Global Exponential Stability of Almost Periodic Solutions for SICNNs with Continuously Distributed Leakage Delays

Abstract

Shunting inhibitory cellular neural networks (SICNNs) are considered with the introduction of continuously distributed delays in the leakage (or forgetting) terms. By using the Lyapunov functional method and differential inequality techniques, some sufficient conditions for the existence and exponential stability of almost periodic solutions are established. Our results complement with some recent ones.

1. Introduction

It is well known that a neural network usually has a spatial nature due to the presence of an amount of parallel pathways of a variety of axon sizes and lengths; it is desired to model them by introducing continuously distributed delays over a certain duration of time [1–4]. In particular, shunting inhibitory cellular neural networks (SICNNs) with continuously distributed delays can be described by 𝑥^′_𝑖𝑗(𝑡)=−𝑎_𝑖𝑗(𝑡)𝑥_𝑖𝑗(𝑡)−_{𝐶𝑘𝑙∈𝑁𝑟𝑖,𝑗}𝐶^𝑘𝑙_𝑖𝑗(𝑡)𝑥_𝑘𝑙(𝑡−𝜏(𝑡))𝑥_𝑖𝑗(𝑡)−_{𝐶𝑘𝑙∈𝑁𝑞(𝑖,𝑗)}𝐵^𝑘𝑙_𝑖𝑗(𝑡)⋅^∞₀𝐾_𝑖𝑗(𝑢)𝑔𝑥_𝑘𝑙(𝑡−𝑢)𝑑𝑢𝑥_𝑖𝑗(𝑡)+𝐿_𝑖𝑗(𝑡),𝑖=1,2,…,𝑚,𝑗=1,2,…,𝑛, (1) where 𝐶_𝑖𝑗 denotes the cell at the (𝑖,𝑗) position of the lattice. The 𝑟-neighborhood 𝑁_𝑟(𝑖,𝑗) of is given as 𝑁_𝑟𝑖,𝑗=𝐶_𝑘𝑙||,,∶max𝑘−𝑖𝑙−𝑗≤𝑟,1≤𝑘≤𝑚,1≤𝑙≤𝑛 (2) where 𝑁_𝑞(𝑖,𝑗) is similarly specified, 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(𝑡) and 𝐵^𝑘𝑙_𝑖𝑗(𝑡) are the connection or coupling strengths of postsynaptic activity of the cell transmitted to the cell 𝐶_𝑖𝑗, the activity functions 𝑓(⋅) and 𝑔(⋅) are continuous functions representing the output or firing rate of the cell 𝐶_𝑘𝑙, and 𝜏(𝑡)≥0 corresponds to the transmission delay.

Since SICNNs (1) have been introduced as a new cellular neural networks (CNNs) in Bouzerdout and Pinter in [5–7], it has been extensively applied in psychophysics, speech, perception, robotics, adaptive pattern recognition, vision, and image processing. Hence, there have been extensive results on the problem of the existence and stability of the equilibrium point, periodic and almost periodic solutions of SICNNs with continuously distributed delays in the literature. We refer the reader to [8–12] and references cited therein.

As pointed out in Gopalsamy [13], the first term in each of the right side of (1) corresponds to a stabilizing negative feedback of the system which acts instantaneously without time delay; these terms are variously known as “forgettin” or leakage terms (see, e.g., Kosko [14] and Haykin [15]). It is known from the literature on population dynamics and neural networks dynamics (see Gopalsamy [16]) that time delays in the stabilizing negative feedback terms will have a tendency to destabilize a system. Therefore, the authors of [17–21] dealt with the existence and stability of equilibrium and periodic solutions for neuron networks model involving leakage delays. Since leakage delays can have a destabilizing influence on the dynamical behaviors of neural networks and the incorporation of time delays in the leakage terms are usually not easy to handle, it necessary to investigate leakage delay effects on the stability of neural networks. On the other hand, as pointed out in [22, 23], periodically varying environment and almost periodically varying environment are foundations for the theory of nature selection. Compared with periodic effects, almost periodic effects are more frequent. Hence, the effects of the almost periodic environment on the evolutionary theory have been the object of intensive analysis by numerous authors, and some of these results can be found in [8, 9, 11] and references cited therein. However, to the best of our knowledge, few authors have considered the existence and exponential stability of almost periodic solutions of SICNNs with continuously distributed delays in the leakage terms. Motivated by the above discussions, in this present paper, we will consider the following SICNNs with continuously distributed leakage delays: 𝑥^′_𝑖𝑗(𝑡)=−𝑎_𝑖𝑗(𝑡)^∞₀ℎ_𝑖𝑗(𝑠)𝑥_𝑖𝑗(𝑡−𝑠)𝑑𝑠−_{𝐶𝑘𝑙∈𝑁𝑟𝑖,𝑗}𝐶^𝑘𝑙_𝑖𝑗(𝑡)𝑓𝑥_𝑘𝑙(𝑡−𝜏(𝑡))𝑥_𝑖𝑗(𝑡)−_{𝐶𝑘𝑙∈𝑁𝑞𝑖,𝑗}𝐵^𝑘𝑙_𝑖𝑗(𝑡)^∞₀𝐾_𝑖𝑗(𝑢)𝑔𝑥_𝑘𝑙(𝑡−𝑢)𝑑𝑢𝑥_𝑖𝑗(𝑡)+𝐿_𝑖𝑗(𝑡), (3) where 𝑖=1,2,…,𝑚,𝑗=1,2,…,𝑛,𝑎_𝑖𝑗∶𝑅→(0+∞), 𝜏∶𝑅→[0+∞), and 𝐿_𝑖𝑗,𝐶^𝑘𝑙_𝑖𝑗,𝐵^𝑘𝑙_𝑖𝑗∶𝑅→𝑅 are almost periodic functions, 𝜏(𝑡) denotes transmission delay, the leakage delay kernels ℎ_𝑖𝑗∶[0,∞)→[0,∞) are continuous and integrable, respectively, and the delay kernels 𝐾_𝑖𝑗∶[0,∞)→[0,∞) are continuous and integrable.

The main purpose of this paper is to give the conditions for the existence and exponential stability of the almost periodic solutions for system (3). By applying the Lyapunov functional method and differential inequality techniques, we derive some new sufficient conditions ensuring the existence, uniqueness, and exponential stability of the almost periodic solution for system (3), which are new and complement previously known results. Moreover, an example is also provided to illustrate the effectiveness of our results.

Throughout this paper, for 𝑖𝑗∈𝐽∶={11,12,…,1𝑛,21,22,…,2𝑛,…,𝑚1,𝑚2,…,𝑚𝑛}, delay kernels ℎ_𝑖(𝑠) and 𝐾_𝑖𝑗(𝑢) are continuous functions, and there exist constants 𝑎⁺_𝑖𝑗 and 𝜂⁺_𝑖𝑗 such that 𝑎⁺_𝑖𝑗=sup_𝑡∈𝑅𝑎_𝑖𝑗(𝑡),𝜂⁺_𝑖𝑗=^∞₀𝑠ℎ_𝑖𝑗(𝑠)𝑑𝑠. (4) From the theory of almost periodic functions in [22, 23], it follows that, for any 𝜖>0, it is possible to find a real number 𝑙=𝑙(𝜖)>0, for any interval with length 𝑙(𝜖), and there exists a number 𝛿=𝛿(𝜖) in this interval such that 𝑎_𝑖𝑗(𝑡+𝛿)−𝑎_𝑖𝑗(𝑡𝐶)<𝜖,_{𝑘𝑙𝑖𝑗}(𝑡+𝛿)−𝐶^𝑘𝑙_𝑖𝑗(𝑡𝐵)<𝜖,_{𝑘𝑙𝑖𝑗}(𝑡+𝛿)−𝐵^𝑘𝑙_𝑖𝑗(𝑡|𝜏()(𝑡𝐿)<𝜖,𝑡+𝛿−𝜏)|<𝜖,_𝑖𝑗(𝑡+𝛿)−𝐿_𝑖𝑗(𝑡)<𝜖, (5) for all 𝑡∈𝑅,𝑘𝑙,𝑖𝑗∈𝐽.

We set 𝑥_𝑖𝑗(𝑡)=𝑥₁₁(𝑡),…,𝑥_1𝑛(𝑡),…,𝑥_𝑖1(𝑡),…,𝑥_𝑖𝑛(𝑡)𝑥,…,_𝑚1(𝑡),…,𝑥_𝑚𝑛(𝑡)∈𝑅^𝑚×𝑛. (6) For any 𝑥(𝑡)={𝑥_𝑖𝑗(𝑡)}∈𝑅^𝑚×𝑛, we define the norm ‖𝑥(𝑡)‖=max_(𝑖,𝑗){|𝑥_𝑖𝑗(𝑡)|}. We also assume that the following conditions (𝑇₁) and (𝑇₂) hold.

(𝑇₁)𝑓∶𝑅→𝑅 and 𝑔∶𝑅→𝑅 are nonincreasing functions on [0,+∞), and there exist constants 𝑀_𝑓, 𝑀_𝑔, 𝜇_𝑓, and 𝜇_𝑔 such that 𝑓(𝑢)−𝑓(𝑣)≤𝜇_𝑓||,𝑓(𝑢𝑢−𝑣)≤𝑀_𝑓,𝑔(𝑢)−𝑔(𝑣)≤𝜇_𝑔||,𝑔(𝑢𝑢−𝑣)≤𝑀_𝑔,∀𝑢,𝑣∈𝑅. (7)

(𝑇₂) For 𝑖𝑗∈𝐽,1−𝑎⁺_𝑖𝑗𝜂⁺_𝑖𝑗>0, 𝛿_𝑖𝑗(𝑡)=𝑎_𝑖𝑗(𝑡)^∞₀ℎ_𝑖𝑗(𝑠)𝑑𝑠1−2𝑎⁺_𝑖𝑗𝜂⁺_𝑖𝑗−^∞₀ℎ_𝑖𝑗(𝑠)𝑎_𝑖𝑗(𝑡)−𝑎_𝑖𝑗(𝑡−𝑠)𝑑𝑠−_{𝐶𝑘𝑙∈𝑁𝑟𝑖,𝑗}𝐶^𝑘𝑙_𝑖𝑗(𝑡)𝑀_𝑓−_{𝐶𝑘𝑙∈𝑁𝑞𝑖,𝑗}𝐵^𝑘𝑙_𝑖𝑗(𝑡)^∞₀𝐾_𝑖𝑗(𝑢)𝑑𝑢𝑀_𝑔×11−𝑎⁺_𝑖𝑗𝜂⁺_𝑖𝑗>0, (8) and there exist positive constants 𝜂  and 𝜆 such that 𝜆<𝑎_𝑖𝑗(𝑡)^∞₀ℎ_𝑖𝑗(𝑠)𝑑𝑠,_∞0ℎ_𝑖𝑗(𝑠)𝑒^𝜆𝑠𝑑𝑠<+∞,_∞0𝐾_𝑖𝑗(𝑠)𝑒^𝜆𝑠−𝑎𝑑𝑠<+∞,_𝑖𝑗(𝑡)^∞₀ℎ_𝑖𝑗(𝑠)𝑑𝑠−𝜆1−2𝑎⁺_𝑖𝑗𝜂⁺_𝑖𝑗−^∞₀ℎ_𝑖𝑗(𝑠)𝑎_𝑖𝑗(𝑡)𝑒^𝜆𝑠−𝑎_𝑖𝑗(1𝑡−𝑠)𝑑𝑠1−𝑎⁺_𝑖𝑗𝜂⁺_𝑖𝑗+_{𝐶𝑘𝑙∈𝑁𝑟(𝑖,𝑗)}𝐶^𝑘𝑙_𝑖𝑗(𝑡)𝜇_𝑓𝑒^{𝜆𝜏(𝑡)}11−𝑎⁺_𝑘𝑙𝜂⁺_𝑘𝑙11−𝑎⁺_𝑖𝑗𝜂⁺_𝑖𝑗𝐿_𝑖𝑗𝛿_𝑖𝑗⁺+_{𝐶𝑘𝑙∈𝑁𝑟(𝑖,𝑗)}𝐶^𝑘𝑙_𝑖𝑗(𝑡)𝑀_𝑓11−𝑎⁺_𝑖𝑗𝜂⁺_𝑖𝑗+_{𝐶𝑘𝑙∈𝑁𝑞(𝑖,𝑗)}𝐵^𝑘𝑙_𝑖𝑗(𝑡)^∞₀𝑒^𝜆𝑢𝐾_𝑖𝑗(𝑢)𝑑𝑢×𝜇_𝑔11−𝑎⁺_𝑘𝑙𝜂⁺_𝑘𝑙×11−𝑎⁺_𝑖𝑗𝜂⁺_𝑖𝑗𝐿_𝑖𝑗𝛿_𝑖𝑗⁺+_{𝐶𝑘𝑙∈𝑁𝑞(𝑖,𝑗)}𝐵^𝑘𝑙_𝑖𝑗(𝑡)^∞₀𝐾_𝑖𝑗(𝑢)𝑑𝑢𝑀_𝑔11−𝑎⁺_𝑖𝑗𝜂⁺_𝑖𝑗<−𝜂, (9) where 𝑡≥0,  and (𝐿_𝑖𝑗/𝛿_𝑖𝑗)⁺=sup_𝑡∈𝑅|𝐿_𝑖𝑗(𝑡)/𝛿_𝑖𝑗(𝑡)|.

The initial conditions associated with system (3) are of the form 𝑥_𝑖𝑗(𝑠)=𝜑_𝑖𝑗(𝑠),𝑠∈(−∞,0],𝑖𝑗∈𝐽, (10) where 𝜑_𝑖𝑗(⋅) denotes real-valued bounded continuous function defined on (−∞,0].

Definition 1 (see [22, 23]). Let 𝑢(𝑡)∶𝑅→𝑅^𝑚×𝑛 be continuous in 𝑡. 𝑢(𝑡) is said to be almost periodic on 𝑅 if, for any 𝜀>0, the set 𝑇(𝑢,𝜀)={𝛿∶‖𝑢(𝑡+𝛿)−𝑢(𝑡)‖<𝜀,∀𝑡∈𝑅} is relatively dense; that is, for any 𝜀>0, it is possible to find a real number 𝑙=𝑙(𝜀)>0, for any interval with length 𝑙(𝜀), and there exists a number 𝛿=𝛿(𝜀) in this interval such that ‖𝑢(𝑡+𝛿)−𝑢(𝑡)‖<𝜀, for all 𝑡∈𝑅.

The remaining part of this paper is organized as follows. In Section 2, we will derive some new sufficient conditions for checking the existence of bounded solutions. In Section 3, we present some new sufficient conditions for the existence, uniqueness and exponential stability of the positive almost periodic solution of (3). In Section 4, we will give some examples and remarks to illustrate our results obtained in previous sections.

2. Preliminary Results

The following lemmas will be useful to prove our main results in Section 3.

Lemma 2. Let (𝑇₁) and (𝑇₂) hold. Suppose that 𝑥(𝑡)={𝑥_𝑖𝑗(𝑡)} is a solution of system (3) with initial conditions 𝑥_𝑖𝑗(𝑠)=𝜑_𝑖𝑗(𝑠),𝜑_𝑖𝑗(𝑡)−^∞₀ℎ_𝑖𝑗(𝑠)^𝑡_𝑡−𝑠𝑎_𝑖𝑗(𝜃)𝜑_𝑖𝑗(𝜃)<𝐿𝑑𝜃𝑑𝑠^𝑖𝑗𝛿_𝑖𝑗⁺, (11) where  𝑠,𝑡∈(−∞,0],𝑖𝑗∈𝐽. Then 𝑥_𝑖𝑗(𝑡)−^∞₀ℎ_𝑖𝑗(𝑠)^𝑡_𝑡−𝑠𝑎_𝑖𝑗(𝜃)𝑥_𝑖𝑗(𝜃)≤𝐿𝑑𝜃𝑑𝑠_𝑖𝑗𝛿_𝑖𝑗⁺𝑥,∀𝑡≥0,𝑖𝑗∈𝐽,_𝑖𝑗(𝑡≤1)1−𝑎⁺_𝑖𝑗𝜂⁺_𝑖𝑗𝐿_𝑖𝑗𝛿_𝑖𝑗⁺,∀𝑡≥0,𝑖𝑗∈𝐽. (12)(13)

Proof. Assume, by way of contradiction, that (12) does not hold. Then, there exist 𝑖𝑗∈𝐽, 𝛾>(𝐿_𝑖𝑗/𝛿_𝑖𝑗)⁺, and 𝑡_∗>0 such that 𝑋_𝑖𝑗𝑡_∗=𝛾,𝑋_𝑖𝑗(𝑡)<𝛾,∀𝑡∈−∞,𝑡_∗, (14) where 𝑋_𝑖𝑗(𝑡)=𝑥_𝑖𝑗(𝑡)−^∞₀ℎ_𝑖𝑗(𝑠)^𝑡_𝑡−𝑠𝑎_𝑖𝑗(𝜃)𝑥_𝑖𝑗(𝜃)𝑑𝜃𝑑𝑠. (15) It follows that 𝑥_𝑖𝑗(𝑡≤𝑥)_𝑖𝑗(𝑡)−^∞₀ℎ_𝑖𝑗(𝑠)^𝑡_𝑡−𝑠𝑎_𝑖𝑗(𝜃)𝑥_𝑖𝑗(𝜃)+𝑑𝜃𝑑𝑠_∞0ℎ_𝑖𝑗(𝑠)^𝑡_𝑡−𝑠𝑎_𝑖𝑗(𝜃)𝑥_𝑖𝑗(𝜃)𝑑𝜃𝑑𝑠≤𝛾+𝑎⁺_𝑖𝑗𝜂⁺_𝑖𝑗sup_{𝑠∈−∞,𝑡∗}𝑥_𝑖𝑗(𝑠),∀𝑡∈−∞,𝑡_∗. (16) Consequently, in view of (16) and the fact 𝑎⁺_𝑖𝑗𝜂⁺_𝑖𝑗<1(𝑖𝑗∈𝐽), we have 𝑥_𝑖𝑗(𝑡)≤sup_{𝑠∈−∞,𝑡∗}𝑥_𝑖𝑗(𝑠≤1)1−𝑎⁺_𝑖𝑗𝜂⁺_𝑖𝑗𝛾,∀𝑡∈−∞,𝑡_∗. (17)
From system (3), we derive 𝑑𝑥𝑑𝑡^𝑖𝑗(𝑡)−^∞₀ℎ_𝑖𝑗(𝑠)^𝑡_𝑡−𝑠𝑎_𝑖𝑗(𝜃)𝑥_𝑖𝑗(𝜃)𝑑𝜃𝑑𝑠=𝑥^′_𝑖𝑗(𝑡)−𝑎_𝑖𝑗(𝑡)^∞₀ℎ_𝑖𝑗(𝑠)𝑑𝑠𝑥_𝑖𝑗(𝑡)+^∞₀ℎ_𝑖𝑗(𝑠)𝑎_𝑖𝑗(𝑡−𝑠)𝑥_𝑖𝑗(𝑡−𝑠)𝑑𝑠=−𝑎_𝑖𝑗(𝑡)^∞₀ℎ_𝑖𝑗(𝑠)×𝑥𝑑𝑠_𝑖𝑗(𝑡)−^∞₀ℎ_𝑖𝑗(𝑠)^𝑡_𝑡−𝑠𝑎_𝑖𝑗(𝜃)𝑥_𝑖𝑗(𝜃)𝑑𝜃𝑑𝑠−𝑎_𝑖𝑗(𝑡)^∞₀ℎ_𝑖𝑗(𝑠)𝑑𝑠^∞₀ℎ_𝑖𝑗(𝑠)×^𝑡_𝑡−𝑠𝑎_𝑖𝑗(𝜃)𝑥_𝑖𝑗(𝜃)−𝑑𝜃𝑑𝑠_∞0ℎ_𝑖𝑗(𝑠)𝑎_𝑖𝑗(𝑡)−𝑎_𝑖𝑗(𝑡−𝑠)𝑥_𝑖𝑗()−𝑡−𝑠𝑑𝑠_{𝐶𝑘𝑙∈𝑁𝑟𝑖,𝑗}𝐶^𝑘𝑙_𝑖𝑗(𝑡)𝑓𝑥_𝑘𝑙(𝑡−𝜏(𝑡))𝑥_𝑖𝑗(𝑡)−_{𝐶𝑘𝑙∈𝑁𝑞𝑖,𝑗}𝐵^𝑘𝑙_𝑖𝑗(𝑡)^∞₀𝐾_𝑖𝑗(𝑢)𝑔𝑥_𝑘𝑙(𝑡−𝑢)𝑑𝑢𝑥_𝑖𝑗(𝑡)+𝐿_𝑖𝑗(𝑡),𝑖𝑗∈𝐽. (18)
Calculating the upper left derivative of |𝑋_𝑖𝑗(𝑡)|, together with (14), (17), (18), (𝑇₁), and (𝑇₂), we obtain 0≤𝐷⁻𝑋_𝑖𝑗𝑡_∗≤−𝑎_𝑖𝑗𝑡_∗^∞₀ℎ_𝑖𝑗(𝑠)𝑑𝑠𝑋_𝑖𝑗𝑡_∗+−𝑎_𝑖𝑗𝑡_∗^∞₀ℎ_𝑖𝑗(𝑠)𝑑𝑠^∞₀ℎ_𝑖𝑗(𝑠)×^𝑡∗_{𝑡∗−𝑠}𝑎_𝑖𝑗(𝜃)𝑥_𝑖𝑗(𝜃)−𝑑𝜃𝑑𝑠_∞0ℎ_𝑖𝑗(𝑠)𝑎_𝑖𝑗𝑡_∗−𝑎_𝑖𝑗𝑡_∗−𝑠𝑥_𝑖𝑗𝑡_∗−𝑠𝑑𝑠−_{𝐶𝑘𝑙∈𝑁𝑟𝑖,𝑗}𝐶^𝑘𝑙_𝑖𝑗𝑡_∗𝑓𝑥_𝑘𝑙𝑡_∗−𝜏𝑡_∗𝑥_𝑖𝑗𝑡_∗−_{𝐶𝑘𝑙∈𝑁𝑞𝑖,𝑗}𝐵^𝑘𝑙_𝑖𝑗𝑡_∗×^∞₀𝐾_𝑖𝑗(𝑢)𝑔𝑥_𝑘𝑙𝑡_∗−𝑢𝑑𝑢𝑥_𝑖𝑗𝑡_∗+𝐿_𝑖𝑗𝑡_∗≤−𝑎_𝑖𝑗𝑡_∗^∞₀ℎ_𝑖𝑗(𝑠)𝑑𝑠𝛾+𝑎_𝑖𝑗𝑡_∗×^∞₀ℎ_𝑖𝑗(𝑠)𝑑𝑠𝑎⁺_𝑖𝑗𝜂⁺_𝑖𝑗11−𝑎⁺_𝑖𝑗𝜂⁺_𝑖𝑗𝛾+^∞₀ℎ_𝑖𝑗(𝑠)𝑎_𝑖𝑗𝑡_∗−𝑎_𝑖𝑗𝑡_∗𝛾−𝑠𝑑𝑠1−𝑎⁺_𝑖𝑗𝜂⁺_𝑖𝑗+_{𝐶𝑘𝑙∈𝑁𝑟𝑖,𝑗}𝐶^𝑘𝑙_𝑖𝑗𝑡_∗𝑀_𝑓11−𝑎⁺_𝑖𝑗𝜂⁺_𝑖𝑗𝛾+_{𝐶𝑘𝑙∈𝑁𝑞𝑖,𝑗}𝐵^𝑘𝑙_𝑖𝑗𝑡_∗^∞₀𝐾_𝑖𝑗(𝑢)𝑑𝑢𝑀_𝑔11−𝑎⁺_𝑖𝑗𝜂⁺_𝑖𝑗𝛾+𝐿_𝑖𝑗𝑡_∗=−𝑎_𝑖𝑗𝑡_∗^∞₀ℎ_𝑖𝑗(𝑠)𝑑𝑠1−2𝑎⁺_𝑖𝑗𝜂⁺_𝑖𝑗−^∞₀ℎ_𝑖𝑗(𝑠)𝑎_𝑖𝑗𝑡_∗−𝑎_𝑖𝑗𝑡_∗−𝑠𝑑𝑠+_{𝐶𝑘𝑙∈𝑁𝑟𝑖,𝑗}𝐶^𝑘𝑙_𝑖𝑗𝑡_∗𝑀_𝑓+_{𝐶𝑘𝑙∈𝑁𝑞(𝑖,𝑗)}𝐵^𝑘𝑙_𝑖𝑗𝑡_∗^∞₀𝐾_𝑖𝑗(𝑢)𝑑𝑢𝑀_𝑔11−𝑎⁺_𝑖𝑗𝜂⁺_𝑖𝑗𝛾+𝐿_𝑖𝑗𝑡_∗≤−𝛿_𝑖𝑗𝑡_∗𝐿𝛾−^𝑖𝑗𝛿_𝑖𝑗⁺<0. (19) It is a contradiction and shows that (12) holds. Then, using a similar argument as in the proof of (16) and (17), we can show that (13) holds. The proof of Lemma 2 is now completed.

Remark 3. In view of the boundedness of this solution, from the theory of functional differential equations with infinite delay in [21], it follows that the solution of system (3) with initial conditions (11) can be defined on [0,∞).

Lemma 4. Suppose that (𝑇₁) and (𝑇₂) hold. Moreover, assume that 𝑥(𝑡)={𝑥_𝑖𝑗(𝑡)} is a solution of system (3) with initial function 𝜑_𝑖𝑗(⋅) satisfying (11), and 𝜑^′_𝑖𝑗(⋅) is bounded continuous on (−∞,0]. Then, for any 𝜖>0, there exists 𝑙=𝑙(𝜖)>0, such that every interval [𝛼,𝛼+𝑙] contains at least one number 𝛿 for which there exists 𝑁>0 which satisfies ‖𝑥(𝑡+𝛿)−𝑥(𝑡)‖≤𝜖,∀𝑡>𝑁. (20)

Proof. For 𝑖𝑗∈𝐽, set 𝜖_𝑖𝑗(𝛿,𝑡)=−𝑎_𝑖𝑗(𝑡+𝛿)−𝑎_𝑖𝑗(𝑡×_∞0)ℎ_𝑖𝑗(𝑠)𝑥_𝑖𝑗(𝑡+𝛿−𝑠)𝑑𝑠−_{𝐶𝑘𝑙∈𝑁𝑟𝑖,𝑗}𝐶^𝑘𝑙_𝑖𝑗(𝑡+𝛿)−𝐶^𝑘𝑙_𝑖𝑗(𝑡×𝑓𝑥_𝑘𝑙)(𝑡−𝜏(𝑡+𝛿)+𝛿)𝑥_𝑖𝑗(𝑡+𝛿)−_{𝐶𝑘𝑙∈𝑁𝑟𝑖,𝑗}𝐶^𝑘𝑙_𝑖𝑗(𝑡)𝑓𝑥_𝑘𝑙(())𝑥𝑡−𝜏𝑡+𝛿+𝛿−𝑓_𝑘𝑙(𝑡−𝜏(𝑡)+𝛿)×𝑥_𝑖𝑗(𝑡+𝛿)−_{𝐶𝑘𝑙∈𝑁𝑞𝑖,𝑗}𝐵^𝑘𝑙_𝑖𝑗(𝑡+𝛿)−𝐵^𝑘𝑙_𝑖𝑗(𝑡×_∞0)𝐾_𝑖𝑗(𝑢)𝑔𝑥_𝑘𝑙(𝑡+𝛿−𝑢)𝑑𝑢𝑥_𝑖𝑗()+𝐿𝑡+𝛿_𝑖𝑗(𝑡+𝛿)−𝐿_𝑖𝑗(𝑡). (21) By Lemma 2, the solution 𝑥(𝑡)={𝑥_𝑖𝑗(𝑡)} is bounded and 𝑥_𝑖𝑗(𝑡≤1)1−𝑎⁺_𝑖𝑗𝜂⁺_𝑖𝑗𝐿_𝑖𝑗𝛿_𝑖𝑗⁺,∀𝑡∈[0,+∞),𝑖𝑗∈𝐽. (22) Thus, the right side of (3) is also bounded, which implies that 𝑥(𝑡) is uniformly continuous on 𝑅. From (5), for any 𝜖>0, there exists 𝑙=𝑙(𝜖)>0, such that every interval [𝛼,𝛼+𝑙],𝛼∈𝑅, contains a 𝛿 for which 𝜖_𝑖𝑗(≤1𝛿,𝑡)2𝑖̃𝜂min_𝑗∈𝐽1−𝑎⁺_̃𝑖𝑗𝜂⁺_̃𝑖𝑗𝜖,where𝑖𝑗∈𝐽,𝑡∈𝑅. (23)
Let 𝑁₀≥0 be sufficiently large such that 𝑡+𝛿≥0, for 𝑡≥𝑁₀, and denote 𝑢_𝑖𝑗(𝑡)=𝑥_𝑖𝑗(𝑡+𝛿)−𝑥_𝑖𝑗(𝑡). We obtain 𝑑𝑢_𝑖𝑗(𝑡)𝑑𝑡=−𝑎_𝑖𝑗(𝑡)^∞₀ℎ_𝑖𝑗(𝑠)𝑢_𝑖𝑗(𝑡−𝑠)𝑑𝑠−_{𝐶𝑘𝑙∈𝑁𝑟𝑖,𝑗}𝐶^𝑘𝑙_𝑖𝑗(𝑡)𝑓𝑥_𝑘𝑙((𝑡))𝑥𝑡−𝜏+𝛿−𝑓_𝑘𝑙(𝑡−𝜏(𝑡))×𝑥_𝑖𝑗(𝑡+𝛿)−_{𝐶𝑘𝑙∈𝑁𝑟𝑖,𝑗}𝐶^𝑘𝑙_𝑖𝑗(𝑡)𝑓𝑥_𝑘𝑙((𝑡⋅𝑥𝑡−𝜏))_𝑖𝑗(𝑡+𝛿)−𝑥_𝑖𝑗(𝑡)−_{𝐶𝑘𝑙∈𝑁𝑞𝑖,𝑗}𝐵^𝑘𝑙_𝑖𝑗(𝑡)^∞₀𝐾_𝑖𝑗(𝑢)×𝑔𝑥_𝑘𝑙()𝑥𝑡+𝛿−𝑢−𝑔_𝑘𝑙(𝑡−𝑢)𝑑𝑢𝑥_𝑖𝑗(𝑡+𝛿)−_{𝐶𝑘𝑙∈𝑁𝑞𝑖,𝑗}𝐵^𝑘𝑙_𝑖𝑗(𝑡)^∞₀𝐾_𝑖𝑗(𝑢)𝑔𝑥_𝑘𝑙()⋅𝑥𝑡−𝑢𝑑𝑢_𝑖𝑗(𝑡+𝛿)−𝑥_𝑖𝑗(𝑡)+𝜖_𝑖𝑗(𝛿,𝑡),∀𝑡≥𝑁₀,𝑖𝑗∈𝐽, (24) which yields 𝑑𝑒𝑑𝑡^𝜆𝑡𝑢_𝑖𝑗(𝑡)−^∞₀ℎ_𝑖𝑗(𝑠)^𝑡_𝑡−𝑠𝑎_𝑖𝑗(𝜃)𝑒^𝜆𝜃𝑢_𝑖𝑗(𝜃)𝑑𝜃𝑑𝑠=𝜆𝑒^𝜆𝑡𝑢_𝑖𝑗(𝑡)+𝑒^𝜆𝑡𝑢^′_𝑖𝑗(𝑡)−𝑎_𝑖𝑗(𝑡)^∞₀ℎ_𝑖𝑗(𝑠)𝑑𝑠𝑒^𝜆𝑡𝑢_𝑖𝑗(𝑡)+^∞₀ℎ_𝑖𝑗(𝑠)𝑎_𝑖𝑗(𝑡−𝑠)𝑒^{𝜆(𝑡−𝑠)}𝑢_𝑖𝑗()𝑎𝑡−𝑠𝑑𝑠=−_𝑖𝑗(𝑡)^∞₀ℎ_𝑖𝑗(𝑠)×𝑒𝑑𝑠−𝜆_𝜆𝑡𝑢_𝑖𝑗(𝑡)−^∞₀ℎ_𝑖𝑗(𝑠)^𝑡_𝑡−𝑠𝑎_𝑖𝑗(𝜃)𝑒^𝜆𝜃𝑢_𝑖𝑗(𝜃)−𝑎𝑑𝜃𝑑𝑠_𝑖𝑗(𝑡)^∞₀ℎ_𝑖𝑗(𝑠)𝑑𝑠−𝜆^∞₀ℎ_𝑖𝑗(𝑠)×^𝑡_𝑡−𝑠𝑎_𝑖𝑗(𝜃)𝑒^𝜆𝜃𝑢_𝑖𝑗(𝜃)−𝑑𝜃𝑑𝑠_∞0ℎ_𝑖𝑗(𝑠)𝑎_𝑖𝑗(𝑡)𝑒^𝜆𝑠−𝑎_𝑖𝑗(𝑡−𝑠)×𝑒^{𝜆(𝑡−𝑠)}𝑢_𝑖𝑗(𝑡−𝑠)𝑑𝑠+𝑒^𝜆𝑡−_{𝐶𝑘𝑙∈𝑁𝑟𝑖,𝑗}𝐶^𝑘𝑙_𝑖𝑗(𝑡)×𝑓𝑥_𝑘𝑙((𝑡))𝑥𝑡−𝜏+𝛿−𝑓_𝑘𝑙(𝑡−𝜏(𝑡))𝑥_𝑖𝑗(𝑡+𝛿)−_{𝐶𝑘𝑙∈𝑁𝑟𝑖,𝑗}𝐶^𝑘𝑙_𝑖𝑗(𝑡)𝑓𝑥_𝑘𝑙((𝑡⋅𝑥𝑡−𝜏))_𝑖𝑗(𝑡+𝛿)−𝑥_𝑖𝑗(𝑡)−_{𝐶𝑘𝑙∈𝑁𝑞𝑖,𝑗}𝐵^𝑘𝑙_𝑖𝑗(𝑡)^∞₀𝐾_𝑖𝑗(𝑢)×𝑔𝑥_𝑘𝑙()𝑥𝑡+𝛿−𝑢−𝑔_𝑘𝑙(𝑡−𝑢)𝑑𝑢𝑥_𝑖𝑗(𝑡+𝛿)−_{𝐶𝑘𝑙∈𝑁𝑞𝑖,𝑗}𝐵^𝑘𝑙_𝑖𝑗(𝑡)^∞₀𝐾_𝑖𝑗(𝑢)𝑔𝑥_𝑘𝑙()⋅𝑥𝑡−𝑢𝑑𝑢_𝑖𝑗(𝑡+𝛿)−𝑥_𝑖𝑗(𝑡)+𝜖_𝑖𝑗(𝛿,𝑡),∀𝑡≥𝑁₀,𝑖𝑗∈𝐽. (25)
Set 𝑈(𝑡)=𝑈_𝑖𝑗(𝑡), (26) where 𝑈_𝑖𝑗(𝑡)=𝑒^𝜆𝑡𝑢_𝑖𝑗(𝑡)−^∞₀ℎ_𝑖𝑗(𝑠)^𝑡_𝑡−𝑠𝑎_𝑖𝑗(𝜃)𝑒^𝜆𝜃𝑢_𝑖𝑗(𝜃)𝑑𝜃𝑑𝑠,𝑖𝑗∈𝐽. (27) Let (𝑖𝑗)_𝑡 be such an index that 𝑈_{(𝑖𝑗)𝑡}(𝑡)=‖𝑈(𝑡)‖. (28) Calculating the upper left derivative of |𝑈_{(𝑖𝑗)𝑠}(𝑠)| along (25), we have 𝐷⁻𝑈_{(𝑖𝑗)𝑠}(𝑠)_𝑠=𝑡≤−𝑎_{(𝑖𝑗)𝑡}(𝑡)^∞₀ℎ_{(𝑖𝑗)𝑡}(𝑠)𝑑𝑠−𝜆𝑈_{(𝑖𝑗)𝑡}(𝑡+−𝑎_{(𝑖𝑗)𝑡})(𝑡)^∞₀ℎ_{(𝑖𝑗)𝑡}(𝑠)𝑑𝑠−𝜆^∞₀ℎ_{(𝑖𝑗)𝑡}(𝑠)×^𝑡_𝑡−𝑠𝑎_{(𝑖𝑗)𝑡}(𝜃)𝑒^𝜆𝜃𝑢_{(𝑖𝑗)𝑡}(𝜃)−𝑑𝜃𝑑𝑠_∞0ℎ_{(𝑖𝑗)𝑡}(𝑠)𝑎_{(𝑖𝑗)𝑡}(𝑡)𝑒^𝜆𝑠−𝑎_{(𝑖𝑗)𝑡}(𝑡−𝑠)×𝑒^{𝜆(𝑡−𝑠)}𝑢_{(𝑖𝑗)𝑡}(𝑡−𝑠)𝑑𝑠+𝑒^𝜆𝑡−_{𝐶𝑘𝑙∈𝑁𝑟(𝑖,𝑗)𝑡}𝐶^𝑘𝑙_{(𝑖𝑗)𝑡}(𝑡)×𝑓𝑥_𝑘𝑙((𝑡))𝑥𝑡−𝜏+𝛿−𝑓_𝑘𝑙(𝑡−𝜏(𝑡))×𝑥_{(𝑖𝑗)𝑡}(𝑡+𝛿)−_{𝐶𝑘𝑙∈𝑁𝑟(𝑖,𝑗)𝑡}𝐶^𝑘𝑙_{(𝑖𝑗)𝑡}(𝑡)𝑓𝑥_𝑘𝑙((𝑡⋅𝑥𝑡−𝜏))_{(𝑖𝑗)𝑡}(𝑡+𝛿)−𝑥_{(𝑖𝑗)𝑡}(𝑡)−_{𝐶𝑘𝑙∈𝑁𝑞(𝑖,𝑗)𝑡}𝐵^𝑘𝑙_{(𝑖𝑗)𝑡}(𝑡)×^∞₀𝐾_{(𝑖𝑗)𝑡}(𝑢)×𝑔𝑥_𝑘𝑙()𝑥𝑡+𝛿−𝑢−𝑔_𝑘𝑙(𝑡−𝑢)𝑑𝑢𝑥_{(𝑖𝑗)𝑡}(𝑡+𝛿)−_{𝐶𝑘𝑙∈𝑁𝑞(𝑖,𝑗)𝑡}𝐵^𝑘𝑙_{(𝑖𝑗)𝑡}(𝑡)×^∞₀𝐾_{(𝑖𝑗)𝑡}(𝑢)𝑔𝑥_𝑘𝑙()⋅𝑥𝑡−𝑢𝑑𝑢_{(𝑖𝑗)𝑡}(𝑡+𝛿)−𝑥_{(𝑖𝑗)𝑡}(𝑡)+𝜖_{(𝑖𝑗)𝑡}(𝛿,𝑡). (29)
Let 𝑀(𝑡)=sup_𝑠≤𝑡{‖𝑈(𝑠)‖}. (30) It is obvious that ‖𝑈(𝑡)‖≤𝑀(𝑡), and 𝑀(𝑡) is nondecreasing. In particular, 𝑒^𝜆𝜌𝑢_𝑖𝑗𝜌≤𝑒^𝜆𝜌𝑢_𝑖𝑗𝜌−^∞₀ℎ_𝑖𝑗(𝑠)^𝜌_𝜌−𝑠𝑎_𝑖𝑗(𝜃)𝑒^𝜆𝜃𝑢_𝑖𝑗(𝜃)+𝑑𝜃𝑑𝑠_∞0ℎ_𝑖𝑗(𝑠)^𝜌_𝜌−𝑠𝑎_𝑖𝑗(𝜃)𝑒^𝜆𝜃𝑢_𝑖𝑗(𝜃)(𝑡)𝑑𝜃𝑑𝑠≤𝑀+𝑎⁺_𝑖𝑗𝜂⁺_𝑖𝑗sup_{(]𝜃∈−∞,𝑡}𝑒^𝜆𝜃𝑢_𝑖𝑗(𝜃),∀𝑡≥𝜌,𝑖𝑗∈𝐽. (31) Consequently, in view of (31) and the fact 𝑎⁺_𝑖𝑗𝜂⁺_𝑖𝑗<1(𝑖𝑗∈𝐽), we have 𝑒^𝜆𝑠𝑢_𝑖𝑗(𝑠)≤sup_{(]𝜃∈−∞,𝑡}𝑒^𝜆𝜃𝑢_𝑖𝑗(𝜃≤𝑀(𝑡))1−𝑎⁺_𝑖𝑗𝜂⁺_𝑖𝑗,where𝑠∈(−∞,𝑡],𝑖𝑗∈𝐽. (32)
Now, we consider two cases.
Case (i). If 𝑀(𝑡)>‖𝑈(𝑡)‖,∀𝑡≥𝑁₀, (33) then, we claim that 𝑀(𝑡)≡𝑀𝑁₀isaconstant,∀𝑡≥𝑁₀. (34) Assume, by way of contradiction, that (34) does not hold. Then, there exists 𝑡₁>𝑁₀, such that 𝑀(𝑡₁)>𝑀(𝑁₀), since ‖𝑈(𝑡)‖≤𝑀𝑁₀,∀𝑡≤𝑁₀. (35) There must exist 𝛽∈(𝑁₀,𝑡₁) such that 𝑈𝛽=𝑀𝑡₁≥𝑀𝛽, (36) which contradicts (33). This contradiction implies that (34) holds. It follows from (32) that there exists 𝑡₂>𝑁₀ such that ‖𝑢(𝑡)‖=max_{𝑖𝑗∈𝐽}𝑢_𝑖𝑗(𝑡)≤max_{𝑖𝑗∈𝐽}𝑒^−𝜆𝑡𝑀(𝑡)1−𝑎⁺_𝑖𝑗𝜂⁺_𝑖𝑗=max_{𝑖𝑗∈𝐽}𝑒^−𝜆𝑡𝑀𝑁₀1−𝑎⁺_𝑖𝑗𝜂⁺_𝑖𝑗<𝜖,∀𝑡≥𝑡₂. (37)
Case (ii). If there is such a point 𝑡₀≥𝑁₀ that 𝑀(𝑡₀)=‖𝑈(𝑡₀)‖, then, in view of (8), (22), (23), (29), (32), (𝑇₁), and (𝑇₂), we get 0≤𝐷⁻𝑈_{(𝑖𝑗)𝑠}(𝑠)_𝑠=𝑡0≤−𝑎_{(𝑖𝑗)𝑡0}𝑡₀^∞₀ℎ_{(𝑖𝑗)𝑡0}(𝑠)𝑑𝑠−𝜆𝑈_{(𝑖𝑗)𝑡0}𝑡₀+𝑎_{(𝑖𝑗)𝑡0}𝑡₀^∞₀ℎ_{(𝑖𝑗)𝑡0}(𝑠)𝑑𝑠−𝜆^∞₀ℎ_{(𝑖𝑗)𝑡0}(𝑠)×^𝑡0_𝑡0−𝑠𝑎⁺_{(𝑖𝑗)𝑡0}𝑒^𝜆𝜃𝑢_{(𝑖𝑗)𝑡0}(𝜃+)𝑑𝜃𝑑𝑠_∞0ℎ_{(𝑖𝑗)𝑡0}(𝑠)𝑎_{(𝑖𝑗)𝑡0}𝑡₀𝑒^𝜆𝑠−𝑎_{(𝑖𝑗)𝑡0}𝑡₀−𝑠×𝑒^{𝜆(𝑡0−𝑠)}𝑢_{(𝑖𝑗)𝑡0}𝑡₀−𝑠𝑑𝑠+_{𝐶𝑘𝑙∈𝑁𝑟(𝑖,𝑗)𝑡0}𝐶^𝑘𝑙_{(𝑖𝑗)𝑡0}𝑡₀𝑒^𝜆𝑡0×𝑓𝑥_𝑘𝑙𝑡₀−𝜏𝑡₀+𝛿−𝑓𝑥_𝑘𝑙𝑡₀−𝜏𝑡₀×𝑥_{(𝑖𝑗)𝑡0}𝑡₀+𝛿+_{𝐶𝑘𝑙∈𝑁𝑟(𝑖,𝑗)𝑡0}𝐶^𝑘𝑙_{(𝑖𝑗)𝑡0}𝑡₀𝑓𝑥_𝑘𝑙𝑡₀−𝜏𝑡₀⋅𝑒^𝜆𝑡0𝑥_{(𝑖𝑗)𝑡0}𝑡₀+𝛿−𝑥_{(𝑖𝑗)𝑡0}𝑡₀+_{𝐶𝑘𝑙∈𝑁𝑞(𝑖,𝑗)𝑡0}𝐵^𝑘𝑙_{(𝑖𝑗)𝑡0}𝑡₀×^∞₀𝐾_{(𝑖𝑗)𝑡0}(𝑢)×𝑒^𝜆𝑡0𝑔𝑥_𝑘𝑙𝑡₀+𝛿−𝑢−𝑔𝑥_𝑘𝑙𝑡₀×𝑥−𝑢𝑑𝑢_{(𝑖𝑗)𝑡0}𝑡₀+𝛿+_{𝐶𝑘𝑙∈𝑁𝑞(𝑖,𝑗)𝑡0}𝐵^𝑘𝑙_{(𝑖𝑗)𝑡0}𝑡₀×^∞₀𝐾_{(𝑖𝑗)𝑡0}(𝑢)𝑔𝑥_𝑘𝑙𝑡₀−𝑢𝑑𝑢⋅𝑒^𝜆𝑡0𝑥_{(𝑖𝑗)𝑡0}𝑡₀+𝛿−𝑥_{(𝑖𝑗)𝑡0}𝑡₀+𝜖_{(𝑖𝑗)𝑡0}𝛿,𝑡₀𝑒^𝜆𝑡0≤−𝑎_{(𝑖𝑗)𝑡0}𝑡₀^∞₀ℎ_{(𝑖𝑗)𝑡0}(𝑠)𝑑𝑠−𝜆𝑀𝑡₀+𝑎_{(𝑖𝑗)𝑡0}𝑡₀^∞₀ℎ_{(𝑖𝑗)𝑡0}(𝑠)𝑑𝑠−𝜆𝑎⁺_{(𝑖𝑗)𝑡0}𝜂⁺_{(𝑖𝑗)𝑡0}×𝑀𝑡₀1−𝑎⁺_{(𝑖𝑗)𝑡0}𝜂⁺_{(𝑖𝑗)𝑡0}+^∞₀ℎ_{(𝑖𝑗)𝑡0}(𝑠)𝑎_{(𝑖𝑗)𝑡0}𝑡₀𝑒^𝜆𝑠−𝑎_{(𝑖𝑗)𝑡0}𝑡₀×𝑀𝑡−𝑠𝑑𝑠₀1−𝑎⁺_{(𝑖𝑗)𝑡0}𝜂⁺_{(𝑖𝑗)𝑡0}+_{𝐶𝑘𝑙∈𝑁𝑟(𝑖,𝑗)𝑡0}𝐶^𝑘𝑙_{(𝑖𝑗)𝑡0}𝑡₀×𝜇_𝑓𝑒^{𝜆𝜏(𝑡0)}𝑒^{𝜆(𝑡0−𝜏(𝑡0))}𝑢_𝑘𝑙𝑡₀−𝜏𝑡₀×11−𝑎⁺_{(𝑖𝑗)𝑡0}𝜂⁺_{(𝑖𝑗)𝑡0}𝐿_{(𝑖𝑗)𝑡0}𝛿_{(𝑖𝑗)𝑡0}⁺+_{𝐶𝑘𝑙∈𝑁𝑟(𝑖,𝑗)𝑡0}𝐶^𝑘𝑙_{(𝑖𝑗)𝑡0}𝑡₀𝑀_𝑓𝑒^𝜆𝑡0𝑢_{(𝑖𝑗)𝑡0}𝑡₀+_{𝐶𝑘𝑙∈𝑁𝑞(𝑖,𝑗)𝑡0}𝐵^𝑘𝑙_{(𝑖𝑗)𝑡0}𝑡₀×^∞₀𝑒^𝜆𝑢𝐾_{(𝑖𝑗)𝑡0}(𝑢)𝜇_𝑔𝑒^{𝜆(𝑡0−𝑢)}𝑢_𝑘𝑙𝑡₀×1−𝑢𝑑𝑢1−𝑎⁺_{(𝑖𝑗)𝑡0}𝜂⁺_{(𝑖𝑗)𝑡0}𝐿_{(𝑖𝑗)𝑡0}𝛿_{(𝑖𝑗)𝑡0}⁺+_{𝐶𝑘𝑙∈𝑁𝑞(𝑖,𝑗)𝑡0}𝐵^𝑘𝑙_{(𝑖𝑗)𝑡0}𝑡₀×^∞₀𝐾_{(𝑖𝑗)𝑡0}(𝑢)𝑑𝑢𝑀_𝑔𝑒^𝜆𝑡0𝑢_{(𝑖𝑗)𝑡0}𝑡₀+𝜖_{(𝑖𝑗)𝑡0}𝛿,𝑡₀𝑒^𝜆𝑡0≤−𝑎_{(𝑖𝑗)𝑡0}𝑡₀^∞₀ℎ_{(𝑖𝑗)𝑡0}(𝑠)×𝑑𝑠−𝜆1−2𝑎⁺_{(𝑖𝑗)𝑡0}𝜂⁺_{(𝑖𝑗)𝑡0}−^∞₀ℎ_{(𝑖𝑗)𝑡0}(𝑠)𝑎_{(𝑖𝑗)𝑡0}𝑡₀𝑒^𝜆𝑠−𝑎_{(𝑖𝑗)𝑡0}𝑡₀×1−𝑠𝑑𝑠1−𝑎⁺_{(𝑖𝑗)𝑡0}𝜂⁺_{(𝑖𝑗)𝑡0}+_{𝐶𝑘𝑙∈𝑁𝑟(𝑖,𝑗)𝑡0}𝐶^𝑘𝑙_{(𝑖𝑗)𝑡0}𝑡₀𝜇_𝑓𝑒^{𝜆𝜏(𝑡0)}×11−𝑎⁺_𝑘𝑙𝜂⁺_𝑘𝑙11−𝑎⁺_{(𝑖𝑗)𝑡0}𝜂⁺_{(𝑖𝑗)𝑡0}𝐿_{(𝑖𝑗)𝑡0}𝛿_{(𝑖𝑗)𝑡0}⁺+_{𝐶𝑘𝑙∈𝑁𝑟(𝑖,𝑗)𝑡0}𝐶^𝑘𝑙_{(𝑖𝑗)𝑡0}𝑡₀𝑀_𝑓11−𝑎⁺_{(𝑖𝑗)𝑡0}𝜂⁺_{(𝑖𝑗)𝑡0}+_{𝐶𝑘𝑙∈𝑁𝑞(𝑖,𝑗)𝑡0}𝐵^𝑘𝑙_{(𝑖𝑗)𝑡0}𝑡₀^∞₀𝑒^𝜆𝑢𝐾_{(𝑖𝑗)𝑡0}(𝑢)𝑑𝑢×𝜇_𝑔11−𝑎⁺_𝑘𝑙𝜂⁺_𝑘𝑙11−𝑎⁺_{(𝑖𝑗)𝑡0}𝜂⁺_{(𝑖𝑗)𝑡0}𝐿_{(𝑖𝑗)𝑡0}𝛿_{(𝑖𝑗)𝑡0}⁺+_{𝐶𝑘𝑙∈𝑁𝑞(𝑖,𝑗)𝑡0}𝐵^𝑘𝑙_{(𝑖𝑗)𝑡0}𝑡₀×^∞₀𝐾_{(𝑖𝑗)𝑡0}(𝑢)𝑑𝑢𝑀_𝑔11−𝑎⁺_{(𝑖𝑗)𝑡0}𝜂⁺_{(𝑖𝑗)𝑡0}𝑀𝑡₀+𝜂min_{𝑖𝑗∈𝐽}1−𝑎⁺_𝑖𝑗𝜂⁺_𝑖𝑗𝜖𝑒^𝜆𝑡0<−𝜂𝑀𝑡₀+𝜂min_{𝑖𝑗∈𝐽}1−𝑎⁺_𝑖𝑗𝜂⁺_𝑖𝑗𝜖𝑒^𝜆𝑡0, (38) which yields that 𝑈𝑡₀=𝑀𝑡₀<min_{𝑖𝑗∈𝐽}1−𝑎⁺_𝑖𝑗𝜂⁺_𝑖𝑗𝜖𝑒^𝜆𝑡0,𝑢𝑡₀≤max_{𝑖𝑗∈𝐽}𝑒^−𝜆𝑡0𝑀𝑡₀1−𝑎⁺_𝑖𝑗𝜂⁺_𝑖𝑗<𝜖. (39)
For any 𝑡>𝑡₀, by the same approach used in the proof of (39), we have ‖𝑈(𝑡)‖=𝑀(𝑡)<min_{𝑖𝑗∈𝐽}1−𝑎⁺_𝑖𝑗𝜂⁺_𝑖𝑗𝜖𝑒^𝜆𝑡,‖𝑢(𝑡(𝑡)=‖𝑈(𝑡.)‖<𝜖,if𝑀)‖ (40)
On the other hand, if 𝑀(𝑡)>‖𝑈(𝑡)‖ and 𝑡>𝑡₀, we can choose 𝑡₀≤𝑡₃<𝑡 such that 𝑀𝑡₃=𝑈𝑡₃,𝑀(𝑠)>‖𝑈(𝑠)‖,∀𝑠∈𝑡₃,𝑡, (41) which, together with (40), yields that 𝑀𝑡₃=𝑈𝑡₃<min_{𝑖𝑗∈𝐽}1−𝑎⁺_𝑖𝑗𝜂⁺_𝑖𝑗𝜖𝑒^𝜆𝑡3,𝑢𝑡₃<𝜖. (42) Using a similar argument as in the proof of Case (i), we can show that 𝑀(𝑠)≡𝑀𝑡₃isaconstant,∀𝑠∈𝑡₃,𝑡, (43) which implies that ‖𝑢(𝑡)‖≤max_{𝑖𝑗∈𝐽}𝑒^−𝜆𝑡𝑀(𝑡)1−𝑎⁺_𝑖𝑗𝜂⁺_𝑖𝑗=max_{𝑖𝑗∈𝐽}𝑒^−𝜆𝑡𝑀𝑡₃1−𝑎⁺_𝑖𝑗𝜂⁺_𝑖𝑗<max_{𝑖𝑗∈𝐽}𝑒^−𝜆𝑡min_{𝑖𝑗∈𝐽}1−𝑎⁺_𝑖𝑗𝜂⁺_𝑖𝑗𝜖𝑒^𝜆𝑡31−𝑎⁺_𝑖𝑗𝜂⁺_𝑖𝑗<𝜖. (44)
In summary, there must exist 𝑁>max{𝑡₀,𝑁₀,𝑡₂} such that ‖𝑢(𝑡)‖≤𝜖 holds, for all 𝑡>𝑁. The proof of Lemma 4 is now complete.

3. Main Results

In this section, we establish some results for the existence, uniqueness, and exponential stability of the almost periodic solution of (3).

Theorem 5. Suppose that (𝑇₁) and (𝑇₂) are satisfied. Then system (3) has exactly one almost periodic solution 𝑍^∗(𝑡). Moreover, 𝑍^∗(𝑡) is globally exponentially stable.

Proof. Let 𝑣(𝑡)={𝑣_𝑖𝑗(𝑡)} be a solution of system (3) with initial function 𝜑^𝑣_𝑖𝑗(⋅) satisfying (11), and (𝜑^𝑣_𝑖𝑗(⋅))^′ is bounded continuous on (−∞,0].
Set 𝜖_{𝑖𝑗,𝑘}(𝑡)=−𝑎_𝑖𝑗𝑡+𝑡_𝑘−𝑎_𝑖𝑗(𝑡×_∞0)ℎ_𝑖𝑗(𝑠)𝑣_𝑖𝑗𝑡+𝑡_𝑘−𝑠𝑑𝑠−_{𝐶𝑘𝑙∈𝑁𝑟𝑖,𝑗}𝐶^𝑘𝑙_𝑖𝑗𝑡+𝑡_𝑘−𝐶^𝑘𝑙_𝑖𝑗(𝑡×𝑓𝑥_𝑘𝑙)𝑡−𝜏𝑡+𝑡_𝑘+𝑡_𝑘𝑥_𝑖𝑗𝑡+𝑡_𝑘−_{𝐶𝑘𝑙∈𝑁𝑟𝑖,𝑗}𝐶^𝑘𝑙_𝑖𝑗(𝑡)×𝑓𝑥_𝑘𝑙𝑡−𝜏𝑡+𝑡_𝑘+𝑡_𝑘−𝑓𝑥_𝑘𝑙𝑡−𝜏(𝑡)+𝑡_𝑘×𝑥_𝑖𝑗𝑡+𝑡_𝑘−_{𝐶𝑘𝑙∈𝑁𝑞𝑖,𝑗}𝐵^𝑘𝑙_𝑖𝑗𝑡+𝑡_𝑘−𝐵^𝑘𝑙_𝑖𝑗(𝑡×_∞0)𝐾_𝑖𝑗(𝑢)𝑔𝑥_𝑘𝑙𝑡+𝑡_𝑘−𝑢𝑑𝑢𝑥_𝑖𝑗𝑡+𝑡_𝑘+𝐿_𝑖𝑗𝑡+𝑡_𝑘−𝐿_𝑖𝑗(𝑡),𝑖𝑗∈𝐽, (45) where {𝑡_𝑘} is any sequence of real numbers. By Lemma 2, the solution 𝑣(𝑡) is bounded and 𝑣_𝑖𝑗(𝑡≤1)1−𝑎⁺_𝑖𝑗𝜂⁺_𝑖𝑗𝐿_𝑖𝑗𝛿_𝑖𝑗⁺,∀𝑡∈𝑅,𝑖𝑗∈𝐽, (46) which implies that the right side of (3) is also bounded, and 𝑣^′(𝑡) is a bounded function on 𝑅. Thus, 𝑣(𝑡) is uniformly continuous on 𝑅. Then, from the almost periodicity of 𝑎_𝑖𝑗,𝜏,𝐶^𝑘𝑙_𝑖𝑗, and 𝐵^𝑘𝑙_𝑖𝑗, we can select a sequence {𝑡_𝑘}→+∞ such that 𝑎_𝑖𝑗𝑡+𝑡_𝑘−𝑎_𝑖𝑗(𝑡≤1)𝑘,𝐶^𝑘𝑙_𝑖𝑗𝑡+𝑡_𝑘−𝐶^𝑘𝑙_𝑖𝑗(𝑡≤1)𝑘,𝐵^𝑘𝑙_𝑖𝑗𝑡+𝑡_𝑘−𝐵^𝑘𝑙_𝑖𝑗(𝑡≤1)𝑘,𝜏𝑡+𝑡_𝑘(𝑡≤1−𝜏)𝑘,𝜖_{𝑖𝑗,𝑘}(𝑡≤1)𝑘, (47) for all 𝑖𝑗,𝑘𝑙∈𝐽, 𝑡∈𝑅.
Since {𝑣(𝑡+𝑡_𝑘)}^+∞_𝑘=1 is uniformly bounded and equiuniformly continuous, by the Arzala-Ascoli Lemma and diagonal selection principle, we can choose a subsequence {𝑡_𝑘𝑗} of {𝑡_𝑘}, such that 𝑣(𝑡+𝑡_𝑘𝑗)  (for convenience, we still denote by 𝑣(𝑡+𝑡_𝑘)) uniformly converges to a continuous function 𝑍^∗(𝑡)={𝑥^∗_𝑖𝑗(𝑡)} on any compact set of 𝑅, and 𝑥^∗_𝑖𝑗(𝑡≤1)1−𝑎⁺_𝑖𝑗𝜂⁺_𝑖𝑗𝐿_𝑖𝑗𝛿_𝑖𝑗⁺,∀𝑡∈𝑅,𝑖𝑗∈𝐽. (48)
Now, we prove that 𝑍^∗(𝑡) is a solution of (3). In fact, for any 𝑡>0 and Δ𝑡∈𝑅, from (47), we have 𝑥^∗_𝑖𝑗(𝑡+Δ𝑡)−𝑥^∗_𝑖𝑗(𝑡)=lim_𝑘→+∞𝑣_𝑖𝑗𝑡+Δ𝑡+𝑡_𝑘−𝑣_𝑖𝑗𝑡+𝑡_𝑘=lim_{𝑘→+∞𝑡}^𝑡+Δ𝑡−𝑎_𝑖𝑗𝜇+𝑡_𝑘×^∞₀ℎ_𝑖𝑗(𝑠)𝑣_𝑖𝑗𝜇+𝑡_𝑘−−𝑠𝑑𝑠_{𝐶𝑘𝑙∈𝑁𝑟(𝑖,𝑗)}𝐶^𝑘𝑙_𝑖𝑗𝜇+𝑡_𝑘×𝑓𝑣_𝑘𝑙𝜇+𝑡_𝑘−𝜏𝜇+𝑡_𝑘×𝑣_𝑖𝑗𝜇+𝑡_𝑘−_{𝐶𝑘𝑙∈𝑁𝑞(𝑖,𝑗)}𝐵^𝑘𝑙_𝑖𝑗𝜇+𝑡_𝑘×^∞₀𝐾_𝑖𝑗(𝑢)𝑔×𝑣_𝑘𝑙𝜇+𝑡_𝑘−𝑢𝑑𝑢×𝑣_𝑖𝑗𝜇+𝑡_𝑘+𝐿_𝑖𝑗𝜇+𝑡_𝑘𝑑𝜇=lim_{𝑘→+∞𝑡}^𝑡+Δ𝑡−𝑎_𝑖𝑗𝜇^∞₀ℎ_𝑖𝑗(𝑠)𝑣_𝑖𝑗𝜇+𝑡_𝑘−−𝑠𝑑𝑠_{𝐶𝑘𝑙∈𝑁𝑟𝑖,𝑗}𝐶^𝑘𝑙_𝑖𝑗𝜇𝑓×𝑣_𝑘𝑙𝜇−𝜏𝜇+𝑡_𝑘×𝑣_𝑖𝑗𝜇+𝑡_𝑘−_{𝐶𝑘𝑙∈𝑁𝑞(𝑖,𝑗)}𝐵^𝑘𝑙_𝑖𝑗𝜇×^∞₀𝐾_𝑖𝑗(𝑢)𝑔𝑣_𝑘𝑙𝜇+𝑡_𝑘−𝑢×𝑑𝑢𝑣_𝑖𝑗𝜇+𝑡_𝑘+𝐿_𝑖𝑗𝜇+𝜖_{𝑖𝑗,𝑘}𝜇=𝑑𝜇_{𝑡𝑡+Δ𝑡}−𝑎_𝑖𝑗𝜇^∞₀ℎ_𝑖𝑗(𝑠)𝑥^∗_𝑖𝑗𝜇−𝑠𝑑𝑠−_{𝐶𝑘𝑙∈𝑁𝑟𝑖,𝑗}𝐶^𝑘𝑙_𝑖𝑗𝜇𝑓𝑥^∗_𝑘𝑙𝜇−𝜏𝜇𝑥^∗_𝑖𝑗𝜇−_{𝐶𝑘𝑙∈𝑁𝑞(𝑖,𝑗)}𝐵^𝑘𝑙_𝑖𝑗𝜇×^∞₀𝐾_𝑖𝑗(𝑢)𝑔𝑥^∗_𝑘𝑙𝜇−𝑢𝑑𝑢𝑥^∗_𝑖𝑗𝜇+𝐿_𝑖𝑗𝜇𝑑𝜇+lim_{𝑘→+∞𝑡}^𝑡+Δ𝑡𝜖_{𝑖𝑗,𝑘}𝜇=𝑑𝜇_{𝑡𝑡+Δ𝑡}−𝑎_𝑖𝑗𝜇^∞₀ℎ_𝑖𝑗(𝑠)𝑥^∗_𝑖𝑗𝜇−𝑠𝑑𝑠−_{𝐶𝑘𝑙∈𝑁𝑟𝑖,𝑗}𝐶^𝑘𝑙_𝑖𝑗𝜇𝑓𝑥^∗_𝑘𝑙𝜇−𝜏𝜇𝑥^∗_𝑖𝑗𝜇−_{𝐶𝑘𝑙∈𝑁𝑞(𝑖,𝑗)}𝐵^𝑘𝑙_𝑖𝑗𝜇×^∞₀𝐾_𝑖𝑗(𝑢)𝑔𝑥^∗_𝑘𝑙𝜇−𝑢𝑑𝑢𝑥^∗_𝑖𝑗𝜇+𝐿_𝑖𝑗𝜇𝑑𝜇, (49) which implies that 𝑑𝑥𝑑𝑡^∗𝑖𝑗(𝑡)=−𝑎_𝑖𝑗(𝑡)^∞₀ℎ_𝑖𝑗(𝑠)𝑥^∗_𝑖𝑗()−𝑡−𝑠𝑑𝑠_{𝐶𝑘𝑙∈𝑁𝑟𝑖,𝑗}𝐶^𝑘𝑙_𝑖𝑗(𝑡)𝑓𝑥^∗_𝑘𝑙(𝑡−𝜏(𝑡))𝑥^∗_𝑖𝑗(𝑡)−_{𝐶𝑘𝑙∈𝑁𝑞(𝑖,𝑗)}𝐵^𝑘𝑙_𝑖𝑗(𝑡)×^∞₀𝐾_𝑖𝑗(𝑢)𝑔𝑥^∗_𝑘𝑙(𝑡−𝑢)𝑑𝑢𝑥^∗_𝑖𝑗(𝑡)+𝐿_𝑖𝑗(𝑡),𝑖𝑗∈𝐽. (50) Therefore, 𝑍^∗(𝑡) is a solution of (3).
Secondly, we prove that 𝑍^∗(𝑡) is an almost periodic solution of (3). From Lemma 4, for any 𝜀>0, there exists 𝑙=𝑙(𝜀)>0, such that every interval [𝛼,𝛼+𝑙] contains at least one number 𝛿 for which there exists 𝑁>0 which satisfies 𝑣_𝑖𝑗(𝑡+𝛿)−𝑣_𝑖𝑗(𝑡)≤𝜀,∀𝑡>𝑁,𝑖𝑗∈𝐽. (51) Then, for any fixed 𝑠∈𝑅, we can find a sufficiently large positive integer 𝑁₁>𝑁 such that, for any 𝑘>𝑁₁, 𝑠+𝑡_𝑘𝑣>𝑁,_𝑖𝑗𝑠+𝑡_𝑘+𝛿−𝑣_𝑖𝑗𝑠+𝑡_𝑘≤𝜀,𝑖𝑗∈𝐽. (52) Let 𝑘→+∞; we obtain 𝑥^∗_𝑖𝑗(𝑠+𝛿)−𝑥^∗_𝑖𝑗(𝑠)≤𝜀,𝑖𝑗∈𝐽, (53) which implies that 𝑍^∗(𝑡) is an almost periodic solution of (3).
Finally, we prove that 𝑍^∗(𝑡) is globally exponentially stable.
Let 𝑍^∗(𝑡)={𝑥^∗_𝑖𝑗(𝑡)} be the positive almost periodic solution of system (3) with initial value 𝜑^∗={𝜑^∗_𝑖𝑗(𝑡)} and 𝑍(𝑡)={𝑥_𝑖𝑗(𝑡)} an arbitrary solution of system (3) with initial value 𝜑={𝜑_𝑖𝑗(𝑡)}, and set 𝑦(𝑡)={𝑦_𝑖𝑗(𝑡)}={𝑥_𝑖𝑗(𝑡)−𝑥^∗_𝑖𝑗(𝑡)}=𝑍(𝑡)−𝑍^∗(𝑡). Then 𝑦^′_𝑖𝑗(𝑡)=−𝑎_𝑖𝑗(𝑡)^∞₀ℎ_𝑖𝑗(𝑠)𝑦_𝑖𝑗(𝑡−𝑠)𝑑𝑠−_{𝐶𝑘𝑙∈𝑁𝑟(𝑖,𝑗)}𝐶^𝑘𝑙_𝑖𝑗(𝑡)×𝑓𝑥_𝑘𝑙(𝑡−𝜏(𝑡))𝑥_𝑖𝑗(𝑡)−𝑓𝑥^∗_𝑘𝑙(𝑡−𝜏(𝑡))𝑥^∗_𝑖𝑗(𝑡)−_{𝐶𝑘𝑙∈𝑁𝑞(𝑖,𝑗)}𝐵^𝑘𝑙_𝑖𝑗(𝑡)×^∞₀𝐾_𝑖𝑗(𝑢)𝑔𝑥_𝑘𝑙(𝑡−𝑢)𝑑𝑢𝑥_𝑖𝑗(𝑡)−^∞₀𝐾_𝑖𝑗(𝑢)𝑔𝑥^∗_𝑘𝑙(𝑡−𝑢)𝑑𝑢𝑥^∗_𝑖𝑗(𝑡), (54) which yields 𝑑𝑒𝑑𝑡^𝜆𝑡𝑦_𝑖𝑗(𝑡)−^∞₀ℎ_𝑖𝑗(𝑠)^𝑡_𝑡−𝑠𝑎_𝑖𝑗(𝜃)𝑒^𝜆𝜃𝑦_𝑖𝑗(𝜃)𝑑𝜃𝑑𝑠=𝜆𝑒^𝜆𝑡𝑦_𝑖𝑗(𝑡)+𝑒^𝜆𝑡𝑦^′_𝑖𝑗(𝑡)−𝑎_𝑖𝑗(𝑡)^∞₀ℎ_𝑖𝑗(𝑠)𝑑𝑠𝑒^𝜆𝑡𝑦_𝑖𝑗(𝑡)+^∞₀ℎ_𝑖𝑗(𝑠)𝑎_𝑖𝑗(𝑡−𝑠)𝑒^{𝜆(𝑡−𝑠)}𝑦_𝑖𝑗()𝑎𝑡−𝑠𝑑𝑠=−_𝑖𝑗(𝑡)^∞₀ℎ_𝑖𝑗(𝑠)×𝑒𝑑𝑠−𝜆_𝜆𝑡𝑦_𝑖𝑗(𝑡)−^∞₀ℎ_𝑖𝑗(𝑠)^𝑡_𝑡−𝑠𝑎_𝑖𝑗(𝜃)𝑒^𝜆𝜃𝑦_𝑖𝑗(𝜃)−𝑎𝑑𝜃𝑑𝑠_𝑖𝑗(𝑡)^∞₀ℎ_𝑖𝑗(𝑠)𝑑𝑠−𝜆^∞₀ℎ_𝑖𝑗(𝑠)×^𝑡_𝑡−𝑠𝑎_𝑖𝑗(𝜃)𝑒^𝜆𝜃𝑦_𝑖𝑗(𝜃)−𝑑𝜃𝑑𝑠_∞0ℎ_𝑖𝑗(𝑠)𝑎_𝑖𝑗(𝑡)𝑒^𝜆𝑠−𝑎_𝑖𝑗(𝑡−𝑠)×𝑒^{𝜆(𝑡−𝑠)}𝑦_𝑖𝑗(𝑡−𝑠)𝑑𝑠+𝑒^𝜆𝑡−_{𝐶𝑘𝑙∈𝑁𝑟(𝑖,𝑗)}𝐶^𝑘𝑙_𝑖𝑗(𝑡)×𝑓𝑥_𝑘𝑙(𝑡−𝜏(𝑡))−𝑓𝑥^∗_𝑘𝑙(𝑡−𝜏(𝑡))𝑥_𝑖𝑗(𝑡)−_{𝐶𝑘𝑙∈𝑁𝑟𝑖,𝑗}𝐶^𝑘𝑙_𝑖𝑗(𝑡)𝑓𝑥^∗_𝑘𝑙((𝑡×𝑥𝑡−𝜏))_𝑖𝑗(𝑡)−𝑥^∗_𝑖𝑗(𝑡)−_{𝐶𝑘𝑙∈𝑁𝑞(𝑖,𝑗)}𝐵^𝑘𝑙_𝑖𝑗(𝑡)^∞₀𝐾_𝑖𝑗(𝑢)×𝑔𝑥_𝑘𝑙(𝑡−𝑢)−𝑔𝑥^∗_𝑘𝑙(𝑡−𝑢)𝑑𝑢𝑥_𝑖𝑗(𝑡)−_{𝐶𝑘𝑙∈𝑁𝑞(𝑖,𝑗)}𝐵^𝑘𝑙_𝑖𝑗(𝑡)^∞₀𝐾_𝑖𝑗(𝑢)𝑔×𝑥^∗_𝑘𝑙(𝑡−𝑢)𝑑𝑢𝑥_𝑖𝑗(𝑡)−𝑥^∗_𝑖𝑗(𝑡), (55) where 𝑖𝑗∈𝐽.
Let 𝑌_𝑖𝑗(𝑡)=𝑒^𝜆𝑡𝑦_𝑖𝑗(𝑡)−^∞₀ℎ_𝑖𝑗(𝑠)_{𝑡𝑡−𝑠}𝑎_𝑖𝑗(𝜃)𝑒^𝜆𝜃𝑦_𝑖𝑗(𝜃)𝑑𝜃𝑑𝑠,𝑖𝑗∈𝐽. (56)
We define a positive constant 𝑀 as follows: 𝑀=max_{𝑖𝑗∈𝐽}sup_{(]𝑠∈−∞,0}𝑌_𝑖𝑗(𝑠). (57) Let 𝐾 be a positive number such that 𝑌_𝑖𝑗(𝑡)≤𝑀<𝑀+1=𝐾,∀𝑡∈(−∞,0],𝑖𝑗∈𝐽. (58) We claim that 𝑌_𝑖𝑗(𝑡)<𝐾,∀𝑡>0,𝑖=1,2,…,𝑛. (59) Otherwise, there must exist 𝑖𝑗∈𝐽 and 𝜍>0 such that 𝑌_𝑖𝑗(𝜍)=𝐾,𝑌_̃𝑖𝑗(𝑡)<𝐾,∀𝑡∈(−∞,𝜍),𝑖𝑗̃∈𝐽. (60) It follows that 𝑒^𝜆𝑡𝑦_̃𝑖𝑗(𝑡)≤𝑒^𝜆𝑡𝑦_̃𝑖𝑗(𝑡)−^∞₀ℎ_̃𝑖𝑗(𝑠)×^𝑡_𝑡−𝑠𝑎_̃𝑖𝑗(𝜃)𝑒^𝜆𝜃𝑦_̃𝑖𝑗(𝜃)+𝑑𝜃𝑑𝑠_∞0ℎ_̃𝑖𝑗(𝑠)^𝑡_𝑡−𝑠𝑎_̃𝑖𝑗(𝜃)𝑒^𝜆𝜃𝑦_̃𝑖𝑗(𝜃)𝑑𝜃𝑑𝑠≤𝐾+𝑎⁺_̃𝑖𝑗𝜂⁺_̃𝑖𝑗sup_{(]𝑠∈−∞,𝜍}𝑒^𝜆𝑠𝑦_̃𝑖𝑗(𝑠,(],̃)∀𝑡∈−∞,𝜍𝑖𝑗∈𝐽. (61) Consequently, in view of (61) and the fact 𝑎⁺_̃𝑖𝑗𝜂⁺_̃𝑖𝑗<1(𝑖𝑗̃∈𝐽), we have 𝑒^𝜆𝑡𝑦_̃𝑖𝑗(𝑡)≤sup_{(]𝑠∈−∞,𝜍}𝑒^𝜆𝑠𝑦_̃𝑖𝑗(𝑠≤𝐾)1−𝑎⁺_̃𝑖𝑗𝜂⁺_̃𝑖𝑗,∀𝑡∈(−∞,𝜍],𝑖𝑗̃∈𝐽. (62) Calculating the upper left derivative of 𝑌_𝑖𝑗(𝑡), together with (13), (55), (60), (62), (𝑇₁), and (𝑇₂), we obtain 0≤𝐷⁻𝑌_𝑖𝑗(𝜍)≤−𝑎_𝑖𝑗(𝜍)^∞₀ℎ_𝑖𝑗(𝑠)𝑑𝑠−𝜆𝑌_𝑖𝑗(𝜍)+−𝑎_𝑖𝑗(𝜍)^∞₀ℎ_𝑖𝑗(𝑠)×𝑑𝑠−𝜆_∞0ℎ_𝑖𝑗(𝑠)^𝜍_𝜍−𝑠𝑎_𝑖𝑗(𝜃)𝑒^𝜆𝜃𝑦_𝑖𝑗(𝜃)−𝑑𝜃𝑑𝑠_∞0ℎ_𝑖𝑗(𝑠)𝑎_𝑖𝑗(𝜍)𝑒^𝜆𝑠−𝑎_𝑖𝑗(𝜍−𝑠)×𝑒^{𝜆(𝜍−𝑠)}𝑦_𝑖𝑗(𝜍−𝑠)𝑑𝑠+𝑒^𝜆𝜍−_{𝐶𝑘𝑙∈𝑁𝑟𝑖,𝑗}𝐶^𝑘𝑙_𝑖𝑗(𝜍)×𝑓𝑥_𝑘𝑙(𝜍−𝜏(𝜍))−𝑓𝑥^∗_𝑘𝑙(𝜍−𝜏(𝜍))×𝑥_𝑖𝑗(𝜍)−_{𝐶𝑘𝑙∈𝑁𝑟𝑖,𝑗}𝐶^𝑘𝑙_𝑖𝑗(𝜍)𝑓𝑥^∗_𝑘𝑙((𝜍×𝑥𝜍−𝜏))_𝑖𝑗(𝜍)−𝑥^∗_𝑖𝑗(𝜍−_{𝐶𝑘𝑙∈𝑁𝑞𝑖,𝑗})𝐵^𝑘𝑙_𝑖𝑗(𝜍)×^∞₀𝐾_𝑖𝑗(𝑢)×𝑔𝑥_𝑘𝑙(𝜍−𝑢)−𝑔𝑥^∗_𝑘𝑙(𝜍−𝑢)×𝑑𝑢𝑥_𝑖𝑗(𝜍)−_{𝐶𝑘𝑙∈𝑁𝑞𝑖,𝑗}𝐵^𝑘𝑙_𝑖𝑗(𝜍)×^∞₀𝐾_𝑖𝑗(𝑢)𝑔𝑥^∗_𝑘𝑙()×𝑥𝜍−𝑢𝑑𝑢_𝑖𝑗(𝜍)−𝑥^∗_𝑖𝑗(𝜍𝑎)≤−_𝑖𝑗(𝜍)^∞₀ℎ_𝑖𝑗(𝑠)𝑑𝑠−𝜆𝑌_𝑖𝑗(𝜍)+𝑎_𝑖𝑗(𝜍)^∞₀ℎ_𝑖𝑗(𝑠)𝐾𝑑𝑠−𝜆1−𝑎⁺_𝑖𝑗𝜂⁺_𝑖𝑗𝑎⁺_𝑖𝑗𝜂⁺_𝑖𝑗+^∞₀ℎ_𝑖𝑗(𝑠)𝑎_𝑖𝑗(𝜍)𝑒^𝜆𝑠−𝑎_𝑖𝑗(𝜍−𝑠)×𝑒^{𝜆(𝜍−𝑠)}𝑦_𝑖𝑗(𝜍−𝑠)𝑑𝑠+_{𝐶𝑘𝑙∈𝑁𝑟𝑖,𝑗}𝐶^𝑘𝑙_𝑖𝑗(𝜍)𝜇_𝑓𝑒^{𝜆𝜏(𝜍)}𝑒^{𝜆(𝜍−𝜏(𝜍))}×𝑦_𝑘𝑙(𝜍−𝜏(𝜍))𝑥_𝑖𝑗(𝜍+_{𝐶𝑘𝑙∈𝑁𝑟𝑖,𝑗})𝐶^𝑘𝑙_𝑖𝑗(𝜍)𝑀_𝑓𝑒^𝜆𝜍𝑦_𝑖𝑗(𝜍)+_{𝐶𝑘𝑙∈𝑁𝑞𝑖,𝑗}𝐵^𝑘𝑙_𝑖𝑗(𝜍)^∞₀𝑒^𝜆𝑢×𝐾_𝑖𝑗(𝑢)𝜇_𝑔𝑒^{𝜆(𝜍−𝑢)}𝑦_𝑘𝑙(𝜍−𝑢)𝑑𝑢𝑥_𝑖𝑗(𝜍+_{𝐶𝑘𝑙∈𝑁𝑞𝑖,𝑗})𝐵^𝑘𝑙_𝑖𝑗(𝜍×_∞0)𝐾_𝑖𝑗(𝑢)𝑑𝑢×𝑀_𝑔𝑒^𝜆𝜍𝑦_𝑖𝑗(𝜍≤−𝑎_𝑖𝑗)(𝜍)^∞₀ℎ_𝑖𝑗(𝑠)𝑑𝑠−𝜆1−2𝑎⁺_𝑖𝑗𝜂⁺_𝑖𝑗−^∞₀ℎ_𝑖𝑗(𝑠)𝑎_𝑖𝑗(𝜍)𝑒^𝜆𝑠−𝑎_𝑖𝑗(×1𝜍−𝑠)𝑑𝑠1−𝑎⁺_𝑖𝑗𝜂⁺_𝑖𝑗+_{𝐶𝑘𝑙∈𝑁𝑟𝑖,𝑗}𝐶^𝑘𝑙_𝑖𝑗(𝜍)𝜇_𝑓𝑒^{𝜆𝜏(𝜍)}×11−𝑎⁺_𝑘𝑙𝜂⁺_𝑘𝑙11−𝑎⁺_𝑖𝑗𝜂⁺_𝑖𝑗𝐿_𝑖𝑗𝛿_𝑖𝑗⁺+_{𝐶𝑘𝑙∈𝑁𝑟𝑖,𝑗}𝐶^𝑘𝑙_𝑖𝑗(𝜍)𝑀_𝑓11−𝑎⁺_𝑖𝑗𝜂⁺_𝑖𝑗+_{𝐶𝑘𝑙∈𝑁𝑞𝑖,𝑗}𝐵^𝑘𝑙_𝑖𝑗(𝜍)^∞₀𝑒^𝜆𝑢𝐾_𝑖𝑗(𝑢)𝑑𝑢×𝜇_𝑔11−𝑎⁺_𝑘𝑙𝜂⁺_𝑘𝑙11−𝑎⁺_𝑖𝑗𝜂⁺_𝑖𝑗𝐿_𝑖𝑗𝛿_𝑖𝑗⁺+_{𝐶𝑘𝑙∈𝑁𝑞𝑖,𝑗}𝐵^𝑘𝑙_𝑖𝑗(𝜍)^∞₀𝐾_𝑖𝑗(𝑢)𝑑𝑢×𝑀_𝑔11−𝑎⁺_𝑖𝑗𝜂⁺_𝑖𝑗𝐾<−𝜂𝐾<0, (63) which is a contradiction and implies that (59) holds.
Consequently, using a similar argument as in (61)-(62), we know that 𝑦_𝑖𝑗(𝑡)𝑒^𝜆𝑡≤𝐾1−𝑎⁺_𝑖𝑗𝜂⁺_𝑖𝑗,∀𝑡∈𝑅,𝑖𝑗∈𝐽. (64) This completes the proof.

4. An Example

In this section, we give an example to demonstrate the results obtained in the previous sections.

Example 6. Consider the following SICNNs with continuously distributed delays in the leakage terms: 𝑑𝑥_𝑖𝑗𝑑𝑡=−𝑎_𝑖𝑗(𝑡)^∞₀ℎ_𝑖𝑗(𝑠)𝑥_𝑖𝑗(𝑡−𝑠)𝑑𝑠−_{𝐶𝑘𝑙∈𝑁𝑟𝑖,𝑗}𝐶^𝑘𝑙_𝑖𝑗(𝑡)𝑓𝑥_𝑘𝑙(𝑡−𝜏(𝑡))𝑥_𝑖𝑗(𝑡)−_{𝐶𝑘𝑙∈𝑁𝑞𝑖,𝑗}𝐵^𝑘𝑙_𝑖𝑗(𝑡)^∞₀𝐾_𝑖𝑗(𝑢)𝑔𝑥_𝑘𝑙(𝑡−𝑢)𝑑𝑢𝑥_𝑖𝑗(𝑡)+𝐿_𝑖𝑗(𝑡)𝑎,𝑖,𝑗=1,2,3,₁₁(𝑡)𝑎₁₂(𝑡)𝑎₁₃(𝑡)𝑎₂₁(𝑡)𝑎₂₂(𝑡)𝑎₂₃(𝑡)𝑎₃₁(𝑡)𝑎₃₂(𝑡)𝑎₃₃(𝑡)=,𝐵113313313₁₁(𝑡)𝐵₁₂(𝑡)𝐵₁₃(𝑡)𝐵₂₁(𝑡)𝐵₂₂(𝑡)𝐵₂₃(𝑡)𝐵₃₁(𝑡)𝐵₃₂(𝑡)𝐵₃₃(𝑡)=𝐶₁₁(𝑡)𝐶₁₂(𝑡)𝐶₁₃(𝑡)𝐶₂₁(𝑡)𝐶₂₂(𝑡)𝐶₂₃(𝑡)𝐶₃₁(𝑡)𝐶₃₂(𝑡)𝐶₃₃(𝑡)=,𝐿0.10.20.10.200.20.10.20.1₁₁(𝑡)𝐿₁₂(𝑡)𝐿₁₃(𝑡)𝐿₂₁(𝑡)𝐿₂₂(𝑡)𝐿₂₃(𝑡)𝐿₃₁(𝑡)𝐿₃₂(𝑡)𝐿₃₃(𝑡)×=0.05_{0.7+0.24sin2√2𝑡0.41+0.5cos2𝑡0.74+0.2sin2𝑡0.61+0.2cos2𝑡0.67+0.2sin2𝑡0.75+0.2sin2𝑡0.59+0.4cos4𝑡0.5+0.41sin2𝑡0.76+0.2cos2𝑡}. (65)(66) Set 𝜆=0.001, 𝜂=0.05, 𝑟=𝑞=1, ℎ_𝑖𝑗(𝑠)=(1/5)𝑒^−𝑠2, 𝐾_𝑖𝑗(𝑢)=|sin𝑢|𝑒^−𝑢,𝑖=1,2, and 3.𝑗=1,2, and 3, and 𝑓(𝑥)=𝑔(𝑥)=(1/4000)(|𝑥−1|−|𝑥+1|), 𝜏(𝑡)=(1/100)sin²𝑡; clearly, 𝑀_𝑓=𝑀_𝑔=0.0005, 𝜇_𝑓=𝜇_𝑔=0.0005, ∑_{𝐶𝑘𝑙∈𝑁1(1,1)}𝐶^𝑘𝑙₁₁  =  ∑_{𝐶𝑘𝑙∈𝑁1(1,1)}𝐵^𝑘𝑙₁₁=0.5, ∑_{𝐶𝑘𝑙∈𝑁1(1,2)}𝐶^𝑘𝑙₁₂  =  ∑_{𝐶𝑘𝑙∈𝑁1(1,2)}𝐵^𝑘𝑙₁₂=0.8, ∑_{𝐶𝑘𝑙∈𝑁1(1,3)}𝐶^𝑘𝑙₁₃  =  ∑_{𝐶𝑘𝑙∈𝑁1(1,3)}𝐵^𝑘𝑙₁₃=0.5, ∑_{𝐶𝑘𝑙∈𝑁1(2,1)}𝐶^𝑘𝑙₂₁  =  ∑_{𝐶𝑘𝑙∈𝑁1(2,1)}𝐵^𝑘𝑙₂₁=0.8, ∑_{𝐶𝑘𝑙∈𝑁1(2,2)}𝐶^𝑘𝑙₂₂  =  ∑_{𝐶𝑘𝑙∈𝑁1(2,2)}𝐵^𝑘𝑙₂₂=1.2, ∑_{𝐶𝑘𝑙∈𝑁1(2,3)}𝐶^𝑘𝑙₂₃  =  ∑_{𝐶𝑘𝑙∈𝑁1(2,3)}𝐵^𝑘𝑙₂₃=0.8, ∑_{𝐶𝑘𝑙∈𝑁1(3,1)}𝐶^𝑘𝑙₃₁  =  ∑_{𝐶𝑘𝑙∈𝑁1(3,1)}𝐵^𝑘𝑙₃₁=0.5, ∑_{𝐶𝑘𝑙∈𝑁1(3,2)}𝐶^𝑘𝑙₃₂  =  ∑_{𝐶𝑘𝑙∈𝑁1(3,2)}𝐵^𝑘𝑙₃₂=0.8, ∑_{𝐶𝑘𝑙∈𝑁1(3,3)}𝐶^𝑘𝑙₃₃  =  ∑_{𝐶𝑘𝑙∈𝑁1(3,3)}𝐵^𝑘𝑙₃₃=0.5, 1≤𝑎⁺_𝑖𝑗≤3, 𝜂⁺_𝑖𝑗=0.1, 𝑎⁺_𝑖𝑗𝜂⁺_𝑖𝑗≤0.3<1, and 𝑖𝑗∈𝐽={11,12,13,21,22,23,31,32,33}.
Consider, min_{𝑖𝑗∈𝐽}𝛿_𝑖𝑗(𝑡)=min_{𝑖𝑗∈𝐽}𝑎_𝑖𝑗(𝑡)^∞₀ℎ_𝑖𝑗(𝑠)𝑑𝑠1−2𝑎⁺_𝑖𝑗𝜂⁺_𝑖𝑗−^∞₀ℎ_𝑖𝑗(𝑠)𝑎_𝑖𝑗(𝑡)−𝑎_𝑖𝑗(−𝑡−𝑠)𝑑𝑠_{𝐶𝑘𝑙∈𝑁𝑟𝑖,𝑗}𝐶^𝑘𝑙_𝑖𝑗(𝑡)𝑀_𝑓−_{𝐶𝑘𝑙∈𝑁𝑞𝑖,𝑗}𝐵^𝑘𝑙_𝑖𝑗(𝑡)^∞₀𝐾_𝑖𝑗(𝑢)𝑑𝑢𝑀_𝑔×11−𝑎⁺_𝑖𝑗𝜂⁺_𝑖𝑗>0.0715>0,∀𝑡≥0.max_{𝑖𝑗∈𝐽}−𝑎_𝑖𝑗(𝑡)^∞₀ℎ_𝑖𝑗(𝑠)𝑑𝑠−𝜆1−2𝑎⁺_𝑖𝑗𝜂⁺_𝑖𝑗−^∞₀ℎ_𝑖𝑗(𝑠)𝑎_𝑖𝑗(𝑡)𝑒^𝜆𝑠−𝑎_𝑖𝑗(×1𝑡−𝑠)𝑑𝑠1−𝑎⁺_𝑖𝑗𝜂⁺_𝑖𝑗+_{𝐶𝑘𝑙∈𝑁𝑟𝑖,𝑗}𝐶^𝑘𝑙_𝑖𝑗(𝑡)𝜇_𝑓𝑒^{𝜆𝜏(𝑡)}×11−𝑎⁺_𝑘𝑙𝜂⁺_𝑘𝑙11−𝑎⁺_𝑖𝑗𝜂⁺_𝑖𝑗𝐿_𝑖𝑗𝛿_𝑖𝑗⁺+_{𝐶𝑘𝑙∈𝑁𝑟(𝑖,𝑗)}𝐶^𝑘𝑙_𝑖𝑗(𝑡)𝑀_𝑓11−𝑎⁺_𝑖𝑗𝜂⁺_𝑖𝑗+_{𝐶𝑘𝑙∈𝑁𝑞(𝑖,𝑗)}𝐵^𝑘𝑙_𝑖𝑗(𝑡)^∞₀𝑒^𝜆𝑢𝐾_𝑖𝑗(𝑢)𝑑𝑢×𝜇_𝑔11−𝑎⁺_𝑘𝑙𝜂⁺_𝑘𝑙11−𝑎⁺_𝑖𝑗𝜂⁺_𝑖𝑗𝐿_𝑖𝑗𝛿_𝑖𝑗⁺+_{𝐶𝑘𝑙∈𝑁𝑞(𝑖,𝑗)}𝐵^𝑘𝑙_𝑖𝑗(𝑡)^∞₀𝐾_𝑖𝑗(𝑢)𝑑𝑢×𝑀_𝑔11−𝑎⁺_𝑖𝑗𝜂⁺_𝑖𝑗<−0.0556<−0.05=−𝜂,∀𝑡≥0. (67) It follows that system (65) satisfies all the conditions in Theorem 5. Hence, system (65) has exactly one almost periodic solution. Moreover, the almost periodic solution is globally exponentially stable.

Remark 7. Since [1–11] only dealt with SICNNs without leakage delays, [12–21] give no opinions about the problem of almost periodic solutions for SICNNs with leakage delays. One can observe that all the results in these literature and the references therein can not be applicable to prove the existence and exponential stability of almost periodic solutions for SICNNs (65).

Acknowledgments

The authors would like to express the sincere appreciation to the reviewers for their helpful comments in improving the presentation and quality of the paper. In particular, the authors expresses the sincere gratitude to Professor Bingwen Liu for the helpful discussion when this work is carried out. This work was supported by the National Natural Science Foundation of China (Grant no. 11201184), the Natural Scientific Research Fund of Hunan Provincial of China (Grant no. 11JJ6006), the Natural Scientific Research Fund of Hunan Provincial Education Department of China (Grant nos. 11C0916 and 11C0915), the Natural Scientific Research Fund of Zhejiang Provincial of China (Grants nos. Y6110436 and LY12A01018), and the Natural Scientific Research Fund of Zhejiang Provincial Education Department of China (Grant no. Z201122436).