Existence and Global Exponential Stability of Almost Periodic Solutions for SICNNs with Nonlinear Behaved Functions and Mixed Delays

Abstract

By using the Leray-Schauder fixed point theorem and differential inequality techniques, several new sufficient conditions are obtained for the existence and global exponential stability of almost periodic solutions for shunting inhibitory cellular neural networks with discrete and distributed delays. The model in this paper possesses two characters: nonlinear behaved functions and all coefficients are time varying. Hence, our model is general and applicable to many known models. Moreover, our main results are also general and can be easily deduced to many simple cases, including some existing results. An example and its simulation are employed to illustrate our feasible results.

1. Introduction

Consider the following shunting inhibitory cellular neural networks (SICNNs) with discrete and distributed delays (mixed delays):

𝑥^′_𝑖𝑗(𝑡)=−𝑎_𝑖𝑗(𝑡,𝑥_𝑖𝑗(𝑡))+_{𝐵𝑘𝑙∈𝑁𝑟(𝑖,𝑗)}𝐵^𝑘𝑙_𝑖𝑗(𝑡)𝑓_𝑖𝑗(𝑥_𝑘𝑙(𝑡−𝜏_𝑘𝑙(𝑡)))𝑥_𝑖𝑗+(𝑡)_{𝐶𝑘𝑙∈𝑁𝑟(𝑖,𝑗)}𝐶^𝑘𝑙_𝑖𝑗(𝑡)^𝑡_−∞𝑘_𝑖𝑗(𝑡−𝑠)𝑔_𝑖𝑗(𝑥_𝑘𝑙(𝑠))d𝑠𝑥_𝑖𝑗(𝑡)+𝐼_𝑖𝑗(𝑡), (1.1) where 𝑖=1,2,…,𝑛, 𝑗=1,2,…,𝑚.  𝐶_𝑖𝑗(𝑡) denotes the cell at the (𝑖,𝑗) position of the lattice at time 𝑡, the 𝑟-neighborhood 𝑁_𝑟(𝑖,𝑗) of 𝐶_𝑖𝑗(𝑡) is

𝑁_𝑟(𝑖,𝑗)={𝐶_𝑘𝑙(𝑡)∶max(|𝑘−𝑖|,|𝑙−𝑗|)≤𝑟,1≤𝑘≤𝑚,1≤𝑙≤𝑛}. (1.2)𝑥_𝑖𝑗(𝑡) is the activity of the cell 𝐶_𝑖𝑗(𝑡), 𝐼_𝑖𝑗(𝑡) is the external input to 𝐶_𝑖𝑗(𝑡), 𝑎_𝑖𝑗(𝑡,𝑥_𝑖𝑗(𝑡)) represents an appropriately behaved function of the cell 𝐶_𝑖𝑗(𝑡) at time 𝑡; 𝐵^𝑘𝑙_𝑖𝑗(𝑡) and 𝐶^𝑘𝑙_𝑖𝑗(𝑡) are the connection or coupling strength of postsynaptic activity of the cell transmitted to the cell 𝐶_𝑖𝑗(𝑡) depending upon discrete delays and distributed delays, respectively; the activation functions 𝑓_𝑖𝑗(⋅) and 𝑔_𝑖𝑗(⋅) are continuous representing the output or firing rate of the cell 𝐶_𝑘𝑙(𝑡); 𝜏_𝑖𝑗(𝑡) represent axonal signal transmission delays; 𝐵^𝑘𝑙_𝑖𝑗(𝑡),𝐶^𝑘𝑙_𝑖𝑗(𝑡),𝑓_𝑖𝑗(⋅),𝑔_𝑖𝑗(⋅),𝐼_𝑖𝑗(𝑡),𝜏_𝑖𝑗(𝑡) are all continuous almost periodic functions.

Since Bouzerdoum and Pinter described SICNNs as a new cellular neural networks [1–3], SICNNs have been extensively studied and found many important applications in different areas such as psychophysics, speech, perception, robotics, adaptive pattern recognition, vision, and image processing. There have been some results on the existence of periodic and almost solutions for SICNNs with discrete or distributed delays (distributed delay was first introduced in [4]) [5–21]. We find that all the behaved functions in the models in [5–21] are linear. Actually, 𝑎_𝑖𝑗(𝑡,𝑥_𝑖𝑗(𝑡)) may be nonlinear. Moreover, we find that the models in [5–20] are special cases of (1.1). For example, let 𝑎_𝑖𝑗(𝑡,𝑥_𝑖𝑗(𝑡))=𝑎_𝑖𝑗(𝑡)𝑥_𝑖𝑗(𝑡), then 𝐶^𝑘𝑙_𝑖𝑗(𝑡)≡0 in [5], 𝐵^𝑘𝑙_𝑖𝑗(𝑡)≡0 in [7], 𝐶^𝑘𝑙_𝑖𝑗(𝑡)≡0 in [8], 𝑎_𝑖𝑗(𝑡),𝐶^𝑘𝑙_𝑖𝑗(𝑡) are constants and 𝐵^𝑘𝑙_𝑖𝑗(𝑡)≡0 in [9], 𝑎_𝑖𝑗(𝑡),𝐵^𝑘𝑙_𝑖𝑗(𝑡),𝜏_𝑘𝑙(𝑡) are constants, and 𝐶^𝑘𝑙_𝑖𝑗(𝑡)≡0 in [10, 11]. To the best of our knowledge, few authors have considered the existence and global exponential stability of almost periodic solutions for SICNNs with nonlinear behaved functions, periodic coefficients and mixed delays. Obviously, (1.1) is general and is worth to continue to investigate its dynamical properties such as existence and global exponential stability of almost periodic solutions.

The main purpose of this paper is to get sufficient conditions on the existence and global exponential stability of almost periodic solutions for SICNNs (1.1) by using the Leray-Schauder fixed point theorem and differential inequality techniques. Our results are general and possess infinitely adjustable real parameters and can be deduced to many simple results, including some existing results as special cases. Therefore, our results provide a wider application criteria for neural networks.

The remaining part of this paper is organized as follows. We first state some useful definitions and lemmas in Section 2. In Section 3, we study the existence of almost periodic solutions of system (1.1) by using the Leray-Schauder's fixed point theorem. In Section 4, by using Lemma 2.7, we will derive sufficient conditions for the global exponential stability of the almost periodic solution of system (1.1). A useful corollary is also obtained. An illustrative example and its simulation are given in Section 5.

2. Preliminaries

For convenience, we denote

𝑥={𝑥_𝑖𝑗(𝑡)}=(𝑥₁₁(𝑡),𝑥₁₂(𝑡),…,𝑥_1𝑚(𝑡),…,𝑥_𝑛1(𝑡),𝑥_𝑛2(𝑡),…,𝑥_𝑛𝑚(𝑡))^𝑇. (2.1)

Definition 2.1. The continuous function 𝑥_𝑖𝑗(𝑡)∶ℝ→ℝ is called almost periodic on ℝ, if for any 𝜀>0, it is possible to find a real number 𝑙=𝑙(𝜀)>0 such that, for any interval with length 𝑙, there exists a number 𝜏=𝜏(𝜀) in this interval such that |𝑥_𝑖𝑗(𝑡+𝜏)−𝑥_𝑖𝑗(𝑡)|<𝜀, for any 𝑡∈ℝ.

Definition 2.2 (see [22, page 21]). Let 𝔼 be a Banach space, 𝔻 an open subset in 𝔼 and 𝑓(𝑡,𝑥)∈𝐶(ℝ×𝔻,𝔼). For ∀𝑥∈𝔻, 𝑓(𝑡,𝑥) is called uniformly almost periodic about 𝑡, if for any 𝜀>0 and any compact subset 𝕊⊂𝔻, there exists a real number 𝑙=𝑙(𝜀,𝕊)>0 such that, for any interval with length 𝑙, there exists a number 𝜏=𝜏(𝜀,𝕊) in this interval such that ‖𝑓(𝑡+𝜏,𝑥)−𝑓(𝑡,𝑥)‖<𝜀, for any (𝑡,𝑥)∈ℝ×𝕊.

The initial condition 𝜑={𝜑_𝑖𝑗(𝑠)} of (1.1) is of the form

𝑥_𝑖𝑗(𝑠)=𝜑_𝑖𝑗(𝑠),𝑠∈(−∞,0], (2.2) where 𝜑_𝑖𝑗(𝑠),𝑖=1,2,…,𝑛,𝑗=1,…,𝑚, are continuous almost periodic solutions.

Definition 2.3. Let 𝑥^∗(𝑡) be an almost periodic solution of (1.1) with initial value 𝜑^∗. If there exist constants 𝛼>0 and 𝑃>1 such that for every solution 𝑥(𝑡) of (1.1) with initial value 𝜑|𝑥_𝑖𝑗(𝑡)−𝑥^∗_𝑖𝑗(𝑡)|≤𝑃‖𝜑−𝜑^∗‖𝑒^−𝛼𝑡,∀𝑡>0,𝑖=1,2,…,𝑛,𝑗=1,2,…,𝑚, (2.3) where ‖𝜑−𝜑^∗‖=max_(𝑖,𝑗)sup_𝑠≤0{|𝜑_𝑖𝑗(𝑠)−𝜑^∗_𝑖𝑗(𝑠)|}. Then 𝑥^∗(𝑡) is said to be globally exponentially stable.

Lemma 2.4 (see [22, page 136]). Suppose that 𝑓(𝑡,𝑥) is uniformly continuous on ℝ×𝕊, where 𝕊 is any compact set on ℝ^𝑛, and that there exists a nonsingular matrix 𝑃(𝑡)∈𝐶¹, such that (a)there exists a constant 𝜌 satisfies ‖𝑃(𝑡)‖<𝜌, for all 𝑡∈ℝ, (b)the eigenvalues 𝜆_𝑖(𝑡),𝑖=1,2,…,𝑛, of 𝑃(𝑡) satisfy |𝜆_𝑖(𝑡)|≥𝜗>0, where 𝜗 is a constant, and there are 𝑘 negative eigenvalues, 𝑛−𝑘 nonnegative eigenvalues, (c)all the eigenvalues 𝜆_𝑖(𝑡,𝑥),𝑖=1,2,…,𝑛, of the following symmetric matrix: 𝑀(𝑡,𝑥)=𝑃(𝑡)𝑓(𝑡,𝑥)+𝑓^𝑇(𝑡,𝑥)𝑃(𝑡)+𝑃̇(𝑡) (2.4) satisfy 𝜆_𝑖(𝑡,𝑥)≤−𝛿<0,∀‖𝑥‖≤𝑅₀, (2.5) where 𝛿,𝑅₀ are constants.Then, for any fixed 𝜙(𝑡)∈𝐶(ℝ,ℝ^𝑛) and any 𝑡∈ℝ satisfying ‖𝜙(𝑡)‖<𝑅₀, the linear differential equation 𝑥̇=𝑓(𝑡,𝜙(𝑡))𝑥 (2.6) admits an exponential dichotomy on ℝ: |𝑋_𝜙(𝑡)𝑄_𝜙𝑋_𝜙⁻¹(𝑠)|≤𝑀𝜌𝜗𝑒^{−(𝛿/2𝜌)(𝑡−𝑠)},𝑡≥𝑠,|𝑋_𝜙(𝑡)(𝐼−𝑄_𝜙)𝑋_𝜙⁻¹(𝑠)|≤𝑀𝜌𝜗𝑒^{−(𝛿/2𝜌)(𝑡−𝑠)},𝑠≥𝑡, (2.7) where 𝑋_𝜙(𝑡) is fundamental solution matrix of (2.6) satisfying 𝑋_𝜙(0)=𝐼_𝑛×𝑛, 𝐼_𝑛×𝑛 is the identity matrix, 𝑄_𝜙 is a constant projection, the constant 𝑀 has no relationship with 𝜙(𝑡).

Lemma 2.5 (see [22, page 139]). Suppose that the 𝑛×𝑛 matrix function 𝑓(𝑡,𝑥) and the 𝑛-dimensional vector function 𝑔(𝑡,𝑥) are uniformly almost periodic on ℝ×ℝ^𝑛, and that there exists real symmetric nonsingular matrix 𝑃(𝑡) satisfying the conditions (a)–(c) in Lemma 2.4, then, 𝑥̇=𝑓(𝑡,𝜙)𝑥+𝑔(𝑡,𝜙) (2.8) has a unique almost periodic solution 𝑥(𝑡), where 𝜙∈𝐶(ℝ,ℝ^𝑛) is almost periodic function and 𝑥(𝑡)=^𝑡_−∞𝑋_𝜙(𝑠)𝑄_𝜙𝑋_𝜙⁻¹(𝑠)𝑔(𝑠,𝜙(𝑠))d𝑠−_𝑡^+∞𝑋_𝜙(𝑠)(𝐼−𝑄_𝜙)𝑋_𝜙⁻¹(𝑠)𝑔(𝑠,𝜙(𝑠))d𝑠. (2.9)

Lemma 2.6 (Leray-Schauder). Let 𝔼 be a Banach space, and let the operator Φ∶𝔼→𝔼 be completely continuous. If the set {‖𝑥‖∣𝑥∈𝔼,𝑥=𝜆Φ𝑥,0<𝜆<1} is bounded, then Φ has a fixed point in 𝕋, where 𝕋={𝑥∣𝑥∈𝐸,‖𝑥‖≤𝑅},𝑅=sup{‖𝑥‖∣𝑥=𝜆Φ𝑥,0<𝜆<1}. (2.10)

Lemma 2.7 (see [23]). Let 𝑎≥0, 𝑏_𝑘≥0(𝑘=1,2,…,𝑚), the following inequality holds 𝑎^𝑚_𝑘=1𝑏^𝑞𝑘_𝑘≤1𝑟^𝑚_𝑘=1𝑞_𝑘𝑏^𝑟_𝑘+1𝑟𝑎^𝑟, (2.11) where 𝑞_𝑘>0,(𝑘=1,2,…,𝑚) is some constants, ∑^𝑚_𝑘=1𝑞_𝑘=𝑟−1, and 𝑟>1.

Obviously, inequality (2.11) also holds for 𝑎≥0, 𝑏_𝑘≥0, 𝑟=1, and 𝑞_𝑘=0,𝑘=1,2,…,𝑚. Hence, we always assume that 𝑎≥0, 𝑏_𝑘≥0, 𝑟≥1, and 𝑞_𝑘≥0,𝑘=1,2,…,𝑚 in (2.11) in the later sections of this paper.

Furthermore, throughout this paper, we assume that

(H₁)𝑎_𝑖𝑗(𝑡,𝑢)∈𝐶(ℝ²,ℝ) is continuous almost periodic about the first argument and, there exists a positive continuous almost periodic function 𝜇_𝑖𝑗(𝑡) such that 𝜕𝑎_𝑖𝑗(𝑡,𝑢)/𝜕𝑢≥𝜇_𝑖𝑗(𝑡), 𝑢∈ℝ, and 𝑎_𝑖𝑗(𝑡,0)=0, 𝑖=1,2,…,𝑛,𝑗=1,2,…,𝑚,(H₂)there exist nonnegative constants 𝑀_𝑖𝑗 and 𝑁_𝑖𝑗, 𝑖=1,2,…,𝑛,𝑗=1,2,…,𝑚, such that |𝑓_𝑖𝑗(𝑢)|≤𝑀_𝑖𝑗,|𝑔_𝑖𝑗(𝑢)|≤𝑁_𝑖𝑗,∀𝑢∈ℝ; (2.12)(H₃)the delay kernels 𝑘_𝑖𝑗∶[0,+∞)→ℝ are continuous, integrable and there are positive constants 𝑘_𝑖𝑗 such that ₀^+∞|𝑘_𝑖𝑗(𝑠)|d𝑠≤𝑘_𝑖𝑗,𝑖=1,2,…,𝑛,𝑗=1,2,…,𝑚; (2.13)(H₄)there exists a constant 𝛼₀>0 such that ₀^+∞|𝑘_𝑖𝑗(𝑠)|𝑒^𝛼0𝑠d𝑠<+∞,𝑖=1,2,…,𝑛,𝑗=1,2,…,𝑚. (2.14)(H₅)the following inequality holds: max_(𝑖,𝑗)sup_𝑡∈ℝ{∑_𝐵^𝑘𝑙_{∈𝑁𝑟(𝑖,𝑗)}|𝐵^𝑘𝑙_𝑖𝑗(𝑡)|𝑀_𝑖𝑗+∑_𝐶^𝑘𝑙_{∈𝑁𝑟(𝑖,𝑗)}|𝐶^𝑘𝑙_𝑖𝑗(𝑡)|𝑘_𝑖𝑗𝑁_𝑖𝑗𝜇_𝑖𝑗(𝑡)}=𝜂<1; (2.15)(H₆) there are nonnegative constants 𝛼_𝑖𝑗,𝛽_𝑖𝑗 such that 𝛼_𝑖𝑗=sup_𝑢≠𝑣|𝑓_𝑖𝑗(𝑢)−𝑓_𝑖𝑗(𝑣)𝑢−𝑣|,𝛽_𝑖𝑗=sup_𝑢≠𝑣|𝑔_𝑖𝑗(𝑢)−𝑔_𝑖𝑗(𝑣)𝑢−𝑣| (2.16) for all 𝑢,𝑣∈ℝ,𝑢≠𝑣,𝑖=1,2,…,𝑛,𝑗=1,2,…,𝑚.

3. Existence of Almost Periodic Solutions

Let 𝜉_𝑖𝑗, 𝑖=1,2,…,𝑛, 𝑗=1,2,…,𝑚 be constants. Make the following transformation:

𝑥_𝑖𝑗=𝜉_𝑖𝑗𝑦_𝑖𝑗(𝑡),𝑖=1,2,…,𝑛,𝑗=1,2,…,𝑚, (3.1) then (1.1) can be reformulated as

𝑦^′_𝑖𝑗(𝑡)=−𝜉⁻¹_𝑖𝑗𝑎_𝑖𝑗(𝑡,𝜉_𝑖𝑗𝑦_𝑖𝑗(𝑡))+_{𝐵𝑘𝑙∈𝑁𝑟(𝑖,𝑗)}𝐵^𝑘𝑙_𝑖𝑗(𝑡)𝑓_𝑖𝑗(𝜉_𝑘𝑙𝑦_𝑘𝑙(𝑡−𝜏_𝑘𝑙(𝑡)))𝑦_𝑖𝑗+(𝑡)_{𝐶𝑘𝑙∈𝑁𝑟(𝑖,𝑗)}𝐶^𝑘𝑙_𝑖𝑗(𝑡)^𝑡_−∞𝑘_𝑖𝑗(𝑡−𝑠)𝑔_𝑖𝑗(𝜉_𝑘𝑙𝑦_𝑘𝑙(𝑠))d𝑠𝑦_𝑖𝑗(𝑡)+𝜉⁻¹_𝑖𝑗𝐼_𝑖𝑗(𝑡). (3.2)

System (3.2) can be rewritten as

𝑦^′_𝑖𝑗(𝑡)=−𝑑_𝑖𝑗(𝑡,𝑦_𝑖𝑗(𝑡))𝑦_𝑖𝑗(𝑡)+_{𝐵𝑘𝑙∈𝑁𝑟(𝑖,𝑗)}𝐵^𝑘𝑙_𝑖𝑗(𝑡)𝑓_𝑖𝑗(𝜉_𝑘𝑙𝑦_𝑘𝑙(𝑡−𝜏_𝑘𝑙(𝑡)))𝑦_𝑖𝑗+(𝑡)_{𝐶𝑘𝑙∈𝑁𝑟(𝑖,𝑗)}𝐶^𝑘𝑙_𝑖𝑗(𝑡)^𝑡_−∞𝑘_𝑖𝑗(𝑡−𝑠)𝑔_𝑖𝑗(𝜉_𝑘𝑙𝑦_𝑘𝑙(𝑠))d𝑠𝑦_𝑖𝑗(𝑡)+𝜉⁻¹_𝑖𝑗𝐼_𝑖𝑗(𝑡), (3.3) where 𝑑_𝑖𝑗(𝑡,𝑦_𝑖𝑗(𝑡))≜(𝜕𝑎_𝑖𝑗(𝑡,𝑧)/𝜕𝑧)|_{𝑧=𝑒𝑖𝑗}, 𝑒_𝑖𝑗 is between 0 and 𝜉_𝑖𝑗𝑦_𝑖𝑗(𝑡), 𝑒_𝑖𝑗∈ℝ. By (H₁), we know that 𝑎_𝑖𝑗(𝑡,𝜉_𝑖𝑗𝑦_𝑖𝑗) is strictly monotone increasing about 𝑦_𝑖𝑗. Hence, 𝑑_𝑖𝑗(𝑡,𝑦_𝑖𝑗(𝑡)) is unique for any 𝑦_𝑖𝑗(𝑡). Obviously, 𝑑_𝑖𝑗(𝑡,𝑦_𝑖𝑗(𝑡)) is continuous almost periodic about the first argument and 𝑑_𝑖𝑗(𝑡,𝑦_𝑖𝑗(𝑡))≥𝜇_𝑖𝑗(𝑡).

Take 𝕏={𝜙={𝜙_𝑖𝑗(𝑡)}∣𝜙_𝑖𝑗∶ℝ→ℝ is an almost periodic function, 𝑖=1,…,𝑛, 𝑗=1,…,𝑚}. Then 𝕏 is a Banach space with the norm

‖𝜙‖=max_(𝑖,𝑗){|𝜙_𝑖𝑗|₀},|𝜙_𝑖𝑗|₀=sup_𝑡∈ℝ|𝜙_𝑖𝑗(𝑡)|,𝑖=1,…,𝑛,𝑗=1,…,𝑚. (3.4)

For for all 𝜙∈𝕏, we consider the following auxiliary equation:

𝑦^′_𝑖𝑗(𝑡)=−𝑑_𝑖𝑗(𝑡,𝜙_𝑖𝑗(𝑡))𝑦_𝑖𝑗(𝑡)+_{𝐵𝑘𝑙∈𝑁𝑟(𝑖,𝑗)}𝐵^𝑘𝑙_𝑖𝑗(𝑡)𝑓_𝑖𝑗(𝜉_𝑘𝑙𝜙_𝑘𝑙(𝑡−𝜏_𝑘𝑙(𝑡)))𝜙_𝑖𝑗+(𝑡)_{𝐶𝑘𝑙∈𝑁𝑟(𝑖,𝑗)}𝐶^𝑘𝑙_𝑖𝑗(𝑡)^𝑡_−∞𝑘_𝑖𝑗(𝑡−𝑠)𝑔_𝑖𝑗(𝜉_𝑘𝑙𝜙_𝑘𝑙(𝑠))d𝑠𝜙_𝑖𝑗(𝑡)+𝜉⁻¹_𝑖𝑗𝐼_𝑖𝑗(𝑡). (3.5) From (H₁), we know that

𝑑_𝑖𝑗(𝑡,𝑦_𝑖𝑗),𝑖=1,2,…,𝑛,𝑗=1,2,…,𝑚, (3.6) are uniformly almost periodic functions on ℝ×ℝ. Since 𝜇_𝑖𝑗(𝑡),𝑖=1,2,…,𝑛,𝑗=1,2,…,𝑚, are positive continuous almost periodic functions, there exists a positive constant 𝛿 such that

𝑑_𝑖𝑗(𝑡,𝜙_𝑖𝑗(𝑡))≥𝜇_𝑖𝑗(𝑡)≥𝛿>0. (3.7) Hence, the conditions in Lemma 2.5 are satisfied (take 𝑃(𝑡)=𝐼_{𝑛𝑚×𝑛𝑚}).

According to the fact that 𝐵^𝑘𝑙_𝑖𝑗(𝑡),𝜙_𝑖𝑗(𝑡),𝐶^𝑘𝑙_𝑖𝑗(𝑡),𝐼_𝑖𝑗(𝑡) are almost periodic functions, in view of Lemma 2.5, we know that system (3.5) has a unique almost periodic solution

𝑦^𝜙(𝑡)={^𝑡_−∞𝑒^{−∫𝑡𝑠𝑑𝑖𝑗(𝑢,𝜙𝑖𝑗(𝑢))d𝑢}×[_{𝐵𝑘𝑙∈𝑁𝑟(𝑖,𝑗)}𝐵^𝑘𝑙_𝑖𝑗(𝑠)𝑓_𝑖𝑗(𝜉_𝑘𝑙𝜙_𝑘𝑙(𝑠−𝜏_𝑘𝑙(𝑠)))𝜙_𝑖𝑗(𝑠)+_{𝐶𝑘𝑙∈𝑁𝑟(𝑖,𝑗)}𝐶^𝑘𝑙_𝑖𝑗(𝑠)^𝑠_−∞𝑘_𝑖𝑗(𝑠−𝑢)𝑔_𝑖𝑗(𝜉_𝑘𝑙𝜙_𝑘𝑙(𝑢))d𝑢𝜙_𝑖𝑗(𝑠)+𝜉⁻¹_𝑖𝑗𝐼_𝑖𝑗(𝑠)]d𝑠}. (3.8)

Set a mapping Φ∶𝕏→𝕏 by setting

(Φ𝜙)(𝑡)=𝑦^𝜙(𝑡),∀𝜙∈𝕏. (3.9)

Before using Lemma 2.6 to obtain conditions of the existence of almost periodic solution for (1.1), we have to prove the following lemma.

Lemma 3.1. Suppose that (H₁)–( H₅) hold. Then Φ∶𝕏→𝕏 is completely continuous.

Proof. Under our assumptions, it is clear that the operator Φ is continuous. Next, we show that Φ is compact.
For any constant 𝐷>0, let Ω={𝜙∣𝜙∈𝕏,‖𝜙‖<𝐷}. Then, for any 𝜙∈Ω, we have
‖(Φ𝜙)‖=max_(𝑖,𝑗)sup_𝑡∈ℝ{|^𝑡_−∞𝑒^{−∫𝑡𝑠𝑑𝑖𝑗(𝑢,𝜙𝑖𝑗(𝑢))d𝑢}×[_{𝐵𝑘𝑙∈𝑁𝑟(𝑖,𝑗)}𝐵^𝑘𝑙_𝑖𝑗(𝑠)𝑓_𝑖𝑗(𝜉_𝑘𝑙𝜙_𝑘𝑙(𝑠−𝜏_𝑘𝑙(𝑠)))𝜙_𝑖𝑗(𝑠)+_{𝐶𝑘𝑙∈𝑁𝑟(𝑖,𝑗)}C^𝑘𝑙_𝑖𝑗(𝑠)^𝑠_−∞𝑘_𝑖𝑗(𝑠−𝑢)𝑔_𝑖𝑗(𝜉_𝑘𝑙𝜙_𝑘𝑙(𝑢))d𝑢𝜙_𝑖𝑗(𝑠)+𝜉⁻¹_𝑖𝑗𝐼_𝑖𝑗(𝑠)]d𝑠|}≤max_(𝑖,𝑗)sup_𝑡∈ℝ{^𝑡_−∞𝑒^{−∫𝑡𝑠𝜇𝑖𝑗(𝑢)d𝑢}[(_{𝐵𝑘𝑙∈𝑁𝑟(𝑖,𝑗)}|𝐵^𝑘𝑙_𝑖𝑗(𝑠)|𝑀_𝑖𝑗+_{𝐶𝑘𝑙∈𝑁𝑟(𝑖,𝑗)}|𝐶^𝑘𝑙_𝑖𝑗(𝑠)|𝑘_𝑖𝑗𝑁_𝑖𝑗)×|𝜙_𝑖𝑗(𝑠)|+𝜉⁻¹_𝑖𝑗|𝐼_𝑖𝑗(𝑠)|]d𝑠}≤max_(𝑖,𝑗)sup_𝑡∈ℝ{^𝑡_−∞𝑒^{−∫𝑡𝑠𝜇𝑖𝑗(𝑢)d𝑢}𝜇_𝑖𝑗(𝑠)d𝑠}𝜂‖𝜙‖+max_(𝑖,𝑗){𝐼_𝑖𝑗𝜉_𝑖𝑗𝜇_𝑖𝑗}<𝜂𝐷+max_(𝑖,𝑗){𝐼_𝑖𝑗𝜉_𝑖𝑗𝜇_𝑖𝑗}, (3.10)(1)(2)(3)(4)(5)(6) where 𝐼_𝑖𝑗=max_𝑡∈𝑅|𝐼_𝑖𝑗(𝑡)|, 𝜇_𝑖𝑗=inf_𝑡∈ℝ𝜇_𝑖𝑗(𝑡). Hence, Φ(Ω) is uniformly bounded.
By the definition of Φ, we get
(Φ𝜙)^′_𝑖𝑗d(𝑡)=(d𝑡^𝑡_−∞𝑒^{−∫𝑡𝑠𝑑𝑖𝑗(𝑢,𝜙𝑖𝑗(𝑢))d𝑢}×[_{𝐵𝑘𝑙∈𝑁𝑟(𝑖,𝑗)}𝐵^𝑘𝑙_𝑖𝑗(𝑠)𝑓_𝑖𝑗(𝜉_𝑘𝑙𝜙_𝑘𝑙(𝑠−𝜏_𝑘𝑙(𝑠)))𝜙_𝑖𝑗(𝑠)+_{𝐶𝑘𝑙∈𝑁𝑟(𝑖,𝑗)}𝐶^𝑘𝑙_𝑖𝑗(𝑠)^𝑠_−∞𝑘_𝑖𝑗(𝑠−𝑢)𝑔_𝑖𝑗(𝜉_𝑘𝑙𝜙_𝑘𝑙(𝑢))d𝑢𝜙_𝑖𝑗(𝑠)+𝜉⁻¹_𝑖𝑗𝐼_𝑖𝑗(𝑠)]d𝑠)=−𝑑_𝑖𝑗(𝑡,𝜙_𝑖𝑗(𝑡))(Φ𝜙_𝑖𝑗)(𝑡)+_{𝐵𝑘𝑙∈𝑁𝑟(𝑖,𝑗)}𝐵^𝑘𝑙_𝑖𝑗(𝑡)𝑓_𝑖𝑗(𝜉_𝑘𝑙𝜙_𝑘𝑙(𝑡−𝜏_𝑘𝑙(𝑡)))𝜙_𝑖𝑗+(𝑡)_{𝐶𝑘𝑙∈𝑁𝑟(𝑖,𝑗)}𝐶^𝑘𝑙_𝑖𝑗(𝑡)^𝑡_−∞𝑘_𝑖𝑗(𝑡−𝑠)𝑔_𝑖𝑗(𝜉_𝑘𝑙𝜙_𝑘𝑙(𝑠))d𝑠𝜙_𝑖𝑗(𝑡)+𝜉⁻¹_𝑖𝑗𝐼_𝑖𝑗(𝑡). (3.11) Since 𝑑_𝑖𝑗(𝑡,𝜙_𝑖𝑗(𝑡)),𝑖=1,2,…,𝑛,𝑗=1,2,…,𝑚, are uniformly almost periodic functions on ℝ×Ω, there exists a positive constant Θ such that |𝑑_𝑖𝑗(𝑡,𝜙_𝑖𝑗(𝑡))|≤Θ,for𝑡∈ℝ,𝜙_𝑖𝑗(𝑡)∈Ω,𝑖=1,2,…,𝑛,𝑗=1,2,…,𝑚. (3.12) Hence, ‖(Φ𝜙)^′_𝑖𝑗(𝑡)‖≤Θ(𝜂𝐷+max_(𝑖,𝑗){𝐼_𝑖𝑗𝜉_𝑖𝑗𝜇_𝑖𝑗})+max_(𝑖,𝑗){_{𝐵𝑘𝑙∈𝑁𝑟(𝑖,𝑗)}𝐵^𝑘𝑙_𝑖𝑗𝑀_𝑖𝑗𝐷+_{𝐶𝑘𝑙∈𝑁𝑟(𝑖,𝑗)}𝐶^𝑘𝑙_𝑖𝑗𝑘_𝑖𝑗𝑁_𝑖𝑗𝐷+𝜉⁻¹_𝑖𝑗𝐼_𝑖𝑗}, (3.13) where 𝐵^𝑘𝑙_𝑖𝑗=sup_𝑡∈𝑅|𝐵^𝑘𝑙_𝑖𝑗(𝑡)|, 𝐶^𝑘𝑙_𝑖𝑗=sup_𝑡∈ℝ|𝐶^𝑘𝑙_𝑖𝑗(𝑡)|. So, Φ(Ω)⊆𝕏 is a family of uniformly bounded and equicontinuous subsets. By using the Arzela-Ascoli theorem, Φ∶𝕏→𝕏 is compact. Therefore, Φ∶𝕏→𝕏 is completely continuous. This completes the proof.

Theorem 3.2. Suppose that (H₁)-(H₅) hold. Let 𝜉_𝑖𝑗,𝑖=1,2,…,𝑛,𝑗=1,2,…,𝑚, be constants. Then system (1.1) has an almost periodic solution 𝑥^∗(𝑡) with ‖𝑥^∗‖≤max_(𝑖,𝑗){𝜉_𝑖𝑗}𝑅≜𝑅₀, where 𝑅=max_(𝑖,𝑗){𝐼_𝑖𝑗/𝜉_𝑖𝑗𝜇_𝑖𝑗}.1−𝜂 (3.14)

Proof. Let 𝜙∈𝕏. From Lemma 3.1, we get that Φ∶𝕏→𝕏 is completely continuous. Consider the following operator equation: 𝜙=𝜆Φ𝜙,𝜆∈(0,1). (3.15) If 𝜙 is a solution of (3.15), we obtain ‖𝜙‖≤‖Φ𝜙‖≤𝜃‖𝜙‖+max_(𝑖,𝑗){𝐼_𝑖𝑗𝜉_𝑖𝑗𝜇_𝑖𝑗}. (3.16) This and (H₅) imply that ‖𝜙‖≤𝑅. (3.17) In view of Lemma 2.6, we obtain that Φ has a fixed point 𝜙^∗(𝑡) with ‖𝜙^∗‖≤𝑅. From (3.5) and (3.8), we know that 𝜙^∗ satisfies (3.3) and (3.2). Hence, system (3.2) has an almost periodic solution 𝜙^∗(𝑡)={𝜙^∗_𝑖𝑗(𝑡)} with ‖𝜙^∗‖≤𝑅. It follows from (3.1) that 𝑥^∗(𝑡)={𝑥^∗_𝑖𝑗(𝑡)}={𝜉_𝑖𝑗𝜙^∗_𝑖𝑗(𝑡)} is one almost periodic solution of (1.1) with ‖𝑥^∗‖≤max_(𝑖,𝑗){𝜉_𝑖𝑗}𝑅≜𝑅₀. (3.18) This completes the proof.

4. Stability of Almost Periodic solution

In this section, we prove that, under suitable conditions, the almost periodic solution obtained in Theorem 3.2 is globally exponentially stable.

Theorem 4.1. Assume that (H₁)-(H₄) and (H₆) hold and (H₇)there are constants 𝑜_𝑠,𝜀_𝑠,𝑙_𝑠,𝛾_𝑠,𝛿_𝑠,𝑑_𝑠∈ℝ,𝑞≥1, 𝑝_𝑠≥0, 𝜎>0, 𝜉_𝑖𝑗>0,𝑠=1,2,…,𝑚^′+1, 𝑖=1,2,…,𝑛,𝑗=1,2,…,𝑚, such that Λ=max_(𝑖,𝑗)sup_𝑡∈𝑅{(−𝑞𝜇_𝑖𝑗(𝑡)+𝑞_{𝐵𝑘𝑙∈𝑁𝑟(𝑖,𝑗)}|𝐵^𝑘𝑙_𝑖𝑗(𝑡)|𝑀_𝑖𝑗+𝑞_{𝐶𝑘𝑙∈𝑁𝑟(𝑖,𝑗)}|𝐶^𝑘𝑙_𝑖𝑗(𝑡)|𝑘_𝑖𝑗𝑁_𝑖𝑗+_{𝐵𝑘𝑙∈𝑁𝑟}^𝑚_(𝑖,𝑗)^′_𝑠=1𝑝_𝑠|𝐵^𝑘𝑙_𝑖𝑗(𝑡)|^𝑞𝑜𝑠𝛼^𝑞𝜀𝑠_𝑖𝑗𝑅^𝑞𝑙𝑠₀+_{𝐶𝑘𝑙∈𝑁𝑟}^𝑚_(𝑖,𝑗)^′_𝑠=1𝑝_𝑠|𝐶^𝑘𝑙_𝑖𝑗(𝑡)|^𝑞𝛾𝑠𝛽^𝑞𝛿𝑠_𝑖𝑗𝑅^𝑞𝑑𝑠₀)𝜉_𝑖𝑗+_{𝐵𝑘𝑙∈𝑁𝑟(𝑖,𝑗)}|𝐵^𝑘𝑙_𝑖𝑗(𝑡)|^{𝑞𝑜𝑚′+1}𝛼^{𝑞𝜀𝑚′+1}_𝑖𝑗𝑅^{𝑞𝑙𝑚′+1}₀𝜉_𝑘𝑙+_{𝐶𝑘𝑙∈𝑁𝑟(𝑖,𝑗)}|𝐶^𝑘𝑙_𝑖𝑗(𝑡)|^{𝑞𝛾𝑚′+1}𝛽^{𝑞𝛿𝑚′+1}_𝑖𝑗𝑅^{𝑞𝑑𝑚′+1}₀𝑘^𝑞_𝑖𝑗𝜉_𝑘𝑙}≤−𝜎<0, (4.1) where, 𝑜_𝑚′+1+∑^𝑚′_𝑠=1𝑜_𝑠𝑝_𝑠=𝜀_𝑚′+1+∑^𝑚′_𝑠=1𝜀_𝑠𝑝_𝑠=𝑙_𝑚′+1+∑^𝑚′_𝑠=1𝑙_𝑠𝑝_𝑠=𝛾_𝑚′+1+∑^𝑚′_𝑠=1𝛾_𝑠𝑝_𝑠=𝛿_𝑚′+1+∑^𝑚′_𝑠=1𝛿_𝑠𝑝_𝑠=𝑑_𝑚′+1+∑^𝑚′_𝑠=1𝑑_𝑠𝑝_𝑠=1, ∑^𝑚′_𝑠=1𝑝_𝑠=𝑞−1. Then (1.1) has unique almost periodic solution, which is globally exponentially stable.

Proof. Obviously, that (H₇) holds implies that (H₅) holds. By Theorem 3.2, there exists an almost periodic solution 𝑥^∗(𝑡) of (1.1) with initial value 𝜑^∗(𝑡) and ‖𝑥^∗‖≤𝑅₀. Suppose that 𝑥(𝑡) is an arbitrary solution of (1.1) with initial value 𝜑(𝑡). Set 𝑧(𝑡)=𝑥(𝑡)−𝑥^∗(𝑡). Then, from (1.1), we have 𝑧^′_𝑖𝑗(𝑡)=−[𝑎_𝑖𝑗(𝑡,𝑥_𝑖𝑗(𝑡))−𝑎_𝑖𝑗(𝑡,𝑥^∗_𝑖𝑗(𝑡))]+_{𝐵𝑘𝑙∈𝑁𝑟(𝑖,𝑗)}𝐵^𝑘𝑙_𝑖𝑗(𝑡)(𝑓_𝑖𝑗(𝑥_𝑘𝑙(𝑡−𝜏_𝑘𝑙(𝑡)))𝑥_𝑖𝑗(𝑡)−𝑓_𝑖𝑗(𝑥^∗_𝑘𝑙(𝑡−𝜏_𝑘𝑙(𝑡)))𝑥^∗_𝑖𝑗+(𝑡))_{𝐶𝑘𝑙∈𝑁𝑟(𝑖,𝑗)}𝐶^𝑘𝑙_𝑖𝑗(𝑡)(^𝑡_−∞𝑘_𝑖𝑗(𝑡−𝑠)𝑔_𝑖𝑗(𝑥_𝑘𝑙(𝑠))d𝑠𝑥_𝑖𝑗(𝑡)−^𝑡_−∞𝑘_𝑖𝑗(𝑡−𝑠)𝑔_𝑖𝑗(𝑥^∗_𝑘𝑙(𝑠))d𝑠𝑥^∗_𝑖𝑗(𝑡)),𝑖=1,2,…,𝑛,𝑗=1,2,…,𝑚. (4.2) Set Γ_𝑖𝑗(𝜆,𝑡)=(𝜆−𝑞𝜇_𝑖𝑗(𝑡)+𝑞_{𝐵𝑘𝑙∈𝑁𝑟(𝑖,𝑗)}|𝐵^𝑘𝑙_𝑖𝑗(𝑡)|𝑀_𝑖𝑗+𝑞_{𝐶𝑘𝑙∈𝑁𝑟(𝑖,𝑗)}|𝐶^𝑘𝑙_𝑖𝑗(𝑡)|𝑘_𝑖𝑗𝑁_𝑖𝑗+_{𝐵𝑘𝑙∈𝑁𝑟}^𝑚_(𝑖,𝑗)^′_𝑠=1𝑝_𝑠|𝐵^𝑘𝑙_𝑖𝑗(𝑡)|^𝑞𝑜𝑠𝛼^𝑞𝜀𝑠_𝑖𝑗𝑅^𝑞𝑙𝑠₀+_{𝐶𝑘𝑙∈𝑁𝑟}^𝑚_(𝑖,𝑗)^′_𝑠=1𝑝_𝑠|𝐶^𝑘𝑙_𝑖𝑗(𝑡)|^𝑞𝛾𝑠𝛽^𝑞𝛿𝑠_𝑖𝑗𝑅^𝑞𝑑𝑠₀)𝜉_𝑖𝑗+_{𝐵𝑘𝑙∈𝑁𝑟(𝑖,𝑗)}|𝐵^𝑘𝑙_𝑖𝑗(𝑡)|^{𝑞𝑜𝑚′+1}𝛼^{𝑞𝜀𝑚′+1}_𝑖𝑗𝑅^{𝑞𝑙𝑚′+1}₀𝑒^{𝜆𝜏𝑘𝑙(𝑡)}𝜉_𝑘𝑙+_{𝐶𝑘𝑙∈𝑁𝑟(𝑖,𝑗)}|𝐶^𝑘𝑙_𝑖𝑗(𝑡)|^{𝑞𝛾𝑚′+1}𝛽^{𝑞𝛿𝑚′+1}_𝑖𝑗𝑅^{𝑞𝑑𝑚′+1}₀(₀^+∞|𝑘_𝑖𝑗(𝑠)|𝑒^{(𝜆/𝑞)𝑠}d𝑠)^𝑞𝜉_𝑘𝑙,𝑖=1,2,…,𝑛,𝑗=1,2,…,𝑚. (4.3) Clearly, Γ_𝑖𝑗(𝜆,𝑡), 𝑖=1,2,…,𝑛, 𝑗=1,2,…,𝑚, are continuous functions about 𝜆 and almost periodic about 𝑡 on ℝ.
From (H₇), we get
Γ_𝑖𝑗(0,𝑡)≤Λ≤−𝜎<0,𝑖=1,2,…,𝑛,𝑗=1,2,…,𝑚. (4.4) In addition to 𝜕Γ_𝑖𝑗(𝜆,𝑡)𝜕𝜆=𝜉_𝑖𝑗+_{𝐵𝑘𝑙∈𝑁𝑟(𝑖,𝑗)}|𝐵^𝑘𝑙_𝑖𝑗(𝑡)|^{𝑞𝑜𝑚′+1}𝛼^{𝑞𝜀𝑚′+1}_𝑖𝑗𝑅^{𝑞𝑙𝑚′+1}₀𝜏_𝑘𝑙(𝑡)𝑒^{𝜆𝜏𝑘𝑙(𝑡)}𝜉_𝑘𝑙+_{𝐶𝑘𝑙∈𝑁𝑟(𝑖,𝑗)}|𝐶^𝑘𝑙_𝑖𝑗(𝑡)|^{𝑞𝛾𝑚′+1}𝛽^{𝑞𝛿𝑚′+1}_𝑖𝑗𝑅^{𝑞𝑑𝑚′+1}₀(₀^+∞|𝑘_𝑖𝑗(𝑠)|𝑠𝑒^{(𝜆/𝑞)𝑠}d𝑠)^𝑞𝜉_𝑘𝑙>0 (4.5) and Γ_𝑖𝑗(+∞,𝑡)=+∞ for ∀𝑡∈𝑅, we obtain that Γ_𝑖𝑗(𝜆,𝑡),𝑖=1,2,…,𝑛, 𝑗=1,2,…,𝑚, are strictly monotone increasing functions about 𝜆. Therefore, for any 𝑖∈{1,2,…,𝑛}, 𝑗∈{1,2,…,𝑚}, there is unique real number 𝜆₀^𝑖𝑗 such that sup_𝑡∈𝑅Γ_𝑖𝑗(𝜆₀^𝑖𝑗,𝑡)=0. (4.6) In view of (4.4), we have 𝜆₀^𝑖𝑗>0. Let 𝜃=min{𝜆₀¹¹,𝜆₀¹²,…,𝜆₀^𝑛𝑚}, we obtain sup_𝑡∈𝑅Γ_𝑖𝑗(𝜃,𝑡)≤0,𝑖=1,2,…,𝑛,𝑗=1,2,…,𝑚. (4.7) Now, we choose a positive constant 𝑑 such that 𝑞𝑑𝜉_𝑖𝑗≥1,𝑖=1,2,…,𝑛,𝑗=1,2,…,𝑚. It is obvious that |𝑧_𝑖(𝑡)|^𝑞≤‖𝜑−𝜑^∗‖^𝑞≤𝑞𝑑𝜉_𝑖‖𝜑−𝜑^∗‖^𝑞𝑒^−𝜃𝑡,for∈(−∞,0],𝑖=1,2,…,𝑛, (4.8) where ‖𝜑−𝜑^∗‖ is defined as that in Definition 2.3.
Let 𝑉_𝑖𝑗(𝑡)=(1/𝑞)𝑒^𝜃𝑡|𝑧_𝑖𝑗(𝑡)|^𝑞,𝑖=1,2,…,𝑛,𝑗=1,2,…,𝑚. In view of (4.2) and (2.11), for 𝑡≥0, we obtain
d⁺𝑉_𝑖𝑗(𝑡)d𝑡=𝑒^𝜃𝑡|𝑧_𝑖𝑗(𝑡)|^𝑞−1×sgn𝑧_𝑖𝑗{−[𝑎_𝑖𝑗(𝑡,𝑥_𝑖𝑗(𝑡))−𝑎_𝑖𝑗(𝑡,𝑥^∗_𝑖𝑗+(𝑡))]_{𝐵𝑘𝑙∈𝑁𝑟(𝑖,𝑗)}𝐵^𝑘𝑙_𝑖𝑗(𝑡)(𝑓_𝑖𝑗(𝑥_𝑘𝑙(𝑡−𝜏_𝑘𝑙(𝑡)))𝑥_𝑖𝑗(𝑡)−𝑓_𝑖𝑗(𝑥^∗_𝑘𝑙(𝑡−𝜏_𝑘𝑙(𝑡)))𝑥^∗_𝑖𝑗(𝑡))+_{𝐶𝑘𝑙∈𝑁𝑟(𝑖,𝑗)}𝐶^𝑘𝑙_𝑖𝑗(𝑡)(^𝑡_−∞𝑘_𝑖𝑗(𝑡−𝑠)𝑔_𝑖𝑗(𝑥_𝑘𝑙(𝑠))d𝑠𝑥_𝑖𝑗−(𝑡)_𝑡−∞𝑘_𝑖𝑗(𝑡−𝑠)𝑔_𝑖𝑗(𝑥^∗_𝑘𝑙(𝑠))d𝑠𝑥^∗_𝑖𝑗1(𝑡))}+𝑞𝜃𝑒^𝜃𝑡|𝑧_𝑖𝑗(𝑡)|^𝑞≤𝑒^𝜃𝑡|𝑧_𝑖𝑗(𝑡)|^𝑞−1×{−𝜇_𝑖𝑗(𝑡)|𝑧_𝑖𝑗(𝑡)|+_{𝐵𝑘𝑙∈𝑁𝑟(𝑖,𝑗)}|𝐵^𝑘𝑙_𝑖𝑗(𝑡)|×(|𝑓_𝑖𝑗(𝑥_𝑘𝑙(𝑡−𝜏_𝑘𝑙(𝑡)))||𝑧_𝑖𝑗(𝑡)|+|𝑓_𝑖𝑗(𝑥_𝑘𝑙(𝑡−𝜏_𝑘𝑙(𝑡)))−𝑓_𝑖𝑗(𝑥^∗_𝑘𝑙(𝑡−𝜏_𝑘𝑙(𝑡)))||𝑥^∗_𝑖𝑗(𝑡)|)+_{𝐶𝑘𝑙∈𝑁𝑟(𝑖,𝑗)}|𝐶^𝑘𝑙_𝑖𝑗(𝑡)|(^𝑡_−∞|𝑘_𝑖𝑗(𝑡−𝑠)||𝑔_𝑖𝑗(𝑥_𝑘𝑙(𝑠))|d𝑠|𝑧_𝑖𝑗+(𝑡)|_𝑡−∞|𝑘_𝑖𝑗(𝑡−𝑠)||𝑔_𝑖𝑗(𝑥_𝑘𝑙(𝑠))−𝑔_𝑖𝑗(𝑥^∗_𝑘𝑙(𝑠))|d𝑠|𝑥^∗_𝑖𝑗1(𝑡)|)}+𝑞𝜃𝑒^𝜃𝑡|𝑧_𝑖𝑗(𝑡)|^𝑞≤𝑒^𝜃𝑡|𝑧_𝑖𝑗(𝑡)|^𝑞−1×{−𝜇_𝑖𝑗(𝑡)|𝑧_𝑖𝑗(𝑡)|+_{𝐵𝑘𝑙∈𝑁𝑟(𝑖,𝑗)}|𝐵^𝑘𝑙_𝑖𝑗(𝑡)|(𝑀_𝑖𝑗|𝑧_𝑖𝑗(𝑡)|+𝛼_𝑖𝑗𝑅₀|𝑧_𝑘𝑙(𝑡−𝜏_𝑘𝑙(𝑡))|)+_{𝐶𝑘𝑙∈𝑁𝑟(𝑖,𝑗)}|𝐶^𝑘𝑙_𝑖𝑗(𝑡)|(𝑘_𝑖𝑗𝑁_𝑖𝑗|𝑧_𝑖𝑗(𝑡)|+𝛽_𝑖𝑗𝑅₀^𝑡_−∞|𝑘_𝑖𝑗(𝑡−𝑠)||𝑧_𝑘𝑙+1(𝑠)|d𝑠)}𝑞𝜃𝑒^𝜃𝑡|𝑧_𝑖𝑗(𝑡)|^𝑞≤(𝜃−𝑞𝜇_𝑖𝑗(𝑡)+𝑞_{𝐵𝑘𝑙∈𝑁𝑟(𝑖,𝑗)}|𝐵^𝑘𝑙_𝑖𝑗(𝑡)|𝑀_𝑖𝑗+𝑞_{𝐶𝑘𝑙∈𝑁𝑟(𝑖,𝑗)}|𝐶^𝑘𝑙_𝑖𝑗(𝑡)|𝑘_𝑖𝑗𝑁_𝑖𝑗+_{𝐵𝑘𝑙∈𝑁𝑟}^𝑚_(𝑖,𝑗)^′_𝑠=1𝑝_𝑠|𝐵^𝑘𝑙_𝑖𝑗(𝑡)|^𝑞𝑜𝑠𝛼^𝑞𝜀𝑠_𝑖𝑗𝑅^𝑞𝑙𝑠₀+_{𝐶𝑘𝑙∈𝑁𝑟}^𝑚_(𝑖,𝑗)^′_𝑠=1𝑝_𝑠|𝐶^𝑘𝑙_𝑖𝑗(𝑡)|^𝑞𝛾𝑠𝛽^𝑞𝛿𝑠_𝑖𝑗𝑅^𝑞𝑑𝑠₀)𝑉_𝑖𝑗(𝑡)+_{𝐵𝑘𝑙∈𝑁𝑟(𝑖,𝑗)}|𝐵^𝑘𝑙_𝑖𝑗(𝑡)|^{𝑞𝑜𝑚′+1}𝛼^{𝑞𝜀𝑚′+1}_𝑖𝑗𝑅^{𝑞𝑙𝑚′+1}₀𝑒^{𝜃𝜏𝑘𝑙(𝑡)}𝑉_𝑘𝑙(𝑡−𝜏_𝑘𝑙(𝑡))+_{𝐶𝑘𝑙∈𝑁𝑟(𝑖,𝑗)}|𝐶^𝑘𝑙_𝑖𝑗(𝑡)|^{𝑞𝛾𝑚′+1}𝛽^{𝑞𝛿𝑚′+1}_𝑖𝑗𝑅^{𝑞𝑑𝑚′+1}₀(^𝑡_−∞|𝑘_𝑖𝑗(𝑡−𝑠)|𝑒^{(𝜃/𝑞)(𝑡−𝑠)}𝑉_𝑘𝑙(𝑠)^1/𝑞d𝑠)^𝑞. (4.9)(7)(8)(9)(10)(11)(12)(13)(14)(15)(16)(17)(18)(19)(20)(21)(22)(23)(24)(25)
We now prove that the following statement is true:
𝑉_𝑖𝑗1(𝑡)=𝑞𝑒^𝜃𝑡|𝑧_𝑖𝑗(𝑡)|^𝑞≤𝑑𝜉_𝑖𝑗‖𝜙−𝜙^∗‖^𝑞,𝑖=1,2,…,𝑛,𝑗=1,2,…,𝑚,∀𝑡>0. (4.10) Contrarily, there must exist some 𝑖∈{1,2,…,𝑛}, 𝑗∈{1,2,…,𝑚}, and 𝑡̃>0, such that 𝑉_𝑖𝑗(𝑡)̃=𝑑𝜉_𝑖𝑗‖𝜙−𝜙^∗‖^𝑞,d⁺𝑉_𝑖𝑗(𝑡)̃d𝑡>0,𝑉_𝑘𝑙(𝑡)≤𝑑𝜉_𝑘𝑙‖𝜙−𝜙^∗‖^𝑞, (4.11) for all 𝑡∈(−∞,𝑡]̃,𝑘=1,2,…,𝑛,𝑙=1,2,…,𝑚. Together with (4.9), (4.11), and (4.7), we obtain d0<⁺𝑉_𝑖𝑗(𝑡)̃d𝑡≤𝑑‖𝜙−𝜙^∗‖^𝑞{(𝜃−𝑞𝜇_𝑖𝑗(𝑡)̃+𝑞_{𝐵𝑘𝑙∈𝑁𝑟(𝑖,𝑗)}|𝐵^𝑘𝑙_𝑖𝑗(𝑡)̃|𝑀_𝑖𝑗+𝑞_{𝐶𝑘𝑙∈𝑁𝑟(𝑖,𝑗)}|𝐶^𝑘𝑙_𝑖𝑗(𝑡)̃|𝑘_𝑖𝑗𝑁_𝑖𝑗+_{𝐵𝑘𝑙∈𝑁𝑟}^𝑚_(𝑖,𝑗)^′_𝑠=1𝑝_𝑠|𝐵^𝑘𝑙_𝑖𝑗(𝑡)̃|^𝑞𝑜𝑠𝛼^𝑞𝜀𝑠_𝑖𝑗𝑅^𝑞𝑙𝑠₀+_{𝐶𝑘𝑙∈𝑁𝑟}^𝑚_(𝑖,𝑗)^′_𝑠=1𝑝_𝑠|𝐶^𝑘𝑙_𝑖𝑗(𝑡)̃|^𝑞𝛾𝑠𝛽^𝑞𝛿𝑠_𝑖𝑗𝑅^𝑞𝑑𝑠₀)𝜉_𝑖𝑗+_{𝐵𝑘𝑙∈𝑁𝑟(𝑖,𝑗)}|𝐵^𝑘𝑙_𝑖𝑗(𝑡)̃|^{𝑞𝑜𝑚′+1}𝛼^{𝑞𝜀𝑚′+1}_𝑖𝑗𝑅^{𝑞𝑙𝑚′+1}₀𝑒^{𝜃𝜏𝑘𝑙(}𝑡₎̃𝜉_𝑘𝑙+_{𝐶𝑘𝑙∈𝑁𝑟(𝑖,𝑗)}|𝐶^𝑘𝑙_𝑖𝑗(𝑡)̃|^{𝑞𝛾𝑚′+1}𝛽^{𝑞𝛿𝑚′+1}_𝑖𝑗𝑅^{𝑞𝑑𝑚′+1}₀(₀^+∞|𝑘_𝑖𝑗(𝑠)|𝑒^{(𝜃/𝑞)𝑠}d𝑠)^𝑞𝜉_𝑘𝑙}≤0, (4.12) which is a contradiction. Hence, (4.10) holds. It follows that |𝑥_𝑖𝑗(𝑡)−𝑥^∗_𝑖𝑗(𝑡)|=|𝑧_𝑖𝑗(𝑡)|≤(𝑞𝑑𝜉_𝑖𝑗)^1/𝑞‖𝜑−𝜑^∗‖𝑒^{−(𝜃/𝑞)𝑡},∀𝑡>0,𝑖=1,2,…,𝑛,𝑗=1,2,…,𝑚. (4.13) Let 𝑃=max_(𝑖,𝑗){(𝑞𝑑𝜉_𝑖𝑗)^1/𝑞+1}, then |𝑥_𝑖𝑗(𝑡)−𝑥^∗_𝑖𝑗(𝑡)|≤𝑃‖𝜑−𝜑^∗‖𝑒^{−(𝜃/𝑞)𝑡},∀𝑡>0,𝑖=1,2,…,𝑛,𝑗=1,2,…,𝑚. (4.14) By means of Definition 2.3, the almost periodic solution 𝑥^∗(𝑡) of (1.1) is globally exponentially stable. This completes the proof.

Since Theorem 3.2 is general, the expression of (H₇) is complex. In order to check the applicability of the results easily, we give the following corollary.

Corollary 4.2. Assume that (H₁)-(H₄) and (H₆) hold. Furthermore, if one of the following conditions holds (A₁) there are constants 𝜎>0, 𝑞≥1, 𝜉_𝑖𝑗>0, 𝑖=1,2,…,𝑛, 𝑗=1,2,…,𝑚, such that max_(𝑖,𝑗)sup_𝑡∈𝑅{(−𝑞𝜇_𝑖𝑗(𝑡)+𝑞_{𝐵𝑘𝑙∈𝑁𝑟(𝑖,𝑗)}|𝐵^𝑘𝑙_𝑖𝑗(𝑡)|𝑀_𝑖𝑗+𝑞_{𝐶𝑘𝑙∈𝑁𝑟(𝑖,𝑗)}|𝐶^𝑘𝑙_𝑖𝑗(𝑡)|𝑘_𝑖𝑗𝑁_𝑖𝑗+(𝑞−1)_{𝐵𝑘𝑙∈𝑁𝑟(𝑖,𝑗)}|𝐵^𝑘𝑙_𝑖𝑗(𝑡)|+(𝑞−1)_{𝐶𝑘𝑙∈𝑁𝑟(𝑖,𝑗)}|𝐶^𝑘𝑙_𝑖𝑗(𝑡)|)𝜉_𝑖𝑗+_{𝐵𝑘𝑙∈𝑁𝑟(𝑖,𝑗)}|𝐵^𝑘𝑙_𝑖𝑗(𝑡)|𝛼^𝑞_𝑖𝑗𝑅^𝑞₀𝜉_𝑘𝑙+_{𝐶𝑘𝑙∈𝑁𝑟(𝑖,𝑗)}|𝐶^𝑘𝑙_𝑖𝑗(𝑡)|𝛽^𝑞_𝑖𝑗𝑅^𝑞₀𝑘^𝑞_𝑖𝑗𝜉_𝑘𝑙}≤−𝜎<0, (4.15)(A₂) there are constants 𝜎>0, 𝜉_𝑖𝑗>0, 𝑖=1,2,…,𝑛, 𝑗=1,2,…,𝑚, such that max_(𝑖,𝑗)sup_𝑡∈𝑅{(−2𝜇_𝑖𝑗(𝑡)+2_{𝐵𝑘𝑙∈𝑁𝑟(𝑖,𝑗)}|𝐵^𝑘𝑙_𝑖𝑗(𝑡)|𝑀_𝑖𝑗+2_{𝐶𝑘𝑙∈𝑁𝑟(𝑖,𝑗)}|𝐶^𝑘𝑙_𝑖𝑗(𝑡)|𝑘_𝑖𝑗𝑁_𝑖𝑗+_{𝐵𝑘𝑙∈𝑁𝑟(𝑖,𝑗)}|𝐵^𝑘𝑙_𝑖𝑗(𝑡)|+_{𝐶𝑘𝑙∈𝑁𝑟(𝑖,𝑗)}|𝐶^𝑘𝑙_𝑖𝑗(𝑡)|)𝜉_𝑖𝑗+_{𝐵𝑘𝑙∈𝑁𝑟(𝑖,𝑗)}|𝐵^𝑘𝑙_𝑖𝑗(𝑡)|𝛼²_𝑖𝑗𝑅²₀𝜉_𝑘𝑙+_{𝐶𝑘𝑙∈𝑁𝑟(𝑖,𝑗)}|𝐶^𝑘𝑙_𝑖𝑗(𝑡)|𝛽²_𝑖𝑗𝑅²₀𝑘²_𝑖𝑗𝜉_𝑘𝑙}≤−𝜎<0, (4.16)(A₃) there are constants 𝜎>0, 𝜉_𝑖𝑗>0, 𝑖=1,2,…,𝑛, 𝑗=1,2,…,𝑚, such that max_(𝑖,𝑗)sup_𝑡∈𝑅{(−𝜇_𝑖𝑗(𝑡)+_{𝐵𝑘𝑙∈𝑁𝑟(𝑖,𝑗)}|𝐵^𝑘𝑙_𝑖𝑗(𝑡)|𝑀_𝑖𝑗+_{𝐶𝑘𝑙∈𝑁𝑟(𝑖,𝑗)}|𝐶^𝑘𝑙_𝑖𝑗(𝑡)|𝑘_𝑖𝑗𝑁_𝑖𝑗)𝜉_𝑖𝑗+_{𝐵𝑘𝑙∈𝑁𝑟(𝑖,𝑗)}|𝐵^𝑘𝑙_𝑖𝑗(𝑡)|𝛼_𝑖𝑗𝑅₀𝜉_𝑘𝑙+_{𝐶𝑘𝑙∈𝑁𝑟(𝑖,𝑗)}|𝐶^𝑘𝑙_𝑖𝑗(𝑡)|𝛽_𝑖𝑗𝑅₀𝑘_𝑖𝑗𝜉_𝑘𝑙}≤−𝜎<0. (4.17)

Then (1.1) has exactly one almost periodic solution, which is globally exponentially stable.

Proof. Let 𝑚^′=1, 𝑜₁=𝑜₂=𝛾₁=𝛾₂=1/𝑞, 𝜀₁=𝑙₁=𝛿₁=𝑑₁=0, and 𝜀₂=𝑙₂=𝛿₂=𝑑₂=1, 𝑖=1,2,…,𝑛, 𝑗=1,2,…,𝑚, in (H₇), respectively, then (H₇) turns to (A₁).
Let 𝑞=2 and 𝑞=1 in (A₁), respectively, then we get (A₂) and (A₃). This completes the proof.

Remark 4.3. Corollary 4.2 can still be simplified into many simple results, which include the results in [7–11]. For example, authors of [9] considered a special case of our model (1.1) and obtained sufficient conditions of existence and exponential stability of almost periodic solutions for their model. Although the results of [9] and ours are completely different, we claim that the results in [9] are also special cases of the results in this paper. The model considered in [9] is 𝑥^′_𝑖𝑗(𝑡)=−𝑎_𝑖𝑗𝑥_𝑖𝑗(𝑡)+_{𝐶𝑘𝑙∈𝑁𝑟(𝑖,𝑗)}𝐶^𝑘𝑙_𝑖𝑗^∞₀𝑘_𝑖𝑗(𝑠)𝑔(𝑥_𝑘𝑙(𝑡−𝑠))d𝑠𝑥_𝑖𝑗(𝑡)+𝐼_𝑖𝑗(𝑡), (4.18) where 𝑎_𝑖𝑗,𝐶^𝑘𝑙_𝑖𝑗≥0,|𝑔(𝑥)|≤𝑀, and |𝑔(𝑢)−𝑔(v)|≤𝜇|𝑢−𝑣| for 𝑢,𝑣∈𝑅. The other arguments are same as this paper. Obviously, system (4.18) is a special case of system (1.1). The main theorem in [9] is the following.

Corollary 4.4. Assume that there exist nonnegative constants 𝐼,𝑞, and 𝜂 such thatmax_(𝑖,𝑗){∑_𝐶^𝑘𝑙_{∈𝑁𝑟(𝑖,𝑗)}𝐶^𝑘𝑙_𝑖𝑗𝑘_𝑖𝑗𝑀/𝑎_𝑖𝑗} = 𝜂<1, 𝐼=max_(𝑖,𝑗){𝐼_𝑖𝑗/𝑎_𝑖𝑗}, 𝑞=𝜂(1+𝜇𝐼/(1−𝜂)𝑀)<1. Then system (4.18) has exactly one almost periodic solution, which is globally exponentially stable.

Now we prove that Corollary 4.4 is a special case of Corollary 4.2. Comparing (1.1) with (4.18), we have 𝜇_𝑖𝑗(𝑡)=𝑎_𝑖𝑗, |𝐶^𝑘𝑙_𝑖𝑗(𝑡)|=𝐶^𝑘𝑙_𝑖𝑗, |𝐵^𝑘𝑙_𝑖𝑗(𝑡)|=0, 𝑁_𝑖𝑗=𝑀. Let 𝜉_𝑖𝑗=1,𝑖=1,2…,𝑛,𝑗=1,2…,𝑚. According to the proof of Theorem 3.2, we have 𝑅₀=𝐼/(1−𝜂). So, 𝑞=𝜂(1+𝜇𝑅₀/𝑀)<1⇔𝜂((𝑀+𝜇𝑅₀)/𝑀)<1. In view of the condition ∑_𝐶^𝑘𝑙_{∈𝑁𝑟(𝑖,𝑗)}𝐶^𝑘𝑙_𝑖𝑗𝑘_𝑖𝑗𝑀/𝑎_𝑖𝑗≤𝜂, we have

∑_{𝐶𝑘𝑙∈𝑁𝑟(𝑖,𝑗)}𝐶^𝑘𝑙_𝑖𝑗𝑘_𝑖𝑗𝑎_𝑖𝑗(𝑀+𝜇𝑅₀)<1, (4.19) which is equivalent to

−𝑎_𝑖𝑗+𝑀_{𝐶𝑘𝑙∈𝑁𝑟(𝑖,𝑗)}𝐶^𝑘𝑙_𝑖𝑗𝑘_𝑖𝑗+𝜇𝑅_{0𝐶𝑘𝑙∈𝑁𝑟(𝑖,𝑗)}𝐶^𝑘𝑙_𝑖𝑗𝑘_𝑖𝑗<0. (4.20) The expression is a special case of (A₃) in Corollary 4.2.

Similarly, from Corollary 4.2, we can get the main results of [5, 11] by the condition (A₃) (restrict in the case of periodic system in [5]) and the main results of [10] by condition (A₂). Hence, the results of this paper are useful and important.

Remark 4.5. In (1.1), if 𝑓_𝑖𝑗(𝑢),𝑔_𝑖𝑗(𝑢),𝑢∈ℝ are replaced by 𝑓_𝑖𝑗(𝑡,𝑢),𝑔_𝑖𝑗(𝑡,𝑢),𝑡,𝑢∈ℝ, where 𝑓_𝑖𝑗(𝑡,𝑢),𝑔_𝑖𝑗(𝑡,𝑢) are bounded and almost periodic functions about the first argument, then, by the same methods used in this paper, we can also get some general results similar to Theorem 4.1 and Corollary 4.2 and extend and improve the main results in [12].

Remark 4.6. Note that condition (H₇) in Theorem 4.1 possesses infinitely adjustable real parameters. This enables neural networks workers to obtain many special conditions on the existence and global exponential stability of almost periodic solutions and periodic solutions in the designs and applications of neural networks. We can offer many different corollaries by changing the parameters in (H₇), not only Corollary 4.2. Hence, our results provide many criteria for neural network's application.

5. A Numerical Example

Consider the following SICNNs with mixed delays:

𝑥^′_𝑖𝑗(t)=−𝑎_𝑖𝑗(𝑡,𝑥_𝑖𝑗(𝑡))+_{𝐵𝑘𝑙∈𝑁𝑟(𝑖,𝑗)}𝐵^𝑘𝑙_𝑖𝑗(𝑡)𝑓_𝑖𝑗(𝑥_𝑘𝑙(𝑡−𝜏_𝑘𝑙(𝑡)))𝑥_𝑖𝑗+(𝑡)_{𝐶𝑘𝑙∈𝑁𝑟(𝑖,𝑗)}𝐶^𝑘𝑙_𝑖𝑗(𝑡)^𝑡_−∞𝑘_𝑖𝑗(𝑡−𝑠)𝑔_𝑖𝑗(𝑥_𝑘𝑙(𝑠))d𝑠𝑥_𝑖𝑗(𝑡)+𝐼_𝑖𝑗(𝑡), (5.1) where 𝑟=1, 𝜏_𝑖𝑗(𝑡)=(cos𝑡)²,𝑖,𝑗=1,2,3. Take 𝑎₁₁(𝑡)=6𝑥−cos𝑥+𝑥sin𝑡,𝑎₁₂(𝑡)=6𝑥+0.5sin𝑥+𝑥cos𝑡,𝑎₁₃(𝑡)=5𝑥+cos𝑥+𝑥sin𝑡,𝑎₂₁(𝑡)=7𝑥+sin𝑥+𝑥sin𝑡,𝑎₂₂(𝑡)=7𝑥−cos𝑥−𝑥sin𝑡,𝑎₂₃(𝑡)=5𝑥−cos𝑥+𝑥cos𝑡,𝑎₃₁(𝑡)=6𝑥+sin𝑥+𝑥cos𝑡,𝑎₃₂(𝑡)=6𝑥+cos𝑥+𝑥sin𝑡,𝑎₃₃(𝑡)=6𝑥−sin𝑥+𝑥cos𝑡,𝑓_𝑖𝑗(𝑥)=𝑔_𝑖𝑗(𝑥)=0.1sin𝑥, 𝑘_𝑖𝑗(𝑡)=𝑒^−𝑡sin𝑡, and

𝐵₁₁(𝑡)𝐵₁₂(𝑡)𝐵₁₃𝐵(𝑡)₂₁(𝑡)𝐵₂₂(𝑡)𝐵₂₃𝐵(𝑡)₃₁(𝑡)𝐵₃₂(𝑡)𝐵₃₃=,𝐶(𝑡)0.1cos𝑡0.2sin𝑡00.3cos𝑡00.3sin𝑡00.4cos𝑡0.2sin𝑡₁₁(𝑡)𝐶₁₂(𝑡)𝐶₁₃𝐶(𝑡)₂₁(𝑡)𝐶₂₂(𝑡)𝐶₂₃𝐶(𝑡)₃₁(𝑡)𝐶₃₂(𝑡)𝐶₃₃=,𝐼(𝑡)0.2cos𝑡0.4sin𝑡0.3cos𝑡0.6cos𝑡00.5sin𝑡0.5sin𝑡0.6cos𝑡0.5sin𝑡₁₁(𝑡)𝐼₁₂(𝑡)𝐼₁₃𝐼(𝑡)₂₁(𝑡)𝐼₂₂(𝑡)𝐼₂₃𝐼(𝑡)₃₁(𝑡)𝐼₃₂(𝑡)𝐼₃₃=√(𝑡)2cos2𝑡sin√3𝑡1+2cos√√2𝑡1+cos2t2cos√2𝑡1−sin√√3𝑡2sin2𝑡1−cos√3𝑡3sin√.2𝑡 (5.2) Then 𝜇₁₁(𝑡)=5+sin𝑡,𝜇₁₂(𝑡)=5.5+cos𝑡,𝜇₁₃(𝑡)=4+sin𝑡,𝜇₂₁(𝑡)=6+sin𝑡,𝜇₂₂(𝑡)=6−sin𝑡,𝜇₂₃(𝑡)=4+cos𝑡,𝜇₃₁(𝑡)=5+cos𝑡,𝜇₃₂(𝑡)=5+sin𝑡,𝜇₃₃(𝑡)=5+cos𝑡, 𝛼_𝑖𝑗(𝑡)=𝛽_𝑖𝑗(𝑡)=𝑀_𝑖𝑗=𝑁_𝑖𝑗=0.1.

_{𝐵𝑘𝑙∈𝑁𝑟(1,1)}|𝐵^𝑘𝑙₁₁(𝑡)|=0.4|cos𝑡|+0.2|sin𝑡|,_{𝐵𝑘𝑙∈𝑁𝑟(1,2)}|𝐵^𝑘𝑙₁₂(𝑡)|=0.4|cos𝑡|+0.5|sin𝑡|,_{𝐵𝑘𝑙∈𝑁𝑟(1,3)}|𝐵^𝑘𝑙₁₃(𝑡)|=0.5|sin𝑡|,_{𝐵𝑘𝑙∈𝑁𝑟(2,1)}|𝐵^𝑘𝑙₂₁(𝑡)|=0.4|cos𝑡|+0.6|sin𝑡|,_{𝐵𝑘𝑙∈𝑁𝑟(2,2)}|𝐵^𝑘𝑙₂₂(𝑡)|=0.8|cos𝑡|+0.7|sin𝑡|,_{𝐵𝑘𝑙∈𝑁𝑟(2,3)}|𝐵^𝑘𝑙₂₃(𝑡)|=0.7|cos𝑡|+0.4|sin𝑡|,_{𝐵𝑘𝑙∈𝑁𝑟(3,1)}|𝐵^𝑘𝑙₃₁(𝑡)|=0.3|cos𝑡|+0.4|sin𝑡|,_{𝐵𝑘𝑙∈𝑁𝑟(3,2)}|𝐵^𝑘𝑙₃₂(𝑡)|=0.7|cos𝑡|+0.5|sin𝑡|,_{𝐵𝑘𝑙∈𝑁𝑟(3,3)}|𝐵^𝑘𝑙₃₃(𝑡)|=0.4|cos𝑡|+0.5|sin𝑡|,_{𝐶𝑘𝑙∈𝑁𝑟(1,1)}|𝐶^𝑘𝑙₁₁(𝑡)|=0.8|cos𝑡|+0.4|sin𝑡|,_{𝐶𝑘𝑙∈𝑁𝑟(1,2)}|𝐶^𝑘𝑙₁₂(𝑡)|=1.1|cos𝑡|+0.9|sin𝑡|,_{𝐶𝑘𝑙∈𝑁𝑟(1,3)}|𝐶^𝑘𝑙₁₃(𝑡)|=0.3|cos𝑡|+0.9|sin𝑡|,_{𝐶𝑘𝑙∈𝑁𝑟(2,1)}|𝐶^𝑘𝑙₂₁(𝑡)|=1.4|cos𝑡|+0.9|sin𝑡|,_{𝐶𝑘𝑙∈𝑁𝑟(2,2)}|𝐶^𝑘𝑙₂₂(𝑡)|=1.7|cos𝑡|+1.9|sin𝑡|,_{𝐶𝑘𝑙∈𝑁𝑟(2,3)}|𝐶^𝑘𝑙₂₃(𝑡)|=0.9|cos𝑡|+1.4|sin𝑡|,_{𝐶𝑘𝑙∈𝑁𝑟(3,1)}|𝐶^𝑘𝑙₃₁(𝑡)|=1.2|cos𝑡|+0.5|sin𝑡|,_{𝐶𝑘𝑙∈𝑁𝑟(3,2)}|𝐶^𝑘𝑙₃₂(𝑡)|=1.2|cos𝑡|+1.5|sin𝑡|,_{𝐶𝑘𝑙∈𝑁𝑟(3,3)}|𝐶^𝑘𝑙₃₃(𝑡)|=0.6|cos𝑡|+|sin𝑡|. (5.3)(26)(27)(28)(29)(30)(31)(32)(33)(34)

Take 𝜉_𝑖𝑗=1, 𝑖,𝑗=1,2,3. Computing by Matlab toolbox, we have 𝜂≈0.0721<1, 𝑅₀=𝐼₁₃/𝑎₁₃(1−𝜂)≈1.0777, and Λ≈−2.6849<0. It is easy to check that all the conditions needed in Corollary 4.2 with (A₃) are satisfied. Therefore, system (5.1) has a unique almost periodic solution 𝑥^∗(𝑡) with ‖𝑥^∗‖≤1.0777, which is globally exponentially stable. See Figures 1 and 2.

Figure 1 

States of networks (5.1) with 𝑡∈[0,4].

Figure 2 

States of networks (5.1) with 𝑡∈[0,40].

Remark 5.1. In (5.1), the behaved functions are all nonlinear. So, all the results in [5–20] and the references therein are inapplicable for (5.1). This means that the results of this paper are new and useful.

Acknowledgments

The authors would like to thank the editor Allan C Peterson and the reviewers for their valuable comments and constructive suggestions, which considerably improve the presentation of this paper. This work was jointly supported by the National Natural Science Foundation of China under Grant no. 60874088, the Foundation of Chinese Society for Electrical Engineering (2008), the China Postdoctoral Science Foundation under Grant no. 20070410300, 200801336, the Hunan Provincial Natural Science Foundation of China under Grant no. 07JJ4001, the Scientific Research Fund of Yunnan Provincial Education Department under Grant no. 07Y10085, and the Scientific Research Fund of Yunnan Province under Grant no. 2008CD186.