Pseudo Almost-Periodic Solution of Shunting Inhibitory Cellular Neural Networks with Delay

Abstract

Shunting inhibitory cellular neural networks are studied. Some sufficient criteria are obtained for the existence and uniqueness of pseudo almost-periodic solution of this system. Our results improve and generalize those of the previous studies. This is the first paper considering the pseudo almost-periodic SICNNs. Furthermore, several methods are applied to establish sufficient criteria for the globally exponential stability of this system. The approaches are based on constructing suitable Lyapunov functionals and the well-known Banach contraction mapping principle.

1. Introduction

It is well known that the cellular neural networks (CNNs) are widely applied in signal processing, image processing, pattern recognition, and so on. The theoretical and applied studies of CNNs have been a new focus of studies worldwide (see [1–12]). Bouzerdoum and Pinter in [1] have introduced a new class of CNNs, namely, the shunting inhibitory CNNs (SICNNs). Shunting neural networks have been extensively applied in psychophysics, speech, perception, robotics, adaptive pattern recognition, vision, and image processing. Recently, Chen and Cao [9] have studied the existence of almost-periodic solutions of the following system of SICNNs: 𝑥̇_𝑖𝑗(𝑡)=−𝑎_𝑖𝑗𝑥_𝑖𝑗(𝑡)−_{𝐶𝑘𝑙∈𝑁𝑟(𝑖,𝑗)}𝑐^𝑘𝑙_𝑖𝑗𝑓(𝑥_𝑖𝑗(𝑡−𝜏))𝑥_𝑖𝑗(𝑡)+𝐿_𝑖𝑗(𝑡), (1.1) where 𝐶_𝑖𝑗 denotes the cell at the (𝑖,𝑗) position of the lattice, the 𝑟-neighborhood 𝑁_𝑟(𝑖,𝑗) of 𝐶_𝑖𝑗 is 𝑁_𝑟(𝑖,𝑗)={𝐶^𝑘𝑙_𝑖𝑗∶max(|𝑘−𝑙|,|𝑙−𝑗|≤𝑟),1≤𝑘≤𝑚,1≤𝑙≤𝑛}, (1.2)𝑥_𝑖𝑗 is the activity of the cell 𝐶_𝑖𝑗, 𝐿_𝑖𝑗(𝑡) is the external input to 𝐶_𝑖𝑗, the constant 𝑎_𝑖𝑗>0 represents the passive decay rate of the cell activity, 𝐶^𝑘𝑙_𝑖𝑗≥0 is the connection or coupling strength of postsynaptic activity of the cell transmitted to the cell 𝐶_𝑖𝑗, and the activation function 𝑓(𝑥_𝑘𝑙) is a positive continuous function representing the output or firing rate of the cell 𝐶_𝑖𝑗. Since studies on neural dynamic systems not only involve a discussion of stability properties, but also involve many dynamic properties such as periodic oscillatory behavior, almost-periodic oscillatory properties, chaos, and bifurcation. To the best of our knowledge, few authors have studied almost-periodic solutions for SINNs with delays and variable coefficients, and most of them discuss the stability, periodic oscillation in the case of constant coefficients. In their paper, they investigated the existence and stability of periodic solutions of SINNs with delays and variable coefficients. They considered the SICNNs with delays and variable coefficients 𝑥̇_𝑖𝑗(𝑡)=−𝑎_𝑖𝑗(𝑡)𝑥_𝑖𝑗(𝑡)−_{𝐵𝑘𝑙∈𝑁𝑟(𝑖,𝑗)}𝐵^𝑘𝑙_𝑖𝑗(𝑡)𝑓_𝑖𝑗(𝑥_𝑘𝑙(𝑡))𝑥_𝑖𝑗(𝑡)−_{𝐶𝑘𝑙∈𝑁𝑟(𝑖,𝑗)}𝐶^𝑘𝑙_𝑖𝑗(𝑡)𝑔_𝑖𝑗(𝑥_𝑘𝑙(𝑡−𝜏_𝑘𝑙))𝑥_𝑖𝑗(𝑡)+𝐿_𝑖𝑗(𝑡), (1.3) where, for each 𝑖=1,2,…,𝑛,𝑗=1,2,…,𝑚,𝑎_𝑖𝑗(𝑡),𝐵^𝑘𝑙_𝑖𝑗(𝑡),𝐶^𝑘𝑙_𝑖𝑗(𝑡),and 𝐿_𝑖𝑗(𝑡) are all continuous 𝜔− periodic functions and 𝑎_𝑖𝑗(𝑡)>0,𝐵^𝑘𝑙_𝑖𝑗(𝑡)≥0,𝐶^𝑘𝑙_𝑖𝑗(𝑡)≥0, and 𝜏_𝑖j is a positive constant.

In this paper, we consider the following more general SICNNs: 𝑥̇_𝑖𝑗(𝑡)=−𝑎_𝑖𝑗(𝑡)𝑥_𝑖𝑗(𝑡)−_{𝐵𝑘𝑙∈𝑁𝑟(𝑖,𝑗)}𝐵^𝑘𝑙_𝑖𝑗(𝑡)𝑓_𝑖𝑗(𝑥_𝑘𝑙(𝑡))𝑥_𝑖𝑗(𝑡)−_{𝐶𝑘𝑙∈𝑁𝑟(𝑖,𝑗)}𝐶^𝑘𝑙_𝑖𝑗(𝑡)𝑔_𝑖𝑗(𝑥_𝑘𝑙(𝑡−𝜏_𝑘𝑙(𝑡)))𝑥_𝑖𝑗(𝑡)+𝐿_𝑖𝑗(𝑡). (1.4) By using the Lyapunov functional and contraction mapping, a set of criteria are established for the globally exponential stability, the existence, and uniqueness of pseudo almost-periodic solution for the SICNNs. This is the first paper considering the pseudo almost-periodic solution of SICNNs. Since the nature is full of all kinds of tiny perturbations, either the periodicity assumption or the almost-periodicity assumption is just approximation of some degree of the natural perturbations. A well-known extension of almost periodicity is the asymptotically almost periodicity, which was introduced by Frechet. In 1992, Zhang [13, 14] introduced a more general extension of the concept of asymptotically almost periodicity, the so-called pseudo almost periodicity, which has been widely applied in the theory of ODEs and PDEs. However, it is rarely applied in the theory of neural networks or mathematical biology. This paper is expected to establish criteria that provide much flexibility in the designing and training of neural networks and to shed some new light on the application of pseudo almost periodicity in neural networks, population dynamics, and the theory of differential equations.

Throughout this paper, we will use the notations 𝑔^𝑀=sup_𝑡∈𝑅𝑔(𝑡),𝑔^𝐿=sup_𝑡∈𝑅𝑔(𝑡), where 𝑔(𝑡) is a bounded continuous function on 𝑅.

In this paper, we always use 𝑖=1,2,…,𝑚;𝑗=1,2,…,𝑛; unless otherwise stated.

In this paper, we always consider system (1.4) together with the following assumptions.(𝐴₁)𝑎_𝑖𝑗(𝑡) are almost periodic on 𝑅 with 𝑎_𝑖𝑗(𝑡)>0, and 𝐿_𝑖𝑗(𝑡) and 𝜏_𝑖𝑗(𝑡) are pseudo almost periodic on 𝑅 with 𝐿_𝑖𝑗>0.(𝐴₂)𝜏_𝑖𝑗(𝑡) is bounded, continuous and differentiable with 0≤𝜏_𝑖𝑗≤𝜏,1−𝜏>0 for 𝑡∈𝑅, where 𝜏 is constant.(𝐴₃)𝑓_𝑖𝑗,𝑔_𝑖𝑗∈𝐶(𝑅,𝑅) are bounded, and continuous, and there exist positive numbers 𝜇_𝑖𝑗,𝜈_𝑖𝑗 such that |𝑓_𝑖𝑗(𝑥)−𝑓_𝑖𝑗(𝑦)|≤𝜇_𝑖𝑗|𝑥−𝑦|,|𝑔_𝑖𝑗(𝑥)−𝑔_𝑖𝑗(𝑦)|≤𝜈_𝑖𝑗|𝑥−𝑦|, for all 𝑥,𝑦∈𝑅.

2. Preliminaries and Basic Results of Pseudo Almost-Periodic Function

In this section, we explore the existence of pseudo almost-periodic solution of (1.4). First, we would like to recall some basic notations and results of almost periodicity and pseudo almost periodicity [15, 16] which will come into play later on.

Let Ω⊂𝐶^𝑛 be close and let ℒ(𝑅) (resp., ℒ(𝑅×Ω)) denote the 𝐶^∗-algebra of bounded continuous complex-valued functions on 𝑅 (respectively, 𝑅×Ω) with supremum norm |⋅|_∞ denoting the Euclidean norm in 𝐶^𝑛; that is, |𝑥|_∞=max_{1≤𝑖≤𝑛}|𝑥_𝑖|, 𝑥∈𝐶^𝑛.

Definition 2.1. A function 𝑔∈ℒ(𝑅) is called almost periodic if, for each 𝜖>0, there exists an 𝑙_𝜖>0 such that every interval of length 𝑙_𝜖 contains a number 𝜏 with the property that |𝑔(𝑡+𝜏)−𝑔(𝑡)|<𝜖,𝑡∈𝑅.

Definition 2.2. A function 𝑔∈ℒ(𝑅×Ω) is called almost periodic in 𝑡∈𝑅, uniformly in 𝑍∈Ω, if, for each 𝜖>0 and any compact set 𝑀 of Ω, there exists an 𝑙_𝜖>0 such that every interval of length 𝑙_𝜖 contains a number 𝜏 with the property that |𝑔(𝑡+𝜏,𝑧)−𝑔(𝑡,𝑧)|<𝜖,𝑡∈𝑅,𝑍∈Ω. The number 𝜏 is called an 𝜖-translation number of 𝑔. Denote by 𝒜𝒫(𝑅)(𝒜𝒫(𝑅×Ω)) the set of all such function.
Set𝒫𝒜𝒫₀(𝑅)={𝜑∈ℒ(𝑅)∶lim_𝑡→∞12𝑡^𝑡_−𝑡|𝜑(𝑠)|𝑑𝑠=0},𝒫𝒜𝒫₀(𝑅×Ω)={𝜑∈ℒ(𝑅×Ω)∶lim_𝑡→∞12𝑡^𝑡_−𝑡|𝜑(𝑠,𝑍)|𝑑𝑠=0,uniformlyin𝑍∈Ω}. (2.1)

Definition 2.3. A function 𝑓∈ℒ(𝑅)(ℒ(𝑅×Ω)) is called pseudo almost periodic (pseudo almost periodic in 𝑡∈𝑅, uniformly in 𝑍∈Ω), if 𝑓=𝑔+𝜑, where 𝑔∈𝒫𝒜(𝑅)(𝒫𝒜(𝑅×Ω)) and 𝜑∈𝒫𝒜𝒫₀(𝑅)(𝒫𝒜𝒫₀(𝑅×Ω)). The function 𝑔 and 𝜑 are called the almost periodic component and the erigodic perturbation, respectively, of the function 𝑓. Denote by 𝒫𝒜𝒫(𝑅)(𝒫𝒜𝒫(𝑅×Ω)) the set of all such functions 𝑓.

Define ‖𝑥‖_∞=sup_𝑡∈𝑅|𝑥(𝑡)|_∞,𝑥∈𝒫𝒜𝒫. It is trivial to show that 𝒫𝒜𝒫 is a Banach space with ‖⋅‖_∞.

Let 𝐴(𝑡)=(𝑎_𝑖𝑗(𝑡)) be a complex 𝑛×𝑛 matrix-valued function with elements (entries) which are continuous on 𝑅. We consider the homogeneous linear ODE and nonhomogeneous linear ODE as follows:𝑑𝑥𝑑𝑡=𝐴(𝑡)𝑥,𝑑𝑥𝑑𝑡=𝐴(𝑡)𝑥+𝑓(𝑡), (2.2)(2.3) where 𝑥 denotes an 𝑛-column vector.

Definition 2.4 (see [15, 16]). The homogeneous linear ODE (2.2) is said to admit an exponential dichotomy if there exist a linear projection 𝑝(i.e.,𝑝²=𝑃) on 𝐶^𝑛 and positive constants 𝑘,𝛼,𝛽 such that ‖𝑋(𝑡)𝑃𝑋⁻¹(𝑠)‖≤𝑘𝑒^{−𝛼(𝑡−𝑠)},𝑡≥𝑠,‖𝑋(𝑡)(𝐼−𝑃)𝑋⁻¹(𝑠)‖≤𝑘𝑒^{−𝛽(𝑠−𝑡)},𝑡≤𝑠, (2.4) where 𝑋(𝑡) is a fundamental matrix of (2.2) with 𝑋(0)=𝐸; 𝐸 is the 𝑛×𝑛 identity matrix.

Definition 2.5 (see [15, 16]). The matrix 𝐴(𝑡) is said to be row dominant if there exists a number 𝛿>0 such that |Re𝑎_𝑖𝑗(𝑡)|≥∑^𝑛_{𝑗=1,𝑗≠𝑖}|𝑎_𝑖𝑗(𝑡)|+𝛿 for all 𝑡∈(−∞,∞) and 𝑖=1,2,…,𝑛.

Lemma 2.6 (see [15, 16]). If 𝐴(𝑡) is a bounded, continuous, and row-dominant 𝑛×𝑛 matrix function on 𝑅, and there exists 𝑘≤𝑛 such that Re𝑎_𝑖𝑗<0(𝑖=1,2,…,𝑘). Then (2.2) has a fundamental matrix solution 𝑋(𝑡) satisfying ‖𝑋(t)𝑃𝑋⁻¹(𝑠)‖≤𝑘𝑒^{−𝛿(𝑡−𝑠)},𝑡≥𝑠,‖𝑋(𝑡)(𝐼−𝑃)𝑋⁻¹(𝑠)‖≤𝑘𝑒^{−𝛿(𝑠−𝑡)},𝑡≤𝑠, (2.5) where 𝐾 is a positive constant, and 𝑃=diag(𝐸_𝑘,0) with 𝐸_𝑘 being a 𝑘×𝑘 identity matrix.
For 𝐻=(ℎ₁,ℎ₂,…,ℎ_𝑛)∈ℒ(𝑅)^𝑛, suppose that 𝐻(𝑡)∈Ω for all 𝑡∈𝑅. Define 𝐻×𝑙∶𝑅→Ω×𝑅 by 𝐻×𝑙(𝑡)=(ℎ₁(𝑡),ℎ₂(𝑡),…,ℎ_𝑛(𝑡),𝑡)(𝑡∈𝑅).

Lemma 2.7 (see [15, 16]). Assume that the function 𝑓(𝑡,𝑧)∈𝒫𝒜𝒫(𝑅×Ω) is continuous in 𝑧∈𝑀 uniformly in 𝑡∈𝑅 for all compact subsets 𝑀⊂Ω and 𝐹∈𝒫𝒜𝒫(𝑅)^𝑛 such that 𝐹(𝑅)⊂Ω. Then 𝑓∘(𝐹×𝑙)∈𝒫𝒜𝒫(𝑅).
It is obvious that if 𝑓 satisfies a Lipschitz condition; that is, there is an 𝐿>0 such that |𝑓(𝑧^′,𝑡)−𝑓(𝑧,𝑡)|≤𝐿|𝑧^′−𝑧|(𝑧^′,𝑧∈𝑀,𝑡∈𝑅), (2.6) then 𝑓 is continuous in 𝑧∈𝑀 uniformly in 𝑡∈𝑅. Obviously, if 𝑓(𝑡)∈𝒫𝒜𝒫(𝑅) is uniformly continuous in 𝑡∈𝑅 and 𝜑∈𝒫𝒜𝒫(𝑅) such that 𝜑(𝑅)⊂Im𝑓, then 𝑓∘𝜑∈𝒫𝒜𝒫(𝑅).

Lemma 2.8 (see [15, 16]). Assume that 𝐴(𝑡) is an almost-periodic matrix function and 𝑓(𝑡)∈𝒫𝒜𝒫(𝑅^𝑛). If (2.2) satisfies an exponential dichotomy, then (2.3) has unique pseudo almost periodic solution 𝑥(𝑡) reading 𝑥(𝑡)=^𝑡_−∞𝑋(𝑡)𝑃𝑋⁻¹(𝑠)𝑓(𝑠)𝑑𝑠−^∞_𝑡𝑋(𝑡)(𝐸−𝑃)𝑋⁻¹(𝑠)𝑓(𝑠)𝑑𝑠 (2.7) and satisfying ‖𝑥‖≤(𝐾/𝛼+𝐾/𝛽)‖𝑓‖, where 𝑋(𝑡) is a fundamental matrix solution of (2.2).

Definition 2.9. System (1.4) is said to be globally exponentially stable (GES), if for any two solutions 𝑥(𝑡) and 𝑦(𝑡) of (1.4), there exist positive numbers 𝑀 and 𝜀 such that |𝑥(𝑡)−𝑦(𝑡)|_𝑝≤𝑀𝑒^{−𝜀(𝑡−𝑡0)}‖𝜑−𝜓‖_𝑝,𝑡>𝑡₀, (2.8) where 𝑥(𝑡)=𝑥(𝑡,𝜑) and 𝑦(𝑡)=𝑦(𝑡,𝜓) denoting the solution of (1.4) through (𝑡₀,𝜑) and (𝑡₀,𝜓) respectively. Here 𝜀 is called the Lyapunov exponent of (1.4).

3. Existence and Stability of Pseudo Almost-Periodic Solution

Theorem 3.1. Assume that (𝐴₁)–(𝐴₃) hold and (𝐴₄)𝑟=sup_𝑡∈𝑅max_(𝑖,𝑗){^𝑡_−∞𝑒^{−∫𝑡𝑠𝑎𝑖𝑗(𝜂)𝑑𝜂}(_{𝐵ℎ𝑙∈𝑁𝑟(𝑖,𝑗)}𝜇_𝑖𝑗𝐵^ℎ𝑙_𝑖𝑗(𝑠)+_{𝐶ℎ𝑙∈𝑁𝑟(𝑖,𝑗)}𝜈_𝑖𝑗𝐶^ℎ𝑙_𝑖𝑗(𝑠))𝑑𝑠}<1. (3.1)Then (1.4) has a unique pseudo almost-periodic solution, say 𝑥^∗(𝑡), satisfying ‖𝑥^∗‖_∞≤𝐿/(1−𝑟).

Proof. For any 𝜑∈𝒫𝒜𝒫(𝑅), consider 𝑥̇_𝑖𝑗=−𝑎_𝑖𝑗(𝑡)𝑥_𝑖𝑗(𝑡)−_{𝐵ℎ𝑙∈𝑁𝑟(𝑖𝑗)}𝐵^ℎ𝑙_𝑖𝑗(𝑡)𝑓_𝑖𝑗(𝜑_ℎ𝑙(𝑡))𝜑_𝑖𝑗(𝑡)−_{𝐶ℎ𝑙∈𝑁𝑟(𝑖𝑗)}𝐶^ℎ𝑙_𝑖𝑗(𝑡)𝑔_𝑖𝑗(𝜑_ℎ𝑙(𝑡−𝜏_ℎ𝑙(𝑡)))𝜑_𝑖𝑗(𝑡)+𝐿_𝑖𝑗(𝑡). (3.2) Since −𝑎_𝑖𝑗(𝑡)<0, from Lemmas 2.6, 2.7, and 2.8, it follows that (3.2) has a unique pseudo almost-periodic solution, which is given by 𝑥_𝜑(𝑡)=(^𝑡_−∞𝑒^{−∫𝑡𝑠𝑎𝑖𝑗(𝜂)𝑑𝜂}[−_{𝐵ℎ𝑙∈𝑁𝑟(𝑖,𝑗)}𝐵^ℎ𝑙_𝑖𝑗(𝑠)𝑓_𝑖𝑗(𝜑_ℎ𝑙(𝑠))𝜑_𝑖𝑗(𝑠)−_{𝐶ℎ𝑙∈𝑁𝑟(𝑖,𝑗)}𝐶^ℎ𝑙_𝑖𝑗(𝑠)𝑔_𝑖𝑗(𝜑_ℎ𝑙(𝑠−𝜏_ℎ𝑙(𝑠)))𝜑_𝑖𝑗(𝑠)+𝐿_𝑖𝑗(𝑠)]𝑑𝑠)_𝑚𝑛×1. (3.3) Define the mapping 𝒯∶𝒫𝒜𝒫(𝑅)→𝒫𝒜𝒫(𝑅) by 𝒯(𝜑)(𝑡)=𝑥_𝜑(𝑡),𝜑∈𝒫𝒜𝒫(𝑅).
Let 𝐵^∗={𝜑∣𝜑∈𝒫𝒜𝒫(𝑅),‖𝜑−𝜑₀‖_∞≤(𝛾/(1−𝛾))𝐿}, where 𝜑₀(𝑡)=(∫^𝑡_−∞𝑒^{−∫𝑡𝑠𝑎11(𝜂)𝑑𝜂}𝐿₁₁(𝑠)𝑑𝑠,…,∫^𝑡_−∞𝑒^{−∫𝑡𝑠𝑎𝑖𝑗(𝜂)𝑑𝜂}𝐿_𝑖𝑗(𝑠)𝑑𝑠,…,∫^𝑡_−∞𝑒^{−∫𝑡𝑠𝑎𝑚𝑛(𝜂)𝑑𝜂}𝐿_𝑚𝑛(𝑠)𝑑𝑠)^𝑇.
Clearly, 𝐵^∗ is closed and convex in 𝐵. Note that‖𝜑₀‖_∞=sup_𝑡∈𝑅max_(𝑖,𝑗){|^𝑡_−∞𝑒^{−∫𝑡𝑠𝑎𝑖𝑗(𝜂)𝑑𝜂}𝐿_𝑖𝑗(𝑠)𝑑𝑠|}≤sup_𝑡∈𝑅max_(𝑖,𝑗){|^𝑡_−∞𝑒^{−𝑎−𝑖𝑗(𝑠−𝑡)}𝐿⁺_𝑖𝑗𝑑𝑠|}≤sup_𝑡∈𝑅max_(𝑖,𝑗){𝐿⁺_𝑖𝑗𝑎⁻_𝑖𝑗}=max_(𝑖,𝑗){𝐿⁺_𝑖𝑗𝑎⁻_𝑖𝑗}=𝐿. (3.4) Therefore, for any 𝜑∈𝐵^∗, we have ‖𝜑‖≤‖𝜑−𝜑₀‖+‖𝜑₀𝛾‖≤𝐿1−𝛾𝐿+𝐿=.1−𝛾 (3.5) Now, we will show that 𝒯 maps 𝐵^∗ into itself. In fact, for any 𝜑∈𝐵^∗, by using 𝐿/(1−𝛾)≤1, we have ‖𝒯𝜑−𝜑₀‖_∞=sup_𝑡∈𝑅max_(𝑖,𝑗){|^𝑡_−∞𝑒^{−∫𝑡𝑠𝑎𝑖𝑗(𝜂)𝑑𝜂}[−_{𝐵ℎ𝑙∈𝑁𝑟(𝑖,𝑗)}𝐵^ℎ𝑙_𝑖𝑗(𝑠)𝑓_𝑖𝑗(𝜑_ℎ𝑙(𝑠))𝜑_𝑖𝑗(𝑠)−_{𝐶ℎ𝑙∈𝑁𝑟(𝑖,𝑗)}𝐶^ℎ𝑙_𝑖𝑗(𝑠)𝑔_𝑖𝑗(𝜑_ℎ𝑙(𝑠−𝜏_ℎ𝑙(𝑠)))𝜑_𝑖𝑗(𝑠)]𝑑𝑠|)≤sup_𝑡∈𝑅max_(𝑖,𝑗){^𝑡_−∞𝑒^{−∫𝑡𝑠𝑎𝑖𝑗(𝜂)𝑑𝜂}[𝜇_{𝑖𝑗𝐵ℎ𝑙∈𝑁𝑟(𝑖,𝑗)}𝐵^ℎ𝑙_𝑖𝑗(𝑠)|(𝜑_ℎ𝑙(𝑠))||𝜑_𝑖𝑗(𝑠)|+𝜈_{𝑖𝑗𝐶ℎ𝑙∈𝑁𝑟(𝑖,𝑗)}𝐶^ℎ𝑙_𝑖𝑗(𝑠)|𝜑_ℎ𝑙(𝑠−𝜏_ℎ𝑙(𝑠))||𝜑_𝑖𝑗(𝑠)|]𝑑𝑠)≤max_(𝑖,𝑗){^𝑡_−∞𝑒^{−∫𝑡𝑠𝑎𝑖𝑗(𝜂)𝑑𝜂}(𝜇_{𝑖𝑗𝐵ℎ𝑙∈𝑁𝑟(𝑖,𝑗)}𝐵^ℎ𝑙_𝑖𝑗(𝑠)+𝜈_{𝑖𝑗𝐶ℎ𝑙∈𝑁𝑟(𝑖,𝑗)}𝐶^ℎ𝑙_𝑖𝑗(𝑠))𝑑𝑠}‖𝜑‖².≤𝛾‖𝜑‖≤𝛾‖𝜑‖≤𝛾𝐿.1−𝛾 (3.6) For any 𝜑,𝜙∈𝐵^∗, it follows from 𝐿/(1−𝛾)≤1 that ‖𝒯𝜑−𝒯𝜓‖_∞=sup_𝑡∈𝑅max_(𝑖,𝑗){|^𝑡_−∞𝑒^{−∫𝑡𝑠𝑎𝑖𝑗(𝜂)𝑑𝜂}×[_{𝐵ℎ𝑙∈𝑁𝑟(𝑖,𝑗)}𝐵^ℎ𝑙_𝑖𝑗(𝑠)|𝑓_𝑖𝑗(𝜑_ℎ𝑙(𝑠))𝜑_𝑖𝑗(𝑠)−𝑓_𝑖𝑗(𝜓_ℎ𝑙(𝑠))𝜓_𝑖𝑗(𝑠)|+_{𝐶ℎ𝑙∈𝑁𝑟(𝑖,𝑗)}𝐶^ℎ𝑙_𝑖𝑗(𝑠)|𝑔_𝑖𝑗(𝜑_ℎ𝑙(𝑠−𝜏_ℎ𝑙(𝑠)))𝜑_𝑖𝑗(𝑠)−𝑔_𝑖𝑗(𝜓_ℎ𝑙(𝑠−𝜏_ℎ𝑙(𝑠)))𝜓_𝑖𝑗(𝑠)|]𝑑𝑠|)≤sup_𝑡∈𝑅max_(𝑖,𝑗){^𝑡_−∞𝑒^{−∫𝑡𝑠𝑎𝑖𝑗(𝜂)𝑑𝜂}×[_{𝐵ℎ𝑙∈𝑁𝑟(𝑖,𝑗)}𝐵^ℎ𝑙_𝑖𝑗(𝑠)(|𝑓_𝑖𝑗(𝜑_ℎ𝑙(𝑠))|⋅|𝜑_𝑖𝑗(𝑠)−𝜓_𝑖𝑗(𝑠)|+|𝑓_𝑖𝑗(𝜑_ℎ𝑙(𝑠))−𝑓_𝑖𝑗(𝜓_ℎ𝑙(𝑠))|⋅|𝜓_𝑖𝑗(𝑠)|)+_{𝐶ℎ𝑙∈𝑁𝑟(𝑖,𝑗)}𝐶^ℎ𝑙_𝑖𝑗(𝑠)(|𝑔_𝑖𝑗(𝜑_ℎ𝑙(𝑠−𝜏_ℎ𝑙(𝑠)))|×|𝜑_𝑖𝑗(𝑠)−𝜓_𝑖𝑗(𝑠)|+|𝑔_𝑖𝑗(𝜑_ℎ𝑙(𝑠−𝜏_ℎ𝑙(𝑠)))−𝑔_𝑖𝑗(𝜓_ℎ𝑙(𝑠−𝜏_ℎ𝑙(𝑠)))|⋅|𝜓_𝑖𝑗(𝑠)|)]𝑑𝑠}≤max_(𝑖,𝑗){^𝑡_−∞𝑒^{−∫𝑡𝑠𝑎𝑖𝑗(𝜂)𝑑𝜂}[𝜇_{𝑖𝑗𝐵ℎ𝑙∈𝑁𝑟(𝑖,𝑗)}𝐵^ℎ𝑙_𝑖𝑗(𝑠)+𝜈_{𝑖𝑗𝐶ℎ𝑙∈𝑁𝑟(𝑖,𝑗)}𝐶^ℎ𝑙_𝑖𝑗(𝑠)](‖𝜑‖_∞+‖𝜓‖_∞)‖𝜑−𝜓‖𝑑𝑠}=𝛾⋅2𝐿1−𝛾‖𝜑−𝜓‖=2𝐿𝛾1−𝛾‖𝜑−𝜓‖=𝛿‖𝜑−𝜓‖. (3.7)(1)(2)(3)(4)(5)(6)(7)(8)(9) Since 𝛿<1, 𝒯 is a contraction mapping. Therefore, there exists a unique fixed point 𝑥^∗∈𝐵^∗ such that 𝒯𝑥^∗=𝑥^∗. That is, system (1.4) has a unique pseudo almost-periodic solution 𝑥^∗∈𝐵^∗ with ‖𝑥^∗−𝜑₀‖≤(𝛾/(1−𝛾))𝐿.

Now we go ahead with the GES of (1.4). The approaches involve constructing suitable Lyapunov functions and application of a generalized Halanay's delay differential inequality. We will stop here to see our first criteria for the globally exponential stability of (1.4), which is delay dependent.

Theorem 3.2. In addition to (𝐴₁)–(𝐴₄), if one further assumes that(𝐴₅)𝑐=min_(𝑖,𝑗)inf_(𝑖,𝑗){𝛽_𝑖𝑗(2𝑎_𝑖𝑗(𝑡)−3𝛽_𝑖𝑗𝐿1−𝛾[𝜇_{𝑖𝑗𝐵ℎ𝑙∈𝑁𝑟(𝑖,𝑗)}𝐵^ℎ𝑙_𝑖𝑗(𝑡)+𝜈_{𝑖𝑗𝐶ℎ𝑙∈𝑁𝑟(𝑖,𝑗)}𝐶^ℎ𝑙_𝑖𝑗(𝑡)])+𝛽_ℎ𝑙(𝜇_ℎ𝑙𝐿1−𝛾_{𝐵𝑖𝑗∈𝑁𝑟(ℎ,𝑙)}𝐵^𝑖𝑗_ℎ𝑙𝐿(𝑡)+𝜈1−𝛾^ℎ𝑙_{𝐶ℎ𝑙∈𝑁𝑟(𝑖,𝑗)}𝐶^𝑖𝑗_ℎ𝑙(𝜉⁻¹_𝑖𝑗(𝑡))1−𝜏_𝑖𝑗(𝜉⁻¹_𝑖𝑗(𝑡)))}>0 (3.8) or(𝐴₆)𝑐=inf_(𝑖,𝑗){𝛽_𝑖𝑗(𝑎_𝑖𝑗(𝑡)−𝛽_𝑖𝑗𝐿1−𝛾[𝜇_{𝑖𝑗𝐵ℎ𝑙∈𝑁𝑟(𝑖,𝑗)}𝐵^ℎ𝑙_𝑖𝑗(𝑡)+𝜈_{𝑖𝑗𝐶ℎ𝑙∈𝑁𝑟(𝑖,𝑗)}𝐶^ℎ𝑙_𝑖𝑗(𝑡)])−𝛽_ℎ𝑙(𝜇_ℎ𝑙𝐿1−𝛾_{𝐵𝑖𝑗∈𝑁𝑟(ℎ,𝑙)}𝐵^𝑖𝑗_ℎ𝑙𝐿(𝑡)+𝜈1−𝛾^ℎ𝑙_{𝐶ℎ𝑙∈𝑁𝑟(𝑖,𝑗)}𝐶^𝑖𝑗_ℎ𝑙(𝜉⁻¹_𝑖𝑗(𝑡))1−𝜏_𝑖𝑗(𝜉⁻¹_𝑖𝑗(𝑡)))}>0. (3.9) Then there exists a unique pseudo-almost periodic solution of system (1.4) and all other solutions converge exponentially to the (pseudo) almost-periodic attractor.

Proof. By Theorem (3.2), there exists a unique pseudo almost-periodic solution, namely, 𝑥(𝑡)=𝑥(𝑡,𝜑). Let 𝑦=𝑦(𝑡,𝜑) be any other solution of (1.4) through (𝑡₀,𝜑). Assume that (𝐴₅) is satisfied and consider the auxiliary functions 𝐹_𝑖𝑗(𝜀) defined on [0,+∞] as follows: 𝐹_𝑖𝑗(𝜀)=inf_𝑡∈𝑅{𝛽_𝑖𝑗(2𝑎_𝑖𝑗(𝑡)−𝜀)−3𝛽_𝑖𝑗𝐿1−𝛾[𝜇_{𝑖𝑗𝐵ℎ𝑙∈𝑁𝑟(𝑖,𝑗)}𝐵^ℎ𝑙_𝑖𝑗(𝑡)+𝜈_{𝑖𝑗𝐶ℎ𝑙∈𝑁𝑟(𝑖,𝑗)}𝐶^ℎ𝑙_𝑖𝑗(𝑡)]+𝛽_ℎ𝑙(𝜇_ℎ𝑙𝐿1−𝛾_{𝐵𝑖𝑗∈𝑁𝑟(ℎ,𝑙)}𝐵^𝑖𝑗_ℎ𝑙𝐿(𝑡)+𝜈1−𝛾^ℎ𝑙_{𝐶ℎ𝑙∈𝑁𝑟(𝑖,𝑗)}𝐶^𝑖𝑗_ℎ𝑙(𝜉⁻¹_𝑖𝑗(𝑡))1−𝜏_𝑖𝑗(𝜉⁻¹_𝑖𝑗(𝑡)))}. (3.10) From(𝐴₁)–(𝐴₃), one can easily show that 𝐹_𝑖𝑗(𝜀) is well defined and is continuous. From (𝐴₅), it follows that 𝐹_𝑖𝑗(0)>0,𝐹_𝑖𝑗(𝜀)→−∞ as 𝜀→∞ it follows that there exists an 𝜀_𝑖𝑗>0 such that 𝐹_𝑖𝑗(𝜀_𝑖𝑗)>0. Let 𝜀=min_𝑖,𝑗𝜀_𝑖𝑗. Then we have 𝐹_𝑖𝑗(𝜀)>0,1≤𝑖≤𝑛,1≤𝑗≤𝑚.
Consider the Lyapunov functional defined by1𝑉(𝑡)=2_(𝑖,𝑗)𝛽_𝑖𝑗[(𝑥_𝑖𝑗(𝑡)−𝑦_𝑖𝑗(𝑡))²𝑒^𝜀𝑡+_{𝐶ℎ𝑙∈𝑁𝑟(𝑖,𝑗)}𝑉_𝑖𝑗𝐿1−𝛾^𝑡_{𝑡−𝜏ℎ𝑙(𝑡)}𝐶^ℎ𝑙(𝜉⁻¹_ℎ𝑙(𝑡))1−𝜏_ℎ𝑙(𝜉⁻¹_ℎ𝑙)(𝑡)(𝑥_ℎ𝑙(𝑠)−𝑦_ℎ𝑙(𝑠))²𝑒^{𝜀(𝑠+𝜏𝑀ℎ𝑙)}𝑑𝑠]. (3.11) Calculating the upper-right derivative of 𝑉(𝑡) and using the inequality 2𝑎𝑏≤𝑎²+𝑏², one has 𝑉^′(𝑡)≤_(𝑖,𝑗)𝛽_𝑖𝑗{12(𝑥_𝑖𝑗(𝑡)−𝑦_𝑖𝑗(𝑡))²𝜀𝑒^𝜀𝑡−𝑒^𝜀𝑡𝑎_𝑖𝑗(𝑡)(𝑥_𝑖𝑗−𝑦_𝑖𝑗(𝑡))²+𝑒^𝜀𝑡_{𝐵ℎ𝑙∈𝑁𝑟(𝑖,𝑗)}𝐵^ℎ𝑙_𝑖𝑗(𝑡)|𝑥_𝑖𝑗(𝑡)−𝑦_𝑖𝑗(𝑡)|⋅|𝑓_𝑖𝑗(𝑥_ℎ𝑙(𝑡))𝑥_𝑖𝑗(𝑡)−𝑓_𝑖𝑗(𝑦_ℎ𝑙(𝑡))𝑦_𝑖𝑗(𝑡)|+𝑒^𝜀𝑡_{𝐶ℎ𝑙∈𝑁𝑟(𝑖,𝑗)}𝐶^ℎ𝑙_𝑖𝑗(𝑡)|𝑥_𝑖𝑗(𝑡)−𝑦_𝑖𝑗(𝑡)|⋅|𝑔_𝑖𝑗(𝑥_ℎ𝑙(𝑡−𝜏_ℎ𝑙(𝑡)))𝑥_𝑖𝑗(𝑡)−𝑔_𝑖𝑗(𝑦_ℎ𝑙(𝑡−𝜏_ℎ𝑙(𝑡)))𝑦_𝑖𝑗+1(𝑡)|2_{𝐶ℎ𝑙∈𝑁𝑟(𝑖,𝑗)}𝜈_𝑖𝑗𝐿𝐶1−𝛾^{ℎ𝑙𝑖𝑗}(𝜉⁻¹_ℎ𝑙(𝑡))1−𝜏_ℎ𝑙(𝜉⁻¹_ℎ𝑙(𝑡))(𝑥_ℎ𝑙(𝑡)−𝑦_ℎ𝑙(𝑡))²𝑒^{𝜀(𝑡+𝜏𝑀ℎ𝑙)}−12_{𝐶ℎ𝑙∈𝑁𝑟(𝑖,𝑗)}𝜈_𝑖𝑗𝐿𝐶1−𝛾^{ℎ𝑙𝑖𝑗}(𝑡)(𝑥_ℎ𝑙(𝑡−𝜏_ℎ𝑙(𝑡))−𝑦_ℎ𝑙(𝑡−𝜏_ℎ𝑙(𝑡)))²𝑒^{𝜀(𝑡−𝜏ℎ𝑙(𝑡)+𝜏𝑀ℎ𝑙)}≤_(𝑖,𝑗)}𝛽_𝑖𝑗{12(𝑥_𝑖𝑗(𝑡)−𝑦_𝑖𝑗(𝑡))²𝜀𝑒^𝜀𝑡−𝑒^𝜀𝑡𝑎_𝑖𝑗(𝑡)(𝑥_𝑖𝑗−𝑦_𝑖𝑗(𝑡))²+𝑒^𝜀𝑡_{𝐵ℎ𝑙∈𝑁𝑟(𝑖,𝑗)}𝐵^ℎ𝑙_𝑖𝑗(𝑡)|𝑥_𝑖𝑗(𝑡)−𝑦_𝑖𝑗(𝑡)|⋅|𝑓_𝑖𝑗(𝑥_ℎ𝑙(𝑡))𝑥_𝑖𝑗(𝑡)−𝑓_𝑖𝑗(𝑦_ℎ𝑙(𝑡))𝑦_𝑖𝑗(𝑡)|+𝑒^𝜀𝑡_{𝐶ℎ𝑙∈𝑁𝑟(𝑖,𝑗)}𝐶^ℎ𝑙_𝑖𝑗(𝑡)|𝑥_𝑖𝑗(𝑡)−𝑦_𝑖𝑗(𝑡)|⋅|𝑔_𝑖𝑗(𝑥_ℎ𝑙(𝑡−𝜏_ℎ𝑙(𝑡)))𝑥_𝑖𝑗(𝑡)−𝑔_𝑖𝑗(𝑦_ℎ𝑙(𝑡−𝜏_ℎ𝑙(𝑡)))𝑦_𝑖𝑗+1(𝑡)|2_{𝐶ℎ𝑙∈𝑁𝑟(𝑖,𝑗)}𝜈_𝑖𝑗𝐿𝐶1−𝛾^{ℎ𝑙𝑖𝑗}(𝜉⁻¹_ℎ𝑙(𝑡))1−𝜏_ℎ𝑙(𝜉⁻¹_ℎ𝑙(𝑡))(𝑥_ℎ𝑙(𝑡)−𝑦_ℎ𝑙(𝑡))²𝑒^{𝜀(𝑡+𝜏𝑀ℎ𝑙)}−12_{𝐶ℎ𝑙∈𝑁𝑟(𝑖,𝑗)}𝜈_𝑖𝑗𝐿𝐶1−𝛾^{ℎ𝑙𝑖𝑗}(𝑡)(𝑥_ℎ𝑙(𝑡−𝜏_ℎ𝑙(𝑡))−𝑦_ℎ𝑙(𝑡−𝜏_ℎ𝑙(𝑡)))²𝑒^𝜀(𝑡)}≤𝑒^𝜀(𝑡)_(𝑖,𝑗)𝛽_𝑖𝑗𝜀{(2−𝑎_𝑖𝑗(𝑡))(𝑥_𝑖𝑗(𝑡)−𝑦_𝑖𝑗(𝑡))²+_{𝐵ℎ𝑙∈𝑁𝑟(𝑖,𝑗)}𝐵^ℎ𝑙_𝑖𝑗(𝑡)|𝑥_𝑖𝑗(𝑡)−𝑦_𝑖𝑗(𝑡)|[|𝑓_𝑖𝑗(𝑥_ℎ𝑙(𝑡))||𝑥_𝑖𝑗(𝑡)−𝑦_𝑖𝑗(𝑡)|+|𝑓_𝑖𝑗(𝑥_ℎ𝑙(𝑡))−𝑓_𝑖𝑗(𝑦_ℎ𝑙(𝑡))||𝑦_𝑖𝑗+(𝑡)|]_{𝐶ℎ𝑙∈𝑁𝑟(𝑖,𝑗)}𝐶^ℎ𝑙_𝑖𝑗(𝑡)|𝑥_𝑖𝑗(𝑡)−𝑦_𝑖𝑗(𝑡)|×[|𝑔_𝑖𝑗(𝑥_ℎ𝑙(𝑡−𝜏_ℎ𝑙(𝑡)))||𝑥_𝑖𝑗(𝑡)−𝑦_𝑖𝑗(𝑡)|+|𝑔_𝑖𝑗(𝑥_ℎ𝑙(𝑡−𝜏_ℎ𝑙(𝑡)))−𝑔_𝑖𝑗(𝑦_ℎ𝑙(𝑡−𝜏_ℎ𝑙(𝑡)))||𝑦_𝑖𝑗+1(𝑡)|]2_{𝐶ℎ𝑙∈𝑁𝑟(𝑖,𝑗)}𝜈_𝑖𝑗𝐿𝐶1−𝛾^{ℎ𝑙𝑖𝑗}(𝜉⁻¹_ℎ𝑙(𝑡))1−𝜏_ℎ𝑙(𝜉⁻¹_ℎ𝑙(𝑡))(𝑥_ℎ𝑙(𝑡)−𝑦_ℎ𝑙(𝑡))²𝑒^{𝜀𝜏𝑀ℎ𝑙}−12_{𝐶ℎ𝑙∈𝑁𝑟(𝑖,𝑗)}𝜈_𝑖𝑗𝐿𝐶1−𝛾^{ℎ𝑙𝑖𝑗}(𝑡)(𝑥_ℎ𝑙(𝑡−𝜏_ℎ𝑙(𝑡))−𝑦_ℎ𝑙(𝑡−𝜏_ℎ𝑙(𝑡)))²}≤𝑒^𝜀(𝑡)_(𝑖,𝑗)𝛽_𝑖𝑗𝜀{(2−𝑎_𝑖𝑗(𝑡))(𝑥_𝑖𝑗(𝑡)−𝑦_𝑖𝑗(𝑡))²+_{𝐵ℎ𝑙∈𝑁𝑟(𝑖,𝑗)}𝐵^ℎ𝑙_𝑖𝑗(𝑡)𝜇_𝑖𝑗𝐿1−𝛾×[(𝑥_𝑖𝑗(𝑡)−𝑦_𝑖𝑗(𝑡))²+|𝑥_𝑖𝑗(𝑡)−𝑦_𝑖𝑗(𝑡)||𝑥_ℎ𝑙(𝑡)−𝑦_ℎ𝑙(𝑡)|]+_{𝐶ℎ𝑙∈𝑁𝑟(𝑖,𝑗)}𝐶^ℎ𝑙_𝑖𝑗(𝑡)𝜈_𝑖𝑗𝐿1−𝛾×[(𝑥_𝑖𝑗(𝑡)−𝑦_𝑖𝑗(𝑡))²+|𝑥_𝑖𝑗(𝑡)−𝑦_𝑖𝑗(𝑡)|×|𝑥_ℎ𝑙(𝑡−𝜏_ℎ𝑙(𝑡))−𝑦_ℎ𝑙(𝑡−𝜏_ℎ𝑙+1(𝑡))|]2_{𝐶ℎ𝑙∈𝑁𝑟(𝑖,𝑗)}𝜈_𝑖𝑗𝐿𝐶1−𝛾^{ℎ𝑙𝑖𝑗}(𝜉⁻¹_ℎ𝑙(𝑡))1−𝜏_ℎ𝑙(𝜉⁻¹_ℎ𝑙(𝑡))(𝑥_𝑖𝑗(𝑡)−𝑦_𝑖𝑗(𝑡))²𝑒^{𝜀𝜏𝑀ℎ𝑙}−12_{𝐶ℎ𝑙∈𝑁𝑟(𝑖,𝑗)}𝜈_𝑖𝑗𝐿𝐶1−𝛾^{ℎ𝑙𝑖𝑗}(𝑡)(𝑥_ℎ𝑙(𝑡−𝜏_ℎ𝑙(𝑡))−𝑦_ℎ𝑙(𝑡−𝜏_ℎ𝑙(𝑡)))²}≤𝑒^𝜀(𝑡)_(𝑖,𝑗)𝛽_𝑖𝑗𝜀{(2−𝑎_𝑖𝑗(𝑡))(𝑥_𝑖𝑗(𝑡)−𝑦_𝑖𝑗(𝑡))²+_{𝐵ℎ𝑙∈𝑁𝑟(𝑖,𝑗)}𝐵^ℎ𝑙_𝑖𝑗(𝑡)𝜇_𝑖𝑗𝐿1−𝛾×[(𝑥_𝑖𝑗(𝑡)−𝑦_𝑖𝑗(𝑡))²+12(𝑥_𝑖𝑗(𝑡)−𝑦_𝑖𝑗(𝑡))²(𝑥_ℎ𝑙(𝑡)−𝑦_ℎ𝑙(𝑡))²]+_{𝐶ℎ𝑙∈𝑁𝑟(𝑖,𝑗)}𝐶^ℎ𝑙_𝑖𝑗(𝑡)𝜈_𝑖𝑗𝐿1−𝛾×[(𝑥_𝑖𝑗(𝑡)−𝑦_𝑖𝑗(𝑡))²+12(𝑥_𝑖𝑗(𝑡)−𝑦_𝑖𝑗(𝑡))²×(𝑥_ℎ𝑙(𝑡−𝜏_ℎ𝑙(𝑡))−𝑦_ℎ𝑙(𝑡−𝜏_ℎ𝑙(𝑡)))²+1]2_{𝐶ℎ𝑙∈𝑁𝑟(𝑖,𝑗)}𝜈_𝑖𝑗𝐿𝐶1−𝛾^{ℎ𝑙𝑖𝑗}(𝜉⁻¹_ℎ𝑙(𝑡))1−𝜏_ℎ𝑙(𝜉⁻¹_ℎ𝑙(𝑡))(𝑥_𝑖𝑗(𝑡)−𝑦_𝑖𝑗(𝑡))²𝑒^{𝜀𝜏𝑀ℎ𝑙}−12_{𝐶ℎ𝑙∈𝑁𝑟(𝑖,𝑗)}𝜈_𝑖𝑗𝐿𝐶1−𝛾^{ℎ𝑙𝑖𝑗}(𝑡)(𝑥_ℎ𝑙(𝑡−𝜏_ℎ𝑙(𝑡))−𝑦_ℎ𝑙(𝑡−𝜏_ℎ𝑙(𝑡)))²}≤𝑒^𝜀(𝑡)_(𝑖,𝑗){𝛽_𝑖𝑗(𝜀2−𝑎_𝑖𝑗(𝑡))(𝑥_𝑖𝑗(𝑡)−𝑦_𝑖𝑗(𝑡))²+32_{𝐵ℎ𝑙∈𝑁𝑟(𝑖,𝑗)}𝐵^ℎ𝑙_𝑖𝑗(𝑡)𝜇_𝑖𝑗𝐿1−𝛾(𝑥_𝑖𝑗(𝑡)−𝑦_𝑖𝑗(𝑡))²+32𝛽_{𝑖𝑗𝐶ℎ𝑙∈𝑁𝑟(𝑖,𝑗)}𝐶_𝑖^ℎ𝑙_j(𝑡)𝜈_𝑖𝑗𝐿1−𝛾(𝑥_𝑖𝑗(𝑡)−𝑦_𝑖𝑗(𝑡))²+12_{𝐵𝑖𝑗∈𝑁𝑟(ℎ,𝑙)}𝛽_ℎ𝑙𝐵^𝑖𝑗_ℎ𝑙(𝑡)𝜇_ℎ𝑙𝐿1−𝛾(𝑥_𝑖𝑗(𝑡)−𝑦_𝑖𝑗(𝑡))²+12_{𝐶𝑖𝑗∈𝑁𝑟(ℎ,𝑙)}𝜈_ℎ𝑙𝐿𝐶1−𝛾^{𝑖𝑗ℎ𝑙}(𝜉⁻¹_𝑖𝑗(𝑡))1−𝜏_𝑖𝑗(𝜉⁻¹_𝑖𝑗(𝑡))(𝑥_𝑖𝑗(𝑡)−𝑦_𝑖𝑗(𝑡))²𝑒^{𝜀𝜏𝑀ℎ𝑙}𝑐}≤−⁰2𝑒^𝜀𝑡(𝑥_𝑖𝑗(𝑡)−𝑦_𝑖𝑗(𝑡))²≤0, (3.12)(10)(11)(12)(13)(14)(15)(16)(17)(18)(19)(20)(21)(22)(23)(24)(25)(26)(27)(28)(29)(30)(31)(32)(33)(34)(35)(36)(37)(38)(39)(40)(41)(42)(43)(44)(45)(46)(47)(48)(49) where 𝑐₀>0 is defined by 𝑐₀=min_(𝑖,𝑗)inf_𝑡∈𝑅{𝛽_𝑖𝑗(2𝑎_𝑖𝑗(𝑡)−𝜀)−3𝛽_𝑖𝑗𝐿1−𝛾[𝜇_{𝑖𝑗𝐵ℎ𝑙∈𝑁𝑟(𝑖,𝑗)}𝐵^ℎ𝑙_𝑖𝑗(𝑡)+𝜈_{𝑖𝑗𝐶ℎ𝑙∈𝑁𝑟(𝑖,𝑗)}𝐶^ℎ𝑙_𝑖𝑗(𝑡)]+𝛽_ℎ𝑙(𝜇_ℎ𝑙𝐿1−𝛾_{𝐵𝑖𝑗∈𝑁𝑟(ℎ,𝑙)}𝐵^𝑖𝑗_ℎ𝑙𝐿(𝑡)+𝜈1−𝛾^ℎ𝑙_{𝐶ℎ𝑙∈𝑁𝑟(𝑖,𝑗)}𝐶^𝑖𝑗_ℎ𝑙(𝜉⁻¹_𝑖𝑗(𝑡))1−𝜏_𝑖𝑗(𝜉⁻¹_𝑖𝑗(𝑡)))}>0. (3.13) From the above, we have 𝑉(𝑡)≤𝑉(𝑡₀),𝑡≥𝑡₀, and 12𝑒^𝜀𝑡min_ℎ,𝑙𝛽^𝑛_{ℎ𝑙𝑖=1}(𝑥_𝑖𝑗(𝑡)−𝑦_𝑖𝑗(𝑡))²≤𝑉(𝑡),𝑡≥𝑡₀,𝑉(𝑡₀1)≤2𝑒^𝜀𝑡0[_(𝑖,𝑗)𝛽_𝑖𝑗+_{(𝑖,𝑗)(ℎ,𝑙)}𝛽_𝑖𝑗𝜈_𝑖𝑗𝐿𝜏1−𝛾^𝜇ℎ𝑙𝑒^{𝜀𝜏𝜇ℎ𝑙}sup_{𝑡∈[𝑡0−𝜏,𝑡0]}𝐶^ℎ𝑙(𝜉⁻¹_ℎ𝑙(𝑡))1−𝜏_ℎ𝑙(𝜉⁻¹_ℎ𝑙(𝑡))]‖𝜑−𝜓‖²₂. (3.14) Thus, it follows that there exists a positive constant 𝑀>1 such that |𝑥(𝑡)−𝑦(𝑡)|₂≤𝑀𝑒^{−(𝜀/2)(𝑡−𝑡0)}‖𝜑−𝜓‖₂,𝑡≥𝑡₀, (3.15) which implies that (1.4) is GES.
Now we assume that (𝐴₆) is satisfied. By carrying out similar arguments as above, one can easily show that there exists an 𝜀>0 such thatinf_(𝑖,𝑗){𝛽_𝑖𝑗(𝑎_𝑖𝑗(𝑡)−𝜀)−𝛽_𝑖𝑗𝐿1−𝛾[𝜇_{𝑖𝑗𝐵ℎ𝑙∈𝑁𝑟(𝑖,𝑗)}𝐵^ℎ𝑙_𝑖𝑗(𝑡)+𝜈_{𝑖𝑗𝐶ℎ𝑙∈𝑁𝑟(𝑖,𝑗)}𝐶^ℎ𝑙_𝑖𝑗(𝑡)]−𝛽_ℎ𝑙(𝜇_ℎ𝑙𝐿1−𝛾_{𝐵𝑖𝑗∈𝑁𝑟(ℎ,𝑙)}𝐵^𝑖𝑗_ℎ𝑙𝐿(𝑡)+𝜈1−𝛾^ℎ𝑙_{𝐶ℎ𝑙∈𝑁𝑟(𝑖,𝑗)}𝐶^𝑖𝑗_ℎ𝑙(𝜉⁻¹_𝑖𝑗(𝑡))1−𝜏_𝑖𝑗(𝜉⁻¹_𝑖𝑗(𝑡)))}>0. (3.16) Consider the Lyapunov function 𝑉(𝑡)=_(𝑖,𝑗)𝛽_𝑖𝑗{|𝑥_𝑖𝑗(𝑡)−𝑦_𝑖𝑗(𝑡)|𝑒^𝜀𝑡+_{𝐶ℎ𝑙∈𝑁𝑟(𝑖,𝑗)}𝜈_𝑖𝑗𝐿×1−𝛾_{𝑡𝑡−𝜏ℎ𝑙(𝑡)}𝐶^ℎ𝑙_𝑖𝑗(𝜉⁻¹_ℎ𝑙(𝑠))1−𝜏_ℎ𝑙(𝜉⁻¹_ℎ𝑙(𝑠))|𝑥_ℎ𝑙(𝑠)−𝑦_ℎ𝑙(𝑠)|𝑒^{𝜀(𝑡+𝜏𝜇ℎ𝑙)}𝑑𝑠}. (3.17) Similar to the above arguments, calculating the upper-right derivative 𝐷⁺𝑉(𝑡) produces 𝐷⁺𝑉(𝑡)≤−𝑐₂𝑒^𝜀𝑡_(𝑖,𝑗)|𝑥_𝑖𝑗(𝑡)−𝑦_𝑖𝑗(𝑡)|≤0, (3.18) where 𝑐₂=min_(𝑖,𝑗)inf_𝑡∈𝑅{𝛽_𝑖𝑗(𝑎_𝑖𝑗(𝑡)−𝜀)−𝛽_𝑖𝑗𝐿1−𝛾[𝜇_{𝑖𝑗𝐵ℎ𝑙∈𝑁𝑟(𝑖,𝑗)}𝐵^ℎ𝑙_𝑖𝑗(𝑡)+𝜈_{𝑖𝑗𝐶ℎ𝑙∈𝑁𝑟(𝑖,𝑗)}𝐶^ℎ𝑙_𝑖𝑗(𝑡)]−𝛽_ℎ𝑙(𝜇_ℎ𝑙𝐿1−𝛾_{𝐵𝑖𝑗∈𝑁𝑟(ℎ,𝑙)}𝐵^𝑖𝑗_ℎ𝑙𝐿(𝑡)+𝜈1−𝛾^ℎ𝑙_{𝐶ℎ𝑙∈𝑁𝑟(𝑖,𝑗)}𝐶^𝑖𝑗_ℎ𝑙(𝜉⁻¹_𝑖𝑗(𝑡))1−𝜏_𝑖𝑗(𝜉⁻¹_𝑖𝑗(𝑡)))}>0. (3.19) Then we have 𝑒^𝜀𝑡(min_(ℎ,𝑙)𝛽_ℎ𝑙)_(𝑖,𝑗)|𝑥_𝑖𝑗(𝑡)−𝑦_𝑖𝑗(𝑡)|≤𝑉(𝑡)≤𝑉(𝑡₀),𝑡≥𝑡₀. (3.20) Note that 𝑉(𝑡₀)≤𝑒^𝜀𝑡0[_(𝑖,𝑗)𝛽_𝑖𝑗+_(𝑖,𝑗)𝛽_{𝑖𝑗𝐶𝑖𝑗∈𝑁𝑟(ℎ,𝑙)}𝜏^𝜇_ℎ𝑙𝑒^{𝜀𝜏𝜇ℎ𝑙}sup_{𝑡∈[𝑡0−𝜏,𝑡0]}𝐶^ℎ𝑙_𝑖𝑗(𝜉⁻¹_ℎ𝑙(𝑡))1−𝜏_ℎ𝑙(𝜉⁻¹_ℎ𝑙(𝑡))]‖𝜑−𝜓‖₁. (3.21)
Then there exists a positive constant 𝑀>1 such that|𝑥(𝑡)−𝑦(𝑡)|₁=^𝑛_𝑖=1|𝑥_𝑖(𝑡)−𝑦_𝑖(𝑡)|≤𝑀‖𝜑−𝜓‖₁𝑒^{𝜀(𝑡−𝑡0)},𝑡≥𝑡₀. (3.22) The proof is complete.