gif
(2.8)

Definition 2.5.

The zero solutions of comparison systems (2.5),(2.6), as a system , are said to be strictly uniformly stable (SUS*), if https://static-content.springer.com/image/art%3A10.1155%2F2008%2F243863/MediaObjects/13660_2007_Article_1780_IEq105_HTML.gif , and https://static-content.springer.com/image/art%3A10.1155%2F2008%2F243863/MediaObjects/13660_2007_Article_1780_IEq106_HTML.gif in (SS*) are independent of https://static-content.springer.com/image/art%3A10.1155%2F2008%2F243863/MediaObjects/13660_2007_Article_1780_IEq107_HTML.gif .

3. Main Results

We first give two Razumikhin-type comparison lemmas on differential inequalities.

Lemma 3.1.

Assume that

(i) https://static-content.springer.com/image/art%3A10.1155%2F2008%2F243863/MediaObjects/13660_2007_Article_1780_IEq108_HTML.gif for each https://static-content.springer.com/image/art%3A10.1155%2F2008%2F243863/MediaObjects/13660_2007_Article_1780_IEq109_HTML.gif ;

(ii)there exists https://static-content.springer.com/image/art%3A10.1155%2F2008%2F243863/MediaObjects/13660_2007_Article_1780_IEq110_HTML.gif , where https://static-content.springer.com/image/art%3A10.1155%2F2008%2F243863/MediaObjects/13660_2007_Article_1780_IEq111_HTML.gif are continuous on https://static-content.springer.com/image/art%3A10.1155%2F2008%2F243863/MediaObjects/13660_2007_Article_1780_IEq112_HTML.gif and https://static-content.springer.com/image/art%3A10.1155%2F2008%2F243863/MediaObjects/13660_2007_Article_1780_IEq113_HTML.gif exist, https://static-content.springer.com/image/art%3A10.1155%2F2008%2F243863/MediaObjects/13660_2007_Article_1780_IEq114_HTML.gif , satisfying
https://static-content.springer.com/image/art%3A10.1155%2F2008%2F243863/MediaObjects/13660_2007_Article_1780_Equ9_HTML.gif
(3.1)
Then
https://static-content.springer.com/image/art%3A10.1155%2F2008%2F243863/MediaObjects/13660_2007_Article_1780_Equ10_HTML.gif
(3.2)
https://static-content.springer.com/image/art%3A10.1155%2F2008%2F243863/MediaObjects/13660_2007_Article_1780_Equ11_HTML.gif
(3.3)
where https://static-content.springer.com/image/art%3A10.1155%2F2008%2F243863/MediaObjects/13660_2007_Article_1780_IEq115_HTML.gif and https://static-content.springer.com/image/art%3A10.1155%2F2008%2F243863/MediaObjects/13660_2007_Article_1780_IEq116_HTML.gif are the minimal and maximal solutions of systems (3.4) and (3.5), respectively,
https://static-content.springer.com/image/art%3A10.1155%2F2008%2F243863/MediaObjects/13660_2007_Article_1780_Equ12_HTML.gif
(3.4)
https://static-content.springer.com/image/art%3A10.1155%2F2008%2F243863/MediaObjects/13660_2007_Article_1780_Equ13_HTML.gif
(3.5)

Proof.

First, we prove that (3.2) holds. Otherwise, there exist https://static-content.springer.com/image/art%3A10.1155%2F2008%2F243863/MediaObjects/13660_2007_Article_1780_IEq117_HTML.gif such that

(a) https://static-content.springer.com/image/art%3A10.1155%2F2008%2F243863/MediaObjects/13660_2007_Article_1780_IEq118_HTML.gif ,

(b) https://static-content.springer.com/image/art%3A10.1155%2F2008%2F243863/MediaObjects/13660_2007_Article_1780_IEq119_HTML.gif and

(c) https://static-content.springer.com/image/art%3A10.1155%2F2008%2F243863/MediaObjects/13660_2007_Article_1780_IEq120_HTML.gif .

By (a), (b), and (ii), applying the classical comparison theorem, we have
https://static-content.springer.com/image/art%3A10.1155%2F2008%2F243863/MediaObjects/13660_2007_Article_1780_Equ14_HTML.gif
(3.6)

which contradicts (c). So (3.2) is correct. Equation (3.3) can be proved in the same way as above. Then Lemma 3.1 holds.

Lemma 3.2.

Assume that (i) in Lemma 3.1 holds. Suppose further that
  1. (ii)
    there exists https://static-content.springer.com/image/art%3A10.1155%2F2008%2F243863/MediaObjects/13660_2007_Article_1780_IEq121_HTML.gif satisfying
    https://static-content.springer.com/image/art%3A10.1155%2F2008%2F243863/MediaObjects/13660_2007_Article_1780_Equ15_HTML.gif
    (3.7)
     
where https://static-content.springer.com/image/art%3A10.1155%2F2008%2F243863/MediaObjects/13660_2007_Article_1780_IEq122_HTML.gif and for any solution https://static-content.springer.com/image/art%3A10.1155%2F2008%2F243863/MediaObjects/13660_2007_Article_1780_IEq123_HTML.gif of (2.1), https://static-content.springer.com/image/art%3A10.1155%2F2008%2F243863/MediaObjects/13660_2007_Article_1780_IEq124_HTML.gif https://static-content.springer.com/image/art%3A10.1155%2F2008%2F243863/MediaObjects/13660_2007_Article_1780_IEq125_HTML.gif implies that
https://static-content.springer.com/image/art%3A10.1155%2F2008%2F243863/MediaObjects/13660_2007_Article_1780_Equ16_HTML.gif
(3.8)
  1. (iii)
    there exists https://static-content.springer.com/image/art%3A10.1155%2F2008%2F243863/MediaObjects/13660_2007_Article_1780_IEq126_HTML.gif satisfying
    https://static-content.springer.com/image/art%3A10.1155%2F2008%2F243863/MediaObjects/13660_2007_Article_1780_Equ17_HTML.gif
    (3.9)
     
where https://static-content.springer.com/image/art%3A10.1155%2F2008%2F243863/MediaObjects/13660_2007_Article_1780_IEq127_HTML.gif , and for any solution https://static-content.springer.com/image/art%3A10.1155%2F2008%2F243863/MediaObjects/13660_2007_Article_1780_IEq128_HTML.gif of (2.1), https://static-content.springer.com/image/art%3A10.1155%2F2008%2F243863/MediaObjects/13660_2007_Article_1780_IEq129_HTML.gif https://static-content.springer.com/image/art%3A10.1155%2F2008%2F243863/MediaObjects/13660_2007_Article_1780_IEq130_HTML.gif implies that
https://static-content.springer.com/image/art%3A10.1155%2F2008%2F243863/MediaObjects/13660_2007_Article_1780_Equ18_HTML.gif
(3.10)
Then
https://static-content.springer.com/image/art%3A10.1155%2F2008%2F243863/MediaObjects/13660_2007_Article_1780_Equ19_HTML.gif
(3.11)
https://static-content.springer.com/image/art%3A10.1155%2F2008%2F243863/MediaObjects/13660_2007_Article_1780_Equ20_HTML.gif
(3.12)

where https://static-content.springer.com/image/art%3A10.1155%2F2008%2F243863/MediaObjects/13660_2007_Article_1780_IEq131_HTML.gif and https://static-content.springer.com/image/art%3A10.1155%2F2008%2F243863/MediaObjects/13660_2007_Article_1780_IEq132_HTML.gif are the minimal and maximal solutions of (2.5), (2.6), respectively.

Proof.

Assume https://static-content.springer.com/image/art%3A10.1155%2F2008%2F243863/MediaObjects/13660_2007_Article_1780_IEq133_HTML.gif . First, we prove that (3.11) holds for https://static-content.springer.com/image/art%3A10.1155%2F2008%2F243863/MediaObjects/13660_2007_Article_1780_IEq134_HTML.gif , that is
https://static-content.springer.com/image/art%3A10.1155%2F2008%2F243863/MediaObjects/13660_2007_Article_1780_Equ21_HTML.gif
(3.13)
Let https://static-content.springer.com/image/art%3A10.1155%2F2008%2F243863/MediaObjects/13660_2007_Article_1780_IEq135_HTML.gif . Equation (3.13) holds obviously by Lemma 3.1 for https://static-content.springer.com/image/art%3A10.1155%2F2008%2F243863/MediaObjects/13660_2007_Article_1780_IEq136_HTML.gif . By (ii), https://static-content.springer.com/image/art%3A10.1155%2F2008%2F243863/MediaObjects/13660_2007_Article_1780_IEq137_HTML.gif . The same proof as for https://static-content.springer.com/image/art%3A10.1155%2F2008%2F243863/MediaObjects/13660_2007_Article_1780_IEq138_HTML.gif leads to
https://static-content.springer.com/image/art%3A10.1155%2F2008%2F243863/MediaObjects/13660_2007_Article_1780_Equ22_HTML.gif
(3.14)

By induction, (3.11) is correct. Similarly, (3.12) can be proved by using Lemma 3.1 and assumption (iii).

Using Lemma 3.2, we can easily get the following theorem about strict stability properties of (2.1).

Theorem 3.3.

Assume that all the conditions of Lemma 3.2 hold. Suppose further that there exist functions https://static-content.springer.com/image/art%3A10.1155%2F2008%2F243863/MediaObjects/13660_2007_Article_1780_IEq139_HTML.gif , such that
  1. (iv)
    https://static-content.springer.com/image/art%3A10.1155%2F2008%2F243863/MediaObjects/13660_2007_Article_1780_IEq140_HTML.gif
    .
     

Then the strict stability properties of comparison systems (2.5), (2.6) imply the corresponding strict stability properties of zero solution of (2.1).

Proof.

First, let us prove strict stability of the zero solution of (2.1). Suppose that https://static-content.springer.com/image/art%3A10.1155%2F2008%2F243863/MediaObjects/13660_2007_Article_1780_IEq141_HTML.gif and https://static-content.springer.com/image/art%3A10.1155%2F2008%2F243863/MediaObjects/13660_2007_Article_1780_IEq142_HTML.gif are given. Assume that (SS*) holds. Then, given https://static-content.springer.com/image/art%3A10.1155%2F2008%2F243863/MediaObjects/13660_2007_Article_1780_IEq143_HTML.gif , there exists https://static-content.springer.com/image/art%3A10.1155%2F2008%2F243863/MediaObjects/13660_2007_Article_1780_IEq144_HTML.gif , and https://static-content.springer.com/image/art%3A10.1155%2F2008%2F243863/MediaObjects/13660_2007_Article_1780_IEq145_HTML.gif satisfying https://static-content.springer.com/image/art%3A10.1155%2F2008%2F243863/MediaObjects/13660_2007_Article_1780_IEq146_HTML.gif such that
https://static-content.springer.com/image/art%3A10.1155%2F2008%2F243863/MediaObjects/13660_2007_Article_1780_Equ23_HTML.gif
(3.15)
By (iv), there exist https://static-content.springer.com/image/art%3A10.1155%2F2008%2F243863/MediaObjects/13660_2007_Article_1780_IEq147_HTML.gif such that for https://static-content.springer.com/image/art%3A10.1155%2F2008%2F243863/MediaObjects/13660_2007_Article_1780_IEq148_HTML.gif ,
https://static-content.springer.com/image/art%3A10.1155%2F2008%2F243863/MediaObjects/13660_2007_Article_1780_Equ24_HTML.gif
(3.16)

Next, choose https://static-content.springer.com/image/art%3A10.1155%2F2008%2F243863/MediaObjects/13660_2007_Article_1780_IEq149_HTML.gif such that https://static-content.springer.com/image/art%3A10.1155%2F2008%2F243863/MediaObjects/13660_2007_Article_1780_IEq150_HTML.gif and https://static-content.springer.com/image/art%3A10.1155%2F2008%2F243863/MediaObjects/13660_2007_Article_1780_IEq151_HTML.gif . We claim that with the choices of https://static-content.springer.com/image/art%3A10.1155%2F2008%2F243863/MediaObjects/13660_2007_Article_1780_IEq152_HTML.gif , and https://static-content.springer.com/image/art%3A10.1155%2F2008%2F243863/MediaObjects/13660_2007_Article_1780_IEq153_HTML.gif , the zero solution of (2.1) is strictly stable. That means that if https://static-content.springer.com/image/art%3A10.1155%2F2008%2F243863/MediaObjects/13660_2007_Article_1780_IEq154_HTML.gif is any solution of (2.1), https://static-content.springer.com/image/art%3A10.1155%2F2008%2F243863/MediaObjects/13660_2007_Article_1780_IEq155_HTML.gif implies that https://static-content.springer.com/image/art%3A10.1155%2F2008%2F243863/MediaObjects/13660_2007_Article_1780_IEq156_HTML.gif . If not, we have either of the following alternatives.

Case 1.

There exists a https://static-content.springer.com/image/art%3A10.1155%2F2008%2F243863/MediaObjects/13660_2007_Article_1780_IEq157_HTML.gif such that
https://static-content.springer.com/image/art%3A10.1155%2F2008%2F243863/MediaObjects/13660_2007_Article_1780_Equ25_HTML.gif
(3.17)
Then clearly https://static-content.springer.com/image/art%3A10.1155%2F2008%2F243863/MediaObjects/13660_2007_Article_1780_IEq158_HTML.gif . Thus, by Lemma 3.2, (i) and (ii) imply that
https://static-content.springer.com/image/art%3A10.1155%2F2008%2F243863/MediaObjects/13660_2007_Article_1780_Equ26_HTML.gif
(3.18)
Using (3.15)–(3.18) and (iv), we get
https://static-content.springer.com/image/art%3A10.1155%2F2008%2F243863/MediaObjects/13660_2007_Article_1780_Equ27_HTML.gif
(3.19)

which is a contradiction.

Case 2.

There exists a https://static-content.springer.com/image/art%3A10.1155%2F2008%2F243863/MediaObjects/13660_2007_Article_1780_IEq159_HTML.gif such that
https://static-content.springer.com/image/art%3A10.1155%2F2008%2F243863/MediaObjects/13660_2007_Article_1780_Equ28_HTML.gif
(3.20)
https://static-content.springer.com/image/art%3A10.1155%2F2008%2F243863/MediaObjects/13660_2007_Article_1780_Equ29_HTML.gif
(3.21)
By (H3), (3.21) yields
https://static-content.springer.com/image/art%3A10.1155%2F2008%2F243863/MediaObjects/13660_2007_Article_1780_Equ30_HTML.gif
(3.22)
Because of (3.20) and (3.22), there exists a https://static-content.springer.com/image/art%3A10.1155%2F2008%2F243863/MediaObjects/13660_2007_Article_1780_IEq160_HTML.gif such that
https://static-content.springer.com/image/art%3A10.1155%2F2008%2F243863/MediaObjects/13660_2007_Article_1780_Equ31_HTML.gif
(3.23)
By Lemma 3.2, (i) and (iii) imply that
https://static-content.springer.com/image/art%3A10.1155%2F2008%2F243863/MediaObjects/13660_2007_Article_1780_Equ32_HTML.gif
(3.24)
From (3.15), (3.23), (3.24), and (iv), we have the following contradiction:
https://static-content.springer.com/image/art%3A10.1155%2F2008%2F243863/MediaObjects/13660_2007_Article_1780_Equ33_HTML.gif
(3.25)

We, therefore, obtain the strict stability of the zero solution of (2.1). If we assume that the zero solutions of comparison systems (2.5), (2.6) are (SUS*), since https://static-content.springer.com/image/art%3A10.1155%2F2008%2F243863/MediaObjects/13660_2007_Article_1780_IEq161_HTML.gif are independent of https://static-content.springer.com/image/art%3A10.1155%2F2008%2F243863/MediaObjects/13660_2007_Article_1780_IEq162_HTML.gif , we obtain, because of (iv), https://static-content.springer.com/image/art%3A10.1155%2F2008%2F243863/MediaObjects/13660_2007_Article_1780_IEq163_HTML.gif and https://static-content.springer.com/image/art%3A10.1155%2F2008%2F243863/MediaObjects/13660_2007_Article_1780_IEq164_HTML.gif in (3.16) are independent of https://static-content.springer.com/image/art%3A10.1155%2F2008%2F243863/MediaObjects/13660_2007_Article_1780_IEq165_HTML.gif , and hence, (SUS) of (2.1) holds.

Using Theorem 3.3, we can get two direct results on strictly uniform stability of zero solution of (2.1) and the first one is Theorem 3.3 in [15].

Corollary 3.4.

In Theorem 3.3, suppose that https://static-content.springer.com/image/art%3A10.1155%2F2008%2F243863/MediaObjects/13660_2007_Article_1780_IEq166_HTML.gif https://static-content.springer.com/image/art%3A10.1155%2F2008%2F243863/MediaObjects/13660_2007_Article_1780_IEq167_HTML.gif , where https://static-content.springer.com/image/art%3A10.1155%2F2008%2F243863/MediaObjects/13660_2007_Article_1780_IEq168_HTML.gif , and https://static-content.springer.com/image/art%3A10.1155%2F2008%2F243863/MediaObjects/13660_2007_Article_1780_IEq169_HTML.gif https://static-content.springer.com/image/art%3A10.1155%2F2008%2F243863/MediaObjects/13660_2007_Article_1780_IEq170_HTML.gif .

Then the zero solution of (2.1) is strictly uniformly stable.

Corollary 3.5.

In Theorem 3.3, suppose that https://static-content.springer.com/image/art%3A10.1155%2F2008%2F243863/MediaObjects/13660_2007_Article_1780_IEq171_HTML.gif , where https://static-content.springer.com/image/art%3A10.1155%2F2008%2F243863/MediaObjects/13660_2007_Article_1780_IEq172_HTML.gif and https://static-content.springer.com/image/art%3A10.1155%2F2008%2F243863/MediaObjects/13660_2007_Article_1780_IEq173_HTML.gif are bounded, https://static-content.springer.com/image/art%3A10.1155%2F2008%2F243863/MediaObjects/13660_2007_Article_1780_IEq174_HTML.gif and https://static-content.springer.com/image/art%3A10.1155%2F2008%2F243863/MediaObjects/13660_2007_Article_1780_IEq175_HTML.gif are just the same as in Corollary 3.4.

Then the zero solution of (2.1) is strictly uniformly stable.

Proof.

Under the given hypotheses, it is easy to obtain the solutions of (2.5) and (2.6):
https://static-content.springer.com/image/art%3A10.1155%2F2008%2F243863/MediaObjects/13660_2007_Article_1780_Equ34_HTML.gif
(3.26)
Since https://static-content.springer.com/image/art%3A10.1155%2F2008%2F243863/MediaObjects/13660_2007_Article_1780_IEq176_HTML.gif are bounded, there exist two positive constants https://static-content.springer.com/image/art%3A10.1155%2F2008%2F243863/MediaObjects/13660_2007_Article_1780_IEq177_HTML.gif such that https://static-content.springer.com/image/art%3A10.1155%2F2008%2F243863/MediaObjects/13660_2007_Article_1780_IEq178_HTML.gif . Also, since https://static-content.springer.com/image/art%3A10.1155%2F2008%2F243863/MediaObjects/13660_2007_Article_1780_IEq179_HTML.gif , it follows that https://static-content.springer.com/image/art%3A10.1155%2F2008%2F243863/MediaObjects/13660_2007_Article_1780_IEq180_HTML.gif https://static-content.springer.com/image/art%3A10.1155%2F2008%2F243863/MediaObjects/13660_2007_Article_1780_IEq181_HTML.gif and https://static-content.springer.com/image/art%3A10.1155%2F2008%2F243863/MediaObjects/13660_2007_Article_1780_IEq182_HTML.gif , obviously https://static-content.springer.com/image/art%3A10.1155%2F2008%2F243863/MediaObjects/13660_2007_Article_1780_IEq183_HTML.gif . Given https://static-content.springer.com/image/art%3A10.1155%2F2008%2F243863/MediaObjects/13660_2007_Article_1780_IEq184_HTML.gif choose https://static-content.springer.com/image/art%3A10.1155%2F2008%2F243863/MediaObjects/13660_2007_Article_1780_IEq185_HTML.gif and for https://static-content.springer.com/image/art%3A10.1155%2F2008%2F243863/MediaObjects/13660_2007_Article_1780_IEq186_HTML.gif , choose https://static-content.springer.com/image/art%3A10.1155%2F2008%2F243863/MediaObjects/13660_2007_Article_1780_IEq187_HTML.gif . Then, if https://static-content.springer.com/image/art%3A10.1155%2F2008%2F243863/MediaObjects/13660_2007_Article_1780_IEq188_HTML.gif , we have
https://static-content.springer.com/image/art%3A10.1155%2F2008%2F243863/MediaObjects/13660_2007_Article_1780_Equ35_HTML.gif
(3.27)

That is, the zero solutions of (2.5), (2.6) are strictly uniformly stable. Hence, by Theorem 3.3, the zero solution of (2.1) is strictly uniformly stable.

Example 3.6.

Consider the system
https://static-content.springer.com/image/art%3A10.1155%2F2008%2F243863/MediaObjects/13660_2007_Article_1780_Equ36_HTML.gif
(3.28)

where https://static-content.springer.com/image/art%3A10.1155%2F2008%2F243863/MediaObjects/13660_2007_Article_1780_IEq189_HTML.gif are continuous on https://static-content.springer.com/image/art%3A10.1155%2F2008%2F243863/MediaObjects/13660_2007_Article_1780_IEq190_HTML.gif . Assume that https://static-content.springer.com/image/art%3A10.1155%2F2008%2F243863/MediaObjects/13660_2007_Article_1780_IEq191_HTML.gif https://static-content.springer.com/image/art%3A10.1155%2F2008%2F243863/MediaObjects/13660_2007_Article_1780_IEq192_HTML.gif with https://static-content.springer.com/image/art%3A10.1155%2F2008%2F243863/MediaObjects/13660_2007_Article_1780_IEq193_HTML.gif https://static-content.springer.com/image/art%3A10.1155%2F2008%2F243863/MediaObjects/13660_2007_Article_1780_IEq194_HTML.gif , and https://static-content.springer.com/image/art%3A10.1155%2F2008%2F243863/MediaObjects/13660_2007_Article_1780_IEq195_HTML.gif .

Let https://static-content.springer.com/image/art%3A10.1155%2F2008%2F243863/MediaObjects/13660_2007_Article_1780_IEq196_HTML.gif , then
https://static-content.springer.com/image/art%3A10.1155%2F2008%2F243863/MediaObjects/13660_2007_Article_1780_Equ37_HTML.gif
(3.29)
For any solution https://static-content.springer.com/image/art%3A10.1155%2F2008%2F243863/MediaObjects/13660_2007_Article_1780_IEq197_HTML.gif of (3.28) such that https://static-content.springer.com/image/art%3A10.1155%2F2008%2F243863/MediaObjects/13660_2007_Article_1780_IEq198_HTML.gif we have
https://static-content.springer.com/image/art%3A10.1155%2F2008%2F243863/MediaObjects/13660_2007_Article_1780_Equ38_HTML.gif
(3.30)
and if https://static-content.springer.com/image/art%3A10.1155%2F2008%2F243863/MediaObjects/13660_2007_Article_1780_IEq199_HTML.gif , we have
https://static-content.springer.com/image/art%3A10.1155%2F2008%2F243863/MediaObjects/13660_2007_Article_1780_Equ39_HTML.gif
(3.31)

By Corollary 3.5, the zero solution of (2.1) is strictly uniformly stable.

Declarations

Acknowledgments

This project is supported by the National Natural Science Foundation of China (60673101) and the Natural Science Foundation of Shandong Province (Y2007G30). The authors are grateful to the referees for their helful comments.

Authors’ Affiliations

(1)
School of Mathematics, Qingdao University
(2)
School of Automation Engineering, Qingdao University

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Copyright

© K. Liu and G. Yang. 2008

This article is published under license to BioMed Central Ltd. 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.