Journal of Applied Mathematics

Journal of Applied Mathematics / 2012 / Article

Research Article | Open Access

Volume 2012 |Article ID 501891 | https://doi.org/10.1155/2012/501891

Yutian Zhang, "Asymptotic Stability of Impulsive Reaction-Diffusion Cellular Neural Networks with Time-Varying Delays", Journal of Applied Mathematics, vol. 2012, Article ID 501891, 17 pages, 2012. https://doi.org/10.1155/2012/501891

Asymptotic Stability of Impulsive Reaction-Diffusion Cellular Neural Networks with Time-Varying Delays

Academic Editor: E. S. Van Vleck
Received27 Jun 2011
Accepted13 Sep 2011
Published22 Nov 2011

Abstract

This work addresses the asymptotic stability for a class of impulsive cellular neural networks with time-varying delays and reaction-diffusion. By using the impulsive integral inequality of Gronwall-Bellman type and Hardy-Sobolev inequality as well as piecewise continuous Lyapunov functions, we summarize some new and concise sufficient conditions ensuring the global exponential asymptotic stability of the equilibrium point. The provided stability criteria are applicable to Dirichlet boundary condition and showed to be dependent on all of the reaction-diffusion coefficients, the dimension of the space, the delay, and the boundary of the spatial variables. Two examples are finally illustrated to demonstrate the effectiveness of our obtained results.

1. Introduction

Cellular neural networks (CNNs), proposed by Chua and Yang in 1988 [1, 2], have been the focus of a number of investigations due to their potential applications in various fields such as optimization, linear and nonlinear programming, associative memory, pattern recognition, and computer vision [3โ€“7]. Moreover, on the ground that time delays are unavoidably encountered for the finite switching speed of neurons and amplifiers in implementation of neural networks, it was followed by the introduction of the delayed cellular neural networks (DCNNs) so as to solve some dynamic image processing and pattern recognition problems. Such applications concerning CNNs and DCNNs depend heavily on the dynamical behaviors such as stability, convergence, and oscillatory [8, 9]. Particularly, stability analysis has been a major concern in the designs and applications of the CNNs and DCNNs. The stability of CNNs and DCNNs is a subject of current interest, and considerable theoretical efforts have been put into this topic with many good results reported (see, e.g., [10โ€“13]).

With reference to neural networks, however, it is noteworthy that the state of electronic networks is actually subject to instantaneous perturbations more often than not. On this account, the networks experience abrupt change at certain instants which may be caused by a switching phenomenon, frequency change, or other sudden noise; that is, the networks often exhibit impulsive effects [14, 15]. For instance, according to Arbib [16] and Haykin [17], when a stimulus from the body or the external environment is received by receptors, the electrical impulses will be conveyed to the neural net and impulsive effects arise naturally in the net. As a consequence, in the past few years, scientists have become increasingly interested in the influence that impulses may have on the CNNs and DCNNs and a large number of stability criteria have been derived (see, e.g. [18โ€“22]).

In reality, besides impulsive effects, diffusion effects are also nonignorable since diffusion is unavoidable when electrons are moving in asymmetric electromagnetic fields. As such, the model of neural networks with both impulses and reaction-diffusion should be more accurate to describe the evolutionary process of the systems in question, and it is necessary to consider the effects of both diffusion and impulses on the stability of CNNs and DCNNs.

In the past years, there have been a few theoretical contributions to the stability of CNNs and DCNNs with impulses and diffusion. For instance, Qiu [23] formulated a mathematical model of impulsive neural networks with time-varying delays and reaction-diffusion terms described by impulsive partial differential equations and studied, via delay impulsive differential inequality, the problem of global exponential stability with some stability criteria presented. Remarkably, all of the obtained stability criteria in [23] are independent of the diffusion. In 2008, Li and song [24] investigated a class of impulsive Cohen-Grossberg networks with time-varying delays and reaction-diffusion terms. By establishing a delay inequality with impulsive initial conditions and M-matrix theory, some sufficient conditions ensuring global exponential stability of the equilibrium points are given. Analogous to [23], the proposed stability criteria in [24] are also independent of the diffusion. More recently Pan et al. [25] investigated a class of impulsive Cohen-Grossberg neural networks with time-varying delays and reaction-diffusion in 2010. By the aid of the delay impulsive differential inequality quoted in [23], several sufficient conditions are exploited ensuring global exponential stability of the equilibrium points. Especially, different from [23, 24], the estimate of the exponential convergence rate depends on reaction-diffusion in [25].

In this paper, unlike the methods of impulsive differential inequalities and Poincarรฉ inequality used in [25], we attempt to adopt the new techniques of the impulsive integral inequality of Gronwall-Bellman type and Hardy-Sobolev inequality to investigate the problem of global exponential asymptotic stability for impulsive cellular neural networks with time-varying delays and reaction-diffusion terms. Different from the existing research, we find, besides the reaction-diffusion coefficients, the dimension of the space and the boundary of the spatial variables do influence the stability.

The rest of the paper is organized as follows. In Section 2, the model of impulsive delayed cellular neural networks with reaction-diffusion terms and Dirichlet boundary condition is outlined, and some facts and lemmas are introduced for later reference. By the new agency of the impulsive integral inequality of Gronwall-Bellman type as well as Hardy-Sobolev inequality, we discuss the global exponential asymptotic stability and develop some new criteria in Section 3. To conclude, two illustrative examples are given to verify the effectiveness of our results in Section 4.

2. Preliminaries

Let ๐‘…๐‘› denote the n-dimensional Euclidean space, and ฮฉโŠ‚๐‘…๐‘š is a bounded open set containing the origin. The boundary of ฮฉ is smooth and mesฮฉ>0. Let ๐‘…+=[0,โˆž) and ๐‘ก0โˆˆ๐‘…+.

We consider the following impulsive neural networks with time delays and reaction-diffusion terms: ๐œ•๐‘ข๐‘–(๐‘ก,๐‘ฅ)=๐œ•๐‘ก๐‘š๎“๐‘ =1๐œ•๐œ•๐‘ฅ๐‘ ๎‚ต๐ท๐‘–๐‘ ๐œ•๐‘ข๐‘–(๐‘ก,๐‘ฅ)๐œ•๐‘ฅ๐‘ ๎‚ถโˆ’๐‘Ž๐‘–๐‘ข๐‘–(๐‘ก,๐‘ฅ)+๐‘›๎“๐‘—=1๐‘๐‘–๐‘—๐‘“๐‘—๎€ท๐‘ข๐‘—๎€ธ+(๐‘ก,๐‘ฅ)๐‘›๎“๐‘—=1๐‘๐‘–๐‘—๐‘“๐‘—๎€ท๐‘ข๐‘—๎€ท๐‘กโˆ’๐œ๐‘—(๐‘ก),๐‘ฅ๎€ธ๎€ธ๐‘กโ‰ฅ๐‘ก0,๐‘กโ‰ ๐‘ก๐‘˜๐‘ข,๐‘ฅโˆˆฮฉ,๐‘–=1,2,โ€ฆ,๐‘›,๐‘˜=1,2,โ€ฆ,(2.1)๐‘–๎€ท๐‘ก๐‘˜๎€ธ+0,๐‘ฅ=๐‘ข๐‘–๎€ท๐‘ก๐‘˜๎€ธ,๐‘ฅ+๐‘ƒ๐‘–๐‘˜๎€ท๐‘ข๐‘–๎€ท๐‘ก๐‘˜,๐‘ฅ๎€ธ๎€ธ๐‘ฅโˆˆฮฉ,๐‘˜=1,2,โ€ฆ,๐‘–=1,2,โ€ฆ,๐‘›,(2.2) where ๐‘› corresponds to the numbers of units in a neural network; ๐‘ฅ=(๐‘ฅ1,โ€ฆ,๐‘ฅ๐‘š)Tโˆˆฮฉ, ๐‘ข๐‘–(๐‘ก,๐‘ฅ), denotes the state of the ith neuron at time ๐‘ก and in space ๐‘ฅ; smooth functions ๐ท๐‘–๐‘ =๐ท๐‘–๐‘ (๐‘ก,๐‘ฅ,๐‘ข)โ‰ฅ0 represent transmission diffusion operators of the ith unit; activation functions ๐‘“๐‘—(๐‘ข๐‘—(๐‘ก,๐‘ฅ)) stand for the output of the jth unit at time ๐‘ก and in space ๐‘ฅ; ๐‘๐‘–๐‘—, ๐‘๐‘–๐‘—, ๐‘Ž๐‘– are constants: ๐‘๐‘–๐‘— indicates the strength of the jth unit on the ith unit at time ๐‘ก and in space ๐‘ฅ, ๐‘๐‘–๐‘— denotes the strength of the jth unit on the ith unit at time ๐‘กโˆ’๐œ๐‘—(๐‘ก) and in space ๐‘ฅ, where ๐œ๐‘—(๐‘ก) corresponds to the transmission delay along the axon of the jth unit and satisfies 0โ‰ค๐œ๐‘—(๐‘ก)โ‰ค๐œ (๐œ=const) as well as โ€ข๐œ๐‘—(๐‘ก)<1โˆ’1/โ„Ž (โ„Ž>0), and ๐‘Ž๐‘–>0 represents the rate with which the ith unit will reset its potential to the resting state in isolation when disconnected from the network and external inputs at time ๐‘ก and in space ๐‘ฅ. The fixed moments ๐‘ก๐‘˜ (๐‘˜=1,2,โ€ฆ) are called impulsive moments satisfying 0โ‰ค๐‘ก0<๐‘ก1<๐‘ก2<โ‹ฏ and lim๐‘˜โ†’โˆž๐‘ก๐‘˜=โˆž; ๐‘ข๐‘–(๐‘ก๐‘˜+0,๐‘ฅ) and ๐‘ข๐‘–(๐‘ก๐‘˜โˆ’0,๐‘ฅ) represent the right-hand and left-hand limit of ๐‘ข๐‘–(๐‘ก,๐‘ฅ) at time ๐‘ก๐‘˜ and in space ๐‘ฅ, respectively. ๐‘ƒ๐‘–๐‘˜(๐‘ข๐‘–(๐‘ก๐‘˜,๐‘ฅ)) stands for the abrupt change of ๐‘ข๐‘–(๐‘ก,๐‘ฅ) at impulsive moment ๐‘ก๐‘˜ and in space ๐‘ฅ.

Denote by ๐‘ข(๐‘ก,๐‘ฅ)=๐‘ข(๐‘ก,๐‘ฅ;๐‘ก0,๐œ‘), ๐‘ขโˆˆ๐‘…๐‘› the solution of system (2.1)-(2.2), satisfying the initial condition ๐‘ข๎€ท๐‘ ,๐‘ฅ;๐‘ก0๎€ธ,๐œ‘=๐œ‘(๐‘ ,๐‘ฅ),๐‘ก0โˆ’๐œโ‰ค๐‘ โ‰ค๐‘ก0,๐‘ฅโˆˆฮฉ(2.3) and Dirichlet boundary condition ๐‘ข๎€ท๐‘ก,๐‘ฅ;๐‘ก0๎€ธ,๐œ‘=0,๐‘กโ‰ฅ๐‘ก0,๐‘ฅโˆˆ๐œ•ฮฉ,(2.4) where the vector-valued function ๐œ‘(๐‘ ,๐‘ฅ)=(๐œ‘1(๐‘ ,๐‘ฅ),โ€ฆ,๐œ‘๐‘›(๐‘ ,๐‘ฅ))๐‘‡ is such that โˆซฮฉโˆ‘๐‘›๐‘–=1๐œ‘2๐‘–(๐‘ ,๐‘ฅ)๐‘‘๐‘ฅ is bounded on [๐‘ก0โˆ’๐œ,๐‘ก0] and ๐œ‘๐‘–(๐‘ ,๐‘ฅ) (๐‘–=1,2,โ€ฆ,๐‘›) is first-order continuous differentiable as to ๐‘  on [๐‘ก0โˆ’๐œ,๐‘ก0].

The solution ๐‘ข(๐‘ก,๐‘ฅ)=๐‘ข(๐‘ก,๐‘ฅ;๐‘ก0,๐œ‘)=(๐‘ข1(๐‘ก,๐‘ฅ;๐‘ก0,๐œ‘),โ€ฆ,๐‘ข๐‘›(๐‘ก,๐‘ฅ;๐‘ก0,๐œ‘))๐‘‡ of problems ((2.5)โ€“(2.8)) is, for the time variable ๐‘ก, a piecewise continuous function with the first kind discontinuity at the points ๐‘ก๐‘˜ (๐‘˜=1,2,โ€ฆ), where it is continuous from the left, that is the following relations are true:๐‘ข๐‘–๎€ท๐‘ก๐‘˜๎€ธโˆ’0,๐‘ฅ=๐‘ข๐‘–๎€ท๐‘ก๐‘˜๎€ธ,๐‘ฅ,๐‘ข๐‘–๎€ท๐‘ก๐‘˜๎€ธ+0,๐‘ฅ=๐‘ข๐‘–๎€ท๐‘ก๐‘˜๎€ธ,๐‘ฅ+๐‘ƒ๐‘–๐‘˜๎€ท๐‘ข๐‘–๎€ท๐‘ก๐‘˜,๐‘ฅ๎€ธ๎€ธ.(2.5)

Throughout this paper, the norm of ๐‘ข(๐‘ก,๐‘ฅ;๐‘ก0,๐œ‘) is governed by โ€–โ€–๐‘ข๎€ท๐‘ก,๐‘ฅ;๐‘ก0๎€ธโ€–โ€–,๐œ‘ฮฉ=โŽ›โŽœโŽœโŽ๐‘›๎“๐‘–=1๎€œฮฉ๐‘ข2๐‘–๎€ท๐‘ก,๐‘ฅ;๐‘ก0๎€ธโŽžโŽŸโŽŸโŽ ,๐œ‘d๐‘ฅ1/2.(2.6)

Before moving on, we introduce two hypotheses as follows.(H1) Activation function ๐‘“๐‘—(๐‘ข๐‘—(๐‘ก,๐‘ฅ)) satisfies ๐‘“๐‘–(0)=0, and there exists constant ๐‘™๐‘–>0 such that |๐‘“๐‘–(๐‘ฆ1)โˆ’๐‘“๐‘–(๐‘ฆ2)|โ‰ค๐‘™๐‘–|๐‘ฆ1โˆ’๐‘ฆ2| holds for all ๐‘ฆ1,๐‘ฆ2โˆˆ๐‘… and ๐‘–=1,2,โ€ฆ,๐‘›.(H2) The functions ๐‘ƒ๐‘–๐‘˜(๐‘ข๐‘–(๐‘ก๐‘˜,๐‘ฅ)) are continuous on ๐‘… and ๐‘ƒ๐‘–๐‘˜(0)=0, ๐‘–=1,2,โ€ฆ,๐‘›, ๐‘˜=1,2,โ€ฆ.

According to (H1) and (H2), it is easy to see that problems ((2.5)โ€“(2.8)) admits an equilibrium point ๐‘ข=0.

Definition 2.1. The equilibrium point ๐‘ข=0 of problems ((2.5)โ€“(2.8)) is said to be globally exponentially stable if there exist constants ๐œ…>0 and ๐‘€โ‰ฅ1 such that โ€–โ€–๐‘ข๎€ท๐‘ก,๐‘ฅ;๐‘ก0๎€ธโ€–โ€–,๐œ‘ฮฉโ‰ค๐‘€โ€–๐œ‘โ€–ฮฉ๐‘’โˆ’๐œ…(๐‘กโˆ’๐‘ก0),๐‘กโ‰ฅ๐‘ก0,(2.7) where โ€–๐œ‘โ€–2ฮฉ=sup๐‘ก0โˆ’๐œโ‰ค๐‘ โ‰ค๐‘ก0โˆ‘๐‘›๐‘–=1โˆซฮฉ๐œ‘2๐‘–(๐‘ ,๐‘ฅ)d๐‘ฅ.

Lemma 2.2 (Gronwall-Bellman-type impulsive integral inequality [26]). Assume that
(A1) the sequence {๐‘ก๐‘˜} satisfies 0โ‰ค๐‘ก0<๐‘ก1<๐‘ก2<โ‹ฏ, with lim๐‘˜โ†’โˆž๐‘ก๐‘˜=โˆž,
(A2) ๐‘žโˆˆ๐‘ƒ๐ถ1[๐‘…+,๐‘…] and ๐‘ž(๐‘ก) is left-continuous at ๐‘ก๐‘˜, ๐‘˜=1,2,โ€ฆ,
(A3) ๐‘โˆˆ๐ถ[๐‘…+,๐‘…+] and for ๐‘˜=1,2,โ€ฆ๎€œ๐‘ž(๐‘ก)โ‰ค๐‘+๐‘ก๐‘ก0๎“๐‘(๐‘ )๐‘ž(๐‘ )d๐‘ +๐‘ก0<๐‘ก๐‘˜<๐‘ก๐œ‚๐‘˜๐‘ž๎€ท๐‘ก๐‘˜๎€ธ,๐‘กโ‰ฅ๐‘ก0,(2.8) where ๐œ‚๐‘˜โ‰ฅ0 and ๐‘=const. Then, ๎‘๐‘ž(๐‘ก)โ‰ค๐‘๐‘ก0<๐‘ก๐‘˜<๐‘ก๎€ท1+๐œ‚๐‘˜๎€ธ๎‚ต๎€œexp๐‘ก๐‘ก0๎‚ถ๐‘(๐‘ )d๐‘ ,๐‘กโ‰ฅ๐‘ก0.(2.9)

Lemma 2.3 (Hardy-Sobolev inequality [27]). Let ฮฉโŠ‚๐‘…๐‘š(๐‘šโ‰ฅ3) be a bounded open set containing the origin and ๐‘ขโˆˆ๐ป1(ฮฉ)={๐œ”โˆฃ๐œ”โˆˆ๐ฟ2(ฮฉ),๐ท๐‘–๐œ”=๐œ•๐œ”/๐œ•๐‘ฅ๐‘–โˆˆ๐ฟ2(ฮฉ),1โ‰ค๐‘–โ‰ค๐‘š}. Then there exists a positive constant ๐ถ๐‘š=๐ถ๐‘š(ฮฉ) such that (๐‘šโˆ’2)24๎€œฮฉ๐‘ข2|๐‘ฅ|2๎€œ๐‘‘๐‘ฅโ‰คฮฉ||||โˆ‡๐‘ข2๐‘‘๐‘ฅ+๐ถ๐‘š๎€œ๐œ•ฮฉ๐‘ข2๐‘‘๐œŽ.(2.10)

Lemma 2.4. If ๐‘Ž>0 and ๐‘>0, then ๐‘Ž๐‘โ‰ค(1/๐œ€)๐‘Ž2+๐œ€๐‘2 holds for any ๐œ€>0.

3. Main Results

Theorem 3.1. Provided that(1)for ๐‘ฅ=(๐‘ฅ1,โ€ฆ,๐‘ฅ๐‘š)๐‘‡โˆˆฮฉ(๐‘šโ‰ฅ3), there exists a constant ๐›ฝ such that |๐‘ฅ|2=โˆ‘๐‘š๐‘ =1๐‘ฅ2๐‘ <๐›ฝ. In addition, there exists a constant ๐ท>0 such that ๐ท๐‘–๐‘ =๐ท๐‘–๐‘ (๐‘ก,๐‘ฅ,๐‘ข)โ‰ฅ๐ท>0. Denote ๐ท(๐‘šโˆ’2)2/2๐›ฝ=๐œ’,(2)๐‘ƒ๐‘–๐‘˜(๐‘ข๐‘–(๐‘ก๐‘˜,๐‘ฅ))=โˆ’๐œƒ๐‘–๐‘˜๐‘ข๐‘–(๐‘ก๐‘˜,๐‘ฅ), 0โ‰ค๐œƒ๐‘–๐‘˜โ‰ค2,(3)there exists a constant ๐›พ satisfying ๐›พ+๐œ†+โ„Ž๐œŒ๐‘’๐›พ๐œ>0 as well as ๐œ†+โ„Ž๐œŒ๐‘’๐›พ๐œ<0, where ๐œ†=max๐‘–=1,โ€ฆ,๐‘›(โˆ’๐œ’โˆ’2๐‘Ž๐‘–+โˆ‘๐‘›๐‘—=1(๐‘2๐‘–๐‘—+๐‘2๐‘–๐‘—))+๐œŒ, ๐œŒ=๐‘›max๐‘–=1,โ€ฆ,๐‘›(๐‘™๐‘–2),then, the equilibrium point ๐‘ข=0 of problems ((2.5)โ€“(2.8)) is globally exponentially stable with convergence rate โ€“ (๐œ†+โ„Ž๐œŒ๐‘’๐›พ๐œ)/2.

Proof. Multiplying both sides of (2.1) by ๐‘ข๐‘–(๐‘ก,๐‘ฅ) and integrating with respect to spatial variable ๐‘ฅ on ฮฉ, we get d๎€ทโˆซฮฉ๐‘ข๐‘–2๎€ธ(๐‘ก,๐‘ฅ)d๐‘ฅd๐‘ก=2๐‘š๎“๐‘ =1๎€œฮฉ๐‘ข๐‘–๐œ•(๐‘ก,๐‘ฅ)๐œ•๐‘ฅ๐‘ ๎‚ต๐ท๐‘–๐‘ ๐œ•๐‘ข๐‘–(๐‘ก,๐‘ฅ)๐œ•๐‘ฅ๐‘ ๎‚ถd๐‘ฅโˆ’2๐‘Ž๐‘–๎€œฮฉ๐‘ข2๐‘–(๐‘ก,๐‘ฅ)d๐‘ฅ+2๐‘›๎“๐‘—=1๐‘๐‘–๐‘—๎€œฮฉ๐‘ข๐‘–(๐‘ก,๐‘ฅ)๐‘“๐‘—๎€ท๐‘ข๐‘—๎€ธ(๐‘ก,๐‘ฅ)d๐‘ฅ+2๐‘›๎“๐‘—=1๐‘๐‘–๐‘—๎€œฮฉ๐‘ข๐‘–(๐‘ก,๐‘ฅ)๐‘“๐‘—๎€ท๐‘ข๐‘—๎€ท๐‘กโˆ’๐œ๐‘—(๐‘ก),๐‘ฅ๎€ธ๎€ธd๐‘ฅ๐‘กโ‰ฅ๐‘ก0,๐‘กโ‰ ๐‘ก๐‘˜,๐‘˜=1,2,โ€ฆ.(3.1)
Regarding the right-hand part of (3.1), the first term becomes by using Green formula, Dirichlet boundary condition, Lemma 2.3, and condition 1 of Theorem 3.12๐‘š๎“๐‘ =1๎€œฮฉ๐‘ข๐‘–๐œ•(๐‘ก,๐‘ฅ)๐œ•๐‘ฅ๐‘ ๎‚ต๐ท๐‘–๐‘ ๐œ•๐‘ข๐‘–(๐‘ก,๐‘ฅ)๐œ•๐‘ฅ๐‘ ๎‚ถd๐‘ฅ=โˆ’2๐‘š๎“๐‘ =1๎€œฮฉ๐ท๐‘–๐‘ ๎‚ต๐œ•๐‘ข๐‘–(๐‘ก,๐‘ฅ)๐œ•๐‘ฅ๐‘ ๎‚ถ2๐ทd๐‘ฅโ‰คโˆ’(๐‘šโˆ’2)22๎€œฮฉ๐‘ข2๐‘–(๐‘ก,๐‘ฅ)|๐‘ฅ|2๐ท๐‘‘๐‘ฅโ‰คโˆ’(๐‘šโˆ’2)2๎€œ2๐›ฝฮฉ๐‘ข2๐‘–๎€œ(๐‘ก,๐‘ฅ)๐‘‘๐‘ฅโ‰œโˆ’๐œ’ฮฉ๐‘ข2๐‘–(๐‘ก,๐‘ฅ)d๐‘ฅ.(3.2)
Moreover, we derive from (H1) that 2๐‘›๎“๐‘—=1๐‘๐‘–๐‘—๎€œฮฉ๐‘ข๐‘–(๐‘ก,๐‘ฅ)๐‘“๐‘—๎€ท๐‘ข๐‘—๎€ธ(๐‘ก,๐‘ฅ)d๐‘ฅโ‰ค2๐‘›๎“๐‘—=1||๐‘๐‘–๐‘—||๎€œฮฉ||๐‘ข๐‘–||||๐‘“(๐‘ก,๐‘ฅ)๐‘—๎€ท๐‘ข๐‘—๎€ธ||(๐‘ก,๐‘ฅ)d๐‘ฅโ‰ค2๐‘›๎“๐‘—=1๎€œฮฉ๐‘™๐‘—||๐‘๐‘–๐‘—||||๐‘ข๐‘–||||๐‘ข(๐‘ก,๐‘ฅ)๐‘—||โ‰ค(๐‘ก,๐‘ฅ)d๐‘ฅ๐‘›๎“๐‘—=1๎€œฮฉ๎€ท๐‘2๐‘–๐‘—๐‘ข2๐‘–(๐‘ก,๐‘ฅ)+๐‘™2๐‘—๐‘ข2๐‘–๎€ธ2(๐‘ก,๐‘ฅ)d๐‘ฅ,๐‘›๎“๐‘—=1๐‘๐‘–๐‘—๎€œฮฉ๐‘ข๐‘–(๐‘ก,๐‘ฅ)๐‘“๐‘—๎€ท๐‘ข๐‘—๎€ท๐‘กโˆ’๐œ๐‘—(๐‘ก),๐‘ฅ๎€ธ๎€ธ๐‘‘๐‘ฅโ‰ค2๐‘›๎“๐‘—=1||๐‘๐‘–๐‘—||๎€œฮฉ||๐‘ข๐‘–||||๐‘“(๐‘ก,๐‘ฅ)๐‘—๎€ท๐‘ข๐‘—๎€ท๐‘กโˆ’๐œ๐‘—||(๐‘ก),๐‘ฅ๎€ธ๎€ธ๐‘‘๐‘ฅโ‰ค2๐‘›๎“๐‘—=1๎€œฮฉ๐‘™๐‘—||๐‘๐‘–๐‘—||||๐‘ข๐‘–||||๐‘ข(๐‘ก,๐‘ฅ)๐‘—๎€ท๐‘กโˆ’๐œ๐‘—๎€ธ||โ‰ค(๐‘ก),๐‘ฅ๐‘‘๐‘ฅ๐‘›๎“๐‘—=1๎€œฮฉ๎€ท๐‘2๐‘–๐‘—๐‘ข2๐‘–(๐‘ก,๐‘ฅ)+๐‘™2๐‘—๐‘ข2๐‘–๎€ท๐‘กโˆ’๐œ๐‘—(๐‘ก),๐‘ฅ๎€ธ๎€ธ๐‘‘๐‘ฅ.(3.3) Consequently, substituting ((2.10)โ€“(3.14)) into (3.1) produces d๎€ทโˆซฮฉ๐‘ข2๐‘–๎€ธ(๐‘ก,๐‘ฅ)d๐‘ฅ๎€œd๐‘กโ‰คโˆ’๐œ’ฮฉ๐‘ข2๐‘–(๐‘ก,๐‘ฅ)d๐‘ฅโˆ’2๐‘Ž๐‘–๎€œฮฉ๐‘ข2๐‘–+(๐‘ก,๐‘ฅ)d๐‘ฅ๐‘›๎“๐‘—=1๎€œฮฉ๎€ท๐‘2๐‘–๐‘—๐‘ข2๐‘–(๐‘ก,๐‘ฅ)+๐‘™2๐‘—๐‘ข2๐‘–๎€ธ+(๐‘ก,๐‘ฅ)d๐‘ฅ๐‘›๎“๐‘—=1๎€œฮฉ๎€ท๐‘2๐‘–๐‘—๐‘ข2๐‘–(๐‘ก,๐‘ฅ)+๐‘™2๐‘—๐‘ข2๐‘–๎€ท๐‘กโˆ’๐œ๐‘—(๐‘ก),๐‘ฅ๎€ธ๎€ธd๐‘ฅ(3.4)for๐‘กโ‰ฅ๐‘ก0,๐‘กโ‰ ๐‘ก๐‘˜,๐‘˜=1,2,โ€ฆ.
We define a Lyapunov function ๐‘‰๐‘–(๐‘ก) as ๐‘‰๐‘–โˆซ(๐‘ก)=ฮฉ๐‘ข2๐‘–(๐‘ก,๐‘ฅ)d๐‘ฅ. It is easy to find that ๐‘‰๐‘–(๐‘ก) is a piecewise continuous function with points of discontinuity of the first kind ๐‘ก๐‘˜ (๐‘˜=1,2,โ€ฆ), where it is continuous from the left, that is, ๐‘‰๐‘–(๐‘ก๐‘˜โˆ’0)=๐‘‰๐‘–(๐‘ก๐‘˜) (๐‘˜=1,2,โ€ฆ). In addition, due to ๐‘‰๐‘–(๐‘ก0+0)โ‰ค๐‘‰๐‘–(๐‘ก0) and the following estimate derived from condition 2 of Theorem 3.1๐‘ข2๐‘–๎€ท๐‘ก๐‘˜๎€ธ=๎€ท+0,๐‘ฅโˆ’๐œƒ๐‘–๐‘˜๐‘ข๐‘–๎€ท๐‘ก๐‘˜๎€ธ,๐‘ฅ+๐‘ข๐‘–๎€ท๐‘ก๐‘˜,๐‘ฅ๎€ธ๎€ธ2=๎€ท1โˆ’๐œƒ๐‘–๐‘˜๎€ธ2๐‘ข2๐‘–๎€ท๐‘ก๐‘˜๎€ธ,๐‘ฅโ‰ค๐‘ข2๐‘–๎€ท๐‘ก๐‘˜๎€ธ,๐‘ฅ(๐‘˜=1,2,โ€ฆ),(3.5) we have๐‘‰๐‘–๎€ท๐‘ก๐‘˜๎€ธ+0โ‰ค๐‘‰๐‘–๎€ท๐‘ก๐‘˜๎€ธ,๐‘˜=0,1,2,โ€ฆ.(3.6) holds for ๐‘ก=๐‘ก๐‘˜ (๐‘˜=0,1,2,โ€ฆ). Put ๐‘กโˆˆ(๐‘ก๐‘˜,๐‘ก๐‘˜+1), ๐‘˜=0,1,2,โ€ฆ. Then for the derivative d๐‘‰๐‘–(๐‘ก)/d๐‘ก of ๐‘‰๐‘– with respect to problems ((2.5)โ€“(2.8)), it results from (3.4) that d๐‘‰๐‘–(๐‘ก)๎€œd๐‘กโ‰คโˆ’๐œ’ฮฉ๐‘ข2๐‘–(๐‘ก,๐‘ฅ)d๐‘ฅโˆ’2๐‘Ž๐‘–๎€œฮฉ๐‘ข2๐‘–(๐‘ก,๐‘ฅ)d๐‘ฅ+๐‘›๎“๐‘—=1๎€œฮฉ๎€ท๐‘2๐‘–๐‘—๐‘ข2๐‘–(๐‘ก,๐‘ฅ)+๐‘™2๐‘—๐‘ข2๐‘–๎€ธ+(๐‘ก,๐‘ฅ)d๐‘ฅ๐‘›๎“๐‘—=1๎€œฮฉ๎€ท๐‘2๐‘–๐‘—๐‘ข2๐‘–(๐‘ก,๐‘ฅ)+๐‘™2๐‘—๐‘ข2๐‘—๎€ท๐‘กโˆ’๐œ๐‘—๎ƒฉ(๐‘ก),๐‘ฅ๎€ธ๎€ธd๐‘ฅโ‰คโˆ’๐œ’โˆ’2๐‘Ž๐‘–+๐‘›๎“๐‘—=1๐‘2๐‘–๐‘—+๐‘›๎“๐‘—=1๐‘2๐‘–๐‘—๎ƒช๐‘‰๐‘–(๐‘ก)+max๐‘–=1,โ€ฆ,๐‘›๎€ท๐‘™2๐‘–๎€ธ๐‘›๎“๐‘—=1๐‘‰๐‘—(๐‘ก)+max๐‘–=1,โ€ฆ,๐‘›๎€ท๐‘™๐‘–2๎€ธ๐‘›๎“๐‘—=1๐‘‰๐‘—๎€ท๐‘กโˆ’๐œ๐‘—๎€ธ๎€ท๐‘ก(๐‘ก)๐‘กโˆˆ๐‘˜,๐‘ก๐‘˜+1๎€ธ,๐‘˜=0,1,2,โ€ฆ.(3.7) Choose ๐‘‰(๐‘ก) of the form โˆ‘๐‘‰(๐‘ก)=๐‘›๐‘–=1๐‘‰๐‘–(๐‘ก). From (3.7), one then reads d๐‘‰(๐‘ก)โ‰ค๎ƒฉd๐‘กmax๐‘–=1,โ€ฆ,๐‘›๎ƒฉโˆ’๐œ’โˆ’2๐‘Ž๐‘–+๐‘›๎“๐‘—=1๎€ท๐‘2๐‘–๐‘—+๐‘2๐‘–๐‘—๎€ธ๎ƒช+๐‘›max๐‘–=1,โ€ฆ,๐‘›๎€ท๐‘™๐‘–2๎€ธ๎ƒช๐‘‰(๐‘ก)+๐‘›max๐‘–=1,โ€ฆ,๐‘›๎€ท๐‘™๐‘–2๎€ธ๐‘›๎“๐‘—=1๐‘‰๐‘—๎€ท๐‘กโˆ’๐œ๐‘—๎€ธ=๐œ†๐‘‰(๐‘ก)+๐œŒ๐‘›๎“๐‘—=1๐‘‰๐‘—๎€ท๐‘กโˆ’๐œ๐‘—๎€ธ๎€ท๐‘ก(๐‘ก)๐‘กโˆˆ๐‘˜,๐‘ก๐‘˜+1๎€ธ,๐‘˜=0,1,2,โ€ฆ.(3.8)
Construct ๐‘‰โˆ—(๐‘ก)=e๐›พ(๐‘กโˆ’๐‘ก0)๐‘‰(๐‘ก), where ๐›พ satisfies ๐›พ+๐œ†+โ„Ž๐œŒ๐‘’๐›พ๐œ>0 and ๐œ†+โ„Ž๐œŒ๐‘’๐›พ๐œ<0. Evidently, ๐‘‰โˆ—(๐‘ก) is also a piecewise continuous function with points of discontinuity of the first kind ๐‘ก๐‘˜ (๐‘˜=1,2,โ€ฆ), in which it is continuous from the left, that is ๐‘‰โˆ—(๐‘ก๐‘˜โˆ’0)=๐‘‰โˆ—(๐‘ก๐‘˜) (๐‘˜=1,2,โ€ฆ). Moreover, at ๐‘ก=๐‘ก๐‘˜ (๐‘˜=0,1,2,โ€ฆ), we find by use of (3.6) ๐‘‰โˆ—๎€ท๐‘ก๐‘˜๎€ธ+0โ‰ค๐‘‰โˆ—๎€ท๐‘ก๐‘˜๎€ธ,๐‘˜=0,1,2,โ€ฆ.(3.9)
Set ๐‘กโˆˆ(๐‘ก๐‘˜,๐‘ก๐‘˜+1), ๐‘˜=0,1,2,โ€ฆ. By virtue of (3.8), one has d๐‘‰โˆ—(๐‘ก)d๐‘ก=๐›พe๐›พ(๐‘กโˆ’๐‘ก0)๐‘‰(๐‘ก)+e๐›พ(๐‘กโˆ’๐‘ก0)d๐‘‰(๐‘ก)d๐‘กโ‰ค๐›พe๐›พ(๐‘กโˆ’๐‘ก0)๎ƒฉ๐‘‰(๐‘ก)+๐œ†๐‘‰(๐‘ก)+๐œŒ๐‘›๎“๐‘—=1๐‘‰๐‘—๎€ท๐‘กโˆ’๐œ๐‘—๎€ธ๎ƒชe(๐‘ก)๐›พ(๐‘กโˆ’๐‘ก0)=(๐›พ+๐œ†)๐‘‰โˆ—(๐‘ก)+๐œŒe๐›พ(๐‘กโˆ’๐‘ก0)๐‘›๎“๐‘—=1๐‘‰๐‘—๎€ท๐‘กโˆ’๐œ๐‘—๎€ธ๎€ท๐‘ก(๐‘ก)๐‘กโˆˆ๐‘˜,๐‘ก๐‘˜+1๎€ธ,๐‘˜=0,1,2,โ€ฆ.(3.10)
Choose small enough ๐œ€>0. Integrating (3.10) from ๐‘ก๐‘˜+๐œ€ to ๐‘ก gives ๐‘‰โˆ—(๐‘ก)โ‰ค๐‘‰โˆ—๎€ท๐‘ก๐‘˜๎€ธ๎€œ+๐œ€+(๐›พ+๐œ†)๐‘ก๐‘ก๐‘˜+๐œ€๐‘‰โˆ—(+๎€œ๐‘ )๐‘‘๐‘ ๐‘ก๐‘ก๐‘˜+๐œ€๐œŒe๐›พ(๐‘ โˆ’๐‘ก0)๐‘›๎“๐‘—=1๐‘‰๐‘—๎€ท๐‘ โˆ’๐œ๐‘—๎€ธ๎€ท๐‘ก(๐‘ )๐‘‘๐‘ ,๐‘กโˆˆ๐‘˜,๐‘ก๐‘˜+1๎€ธ,๐‘˜=0,1,2,โ€ฆ(3.11)
which yields after letting ๐œ€โ†’0 in (3.11) ๐‘‰โˆ—(๐‘ก)โ‰ค๐‘‰โˆ—๎€ท๐‘ก๐‘˜๎€ธ๎€œ+0+(๐›พ+๐œ†)๐‘ก๐‘ก๐‘˜๐‘‰โˆ—(+๎€œ๐‘ )๐‘‘๐‘ ๐‘ก๐‘ก๐‘˜๐œŒe๐›พ(๐‘ โˆ’๐‘ก0)๐‘›๎“๐‘—=1๐‘‰๐‘—๎€ท๐‘ โˆ’๐œ๐‘—๎€ธ๎€ท๐‘ก(๐‘ )๐‘‘๐‘ ,๐‘กโˆˆ๐‘˜,๐‘ก๐‘˜+1๎€ธ,๐‘˜=0,1,2,โ€ฆ.(3.12)
We now proceed to estimate the value of ๐‘‰โˆ—(๐‘ก) at ๐‘ก=๐‘ก๐‘˜+1, ๐‘˜=0,1,2,โ€ฆ. For small enough ๐œ€>0, we put ๐‘ก=๐‘ก๐‘˜+1โˆ’๐œ€. Now an application of (3.12) leads to, for ๐‘˜=0,1,2,โ€ฆ,๐‘‰โˆ—๎€ท๐‘ก๐‘˜+1๎€ธโˆ’๐œ€โ‰ค๐‘‰โˆ—๎€ท๐‘ก๐‘˜๎€ธ๎€œ+0+(๐›พ+๐œ†)๐‘ก๐‘˜+1๐‘กโˆ’๐œ€๐‘˜๐‘‰โˆ—๎€œ(๐‘ )๐‘‘๐‘ +๐‘ก๐‘˜+1๐‘กโˆ’๐œ€๐‘˜๐œŒe๐›พ(๐‘ โˆ’๐‘ก0)๐‘›๎“๐‘—=1๐‘‰๐‘—๎€ท๐‘ โˆ’๐œ๐‘—๎€ธ(๐‘ )๐‘‘๐‘ .(3.13)
If we let ๐œ€โ†’0 in (3.13), there results ๐‘‰โˆ—๎€ท๐‘ก๐‘˜+1๎€ธโˆ’0โ‰ค๐‘‰โˆ—๎€ท๐‘ก๐‘˜๎€ธ๎€œ+0+(๐›พ+๐œ†)๐‘ก๐‘˜+1๐‘ก๐‘˜๐‘‰โˆ—(+๎€œ๐‘ )๐‘‘๐‘ ๐‘ก๐‘˜+1๐‘ก๐‘˜๐œŒe๐›พ(๐‘ โˆ’๐‘ก0)๐‘›๎“๐‘—=1๐‘‰๐‘—๎€ท๐‘ โˆ’๐œ๐‘—๎€ธ(๐‘ )๐‘‘๐‘ ,๐‘˜=0,1,2,โ€ฆ.(3.14)
Note that ๐‘‰โˆ—(๐‘ก๐‘˜+1โˆ’0)=๐‘‰โˆ—(๐‘ก๐‘˜+1) is applicable for ๐‘˜=0,1,2,โ€ฆ. Thus, ๐‘‰โˆ—๎€ท๐‘ก๐‘˜+1๎€ธโ‰ค๐‘‰โˆ—๎€ท๐‘ก๐‘˜๎€ธ๎€œ+0+(๐›พ+๐œ†)๐‘ก๐‘˜+1๐‘ก๐‘˜๐‘‰โˆ—๎€œ(๐‘ )๐‘‘๐‘ +๐‘ก๐‘˜+1๐‘ก๐‘˜๐œŒe๐›พ(๐‘ โˆ’๐‘ก0)๐‘›๎“๐‘—=1๐‘‰๐‘—๎€ท๐‘ โˆ’๐œ๐‘—๎€ธ(๐‘ )๐‘‘๐‘ (3.15)
holds for ๐‘˜=0,1,2,โ€ฆ. By synthesizing (3.12) and (3.15), we then arrive at ๐‘‰โˆ—(๐‘ก)โ‰ค๐‘‰โˆ—๎€ท๐‘ก๐‘˜๎€ธ๎€œ+0+(๐›พ+๐œ†)๐‘ก๐‘ก๐‘˜๐‘‰โˆ—(+๎€œ๐‘ )๐‘‘๐‘ ๐‘ก๐‘ก๐‘˜๐œŒe๐›พ(๐‘ โˆ’๐‘ก0)๐‘›๎“๐‘—=1๐‘‰๐‘—๎€ท๐‘ โˆ’๐œ๐‘—๎€ธ๎€ท๐‘ก(๐‘ )๐‘‘๐‘ ๐‘กโˆˆ๐‘˜,๐‘ก๐‘˜+1๎€ป,๐‘˜=0,1,2,โ€ฆ.(3.16)
This, together with (3.9), results in ๐‘‰โˆ—(๐‘ก)โ‰ค๐‘‰โˆ—๎€ท๐‘ก๐‘˜๎€ธ๎€œ+(๐›พ+๐œ†)๐‘ก๐‘ก๐‘˜๐‘‰โˆ—๎€œ(๐‘ )๐‘‘๐‘ +๐‘ก๐‘ก๐‘˜๐œŒe๐›พ(๐‘ โˆ’๐‘ก0)๐‘›๎“๐‘—=1๐‘‰๐‘—๎€ท๐‘ โˆ’๐œ๐‘—๎€ธ(๐‘ )๐‘‘๐‘ (3.17)for๐‘กโˆˆ(๐‘ก๐‘˜,๐‘ก๐‘˜+1],๐‘˜=0,1,2,โ€ฆ.
Recalling assumptions that 0โ‰ค๐œ๐‘—(๐‘ก)โ‰ค๐œ and โ€ข๐œ๐‘—(๐‘ก)<1โˆ’1/โ„Ž (โ„Ž>0), we have ๎€œ๐‘ก๐‘ก๐‘˜๐œŒe๐›พ(๐‘ โˆ’๐‘ก0)๐‘›๎“๐‘—=1๐‘‰๐‘—๎€ท๐‘ โˆ’๐œ๐‘—๎€ธ((๐‘ ))๐‘‘๐‘ =๐‘›๎“๐‘—=1๎€œ๐‘กโˆ’๐œ๐‘—๐‘ก(๐‘ก)๐‘˜โˆ’๐œ๐‘—๎€ท๐‘ก๐‘˜๎€ธ๐œŒe๐›พ(๐œƒ+๐œ๐‘—(๐‘ )โˆ’๐‘ก0)๐‘‰๐‘—1(๐œƒ)1โˆ’โ€ข๐œ๐‘—(๐‘ )๐‘‘๐œƒโ‰คโ„Ž๐œŒ๐‘’๐‘›๐›พ๐œ๎“๐‘—=1๎€œ๐‘กโˆ’๐œ๐‘—๐‘ก(๐‘ก)๐‘˜โˆ’๐œ๐‘—๎€ท๐‘ก๐‘˜๎€ธe๐›พ(๐œƒโˆ’๐‘ก0)๐‘‰๐‘—(๐œƒ)๐‘‘๐œƒ.(3.18) Hence, ๐‘‰โˆ—(๐‘ก)โ‰ค๐‘‰โˆ—๎€ท๐‘ก๐‘˜๎€ธ๎€œ+(๐›พ+๐œ†)๐‘ก๐‘ก๐‘˜๐‘‰โˆ—(๐‘ )๐‘‘๐‘ +โ„Ž๐œŒ๐‘’๐‘›๐›พ๐œ๎“๐‘—=1๎€œ๐‘กโˆ’๐œ๐‘—๐‘ก(๐‘ก)๐‘˜โˆ’๐œ๐‘—๎€ท๐‘ก๐‘˜๎€ธe๐›พ(๐‘ โˆ’๐‘ก0)๐‘‰๐‘—๎€ท๐‘ก(๐‘ )๐‘‘๐‘ ๐‘กโˆˆ๐‘˜,๐‘ก๐‘˜+1๎€ป,๐‘˜=0,1,2,โ€ฆ.(3.19)
By induction argument, we reach ๐‘‰โˆ—๎€ท๐‘ก๐‘˜๎€ธโ‰ค๐‘‰โˆ—๎€ท๐‘ก๐‘˜โˆ’1๎€ธ๎€œ+(๐›พ+๐œ†)๐‘ก๐‘˜๐‘ก๐‘˜โˆ’1๐‘‰โˆ—(๐‘ )๐‘‘๐‘ +โ„Ž๐œŒ๐‘’๐‘›๐›พ๐œ๎“๐‘—=1๎€œ๐‘ก๐‘˜โˆ’๐œ๐‘—(๐‘ก๐‘˜)๐‘ก๐‘˜โˆ’1โˆ’๐œ๐‘—๎€ท๐‘ก๐‘˜โˆ’1๎€ธe๐›พ(๐‘ โˆ’๐‘ก0)๐‘‰๐‘—โ‹ฎ๐‘‰(๐‘ )๐‘‘๐‘ ,โˆ—๎€ท๐‘ก2๎€ธโ‰ค๐‘‰โˆ—๎€ท๐‘ก1๎€ธ๎€œ+(๐›พ+๐œ†)๐‘ก2๐‘ก1๐‘‰โˆ—(๐‘ )๐‘‘๐‘ +โ„Ž๐œŒ๐‘’๐‘›๐›พ๐œ๎“๐‘—=1๎€œ๐‘ก2โˆ’๐œ๐‘—(๐‘ก2)๐‘ก1โˆ’๐œ๐‘—๎€ท๐‘ก1๎€ธe๐›พ(๐‘ โˆ’๐‘ก0)๐‘‰๐‘—๐‘‰(๐‘ )๐‘‘๐‘ ,โˆ—๎€ท๐‘ก1๎€ธโ‰ค๐‘‰โˆ—๎€ท๐‘ก0๎€ธ๎€œ+(๐›พ+๐œ†)๐‘ก1๐‘ก0๐‘‰โˆ—(๐‘ )๐‘‘๐‘ +โ„Ž๐œŒ๐‘’๐‘›๐›พ๐œ๎“๐‘—=1๎€œ๐‘ก1โˆ’๐œ๐‘—(๐‘ก1)๐‘ก0โˆ’๐œ๐‘—๎€ท๐‘ก0๎€ธe๐›พ(๐‘ โˆ’๐‘ก0)๐‘‰๐‘—(๐‘ )๐‘‘๐‘ .(3.20)
Therefore, ๐‘‰โˆ—(๐‘ก)โ‰ค๐‘‰โˆ—๎€ท๐‘ก0๎€ธ๎€œ+(๐›พ+๐œ†)๐‘ก๐‘ก0๐‘‰โˆ—(๐‘ )๐‘‘๐‘ +โ„Ž๐œŒ๐‘’๐‘›๐›พ๐œ๎“๐‘—=1๎€œ๐‘กโˆ’๐œ๐‘—๐‘ก(๐‘ก)0โˆ’๐œ๐‘—๎€ท๐‘ก0๎€ธe๐›พ(๐‘ โˆ’๐‘ก0)๐‘‰๐‘—(๐‘ )๐‘‘๐‘ โ‰ค๐‘‰โˆ—๎€ท๐‘ก0๎€ธ๎€œ+(๐›พ+๐œ†)๐‘ก๐‘ก0๐‘‰โˆ—(๐‘ )๐‘‘๐‘ +โ„Ž๐œŒ๐‘’๐‘›๐›พ๐œ๎“๐‘—=1๎€œ๐‘ก๐‘ก0โˆ’๐œ๐‘—๎€ท๐‘ก0๎€ธe๐›พ(๐‘ โˆ’๐‘ก0)๐‘‰๐‘—(๐‘ )๐‘‘๐‘ =๐‘‰โˆ—๎€ท๐‘ก0๎€ธ+(๐›พ+๐œ†+โ„Ž๐œŒ๐‘’๐›พ๐œ)๎€œ๐‘ก๐‘ก0๐‘‰โˆ—(๐‘ )๐‘‘๐‘ +โ„Ž๐œŒ๐‘’๐‘›๐›พ๐œ๎“๐‘—=1๎€œ๐‘ก0๐‘ก0โˆ’๐œ๐‘—๎€ท๐‘ก0๎€ธe๐›พ(๐‘ โˆ’๐‘ก0)๐‘‰๐‘—๎€ท๐‘ก(๐‘ )๐‘‘๐‘ ๐‘กโˆˆ๐‘˜,๐‘ก๐‘˜+1๎€ป,๐‘˜=0,1,2,โ€ฆ.(3.21) Since โ„Ž๐œŒ๐‘’๐‘›๐›พ๐œ๎“๐‘—=1๎€œ๐‘ก0๐‘ก0โˆ’๐œ๐‘—๎€ท๐‘ก0๎€ธe๐›พ(๐‘ โˆ’๐‘ก0)๐‘‰๐‘—(๐‘ )๐‘‘๐‘ โ‰คโ„Ž๐œŒ๐‘’๐‘›๐›พ๐œ๎“๐‘—=1๎€œ๐‘ก0๐‘ก0โˆ’๐œ๐‘‰๐‘—(๐‘ )๐‘‘๐‘ =โ„Ž๐œŒ๐‘’๐›พ๐œ๎€œ๐‘ก0๐‘ก0โˆ’๐œ๎ƒฉ๐‘›๎“๐‘—=1๎€œฮฉ๐œ‘2๐‘—๎ƒช(๐‘ ,๐‘ฅ)๐‘‘๐‘ฅ๐‘‘๐‘ โ‰ค๐œโ„Ž๐œŒ๐‘’๐›พ๐œโ€–๐œ‘โ€–2ฮฉ,(3.22) we claim ๐‘‰โˆ—(๐‘ก)โ‰ค๐‘‰โˆ—๎€ท๐‘ก0๎€ธ+๐œโ„Ž๐œŒ๐‘’๐›พ๐œโ€–๐œ‘โ€–2ฮฉ+(๐›พ+๐œ†+โ„Ž๐œŒ๐‘’๐›พ๐œ)๎€œ๐‘ก๐‘ก0๐‘‰โˆ—(๎€ท๐‘ก๐‘ )๐‘‘๐‘ ๐‘กโˆˆ๐‘˜,๐‘ก๐‘˜+1๎€ป,๐‘˜=0,1,2โ€ฆ.(3.23)
According to Lemma 2.2, we claim ๐‘‰โˆ—(๎‚€๐‘‰๐‘ก)โ‰คโˆ—๎€ท๐‘ก0๎€ธ+๐œโ„Ž๐œŒ๐‘’๐›พ๐œโ€–๐œ‘โ€–2ฮฉ๎‚๎€ฝ(exp๐›พ+๐œ†+โ„Ž๐œŒ๐‘’๐›พ๐œ)๎€ท๐‘กโˆ’๐‘ก0๎€ธ๎€พ,๐‘กโ‰ฅ๐‘ก0(3.24) which reduces toโ€–โ€–๐‘ข๎€ท๐‘ก,๐‘ฅ;๐‘ก0๎€ธโ€–โ€–,๐œ‘ฮฉโ‰คโˆš1+๐œโ„Ž๐œŒ๐‘’๐›พ๐œโ€–๐œ‘โ€–ฮฉexp๎‚ป๎‚ต๐œ†+โ„Ž๐œŒ๐‘’๐›พ๐œ2๎‚ถ๎€ท๐‘กโˆ’๐‘ก0๎€ธ๎‚ผ,๐‘กโ‰ฅ๐‘ก0.(3.25) This completes the proof.

Remark 3.2. According to the conditions of Theorem 3.1, we see that the reaction-diffusion term do influence the stability of problem ((2.5)โ€“(2.8)). Moreover, besides the reaction-diffusion coefficients, the dimension of the space and the boundary of spatial variables have also an effect on the stability of the equilibrium point ๐‘ข=0.

Theorem 3.3 .. Providing that(1) for ๐‘ฅ=(๐‘ฅ1,โ€ฆ,๐‘ฅ๐‘š)๐‘‡โˆˆฮฉ(๐‘šโ‰ฅ3), there exist constants ๐›ฝ such that |๐‘ฅ|2=โˆ‘๐‘š๐‘ =1๐‘ฅ2๐‘ <๐›ฝ, in addition, there exists constant ๐ท>0 such that ๐ท๐‘–๐‘ =๐ท๐‘–๐‘ (๐‘ก,๐‘ฅ,๐‘ข)โ‰ฅ๐ท>0; denote ๐ท(๐‘šโˆ’2)2/2๐›ฝ=๐œ’,(2)๐‘ƒ๐‘–๐‘˜(๐‘ข๐‘–(๐‘ก๐‘˜,๐‘ฅ))=โˆ’๐œƒ๐‘–๐‘˜๐‘ข๐‘–(๐‘ก๐‘˜,๐‘ฅ), โˆš1โˆ’1+๐›ผโ‰ค๐œƒ๐‘–๐‘˜โˆšโ‰ค1+1+๐›ผ, ๐›ผโ‰ฅ0,(3)inf๐‘˜=1,2โ€ฆ(๐‘ก๐‘˜โˆ’๐‘ก๐‘˜โˆ’1)>๐œ‡, (4)there exists constant ๐›พ which satisfies ๐›พ+๐œ†+โ„Ž๐œŒ๐‘’๐›พ๐œ>0 and ๐œ†+โ„Ž๐œŒ๐‘’๐›พ๐œ+ln(1+๐›ผ)/๐œ‡<0, where ๐œ†=max๐‘–=1,โ€ฆ,๐‘›(โˆ’๐œ’โˆ’2๐‘Ž๐‘–+โˆ‘๐‘›๐‘—=1(๐‘2๐‘–๐‘—+๐‘2๐‘–๐‘—))+๐œŒ and ๐œŒ=๐‘›max๐‘–=1,โ€ฆ,๐‘›(๐‘™๐‘–2), then, the equilibrium point ๐‘ข=0 of problem ((2.5)โ€“(2.8)) is globally exponentially stable with convergence rate โ€“(1/2)(๐œ†+โ„Ž๐œŒ๐‘’๐›พ๐œ+ln(1+๐›ผ)/๐œ‡).

Proof. Define a Lyapunov function ๐‘‰ of the form โˆ‘๐‘‰(๐‘ก)=๐‘›๐‘–=1๐‘‰๐‘–(๐‘ก), where ๐‘‰๐‘–โˆซ(๐‘ก)=ฮฉ๐‘ข2๐‘–(๐‘ก,๐‘ฅ)d๐‘ฅ. Obviously, ๐‘‰(๐‘ก) is a piecewise continuous function with points of discontinuity of the first kind ๐‘ก๐‘˜, ๐‘˜=1,2,โ€ฆ, where it is continuous from the left, that is, ๐‘‰1(๐‘ก๐‘˜โˆ’0)=๐‘‰1(๐‘ก๐‘˜) (๐‘˜=1,2,โ€ฆ). Furthermore, for ๐‘ก=๐‘ก๐‘˜ (๐‘˜=0,1,2,โ€ฆ), it follows from condition 2 of Theorem 3.3 that ๐‘ข2๐‘–๎€ท๐‘ก๐‘˜๎€ธ+0,๐‘ฅโˆ’๐‘ข2๐‘–๎€ท๐‘ก๐‘˜๎€ธ=๎€ท,๐‘ฅ1โˆ’๐œƒ๐‘–๐‘˜๎€ธ2๐‘ข2๐‘–๎€ท๐‘ก๐‘˜๎€ธ,๐‘ฅโˆ’๐‘ข2๐‘–๎€ท๐‘ก๐‘˜๎€ธ,๐‘ฅโ‰ค๐›ผ๐‘ข2๐‘–๎€ท๐‘ก๐‘˜๎€ธ,๐‘ฅ.(3.26) Thereby, ๐‘‰๎€ท๐‘ก๐‘˜๎€ธ๎€ท๐‘ก+0โ‰ค๐›ผ๐‘‰๐‘˜๎€ธ๎€ท๐‘ก+๐‘‰๐‘˜๎€ธ,๐‘˜=0,1,2,โ€ฆ.(3.27)
Construct another Lyapunov function defined by ๐‘‰โˆ—(๐‘ก)=e๐›พ(๐‘กโˆ’๐‘ก0)๐‘‰(๐‘ก), where ๐›พ satisfies ๐›พ+๐œ†+โ„Ž๐œŒ๐‘’๐›พ๐œ>0 and ๐œ†+โ„Ž๐œŒ๐‘’๐›พ๐œ+(ln(1+๐›ผ)/๐œ‡)<0. Then, ๐‘‰โˆ—(๐‘ก) is also a piecewise continuous function with points of discontinuity of the first kind ๐‘ก๐‘˜, ๐‘˜=1,2,โ€ฆ, where it is continuous from the left, that is ๐‘‰โˆ—(๐‘ก๐‘˜โˆ’0)=๐‘‰โˆ—(๐‘ก๐‘˜) (๐‘˜=1,2,โ€ฆ). And for ๐‘ก=๐‘ก๐‘˜ (๐‘˜=0,1,2,โ€ฆ), it results from (3.27) that ๐‘‰โˆ—๎€ท๐‘ก๐‘˜๎€ธ+0โ‰ค๐›ผ๐‘‰โˆ—๎€ท๐‘ก๐‘˜๎€ธ+๐‘‰โˆ—๎€ท๐‘ก๐‘˜๎€ธ,๐‘˜=0,1,2,โ€ฆ.(3.28)
Set ๐‘กโˆˆ(๐‘ก๐‘˜,๐‘ก๐‘˜+1], ๐‘˜=0,1,2,โ€ฆ. Following the same procedure as in Theorem 3.1, we get ๐‘‰โˆ—(๐‘ก)โ‰ค๐‘‰โˆ—๎€ท๐‘ก๐‘˜๎€ธ๎€œ+0+(๐›พ+๐œ†)๐‘ก๐‘ก๐‘˜๐‘‰โˆ—(๐‘ )๐‘‘๐‘ +โ„Ž๐œŒ๐‘’๐›พ๐œร—๐‘›๎“๐‘—=1๎€œ๐‘กโˆ’๐œ๐‘—๐‘ก(๐‘ก)๐‘˜โˆ’๐œ๐‘—๎€ท๐‘ก๐‘˜๎€ธe๐›พ(๐œƒโˆ’๐‘ก0)๐‘‰๐‘—๎€ท๐‘ก(๐œƒ)๐‘‘๐œƒ๐‘กโˆˆ๐‘˜,๐‘ก๐‘˜+1๎€ป,๐‘˜=0,1,2,โ€ฆ.(3.29)
The relations (3.28) and (3.29) yield ๐‘‰โˆ—(๐‘ก)โˆ’๐‘‰โˆ—๎€ท๐‘ก๐‘˜๎€ธโ‰ค๐›ผ๐‘‰โˆ—๎€ท๐‘ก๐‘˜๎€ธ๎€œ+(๐›พ+๐œ†)๐‘ก๐‘ก๐‘˜๐‘‰โˆ—(๐‘ )๐‘‘๐‘ +โ„Ž๐œŒ๐‘’๐›พ๐œร—๐‘›๎“๐‘—=1๎€œ๐‘กโˆ’๐œ๐‘—๐‘ก(๐‘ก)๐‘˜โˆ’๐œ๐‘—๎€ท๐‘ก๐‘˜๎€ธe๐›พ(๐œƒโˆ’๐‘ก0)๐‘‰๐‘—๎€ท๐‘ก(๐œƒ)๐‘‘๐œƒ๐‘กโˆˆ๐‘˜,๐‘ก๐‘˜+1๎€ป,๐‘˜=0,1,2,โ€ฆ.(3.30)
By induction argument, we arrive at ๐‘‰โˆ—๎€ท๐‘ก๐‘˜๎€ธโˆ’๐‘‰โˆ—๎€ท๐‘ก๐‘˜โˆ’1๎€ธโ‰ค๐›ผ๐‘‰โˆ—๎€ท๐‘ก๐‘˜โˆ’1๎€ธ๎€œ+(๐›พ+๐œ†)๐‘ก๐‘˜๐‘ก๐‘˜โˆ’1๐‘‰โˆ—(๐‘ )๐‘‘๐‘ +โ„Ž๐œŒ๐‘’๐‘›๐›พ๐œ๎“๐‘—=1๎€œ๐‘ก๐‘˜โˆ’๐œ๐‘—(๐‘ก๐‘˜)๐‘ก๐‘˜โˆ’1โˆ’๐œ๐‘—๎€ท๐‘ก๐‘˜โˆ’1๎€ธe๐›พ(๐œƒโˆ’๐‘ก0)๐‘‰๐‘—โ‹ฎ๐‘‰(๐œƒ)๐‘‘๐œƒ,โˆ—๎€ท๐‘ก2๎€ธโˆ’๐‘‰โˆ—๎€ท๐‘ก1๎€ธโ‰ค๐›ผ๐‘‰โˆ—๎€ท๐‘ก1๎€ธ๎€œ+(๐›พ+๐œ†)๐‘ก2๐‘ก1๐‘‰โˆ—(๐‘ )๐‘‘๐‘ +โ„Ž๐œŒ๐‘’๐‘›๐›พ๐œ๎“๐‘—=1๎€œ๐‘ก2โˆ’๐œ๐‘—(๐‘ก2)๐‘ก1โˆ’๐œ๐‘—๎€ท๐‘ก1๎€ธe๐›พ(๐œƒโˆ’๐‘ก0)๐‘‰๐‘—๐‘‰(๐œƒ)๐‘‘๐œƒ,โˆ—๎€ท๐‘ก1๎€ธโˆ’๐‘‰โˆ—๎€ท๐‘ก0๎€ธโ‰ค๐›ผ๐‘‰โˆ—๎€ท๐‘ก0๎€ธ๎€œ+(๐›พ+๐œ†)๐‘ก1๐‘ก0๐‘‰โˆ—(๐‘ )๐‘‘๐‘ +โ„Ž๐œŒ๐‘’๐‘›๐›พ๐œ๎“๐‘—=1๎€œ๐‘ก1โˆ’๐œ๐‘—(๐‘ก1)๐‘ก0โˆ’๐œ๐‘—๎€ท๐‘ก0๎€ธe๐›พ(๐œƒโˆ’๐‘ก0)๐‘‰๐‘—(๐œƒ)๐‘‘๐œƒ.(3.31) Hence, ๐‘‰โˆ—(๐‘ก)โˆ’๐‘‰โˆ—๎€ท๐‘ก0๎€ธโ‰ค๐›ผ๐‘‰โˆ—๎€ท๐‘ก0๎€ธ๎€œ+(๐›พ+๐œ†)๐‘ก๐‘ก0๐‘‰โˆ—(๐‘ )๐‘‘๐‘ +โ„Ž๐œŒ๐‘’๐‘›๐›พ๐œ๎“๐‘—=1๎€œ๐‘กโˆ’๐œ๐‘—๐‘ก(๐‘ก)0โˆ’๐œ๐‘—๎€ท๐‘ก0๎€ธe๐›พ(๐œƒโˆ’๐‘ก0)๐‘‰๐‘—๎“(๐œƒ)๐‘‘๐œƒ+๐›ผ๐‘ก0<๐‘ก๐‘˜<๐‘ก๐‘‰๎€ท๐‘ก๐‘˜๎€ธโ‰ค๐›ผ๐‘‰โˆ—๎€ท๐‘ก0๎€ธ+(๐›พ+๐œ†+โ„Ž๐œŒ๐‘’๐›พ๐œ)๎€œ๐‘ก๐‘ก0๐‘‰โˆ—(๐‘ )๐‘‘๐‘ +โ„Ž๐œŒ๐‘’๐‘›๐›พ๐œ๎“๐‘—=1๎€œ๐‘ก0๐‘ก0โˆ’๐œ๐‘—๎€ท๐‘ก0๎€ธe๐›พ(๐œƒโˆ’๐‘ก0)๐‘‰๐‘—(๎“๐œƒ)๐‘‘๐œƒ+๐›ผ๐‘ก0<๐‘ก๐‘˜<๐‘ก๐‘‰๎€ท๐‘ก๐‘˜๎€ธ๎€ท๐‘ก๐‘กโˆˆ๐‘˜,๐‘ก๐‘˜+1๎€ป,๐‘˜=0,1,2,โ€ฆ.(3.32)
Introducing โ„Ž๐œŒ๐‘’๐›พ๐œโˆ‘๐‘›๐‘—=1โˆซ๐‘ก0๐‘ก0โˆ’๐œ๐‘—(๐‘ก0)e๐›พ(๐œƒโˆ’๐‘ก0)๐‘‰๐‘—(๐œƒ)๐‘‘๐œƒโ‰ค๐œโ„Ž๐œŒ๐‘’๐›พ๐œโ€–๐œ‘โ€–2ฮฉ as shown in the proof of Theorem 3.1 into (3.32), the expression becomes ๐‘‰โˆ—(๐‘ก)โˆ’๐‘‰โˆ—๎€ท๐‘ก0๎€ธโ‰ค๐›ผ๐‘‰โˆ—๎€ท๐‘ก0๎€ธ+๐œโ„Ž๐œŒ๐‘’๐›พ๐œโ€–๐œ‘โ€–2ฮฉ+(๐›พ+๐œ†+โ„Ž๐œŒ๐‘’๐›พ๐œ)๎€œ๐‘ก๐‘ก0๐‘‰โˆ—(๎“๐‘ )๐‘‘๐‘ +๐›ผ๐‘ก0<๐‘ก๐‘˜<๐‘ก๐‘‰๎€ท๐‘ก๐‘˜๎€ธ๎€ท๐‘ก๐‘กโˆˆ๐‘˜,๐‘ก๐‘˜+1๎€ป,๐‘˜=0,1,2โ€ฆ.(3.33)
It then results from Lemma 2.2 that ๐‘‰โˆ—(๎‚€(๐‘ก)โ‰ค๐›ผ+1)๐‘‰โˆ—๎€ท๐‘ก0๎€ธ+๐œโ„Ž๐œŒ๐‘’๐›พ๐œโ€–๐œ‘โ€–2ฮฉ๎‚๎‘๐‘ก0<๐‘ก๐‘˜<๐‘ก(๎€ท(1+๐›ผ)exp๐›พ+๐œ†+โ„Ž๐œŒ๐‘’๐›พ๐œ)๎€ท๐‘กโˆ’๐‘ก0=๎‚€๎€ธ๎€ธ(๐›ผ+1)๐‘‰โˆ—๎€ท๐‘ก0๎€ธ+๐œโ„Ž๐œŒ๐‘’๐›พ๐œโ€–๐œ‘โ€–2ฮฉ๎‚(1+๐›ผ)๐‘˜๎€ทexp(๐›พ+๐œ†+โ„Ž๐œŒ๐‘’๐›พ๐œ)๎€ท๐‘กโˆ’๐‘ก0๎€ธ๎€ธ,๐‘กโ‰ฅ๐‘ก0.(3.34)
On the other hand, since inf๐‘˜=1,2,โ€ฆ(๐‘ก๐‘˜โˆ’๐‘ก๐‘˜โˆ’1)>๐œ‡, one has ๐‘˜<(๐‘ก๐‘˜โˆ’๐‘ก0)/๐œ‡. Thereby, (1+๐›ผ)๐‘˜๎‚ป<๐‘’๐‘ฅ๐‘ln(1+๐›ผ)๐œ‡๎€ท๐‘ก๐‘˜โˆ’๐‘ก0๎€ธ๎‚ผ๎‚ป<๐‘’๐‘ฅ๐‘ln(1+๐›ผ)๐œ‡๎€ท๐‘กโˆ’๐‘ก0๎€ธ๎‚ผ.(3.35)
And (3.34) can be rewritten as ๐‘‰โˆ—๎‚€(๐‘ก)โ‰ค(๐›ผ+1)๐‘‰โˆ—๎€ท๐‘ก0๎€ธ+๐œโ„Ž๐œŒ๐‘’๐›พ๐œโ€–๐œ‘โ€–2ฮฉ๎‚exp๎‚ต๎‚ต๐›พ+๐œ†+โ„Ž๐œŒ๐‘’๐›พ๐œ+ln(1+๐›ผ)๐œ‡๎‚ถ๎€ท๐‘กโˆ’๐‘ก0๎€ธ๎‚ถ(3.36) which implies โ€–โ€–๐‘ข๎€ท๐‘ก,๐‘ฅ;๐‘ก0๎€ธโ€–โ€–,๐œ‘ฮฉโ‰คโˆš(๐›ผ+1+๐œโ„Ž๐œŒ๐‘’๐›พ๐œ)โ€–๐œ‘โ€–ฮฉ๎‚ต1exp2๎‚ต๐œ†+โ„Ž๐œŒ๐‘’๐›พ๐œ+ln(1+๐›ผ)๐œ‡๎‚ถ๎€ท๐‘กโˆ’๐‘ก0๎€ธ๎‚ถ,๐‘กโ‰ฅ๐‘ก0.(3.37) The proof is completed.

Remark 3.4. Theorem 3.1 is in fact the special case of Theorem 3.3 by choosing ๐›ผ=0. Due to Lemma 2.4, we know 2๐‘›๎“๐‘—=1๐‘๐‘–๐‘—๎€œฮฉ๐‘ข๐‘–๎€ท๐‘ข(๐‘ก,๐‘ฅ)๐‘“๐‘—๎€ธ(๐‘ก,๐‘ฅ)d๐‘ฅโ‰ค๐‘›๎“๐‘—=1๎€œฮฉ๎ƒฉ๐œ€1๐‘2๐‘–๐‘—๐‘ข2๐‘–๐‘™(๐‘ก,๐‘ฅ)+2๐‘—๐œ€1๐‘ข2๐‘–๎ƒช2(๐‘ก,๐‘ฅ)d๐‘ฅ,๐‘›๎“๐‘—=1๐‘๐‘–๐‘—๎€œฮฉ๐‘ข๐‘–๎€ท๐‘ข(๐‘ก,๐‘ฅ)๐‘“๐‘—๎€ท๐‘กโˆ’๐œ๐‘—,๐‘ฅ๎€ธ๎€ธd๐‘ฅโ‰ค๐‘›๎“๐‘—=1๎€œฮฉ๎ƒฉ๐œ€2๐‘2๐‘–๐‘—๐‘ข2๐‘–๐‘™(๐‘ก,๐‘ฅ)+2๐‘—๐œ€2๐‘ข2๐‘–๎€ท๐‘กโˆ’๐œ๐‘—๎€ธ๎ƒช,๐‘ฅd๐‘ฅ(3.38) hold for any ๐œ€1,๐œ€2>0.
In the sequel, we follow the same procedures as in Theorems 3.1 and 3.3 to find the following theorems.

Theorem 3.5. Provided that(1)for ๐‘ฅ=(๐‘ฅ1,โ€ฆ,๐‘ฅ๐‘š)๐‘‡โˆˆฮฉ(๐‘šโ‰ฅ3), there exists a constant ๐›ฝ such that |๐‘ฅ|2=โˆ‘๐‘š๐‘ =1๐‘ฅ2๐‘ <๐›ฝ. in addition, there exists a constant ๐ท>0 such that ๐ท๐‘–๐‘ =๐ท๐‘–๐‘ (๐‘ก,๐‘ฅ,๐‘ข)โ‰ฅ๐ท>0; denote ๐ท(๐‘šโˆ’2)2/2๐›ฝ=๐œ’,(2)๐‘ƒ๐‘–๐‘˜(๐‘ข๐‘–(๐‘ก๐‘˜,๐‘ฅ))=โˆ’๐œƒ๐‘–๐‘˜๐‘ข๐‘–(๐‘ก๐‘˜,๐‘ฅ), 0โ‰ค๐œƒ๐‘–๐‘˜โ‰ค2,(3)there exist constants ๐›พ and ๐œ€1,๐œ€2>0 such that ๐›พ+๐œ†+โ„Ž๐œŒ๐‘’๐›พ๐œ>0 and ๐œ†+โ„Ž๐œŒ๐‘’๐›พ๐œ<0, where ๐œ†=max๐‘–=1,โ€ฆ,๐‘›(โˆ’๐œ’โˆ’2๐‘Ž๐‘–+โˆ‘๐‘›๐‘—=1(๐œ€1๐‘2๐‘–๐‘—+๐œ€2๐‘2๐‘–๐‘—))+(๐‘›/๐œ€1)max๐‘–=1,โ€ฆ,๐‘›(๐‘™2๐‘–) and ๐œŒ=(๐‘›/๐œ€2)max๐‘–=1,โ€ฆ,๐‘›(๐‘™2๐‘–), then, the equilibrium point ๐‘ข=0 of problem ((2.5)โ€“(2.8)) is globally exponentially stable with convergence rate โ€“(๐œ†+โ„Ž๐œŒ๐‘’๐›พ๐œ)/2.

Theorem 3.6. Assume that(1)for ๐‘ฅ=(๐‘ฅ1,โ€ฆ,๐‘ฅ๐‘š)๐‘‡โˆˆฮฉ(๐‘šโ‰ฅ3), there exists a constant ๐›ฝ such that |๐‘ฅ|2=โˆ‘๐‘š๐‘ =1๐‘ฅ2๐‘ <๐›ฝ; In addition, there exists a constant ๐ท>0 such that ๐ท๐‘–๐‘ =๐ท๐‘–๐‘ (๐‘ก,๐‘ฅ,๐‘ข)โ‰ฅ๐ท>0; denote ๐ท(๐‘šโˆ’2)2/2๐›ฝ=๐œ’,(2)๐‘ƒ๐‘–๐‘˜(๐‘ข๐‘–(๐‘ก๐‘˜,๐‘ฅ))=โˆ’๐œƒ๐‘–๐‘˜๐‘ข๐‘–(๐‘ก๐‘˜,๐‘ฅ), โˆš1โˆ’1+๐›ผโ‰ค๐œƒ๐‘–๐‘˜โˆšโ‰ค1+1+๐›ผ, ๐›ผโ‰ฅ0,(3)inf๐‘˜=1,2,โ€ฆ(๐‘ก๐‘˜โˆ’๐‘ก๐‘˜โˆ’1)>๐œ‡,(4)there exist constants ๐›พ and ๐œ€1,๐œ€2>0 such that ๐›พ+๐œ†+โ„Ž๐œŒ๐‘’๐›พ๐œ>0 and ๐œ†+โ„Ž๐œŒ๐‘’๐›พ๐œ+ln(1+๐›ผ)/๐œ‡<0, where ๐œ†=max๐‘–=1,โ€ฆ,๐‘›(โˆ’๐œ’โˆ’2๐‘Ž๐‘–+๐‘›โˆ‘๐‘—=1(๐œ€1๐‘2๐‘–๐‘—+๐œ€2๐‘2๐‘–๐‘—))+(๐‘›/๐œ€1)max๐‘–=1,โ€ฆ,๐‘›(๐‘™2๐‘–) and ๐œŒ=(๐‘›/๐œ€2)max๐‘–=1,โ€ฆ,๐‘›(๐‘™2๐‘–).
Then, the equilibrium point ๐‘ข=0 of problem ((2.5)โ€“(2.8)) is globally exponentially stable with convergence rate โˆ’(1/2)(๐œ†+โ„Ž๐œŒ๐‘’๐›พ๐œ+ln(1+๐›ผ)/๐œ‡).
Further, on the condition that |๐‘ƒ๐‘–๐‘˜(๐‘ข๐‘–(๐‘ก๐‘˜,๐‘ฅ))|โ‰ค๐œƒ๐‘–๐‘˜|๐‘ข๐‘–(๐‘ก๐‘˜,๐‘ฅ)|, where ๐œƒ2๐‘–๐‘˜<(๐›ผโˆ’1)/2 and ๐›ผโ‰ฅ1, we obtain, for ๐‘ก=๐‘ก๐‘˜ (๐‘˜=1,2,โ€ฆ), ๐‘ข2๐‘–๎€ท๐‘ก๐‘˜๎€ธ+0,๐‘ฅโˆ’๐‘ข2๐‘–๎€ท๐‘ก๐‘˜๎€ธ=๎€ท๐‘ƒ,๐‘ฅ๐‘–๐‘˜๎€ท๐‘ข๐‘–๎€ท๐‘ก๐‘˜,๐‘ฅ๎€ธ๎€ธ+๐‘ข๐‘–๎€ท๐‘ก๐‘˜,๐‘ฅ๎€ธ๎€ธ2โˆ’๐‘ข2๐‘–๎€ท๐‘ก๐‘˜๎€ธ๎€ท๐‘ข,๐‘ฅโ‰ค2๐‘–๎€ท๐‘ก๐‘˜,๐‘ฅ๎€ธ๎€ธ2๎€ท๐‘ƒ+2๐‘–๐‘˜๎€ท๐‘ข๐‘–๎€ท๐‘ก๐‘˜,๐‘ฅ๎€ธ๎€ธ๎€ธ2โˆ’๐‘ข2๐‘–๎€ท๐‘ก๐‘˜๎€ธโ‰ค๎€ท,๐‘ฅ2+2๐œƒ2๐‘–๐‘˜๐‘ข๎€ธ๎€ท๐‘–๎€ท๐‘ก๐‘˜,๐‘ฅ๎€ธ๎€ธ2โˆ’๐‘ข2๐‘–๎€ท๐‘ก๐‘˜๎€ธ,๐‘ฅโ‰ค๐›ผ๐‘ข2๐‘–๎€ท๐‘ก๐‘˜๎€ธ.,๐‘ฅ(3.39)
Identical with the proof of Theorem 3.3, we present the theorem as follows.

Theorem 3.7. Assume that(1)for ๐‘ฅ=(๐‘ฅ1,โ€ฆ,๐‘ฅ๐‘š)๐‘‡โˆˆฮฉ(๐‘šโ‰ฅ3), there exists a constant ๐›ฝ such that |๐‘ฅ|2=โˆ‘๐‘š๐‘ =1๐‘ฅ2๐‘ <๐›ฝ. in addition, there exists a constant ๐ท>0 such that ๐ท๐‘–๐‘ =๐ท๐‘–๐‘ (๐‘ก,๐‘ฅ,๐‘ข)โ‰ฅ๐ท>0. Denote ๐ท(๐‘šโˆ’2)2/2๐›ฝ=๐œ’,(2)|๐‘ƒ๐‘–๐‘˜(๐‘ข๐‘–(๐‘ก๐‘˜,๐‘ฅ))|โ‰ค๐œƒ๐‘–๐‘˜|๐‘ข๐‘–(๐‘ก๐‘˜,๐‘ฅ)|, where ๐œƒ2๐‘–๐‘˜โ‰ค(๐›ผโˆ’1)/2 and ๐›ผโ‰ฅ1,(3)inf๐‘˜=1,2,โ€ฆ(๐‘ก๐‘˜โˆ’๐‘ก๐‘˜โˆ’1)>๐œ‡,(4)there exist constants๐›พand๐œ€1,๐œ€2>0such that๐›พ+๐œ†+โ„Ž๐œŒ๐‘’๐›พ๐œ>0and๐œ†+โ„Ž๐œŒ๐‘’๐›พ๐œ+ln(1+๐›ผ)/๐œ‡<0,where๐œ†=max๐‘–=1,โ€ฆ,๐‘›(โˆ’๐œ’โˆ’2๐‘Ž๐‘–+โˆ‘๐‘›๐‘—=1(๐œ€1๐‘2๐‘–๐‘—+๐œ€2๐‘2๐‘–๐‘—))+(๐‘›/๐œ€1)max๐‘–=1,โ€ฆ,๐‘›(๐‘™๐‘–2)and๐œŒ=(๐‘›/๐œ€2)max๐‘–=1,โ€ฆ,๐‘›(๐‘™๐‘–2).
Then, the equilibrium point ๐‘ข=0 of problem ((2.5)โ€“(2.8)) is globally exponentially stable with convergence rate โˆ’1/2(๐œ†+โ„Ž๐œŒ๐‘’๐›พ๐œ+ln(1+๐›ผ)/๐œ‡).

Remark 3.8. Different from Theorems 3.1โ€“3.6, the impulsive part in Theorem 3.7 could be nonlinear and this will be of more applicability. Actually, Theorems 3.1โ€“3.6 can be regarded as the special cases of Theorem 3.7.

4. Examples

Example 4.1. Consider the following impulsive reaction-diffusion delayed neural network: ๐œ•๐‘ข๐‘–(๐‘ก,๐‘ฅ)=๐œ•๐‘ก๐‘š๎“๐‘ =1๐œ•๐œ•๐‘ฅ๐‘ ๎‚ต๐ท๐‘–๐‘ ๐œ•๐‘ข๐‘–(๐‘ก,๐‘ฅ)๐œ•๐‘ฅ๐‘ ๎‚ถโˆ’๐‘Ž๐‘–๐‘ข๐‘–(๐‘ก,๐‘ฅ)+๐‘›๎“๐‘—=1๐‘๐‘–๐‘—๐‘“๐‘—๎€ท๐‘ข๐‘—๎€ธ+(๐‘ก,๐‘ฅ)๐‘›๎“๐‘—=1๐‘๐‘–๐‘—๐‘“๐‘—๎€ท๐‘ข๐‘—๎€ท๐‘กโˆ’๐œ๐‘—(๐‘ก),๐‘ฅ๎€ธ๎€ธ๐‘กโ‰ฅ0,๐‘กโ‰ ๐‘ก๐‘˜,๐‘ฅโˆˆฮฉ,๐‘˜=1,2,โ€ฆ,๐‘–=1,โ€ฆ,๐‘›(4.1) with the impulsive effects characterized by ๐‘ข1๎€ท๐‘ก๐‘˜๎€ธ+0,๐‘ฅ=๐‘ข1๎€ท๐‘ก๐‘˜๎€ธ,๐‘ฅ+1.343๐‘ข1๎€ท๐‘ก๐‘˜๎€ธ,๐‘ฅ,๐‘ข2๎€ท๐‘ก๐‘˜๎€ธ+0,๐‘ฅ=๐‘ข2๎€ท๐‘ก๐‘˜๎€ธ,๐‘ฅ+1.343๐‘ข2๎€ท๐‘ก๐‘˜๎€ธ,๐‘ฅ๐‘˜=1,2,โ€ฆ,๐‘ฅโˆˆฮฉ(4.2) and initial condition (2.3) and Dirichlet condition (2.4), where ๐‘›=2, ๐‘š=4, ฮฉ={(๐‘ฅ1,โ€ฆ,๐‘ฅ4)๐‘‡|โˆ‘4๐‘–=1๐‘ฅ2๐‘–<4}, ๐‘Ž1=๐‘Ž2=6.5, (๐ท๐‘–๐‘ )2ร—4=๎€ท1.22.32.53.11.83.22.73.4๎€ธ, (๐‘๐‘–๐‘—)2ร—2=๎€ทโˆ’0.231.3โˆ’0.13๎€ธ, (๐‘๐‘–๐‘—)2ร—2=๎€ทโˆ’0.1โˆ’0.20.1โˆ’0.3๎€ธ, ๐‘“๐‘—(๐‘ข๐‘—)=(1/4)(|๐‘ข๐‘—+1|โˆ’|๐‘ข๐‘—โˆ’1|), 0โ‰ค๐œ๐‘—(๐‘ก)โ‰ค0.5, and โ€ข๐œ๐‘—(๐‘ก)<0. For ๐›ฝ=4 and ๐ท=1.2, we compute ๐œ’=0.6. This, together with the chosen ๐‘™๐‘–=1/2, yields ๐œ†=max๐‘–=1,โ€ฆ,๐‘›๎ƒฉโˆ’๐œ’โˆ’2๐‘Ž๐‘–+๐‘›๎“๐‘—=1๎€ท๐‘2๐‘–๐‘—+๐‘2๐‘–๐‘—๎€ธ๎ƒช+๐‘›max๐‘–=1,โ€ฆ,๐‘›๎€ท๐‘™2๐‘–๎€ธ=โˆ’4,๐œŒ=๐‘›max๐‘–=1,โ€ฆ,๐‘›๎€ท๐‘™2๐‘–๎€ธ=12.(4.3)
By selecting ๐›พ=2.6, ๐œ=0.5 and โ„Ž=1, we estimate ๐›พ+๐œ†+โ„Ž๐œŒ๐‘’๐›พ๐œ1=2.6โˆ’4+2๐‘’1.3>0,๐œ†+โ„Ž๐œŒ๐‘’๐›พ๐œ1=โˆ’4+2๐‘’1.3<0.(4.4)
According to Theorem 3.1, we therefore conclude that the system in Example 4.1 is globally exponential stable.

Example 4.2. Consider the following impulsive reaction-diffusion delayed neural network: ๐œ•๐‘ข๐‘–(๐‘ก,๐‘ฅ)=๐œ•๐‘ก๐‘š๎“๐‘ =1๐œ•๐œ•๐‘ฅ๐‘ ๎‚ต๐ท๐‘–๐‘ ๐œ•๐‘ข๐‘–(๐‘ก,๐‘ฅ)๐œ•๐‘ฅ๐‘ ๎‚ถโˆ’๐‘Ž๐‘–๐‘ข๐‘–(๐‘ก,๐‘ฅ)+๐‘›๎“๐‘—=1๐‘๐‘–๐‘—๐‘“๐‘—๎€ท๐‘ข๐‘—๎€ธ+(๐‘ก,๐‘ฅ)๐‘›๎“๐‘—=1๐‘๐‘–๐‘—๐‘“๐‘—๎€ท๐‘ข๐‘—๎€ท๐‘กโˆ’๐œ๐‘—(๐‘ก),๐‘ฅ๎€ธ๎€ธ๐‘กโ‰ฅ0,๐‘กโ‰ ๐‘ก๐‘˜,๐‘ฅโˆˆฮฉ,๐‘˜=1,2,โ€ฆ,๐‘–=1,โ€ฆ,๐‘›(4.5) with the impulsive effects featured by ๐‘ข1๎€ท๐‘ก๐‘˜๎€ธ+0,๐‘ฅ=๐‘ข1๎€ท๐‘ก๐‘˜๎€ธ๎€ท,๐‘ฅ+arctan0.5๐‘ข1๎€ท๐‘ก๐‘˜,๐‘ฅ๎€ธ๎€ธ,๐‘ข2๎€ท๐‘ก๐‘˜๎€ธ+0,๐‘ฅ=๐‘ข2๎€ท๐‘ก๐‘˜๎€ธ๎€ท,๐‘ฅ+arctan0.5๐‘ข2๎€ท๐‘ก๐‘˜,๐‘ฅ๎€ธ๎€ธ๐‘˜=1,2,โ€ฆ,๐‘ฅโˆˆฮฉ(4.6) and initial condition (2.3) and Dirichlet condition (2.4), where ๐‘›=2, ๐‘š=4, ฮฉ={(๐‘ฅ1,โ€ฆ,๐‘ฅ4)๐‘‡|โˆ‘4๐‘–=1๐‘ฅ2๐‘–<4}, ๐‘Ž1=๐‘Ž2=6.5, (๐ท๐‘–๐‘ )2ร—4=๎€ท1.22.32.53.11.83.22.73.4๎€ธ, (๐‘๐‘–๐‘—)2ร—2=๎€ทโˆ’0.231.3โˆ’0.13๎€ธ, (๐‘๐‘–๐‘—)2ร—2=๎€ทโˆ’0.1โˆ’0.20.1โˆ’0.3๎€ธ, ๐‘“๐‘—(๐‘ข๐‘—)=1/4(|๐‘ข๐‘—+1|โˆ’|๐‘ข๐‘—โˆ’1|), 0โ‰ค๐œ๐‘—(๐‘ก)โ‰ค0.5, โ€ข๐œ๐‘—(๐‘ก)<0, and inf๐‘˜=1,2,โ€ฆ(๐‘ก๐‘˜โˆ’๐‘ก๐‘˜โˆ’1)>1. For ๐›ฝ=4 and ๐ท=1.2, we compute ๐œ’=0.6. This, together with ๐‘™๐‘–=1/2 and ๐œ€1=๐œ€2=1, yields ๐‘›๐œŒ=๐œ€2max๐‘–=1,โ€ฆ,๐‘›๎€ท๐‘™2๐‘–๎€ธ=12,๐œ†=max๐‘–=1,โ€ฆ,๐‘›๎ƒฉโˆ’๐œ’โˆ’2๐‘Ž๐‘–+๐‘›๎“๐‘—=1๎€ท๐œ€1๐‘2๐‘–๐‘—+๐œ€2๐‘2๐‘–๐‘—๎€ธ๎ƒช+๐‘›๐œ€1max๐‘–=1,โ€ฆ,๐‘›๎€ท๐‘™2๐‘–๎€ธ=โˆ’4.(4.7)
Select ๐›ผ=1.5 by setting ๐œƒ๐‘–๐‘˜=0.5. Hence, we compute by letting ๐œ‡=1, ๐›พ=3, ๐œ=0.5, and โ„Ž=1 that ๐›พ+๐œ†+โ„Ž๐œŒ๐‘’๐›พ๐œ1=3โˆ’4+2๐‘’1.5>0,๐œ†+โ„Ž๐œŒ๐‘’๐›พ๐œ+ln(1+๐›ผ)๐œ‡1=โˆ’4+2๐‘’1.5+ln2.5<0.(4.8)
It is then concluded from Theorem 3.7 that the system in Example 4.2 is globally exponentially stable.

Acknowledgment

The work is supported by National Natural Science Foundation of China under Grant 60904028.

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Copyright ยฉ 2012 Yutian Zhang. 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.


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