Global Exponential Stability of a Unique Almost Periodic Solution for Neutral-Type Cohen-Grossberg Neural Networks with Time Delays

Abstract

This paper is concerned with the problem of the existence, uniqueness, and global exponential stability of almost periodic solution for neutral-type Cohen-Grossberg neural networks with time delays. Based on fixed point theory and Lyapunov functional, several sufficient conditions are established for the existence, uniqueness, and global exponential stability of almost periodic solution for the above system. Finally, an example and numerical simulations are given to illustrate the feasibility and effectiveness of our main results.

1. Introduction

In recent years, the Cohen-Grossberg neural networks [1] have been extensively studied because of their immense potentials of application perspective in different areas such as pattern recognition, optimization, and signal and image processing. It is well known that studies on Cohen-Grossberg neural networks not only involve a discussion of stability properties, but also involve many dynamic behaviors such as periodic oscillatory behavior and almost periodic oscillatory properties. In applications, if the various constituent components of the temporally nonuniform environment is with incommensurable (nonintegral multiples) periods, then one has to consider the environment to be almost periodic since there is no a priori reason to expect the existence of periodic solutions. Therefore, if we consider the effects of the environmental factors, almost periodicity is sometimes more realistic and more general than periodicity. In addition, the (almost) periodic oscillatory behavior is of great interest; it has been found that applications in learning theory [2], which is motivated by the fact that learning usually requires repetition. And in pattern recognition or associative memory, the existence of stable almost periodic encoded patterns (including equilibria or periodic orbits) is an important feature [3, 4]. Hence, it is of prime importance to study the existence and stability of (almost) periodic solutions for Cohen-Grossberg neural networks with delays. We refer the reader to [5–18] and references therein.

On the other hand, owing to the complicated dynamic properties of the neural cells in the real world, the existing neural network models in many cases can not characterize the properties of a neural reaction process precisely. It is natural and important that systems will contain some information about the derivative of the past state to further describe and model the dynamics for such complex neural reactions. This new type of neural networks is called neutral neural networks or neural networks of neutral type. The motivation for us to study neural networks of neutral type comes from three aspects. First, based on biochemistry experiments, neural information may transfer across chemical reactivity, which results in a neutral-type process. Second, in view of electronics, it has been shown that neutral phenomena exist in large-scale integrated (LSI) circuits. Last, the key point is that cerebra can be considered as a super LSI circuit with chemical reactivity, which reasonably implies that the neutral dynamic behaviors should be included in neural dynamic systems [19]. To the best of our knowledge, the problem of global exponential stability of almost periodic solution for neutral-type Cohen-Grossberg neural networks (see [20–24]) has not been fully investigated in the literature.

In this papers we consider the following neutral-type Cohen-Grossberg neural networks with time delays: 𝑢̇_𝑖(𝑡)=−𝑎_𝑖𝑢_𝑖(𝑡)×𝑏_𝑖𝑡,𝑢_𝑖(𝑡)−^𝑛_𝑗=1𝑐_𝑖𝑗(𝑡)𝑓_𝑗𝑢_𝑗(𝑡)−^𝑛_𝑗=1𝑑_𝑖𝑗(𝑡)𝑔_𝑗𝑢_𝑗𝑡−𝜇_𝑗(𝑡−_𝑛𝑗=1)𝑣_𝑖𝑗(𝑡)𝑘_𝑗𝑢̇_𝑗𝑡−𝜈_𝑗(𝑡)−𝐼_𝑖(𝑡), (1) where 𝑥_𝑖 denotes the state variable associated to the 𝑖th neuron, 𝑎_𝑖 represents an amplification function, 𝑏_𝑖 is an appropriately behaved function; 𝑓_𝑗, 𝑔_𝑗, and 𝑘_𝑗 denote the normal and the delayed activation function; 𝑐_𝑖𝑗, 𝑑_𝑖𝑗, and 𝑣_𝑖𝑗 denote the connection strengths of the 𝑗th neuron on the 𝑖th neuron at time 𝑡, respectively; 𝜇_𝑗≥0 and 𝜈_𝑗≥0 are the delays caused during the switching and transmission processes; 𝐼_𝑖 denotes external input to the 𝑖th neuron at time 𝑡, 𝑖,𝑗=1,2,…,𝑛.

Let 𝐶(𝕏,𝕐) and 𝐶¹(𝕏,𝕐) be the space of continuous functions and continuously differential functions which map 𝕏 into 𝕐, respectively. In particular, 𝐶(𝕏)∶=𝐶(𝕏,𝕏), 𝐶¹(𝕏)∶=𝐶¹(𝕏,𝕏). For any bounded function, 𝑓∈𝐶(ℝ), 𝑓⁺=sup_𝑠∈ℝ|𝑓(𝑠)|, 𝑓⁻=inf_𝑠∈ℝ|𝑓(𝑠)|.

Throughout this paper, we set 𝑥=𝑥_𝑖=𝑥₁,𝑥₂,…,𝑥_𝑛∈ℝ^𝑛. (2)

For 𝑥={𝑥_𝑖}∈ℝ^𝑛, we define the norm ‖𝑥‖=max_{1≤𝑖≤𝑛}|𝑥_𝑖|.

We list some assumptions which will be used in this paper. (𝐻₁)𝑐_𝑖𝑗, 𝑑_𝑖𝑗, 𝑣_𝑖𝑗, 𝜇_𝑖𝑗, 𝜈_𝑖𝑗, and 𝐼_𝑖𝑗 are continuous almost periodic functions, 𝑖,𝑗=1,2,…,𝑛.(𝐻₂)𝑎_𝑖∈𝐶(ℝ) and there exist positive constants 𝑎_𝑖 and 𝑎_𝑖 such that 0<𝑎_𝑖≤𝑎_𝑖(𝑡)≤𝑎_𝑖, for all 𝑡∈ℝ, 𝑖=1,2,…,𝑛.(𝐻₃)𝑏_𝑖∈𝐶(ℝ²,ℝ) is almost periodic about the first argument, and there are positive constants 𝑝_𝑖 and 𝑞_𝑖 such that 𝑝_𝑖≤(𝜕𝑏_𝑖(𝑡,𝑢)/𝜕𝑢)≤𝑞_𝑖 and 𝑏_𝑖(𝑡,0)=0, for all (𝑡,𝑢)∈(ℝ,ℝ), 𝑖=1,2,…,𝑛.(𝐻₄)There exist positive constants 𝐿^𝑓_𝑗, 𝐿^𝑔_𝑗, 𝐿^𝑘_𝑗, and 𝐿^𝑎_𝑗 such that 𝑓_𝑗(𝑢)−𝑓_𝑗(𝑣)≤𝐿^𝑓_𝑗||,𝑔𝑢−𝑣_𝑗(𝑢)−𝑔_𝑗(𝑣)≤𝐿^𝑔_𝑗||,𝑘𝑢−𝑣_𝑗(𝑢)−𝑘_𝑗(𝑣)≤𝐿^𝑘_𝑗||,𝑎𝑢−𝑣_𝑗(𝑢)−𝑎_𝑗(𝑣)≤𝐿^𝑎_𝑗|𝑢−𝑣|, (3) for all 𝑢,𝑣∈ℝ, 𝑗=1,2,…,𝑛.

The organization of this paper is as follows. In Section 2, we give some basic definitions and necessary lemmas which will be used in later sections. In Sections 3 and 4, by using a fixed point theorem and constructing suitable Lyapunov functional, we obtain some sufficient conditions ensuring the existence, uniqueness, and global exponential stability of almost periodic solution of system (1). Finally, an example and numerical simulations are given to illustrate that our results are feasible.

2. Preliminaries

First of all, we shall transform system (1) and state some notations, which will be used in later sections.

From (𝐻₂), the antiderivative of 1/𝑎_𝑖(𝑢_𝑖) exists. We choose an antiderivative ℎ_𝑖(𝑢_𝑖) of 1/𝑎_𝑖(𝑢_𝑖) that satisfies ℎ_𝑖(0)=0. Obviously, (d/d𝑢_𝑖)ℎ_𝑖(𝑢_𝑖)=1/𝑎_𝑖(𝑢_𝑖). By 𝑎_𝑖(𝑢_𝑖)>0, we obtain that ℎ_𝑖(𝑢_𝑖) is strictly monotone increasing about 𝑢_𝑖. In view of derivative theorem for inverse function, the inverse function ℎ_𝑖⁻¹(𝑢_𝑖) of ℎ_𝑖(𝑢_𝑖) is differential and (d/d𝑢_𝑖)ℎ_𝑖⁻¹(𝑢_𝑖)=𝑎_𝑖(𝑢_𝑖). By (𝐻₃), composition function 𝑏_𝑖(𝑡,ℎ_𝑖⁻¹(𝑧)) is differentiable. Denote 𝑥_𝑖=ℎ_𝑖(𝑢_𝑖). It is easy to see that 𝑥̇_𝑖(𝑡)=𝑢̇_𝑖(𝑡)/𝑎_𝑖(𝑢_𝑖), 𝑢_𝑖(𝑡)=ℎ_𝑖⁻¹(𝑥_𝑖(𝑡)), and 𝑢̇_𝑖(𝑡)=𝑎_𝑖(𝑥_𝑖(𝑡))𝑥̇_𝑖(𝑡). Substituting these equalities into system (1), we get 𝑥̇_𝑖(𝑡)=−𝑏_𝑖𝑡,ℎ_𝑖⁻¹𝑥_𝑖(𝑡+_𝑛𝑗=1)𝑐_𝑖𝑗(𝑡)𝑓_𝑗ℎ_𝑗⁻¹𝑥_𝑗(𝑡)+^𝑛_𝑗=1𝑑_𝑖𝑗(𝑡)𝑔_𝑗ℎ_𝑗⁻¹𝑥_𝑗𝑡−𝜇_𝑗(𝑡)+^𝑛_𝑗=1𝑣_𝑖𝑗(𝑡)𝑘_𝑗𝑎_𝑗𝑥_𝑗𝑡−𝜈_𝑗(𝑡)×𝑥̇_𝑗𝑡−𝜈_𝑗(𝑡)+𝐼_𝑖(𝑡), (4) which can be rewritten as 𝑥̇_𝑖(𝑡)=−𝑤_𝑖𝑡,𝑥_𝑖(𝑡)𝑥_𝑖(𝑡)+^𝑛_𝑗=1𝑐_𝑖𝑗(𝑡)𝑓_𝑗ℎ_𝑗⁻¹𝑥_𝑗(𝑡)+^𝑛_𝑗=1𝑑_𝑖𝑗(𝑡)𝑔_𝑗ℎ_𝑗⁻¹𝑥_𝑗𝑡−𝜇_𝑗(𝑡)+^𝑛_𝑗=1𝑣_𝑖𝑗(𝑡)𝑘_𝑗𝑎_𝑗𝑥_𝑗𝑡−𝜈_𝑗(𝑡×𝑥̇_𝑗)𝑡−𝜈_𝑗(𝑡)+𝐼_𝑖(𝑡), (5) where 𝑤_𝑖(𝑡,𝑥_𝑖(𝑡))=(𝜕𝑏_𝑖(𝑡,ℎ_𝑖⁻¹(𝑧))/𝜕𝑧)|_{𝑧=𝜉𝑖}, 𝜉_𝑖 is between 0 and 𝑥_𝑖, 𝑖=1,2,…,𝑛.

By (𝐻₂), (𝐻₃), and the definition of ℎ_𝑖⁻¹, we obtain that 𝑏_𝑖(𝑡,ℎ_𝑖⁻¹(𝑥_𝑖)) is strictly monotone increasing about 𝑥_𝑖. Hence, 𝑤_𝑖(𝑡,𝑥_𝑖) is unique for any 𝑥_𝑖 and continuous about 𝑥_𝑖; moreover, 𝛼_𝑖=𝑎_𝑖𝑝_𝑖≤𝑤_𝑖𝑡,𝑥_𝑖≤𝑎_𝑖𝑞_𝑖=𝛽_𝑖,𝑖=1,2,…,𝑛. (6)

From the definition of ℎ_𝑖⁻¹, using Lagrange mean-value theorem, one gets ℎ_𝑖⁻¹(𝑢)−ℎ_𝑖⁻¹(𝑣)≤𝑎_𝑖|𝑢−𝑣|,∀𝑢,𝑣∈ℝ,𝑖=1,2,…,𝑛. (7)

The existence and global exponential stability of almost periodic solution for system (1) are equivalent to the existence and global exponential stability of almost periodic solution for system (5), respectively. So, we investigate the existence and global exponential stability of almost periodic solution for system (5).

Let 𝜏∶=max_{1≤𝑖≤𝑛}{𝜇⁺_𝑖,𝜈⁺_𝑖}. The initial conditions associated with system (5) are of the form 𝑥_𝑖(𝑠)=𝜑_𝑖(𝑠),𝑥̇_𝑖(𝑠)=𝜑̇_𝑖(𝑠),∀𝑠∈[−𝜏,0],𝜑_𝑖∈𝐶¹([−𝜏,0],ℝ),𝑖=1,2,…,𝑛. (8)

Now, let us state the following definitions and lemmas, which will be useful in proving our main result.

Definition 1 (see [25]). 𝑥∈𝐶(ℝ,ℝ^𝑛) is called almost periodic, if for any 𝜖>0, it is possible to find a real number 𝑙=𝑙(𝜖)>0, for any interval with length 𝑙(𝜖), there exists a number 𝜏=𝜏(𝜖) in this interval such that ‖𝑥(𝑡+𝜏)−𝑥(𝑡)‖<𝜖, for all 𝑡∈ℝ. The collection of those functions is denoted by 𝐴𝑃(ℝ,ℝ^𝑛).

Definition 2 (see [25]). Let 𝑦∈𝐶(ℝ,ℝ^𝑛) and 𝑃(𝑡) be an 𝑛×𝑛 continuous matrix defined on ℝ. The linear system 𝑦(̇𝑡)=𝑃(𝑡)𝑦(𝑡) (9) is said to be an exponential dichotomy on ℝ if there exist constants 𝑘,𝜆>0, projection 𝑆, and the fundamental matrix 𝑌(𝑡) satisfying 𝑌(𝑡)𝑆𝑌⁻¹(𝑠)≤𝑘𝑒^{−𝜆(𝑡−𝑠)}𝑌(𝑡)𝑌,∀𝑡≥𝑠,)(𝐼−𝑆⁻¹(𝑠)≤𝑘𝑒^{−𝜆(𝑠−𝑡)},∀𝑡≤𝑠. (10)

Lemma 3 (see [25]). If the linear system 𝑦(̇𝑡)=𝑃(𝑡)𝑦(𝑡) has an exponential dichotomy, then almost periodic system 𝑦(̇𝑡)=𝑃(𝑡)𝑦(𝑡)+𝑔(𝑡) (11) has a unique almost periodic solution 𝑦(𝑡) which can be expressed as follows: 𝑦(𝑡)=^𝑡_−∞𝑌(𝑡)𝑆𝑌⁻¹(𝑠)𝑔(𝑠)−d𝑠_∞𝑡𝑌(𝑡)(𝐼−𝑆)𝑌⁻¹(𝑠)𝑔(𝑠)d𝑠. (12)

Lemma 4 (see [25]). Let 𝑎_𝑖(𝑡) be an almost periodic function and 𝑀𝑎_𝑖=lim_𝑇→∞1𝑇_𝑡^𝑡+𝑇𝑎_𝑖(𝑠)d𝑠>0,𝑖=1,2,…,𝑛. (13)
Then the linear system 𝑦(̇𝑡)=−𝐴(𝑡)𝑦(𝑡) admits an exponential dichotomy, where 𝐴(𝑡)=diag{𝑎₁(𝑡),𝑎₂(𝑡),…,𝑎_𝑛(𝑡)}.

Definition 5. The almost periodic solution 𝑥={𝑥_𝑖} of system (5) with the initial value 𝜑={𝜑_𝑖} is said to be globally exponentially stable if there exist constants 𝜔>0 and 𝑀≥1, for any solution 𝑦={𝑦_𝑖} of system (5) with initial value 𝜙={𝜙_𝑖} such that 𝑥−𝑦≤𝑀𝜑−𝜙_𝜏𝑒^−𝜔𝑡,∀𝑡>0, (14) where 𝜑−𝜙_𝜏∶=^𝑛_𝑖=1sup_{𝑠∈[−𝜏,0]}𝜑_𝑖(𝑠)−𝜙_𝑖(𝑠)+𝜑̇_𝑖(𝑠)−𝜙̇_𝑖(𝑠). (15)

Lemma 6 (see [26, 27]). Let (𝕏,‖⋅‖_𝕏) be a Banach space. Assume that 0∈Ω is an open bounded subset of 𝕏 and 𝑇∶Ω→𝕏 is completely continuous satisfying ‖𝑇𝑥‖_𝕏≤‖𝑥‖_𝕏,∀𝑥∈𝜕Ω, (16) then 𝑇 has a fixed point in Ω.

Let 𝕏=𝑥∈𝐴𝑃ℝ,ℝ^𝑛∩𝐶¹ℝ,ℝ^𝑛∶𝑥̇∈𝐴𝑃ℝ,ℝ^𝑛 (17) with the norm ‖𝑥‖_𝕏=max{sup_𝑠∈ℝ‖𝑥(𝑠)‖,sup_𝑠∈ℝ‖𝑥̇(𝑠)‖}. Then 𝕏 is a Banach space with the norm ‖⋅‖_𝕏.

For given positive constant 𝛾, define an open bounded subset Ω_𝛾 in 𝕏 by Ω=𝑥∈𝕏∶‖𝑥‖_𝕏<𝛾. (18)

By Lemmas 3 and 4, system (5) has a unique almost periodic solution 𝑥^𝜑=𝑥^𝜑𝑖_𝑖=𝑥^𝜑1₁,𝑥^𝜑2₂,…,𝑥^𝜑𝑛_𝑛^𝑇 (19) which can be expressed as follows: 𝑥^𝜑𝑖_𝑖=^𝑡_−∞𝑒^{−∫𝑡𝑠𝑤𝑖(𝑟,𝜑𝑖(𝑟))d𝑟}𝐹_𝑖𝑠,𝜑(𝑠)d𝑠, (20) where 𝐹_𝑖𝑡,𝜑(𝑡)=^𝑛_𝑗=1𝑐_𝑖𝑗(𝑡)𝑓_𝑗ℎ_𝑗⁻¹𝜑_𝑗(𝑡+_𝑛𝑗=1)𝑑_𝑖𝑗(𝑡)𝑔_𝑗ℎ_𝑗⁻¹𝜑_𝑗𝑡−𝜇_𝑗(𝑡)+^𝑛_𝑗=1𝑣_𝑖𝑗(𝑡)𝑘_𝑗𝑎_𝑗𝜑_𝑗𝑡−𝜈_𝑗(𝑡)×𝜑̇_𝑗𝑡−𝜈_𝑗(𝑡)+𝐼_𝑖(𝑡), (21) where 𝑖=1,2,…,𝑛, for all 𝑡∈ℝ.

Let the map 𝑇∶𝕏→𝕏 be defined by 𝑇𝜑=𝑇_𝑖𝜑=𝑥^𝜑𝑖_𝑖,∀𝜑=𝜑_𝑖∈𝕏. (22)

Lemma 7. 𝑇∶Ω_𝛾→𝕏 is completely continuous.

Proof. Obviously, 𝑇∶Ω→𝕏 is continuous. Next, we show that 𝑇 maps bounded set into itself. Assume 𝑐 is a positive constant and 𝜑∈Ω_𝑐={𝑥∈Ω∶‖𝑥‖_𝕏≤𝑐}. By the almost periodicity of system (1), there exists a constant 𝐾>0 such that max_{1≤𝑖≤𝑛}sup_{𝑠∈ℝ,𝜑∈Ω𝑐}𝐹_𝑖(𝑠,0),𝐹_𝑖𝑠,𝜑<𝐾. (23)
Assume that 𝜑∈Ω_𝛾, then 𝐹_𝑖𝑡,𝜑(𝑡)−𝐹_𝑖(≤𝑡,0)_𝑛𝑗=1𝑐_𝑖𝑗(𝑡)𝑓_𝑗ℎ_𝑗⁻¹𝜑_𝑗(𝑡)−^𝑛_𝑗=1𝑐_𝑖𝑗(𝑡)𝑓_𝑗ℎ_𝑗⁻¹(0)+^𝑛_𝑗=1𝑑_𝑖𝑗(𝑡)𝑔_𝑗ℎ_𝑗⁻¹𝜑_𝑗𝑡−𝜇_𝑗(𝑡)−^𝑛_𝑗=1𝑑_𝑖𝑗(𝑡)𝑔_𝑗ℎ_𝑗⁻¹(0)+^𝑛_𝑗=1𝑣_𝑖𝑗(𝑡)𝑘_𝑗𝑎_𝑗𝜑_𝑗𝑡−𝜈_𝑗(𝑡)𝜑̇_𝑗𝑡−𝜈_𝑗(𝑡)−^𝑛_𝑗=1𝑣_𝑖𝑗(𝑡)𝑘_𝑗(0)≤^𝑛_𝑗=1𝑐⁺_𝑖𝑗𝐿^𝑓_𝑗𝑎_𝑗+𝑑⁺_𝑖𝑗𝐿^𝑔_𝑗𝑎_𝑗+𝑣⁺_𝑖𝑗𝑀^𝑘_𝑗𝑎_𝑗+𝐿^𝑎_𝑗𝛾𝜑_𝕏, (24) which implies that 𝑇_𝑖𝜑=^𝑡_−∞𝑒^{−∫𝑡𝑠𝑤𝑖(𝑟,𝜑𝑖(𝑟))d𝑟}𝐹_𝑖𝑠,𝜑(𝑠)−𝐹_𝑖(+𝑠,0)d𝑠_𝑡−∞𝑒^{−∫𝑡𝑠𝑤𝑖(𝑟,𝜑𝑖(𝑟))d𝑟}𝐹_𝑖(≤1𝑠,0)d𝑠𝛼_𝑖^𝑛_𝑗=1𝑐⁺_𝑖𝑗𝐿^𝑓_𝑗𝑎_𝑗+𝑑⁺_𝑖𝑗𝐿^𝑔_𝑗𝑎_𝑗+𝑣⁺_𝑖𝑗𝑀^𝑘_𝑗𝑎_𝑗+𝐿^𝑎_𝑗𝛾×𝜑_𝕏+𝐾𝛼_𝑖≤1𝛼_𝑖^𝑛_𝑗=1𝑐⁺_𝑖𝑗𝐿^𝑓_𝑗𝑎_𝑗+𝑑⁺_𝑖𝑗𝐿^𝑔_𝑗𝑎_𝑗+𝑣⁺_𝑖𝑗𝑀^𝑘_𝑗𝑎_𝑗+𝐿^𝑎_𝑗𝛾𝐾×𝑐+𝛼_𝑖,𝑇̇_𝑖𝜑=−𝑤_𝑖𝑡,𝜑_𝑖(𝑡)𝑇_𝑖𝜑+𝐹_𝑖𝑡,𝜑≤𝛽_𝑖1𝛼_𝑖^𝑛_𝑗=1𝑐⁺_𝑖𝑗𝐿^𝑓_𝑗𝑎_𝑗+𝑑⁺_𝑖𝑗𝐿^𝑔_𝑗𝑎_𝑗+𝑣⁺_𝑖𝑗𝑀^𝑘_𝑗𝑎_𝑗+𝐿^𝑎_𝑗𝛾𝐾𝑐+𝛼_𝑖+𝐾, (25) where 𝑖=1,2,…,𝑛. Therefore, 𝑇Ω_𝑐 is a family of uniformly bounded and equicontinuous subsets. Using Arzela-Ascoli theorem, we obtain that 𝑇Ω_𝑐 is relatively compact. Hence, 𝑇 is completely continuous. The proof of this lemma is complete.

3. Existence of Almost Periodic Solution

In this section, we study the existence of almost periodic solutions of system (1).

Theorem 8. Assume that (𝐻₁)–(𝐻₄) hold, suppose further that (𝐻₅)𝜃=max{𝜃₁,𝜃₂}<1, where 𝜃₁=max_{1≤𝑖≤𝑛}1𝛼_𝑖^𝑛_𝑗=1𝑐⁺_𝑖𝑗𝐿^𝑓_𝑗𝑎_𝑗+𝑑⁺_𝑖𝑗𝐿^𝑔_𝑗𝑎_𝑗+𝑣⁺_𝑖𝑗𝐿^𝑘_𝑗𝑎_𝑗,𝜃₂=max_{1≤𝑖≤𝑛}𝛽1+^𝑖𝛼_𝑖^𝑛_𝑗=1𝑐⁺_𝑖𝑗𝐿^𝑓_𝑗𝑎_𝑗+𝑑⁺_𝑖𝑗𝐿^𝑔_𝑗𝑎_𝑗+𝑣⁺_𝑖𝑗𝐿^𝑘_𝑗𝑎_𝑗, (26)then system (1) admits at least one almost periodic solution.

Proof. Define Ω_𝑟0=𝑥∈𝕏∶‖𝑥‖_𝕏<𝑟₀=𝛿1−𝜃, (27) where 𝛿=max{𝛿₁,𝛿₂}, 𝛿₁=max_{1≤𝑖≤𝑛}1𝛼_𝑖^𝑛_𝑗=1𝑐⁺_𝑖𝑗𝑓_𝑗(0)+𝑑⁺_𝑖𝑗𝑔_𝑗(0)+𝑣⁺_𝑖𝑗𝑘_𝑗(0)+𝐼⁺_𝑖𝛼_𝑖,𝛿₂=max_{1≤𝑖≤𝑛}𝛽1+^𝑖𝛼_𝑖^𝑛_𝑗=1𝑐⁺_𝑖𝑗𝑓_𝑗(0)+𝑑⁺_𝑖𝑗𝑔_𝑗(0)+𝑣⁺_𝑖𝑗𝑘_𝑗(0)+𝐼⁺_𝑖. (28)
Consider the following nonlinear operator: 𝑇𝜙=𝑇_𝑖𝜙=𝑥^𝜙𝑖_𝑖,∀𝜙=𝜙_𝑖∈Ω_𝑟0. (29)
For ∀𝜑={𝜑_i}∈𝜕Ω_r0, we have ‖𝜑‖_𝕏=𝑟₀. So, we obtain 𝑇𝜑(𝑡)=max_{1≤𝑖≤𝑛}𝑇_𝑖𝜑(𝑡)=max_{1≤𝑖≤𝑛}^t_−∞𝑒^{−∫𝑡𝑠𝑤𝑖(𝑟,𝜑𝑖(𝑟))d𝑟}×^𝑛_𝑗=1𝑐_𝑖𝑗(𝑠)𝑓_𝑗ℎ_𝑗⁻¹𝜑_𝑗(𝑠)+^𝑛_𝑗=1𝑑_𝑖𝑗(𝑠)𝑔_𝑗×ℎ_𝑗⁻¹𝜑_𝑗𝑠−𝜇_𝑗(𝑠)+^𝑛_𝑗=1𝑣_𝑖𝑗(𝑠)𝑘_𝑗×𝑎_𝑗𝜑_𝑗𝑠−𝜈_𝑗(𝑠)×𝜑̇_𝑗𝑠−𝜈_𝑗(𝑠)+𝐼_𝑖(𝑠)d𝑠≤max_{1≤𝑖≤𝑛}^𝑡_−∞𝑒^{−∫𝑡𝑠𝑤𝑖(𝑟,𝜑𝑖(𝑟))d𝑟}×^𝑛_𝑗=1𝑐⁺_𝑖𝑗𝐿^𝑓_𝑗𝑎_𝑗+𝑑⁺_𝑖𝑗𝐿^𝑔_𝑗𝑎_𝑗+𝑣⁺_𝑖𝑗𝐿^𝑘_𝑗𝑎_𝑗𝜑_𝕏+^𝑛_𝑗=1𝑐⁺_𝑖𝑗𝑓_𝑗(0)+𝑑⁺_𝑖𝑗𝑔_𝑗(0)+𝑣⁺_𝑖𝑗𝑘_𝑗(0)+𝐼⁺_𝑖d𝑠≤max_{1≤𝑖≤𝑛}1𝛼_𝑖^𝑛_𝑗=1𝑐⁺_𝑖𝑗𝐿^𝑓_𝑗𝑎_𝑗+𝑑⁺_𝑖𝑗𝐿^𝑔_𝑗𝑎_𝑗+𝑣⁺_𝑖𝑗𝐿^𝑘_𝑗𝑎_𝑗𝑟₀+max_{1≤𝑖≤𝑛}1𝛼_𝑖^𝑛_𝑗=1𝑐⁺_𝑖𝑗𝑓_𝑗(0)+𝑑⁺_𝑖𝑗𝑔_𝑗(0)+𝑣⁺_𝑖𝑗𝑘_𝑗(0)+𝐼⁺_𝑖𝛼_𝑖∶=𝜃₁𝑟₀+𝛿₁,𝑇̇𝜑(𝑡)=max_{1≤𝑖≤𝑛}𝑇̇_𝑖𝜑(𝑡)=max_{1≤𝑖≤𝑛}−𝑤_𝑖𝑡,𝜑𝑇_𝑖𝜑+𝐹_𝑖𝑡,𝜑(𝑡)≤max_{1≤𝑖≤𝑛}𝛽1+^𝑖𝛼_𝑖×^𝑛_𝑗=1𝑐⁺_𝑖𝑗𝐿^𝑓_𝑗𝑎_𝑗+𝑑⁺_𝑖𝑗𝐿^𝑔_𝑗𝑎_𝑗+𝑣⁺_𝑖𝑗𝐿^𝑘_𝑗𝑎_𝑗𝑟₀+max_{1≤𝑖≤𝑛}𝛽1+^𝑖𝛼_𝑖×^𝑛_𝑗=1𝑐⁺_𝑖𝑗𝑓_𝑗(0)+𝑑⁺_𝑖𝑗𝑔_𝑗(0)+𝑣⁺_𝑖𝑗𝑘_𝑗(0)+𝐼⁺_𝑖∶=𝜃₂𝑟₀+𝛿₂, (30) which yield 𝑇𝜑_𝕏=maxsup_𝑠∈ℝ𝑇𝜑(𝑠),sup_𝑠∈ℝ𝑇̇𝜑(𝑠𝜃)≤max₁,𝜃₂𝑟₀+max𝛿₁,𝛿₂=𝜃𝑟₀≤𝛿+𝛿1−𝜃=𝑟₀=𝜑_𝕏,∀𝜑∈𝜕Ω_𝑟0. (31)
By Lemma 6, there exists at least one fixed point 𝜑₀∈Ω satisfying 𝑇(𝜑₀)=𝜑₀, which implies that system (5) has at least one almost periodic solution 𝜑₀ with ‖𝜑₀‖_𝕏≤𝑟₀. That is, system (1) has at least one almost periodic solution. This completes the proof.

4. Global Exponential Stability of Almost Periodic Solution

Theorem 9. Assume that (𝐻₁)–(𝐻₅) hold, suppose further that (𝐻₆) there exists a positive constant 𝜆<1 such that Θ₁=max_{1≤𝑖≤𝑛}sup_𝑠∈ℝ−𝛼_𝑖+𝜆𝛽_𝑖+()×1+𝜆_𝑛𝑗=1𝑐_𝑖𝑗(𝑠)𝐿^𝑓_𝑗𝑎_𝑗+^𝑛_𝑗=1(1+𝜆)𝑑_𝑖𝑗(𝑠)𝐿^𝑔_𝑗𝑎_𝑗+^𝑛_𝑗=1(1+𝜆)𝑣_𝑖𝑗(𝑠)𝐿^𝑘_𝑗𝐿^𝑎_𝑗𝑟₀Θ<0,₂=max_{1≤𝑖,𝑗≤𝑛}sup_𝑠∈ℝ^𝑛_𝑗=1𝜆𝑣_𝑖𝑗(𝑠)𝐿^𝑘_𝑗𝑎_𝑗+^𝑛_𝑖=1𝑣_𝑖𝑗(𝑠)𝐿^𝑘_𝑗𝑎_𝑗−𝜆1−𝜈̇⁺_𝑗<0, (32) where 𝑟₀ is defined as that in Theorem 8.
Then system (1) has a unique almost periodic solution, which is globally exponentially stable.

Proof. It follows from Theorem 8 that system (5) has at least one almost periodic solution 𝑦={𝑦_𝑖} with initial value 𝜙={𝜙_𝑖}. Next we show that the almost periodic solution 𝑦 is globally exponentially stable.
Make a transformation for system (5): 𝑥_𝑖=𝑧_𝑖−𝑦_𝑖, 𝑖=1,2,…,𝑛, where 𝑧={𝑧_𝑖} is arbitrary solution of system (5) with initial value 𝜓={𝜓_𝑖}.
By (𝐻₆), there exists a small enough positive constant 𝜔 such that Θ_1𝜔=max_{1≤𝑖≤𝑛}sup_𝑠∈ℝ𝜔−𝛼_𝑖+𝜆𝛽_𝑖+()×1+𝜆_𝑛𝑗=1𝑐_𝑖𝑗(𝑠)𝐿^𝑓_𝑗𝑎_𝑗+^𝑛_𝑗=1(1+𝜆)𝑑_𝑖𝑗(𝑠)𝐿^𝑔_𝑗𝑎_𝑗𝑒^{𝜔𝜇+𝑗}+^𝑛_𝑗=1(1+𝜆)𝑣_𝑖𝑗(𝑠)𝐿^𝑘_𝑗𝐿^𝑎_𝑗𝑟₀𝑒^{𝜔𝜈+𝑗}Θ<0,_2𝜔=max_{1≤𝑖,𝑗≤𝑛}sup_𝑠∈ℝ^𝑛_𝑗=1𝜆𝑣_𝑖𝑗(𝑠)𝐿^𝑘_𝑗𝑎_𝑗+^𝑛_𝑖=1𝑣_𝑖𝑗(𝑠)𝐿^𝑘_𝑗𝑎_𝑗−𝜆𝑒^{−𝜔𝜈𝑢𝑗}1−𝜈̇⁺_𝑗<0. (33)
Define 𝑉(𝑡)=𝑉₁(𝑡)+𝑉₂(𝑡), (34) where 𝑉₁(𝑡)=^𝑛_𝑖=1𝑒^𝜔𝑡𝑥_𝑖(𝑡,𝑉₂)(𝑡)=𝜆^𝑛_𝑖=1^𝑡_{𝑡−𝜈𝑖(𝑡)}𝑒^𝜔𝑠𝑥̇_𝑖(𝑠)d𝑠. (35)
In view of system (5), we have 𝐷⁺𝑉₁(𝑡)≤𝜔^𝑛_𝑖=1𝑒^𝜔𝑡𝑥_𝑖(𝑡+_𝑛𝑖=1)𝑒^𝜔𝑡sgn𝑥_𝑖(𝑡)𝑥̇_𝑖(𝑡)≤𝜔^𝑛_𝑖=1𝑒^𝜔𝑡𝑥_𝑖(𝑡+_𝑛𝑖=1)𝑒^𝜔𝑡−𝑏_𝑖𝑡,ℎ_𝑖⁻¹𝑧_𝑖(𝑡)−𝑏_𝑖𝑡,ℎ_𝑖⁻¹𝑦_𝑖(𝑡)×sgn𝑧_𝑖(𝑡)−𝑦_𝑖(𝑡)+^𝑛_𝑗=1𝑐_𝑖𝑗(𝑡)𝑓_𝑗ℎ_𝑗⁻¹𝑧_𝑗(𝑡)−^𝑛_𝑗=1𝑐_𝑖𝑗(𝑡)𝑓_𝑗ℎ_𝑗⁻¹𝑦_𝑗(𝑡)+^𝑛_𝑗=1𝑑_𝑖𝑗(𝑡)𝑔_𝑗ℎ_𝑗⁻¹𝑧_𝑗𝑡−𝜇_𝑗(𝑡)−^𝑛_𝑗=1𝑑_𝑖𝑗(𝑡)𝑔_𝑗ℎ_𝑗⁻¹𝑦_𝑗𝑡−𝜇_𝑗(𝑡)+^𝑛_𝑗=1𝑣_𝑖𝑗(𝑡)𝑘_𝑗×𝑎_𝑗𝑧_𝑗𝑡−𝜈_𝑗(𝑡)𝑧̇_𝑗𝑡−𝜈_𝑗(𝑡)−^𝑛_𝑗=1𝑣_𝑖𝑗(𝑡)𝑘_𝑗×𝑎_𝑗𝑦_𝑗𝑡−𝜈_𝑗(𝑡)×𝑦̇_𝑗𝑡−𝜈_𝑗(𝑡)≤𝜔^𝑛_𝑖=1𝑒^𝜔𝑡𝑥_𝑖(𝑡+_𝑛𝑖=1)𝑒^𝜔𝑡−𝛼_𝑖𝑥_𝑖(𝑡+_𝑛𝑗=1)𝑐_𝑖𝑗(𝑡)𝐿^𝑓_𝑗𝑎_𝑗𝑥_𝑗(𝑡+_𝑛𝑗=1)𝑑_𝑖𝑗(𝑡)𝐿^𝑔_𝑗𝑎_𝑗𝑥_𝑗𝑡−𝜇_𝑗(𝑡)+^𝑛_𝑗=1𝑣_𝑖𝑗(𝑡)𝐿^𝑘_𝑗𝐿^𝑎_𝑗𝑟₀𝑥_𝑗𝑡−𝜈_𝑗(𝑡)+^𝑛_𝑗=1𝑣_𝑖𝑗(𝑡)𝐿^𝑘_𝑗𝑎_𝑗𝑥̇_𝑗𝑡−𝜈_𝑗(𝑡). (36)
Further, it follows from (36) that 𝐷⁺𝑉₂(𝑡)≤𝜆^𝑛_𝑖=1𝑒^𝜔𝑡𝑥̇_𝑖(𝑡×)−𝜆_𝑛𝑖=1𝑒^{𝜔(𝑡−𝜈𝑖(𝑡))}1−𝜈̇_𝑖(𝑡)𝑥̇_𝑖𝑡−𝜈_𝑖(𝑡)≤𝜆^𝑛_𝑖=1𝑒^𝜔𝑡𝛽_𝑖𝑥_𝑖(𝑡+_𝑛𝑗=1)𝑐_𝑖𝑗(𝑡)𝐿^𝑓_𝑗𝑎_𝑗𝑥_𝑗(𝑡+_𝑛𝑗=1)𝑑_𝑖𝑗(𝑡)𝐿^𝑔_𝑗𝑎_𝑗𝑥_𝑗𝑡−𝜇_𝑗(𝑡)+^𝑛_𝑗=1𝑣_𝑖𝑗(𝑡)𝐿^𝑘_𝑗𝐿^𝑎_𝑗𝑟₀𝑥_𝑗𝑡−𝜈_𝑗(𝑡)−𝜆^𝑛_𝑗=1𝑒^𝜔𝑡𝑒^{−𝜔𝜈𝑢𝑗}1−𝜈̇⁺_𝑗−^𝑛_𝑗=1𝑣_𝑖𝑗(𝑡)𝐿^𝑘_𝑗𝑎_𝑗×𝑥̇_𝑗𝑡−𝜈_𝑗(𝑡). (37)
By (36) and (37), we have 𝐷⁺𝑉(𝑡)≤^𝑛_𝑖=1𝜔−𝛼_𝑖+𝜆𝛽_𝑖+()×1+𝜆_𝑛𝑗=1𝑐_𝑖𝑗(𝑡)𝐿^𝑓_𝑗𝑎_𝑗𝑒^𝜔𝑡𝑥_𝑖(𝑡+_{𝑛𝑛𝑖=1𝑗=1})(1+𝜆)𝑑_𝑖𝑗(𝑡)𝐿^𝑔_𝑗𝑎_𝑗𝑒^{𝜔𝜇+𝑗}𝑒^{𝜔(𝑡−𝜇𝑗(𝑡))}×𝑥_𝑗𝑡−𝜇_𝑗(𝑡)+^𝑛𝑛_𝑖=1𝑗=1(1+𝜆)𝑣_𝑖𝑗(𝑡)𝐿^k_𝑗𝐿^𝑎_𝑗𝑟₀𝑒^{𝜔𝜈+𝑗}𝑒^{𝜔(𝑡−𝜇𝑗(𝑡))}×𝑥_𝑗𝑡−𝜈_𝑗(𝑡−_𝑛𝑗=1)𝑒^𝜔𝑡𝜆𝑒^{−𝜔𝜈𝑢𝑗}1−𝜈̇⁺_𝑗−^𝑛_𝑗=1𝜆𝑣_𝑖𝑗(𝑡)𝐿^𝑘_𝑗𝑎_𝑗−^𝑛_𝑖=1𝑣_𝑖𝑗(𝑡)𝐿^𝑘_𝑗𝑎_𝑗𝑥̇_𝑗𝑡−𝜈_𝑗(𝑡). (38)
For 𝑡≤0, note that ^𝑛_𝑖=1𝑒^𝜔𝑡𝑥_𝑖(𝑡)=𝑉₁(𝑡)(𝑡)=≤𝑉_𝑛𝑖=1𝑒^𝜔𝑡𝑥_𝑖(𝑡)+𝜆^𝑛_𝑖=1^𝑡_{𝑡−𝜈𝑖(𝑡)}𝑒^𝜔𝑠𝑥̇_𝑖(𝑠≤)d𝑠1+𝑒^𝜔𝜏𝜓−𝜙_𝜏∶=𝑀𝜓−𝜙_𝜏, (39) where 𝑀=1+𝑒^𝜔𝜏. Next, we claim that 𝑉(𝑡)≤𝑀𝜓−𝜙_𝜏,∀𝑡∈[0,+∞). (40)
Contrarily, there is 𝑡₀∈[0,+∞) such that 𝑉𝑡₀=𝑀𝜓−𝜙_𝜏,𝐷⁺𝑉𝑡₀𝑉(𝑡)>0,≤𝑀𝜓−𝜙_𝜏,∀𝑡∈−𝜏,𝑡₀. (41)
By (38), that we have from (33) 0<𝐷⁺𝑉𝑡₀≤^𝑛_𝑖=1𝜔−𝛼_𝑖+𝜆𝛽_𝑖+()×1+𝜆_𝑛𝑗=1𝑐_𝑖𝑗𝑡₀𝐿^𝑓_𝑗𝑎_𝑗+^𝑛_𝑗=1(1+𝜆)𝑑_𝑖𝑗𝑡₀𝐿^𝑔_𝑗𝑎_𝑗𝑒^{𝜔𝜇+𝑗}+^𝑛_𝑗=1(1+𝜆)𝑣_𝑖𝑗𝑡₀𝐿^𝑘_𝑗𝐿^𝑎_𝑗𝑟₀𝑒^{𝜔𝜈+𝑗}𝑀𝜓−𝜙_𝜏−^𝑛_𝑗=1𝑒^𝜔𝑡0𝜆𝑒^{−𝜔𝜈𝑢𝑗}1−𝜈̇⁺_𝑗−^𝑛_𝑗=1𝜆𝑣_𝑖𝑗𝑡₀𝐿^𝑘_𝑗𝑎_𝑗−^𝑛_𝑖=1𝑣_𝑖𝑗𝑡₀𝐿^𝑘_𝑗𝑎_𝑗𝑥̇_𝑗𝑡₀−𝜈_𝑗𝑡₀<0. (42)
This is a contradiction. So, our claim is valid. Therefore, ^𝑛_𝑖=1𝑥_𝑖(𝑡)≤𝑀𝜓−𝜙_𝜏𝑒^−𝜔𝑡,∀𝑡>0. (43)
Thus, the almost periodic solution of system (5) is globally exponentially stable. That is, the corresponding almost periodic solution of system (1) is globally exponentially stable.
Without loss of generality, we assume that 𝑦={𝑥_𝑖} and 𝑧={𝑧_𝑖} are two almost periodic solutions of system (5). Then 𝑥={𝑥_𝑖}=𝑧−𝑦={𝑧_𝑖−𝑦_𝑖}∈𝐴𝑃(ℝ,ℝ^𝑛). Similar to the above argument, (43) is valid. Hence, lim^𝑛_{𝑡→∞𝑖=1}𝑥_𝑖(𝑡)=0, (44) which implies from 𝑥∈𝐴𝑃(ℝ,ℝ^𝑛) that 𝑥≡0. That is, 𝑦=𝑧. So the almost periodic solution of system (5) is unique. This completes the proof.

5. An Example and Numerical Simulations

Example 1. Consider the following neutral-type Cohen-Grossberg neural networks: 𝑢̇_𝑖(𝑡)=−𝑎_𝑖𝑢_𝑖(𝑡)𝑢_𝑖(𝑡)−²_𝑗=1𝑐_𝑖𝑗(𝑡)𝑓𝑢_𝑗(𝑡)−²_𝑗=1𝑑_𝑖𝑗(𝑡)𝑔𝑢_𝑗𝑡−0.01sin²√−2𝑡_2𝑗=10.1𝑘𝑢̇_𝑗(𝑡−1)−𝐼_𝑖(𝑡), (45) where 𝑖=1,2, 𝑓(𝑠)=𝑔(𝑠)=𝑘(𝑠)=0.1𝑠, 𝑎₁(𝑠)𝑎₂(𝑠)=0.11+0.01sin√√2𝑠0.11+0.01cos,𝐼13𝑠₁(𝑠)𝐼₂(𝑠)=0.01sin√(𝑠),𝑐7𝑠0.01sin₁₁(𝑠)𝑐₁₂(𝑠)𝑐₂₁(𝑠)𝑐₂₂(𝑠)=𝑑₁₁(𝑠)𝑑₁₂(𝑠)𝑑₂₁(𝑠)𝑑₂₂(𝑠)=0.1sin√0√2𝑠0.1cos3𝑠0.1,∀𝑠∈ℝ. (46)
Then system (45) has a unique almost periodic solution, which is globally exponentially stable.

Proof. Corresponding to system (1), 𝑎_𝑖=𝛼_𝑖=0.1, 𝑎_𝑖=𝛽_𝑖=0.12, 𝑐⁺_𝑖𝑗=𝑑⁺_𝑖𝑗=𝑣⁺_𝑖𝑗=0.1, 𝐿^𝑓_𝑗=𝐿^𝑔_𝑗=𝐿^𝑘_𝑗=0.1, 𝐿^𝑎_𝑗=0.05, 𝐼⁺_𝑖=0.01, and 𝑖,𝑗=1,2. Taking 𝜆=0.5, by an easy calculation, we obtain 𝜃=0.072,𝛿=0.1,𝑟₀=0.1. (47)
Further, we also have Θ₁≤−0.03<0,Θ₂≤−0.49<0. (48)
It is easy to verify that all the conditions of Theorem 9 are satisfied and that the result follows from Theorem 9. Numerical simulations show that system (45) has a unique almost periodic solution, which is globally exponentially stable (see Figures 1, 2, and 3). This completes the proof.

Figure 1 

State variables 𝑢1 and 𝑢2 of Example 1.

Figure 2 

Exponential stability of state variable 𝑢1 of Example 1.

Figure 3 

Exponential stability of state variable 𝑢2 of Example 1.

In [28–30], the authors studied the existence and global exponential stability of almost periodic solution for a kind of Cohen-Grossberg neural networks without neutral delays based on the exponential dichotomy with fixed point theorem. Therefore, in some sense, we generalize the results in [28–30].

6. Conclusions

Compared with periodic effects, almost periodic effects are more frequent. In recent years, the study on the almost periodic behavior of neural networks has been the object of intensive analysis by numerous authors, and some of these results can be found in the literature. However, the neural networks of neutral type have not been fully investigated in the literature. In this paper, we give some sufficient conditions to ensure the existence, uniqueness, and global exponential stability of almost periodic solution for neutral-type Cohen-Grossberg neural networks. Numerical simulation is given to illustrate the feasibility of our main result. By using the method in this paper, we could study the existence, uniqueness, and global exponential stability of almost periodic solution of other (impulsive) neural networks (on time scales). We leave these for our future work.

Acknowledgment

The authors would like to thank the reviewers very much for their valuable suggestions on the paper.