Existence and Global Exponential Stability of Equilibrium Solution to Reaction-Diffusion Recurrent Neural Networks on Time Scales

Abstract

The existence of equilibrium solutions to reaction-diffusion recurrent neural networks with Dirichlet boundary conditions on time scales is proved by the topological degree theory and M-matrix method. Under some sufficient conditions, we obtain the uniqueness and global exponential stability of equilibrium solution to reaction-diffusion recurrent neural networks with Dirichlet boundary conditions on time scales by constructing suitable Lyapunov functional and inequality skills. One example is given to illustrate the effectiveness of our results.

1. Introduction

In the past few years, various neural network models have been extensively investigated and successfully applied to signal processing, image processing, pattern classification, quadratic optimization, associative memory, moving object speed detection, and so forth. Such applications heavily depend on the dynamical behaviors of the neural networks. Therefore, the analysis of the dynamical behaviors is a necessary step for practical design of neural networks.

As is well known, both in biological and man-made neural networks, strictly speaking, diffusion effects cannot be avoided in the neural network models when electrons are moving in asymmetric electromagnetic fields, so we must consider that the activations vary in spaces as well as in time. References [1–10] have considered the stability of neural networks with diffusion terms, which are expressed by partial differential equations. It is also common to consider the diffusion effects in biological systems (such as immigration, see, e.g., [11–13]). For more details of the literature related to models of reaction-diffusion neural networks and their applications, the reader is referred to [14–21] and the references cited therein.

In fact, both continuous and discrete systems are very important in implementing and applications. But it is troublesome to study the dynamics behavior for continuous and discrete systems, respectively. Therefore, it is meaningful to study that on time scales which can unify the continuous and discrete situations [22, 23].

To the best of our knowledge, few authors have considered global exponential stability of reaction-diffusion recurrent neural networks with Dirichlet boundary conditions on time scales, which is a very important in theories and applications and also is very challenging problem. Motivated by the above discussion, in this paper, we will investigate the global exponential stability of the following reaction-diffusion recurrent neural network with initial value conditions and Dirichlet boundary conditions on time scales:𝑢^Δ_𝑖(𝑡,𝑥)=^𝑚_𝑘=1𝜕𝜕𝑥_𝑘(𝑎_𝑖𝑘𝜕𝑢_𝑖𝜕𝑥_𝑘)−𝑏_𝑖𝑢_𝑖(𝑡,𝑥)+^𝑛_𝑗=1𝑐_𝑖𝑗𝑓_𝑗(𝑢_𝑗(𝑡,𝑥))+𝐼_𝑖𝑢,(𝑡,𝑥)∈𝕋×Ω,_𝑖(0,𝑥)=𝜙_𝑖𝑢(𝑥),𝑥∈Ω,_𝑖(𝑡,𝑥)=0,(𝑡,𝑥)∈[0,+∞)_𝕋×𝜕Ω, (1.1)(1.2)(1.3) where 𝑖=1,2,…,𝑛,𝕋⊂ℝ is a time scale and 𝕋∩[0,+∞)=[0,+∞)_𝕋 is unbounded, 𝑛 is the number of neurons in the networks, 𝑥=(𝑥₁,𝑥₂,…,𝑥_𝑚)^𝑇∈Ω⊂ℝ^𝑚, and Ω={𝑥=(𝑥₁,𝑥₂,…,𝑥_𝑚)^𝑇∶|𝑥_𝑖|<𝑙_𝑖, 𝑖=1,2,…,𝑚} is a bounded compact set with smooth boundary 𝜕Ω in space ℝ^𝑚, 𝑢(𝑡,𝑥)=(𝑢₁(𝑡,𝑥),𝑢₂(𝑡,𝑥),…,𝑢_𝑛(𝑡,𝑥))^𝑇∶𝕋×Ω→ℝ^𝑛, and 𝑢_𝑖(𝑡,𝑥) is the state of the 𝑖th neurons at time 𝑡 and in space 𝑥. Smooth function 𝑎_𝑖𝑘>0 corresponds to the transmission diffusion operator along with the 𝑖th unit, 𝑏_𝑖>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, 𝑐_𝑖𝑗 denotes the strength of the 𝑗th unit on the 𝑖th unit at time 𝑡 and in space 𝑥, 𝑑_𝑖𝑗 is the synaptic connection strength of 𝑗th unit on 𝑖th unit at time 𝑡 and in space 𝑥, 𝑓_𝑗(⋅) denotes the activation function of the 𝑗th unit at time 𝑡 and in space 𝑥, 𝜙(𝑥)=(𝜙₁(𝑥),𝜙₂(𝑥),…,𝜙_𝑛(𝑥))^𝑇∈𝐶(Ω,ℝ^𝑛), and 𝐼=(𝐼₁,𝐼₂,…,𝐼_𝑛)^𝑇∈ℝ^𝑛 is a constant input vector.

2. Preliminaries

In this section, we first recall some basic definitions and lemmas on time scales which are used in what follows.

Let 𝕋 be a nonempty closed subset (time scale) of ℝ. The forward and backward jump operators 𝜎,𝜌∶𝕋→𝕋 and the graininess 𝜇∶𝕋→ℝ⁺ are defined, respectively, by𝜎(𝑡)=inf{𝑠∈𝕋∶𝑠>𝑡},𝜌(𝑡)=sup{𝑠∈𝕋∶𝑠<𝑡},𝜇(𝑡)=𝜎(𝑡)−𝑡. (2.1)

A point 𝑡∈𝕋 is called left-dense if 𝑡>𝑖𝑛𝑓𝕋 and 𝜌(𝑡)=𝑡, left-scattered if 𝜌(𝑡)<𝑡, right-dense if 𝑡<𝑠𝑢𝑝𝕋 and 𝜎(𝑡)=𝑡, and right-scattered if 𝜎(𝑡)>𝑡. If 𝕋 has a left-scattered maximum 𝑚, then 𝕋^𝑘=𝕋⧵{𝑚}; otherwise 𝕋^𝑘=𝕋. If 𝕋 has a right-scattered minimum 𝑚, then 𝕋_𝑘=𝕋⧵{𝑚}; otherwise 𝕋^𝑘=𝕋.

Definition 2.1 (see [24]). A function 𝑓∶𝕋→ℝ is called regulated provided its right-side limits exist (finite) at all right-side points in 𝕋 and its left-side limits exist (finite) at all left-side points in 𝕋.

Definition 2.2 (see [24]). A function 𝑓∶𝕋→ℝ is called 𝑟𝑑 continuous provided it is continuous at right-dense point in 𝕋 and its left-side limits exist (finite) at left-dense points in 𝕋. The set of 𝑟𝑑 continuous functions 𝑓∶𝕋→ℝ will be denoted by 𝐶_𝑟𝑑=𝐶_𝑟𝑑(𝕋)=𝐶_𝑟𝑑(𝕋,ℝ).

Definition 2.3 (see [24]). Assume 𝑓∶𝕋→ℝ and 𝑡∈𝕋^𝑘. Then we define 𝑓^Δ(𝑡) to be the number (if it exists) with the property that given any 𝜀>0 there exists a neighborhood 𝑈 of 𝑡 (i.e., 𝑈=(𝑡−𝛿,𝑡+𝛿)∩𝕋 for some 𝛿>0) such that |[𝑓(𝜎(𝑡))−𝑓(𝑠)]−𝑓^Δ(𝑡)[𝜎(𝑡)−𝑠]|<𝜀|𝜎(𝑡)−𝑠| (2.2) for all 𝑠∈𝑈. We call 𝑓^Δ(𝑡) the delta (or Hilger) derivative of 𝑓 at 𝑡. The set of functions 𝑓∶𝕋→ℝ that are differentiable and whose derivative is 𝑟𝑑 continuous is denoted by 𝐶¹_𝑟𝑑=𝐶¹_𝑟𝑑(𝕋)=𝐶¹_𝑟𝑑(𝕋,ℝ).

If 𝑓 is continuous, then 𝑓 is 𝑟𝑑 continuous. If 𝑓 is 𝑟𝑑 continuous, the 𝑓 is regulated. If 𝑓 is delta differentiable at 𝑡, then 𝑓 is continuous at 𝑡.

Lemma 2.4 (see [24]). Let 𝑓 be regulated, then there exists a function 𝐹 which is delta differentiable with region of differentiation 𝐷 such that 𝐹^Δ(𝑡)=𝑓(𝑡),∀𝑡∈𝐷. (2.3)

Definition 2.5 (see [24]). Assume 𝑓∶𝕋→ℝ is a regulated function. Any function 𝐹 as in Lemma 2.4 is called a Δ-antiderivative of 𝑓. We define the indefinite integral of a regulated function 𝑓 by 𝑓(𝑡)Δ𝑡=𝐹(𝑡)+𝐶, (2.4) where 𝐶 is an arbitrary constant and 𝐹 is a Δ-antiderivative of 𝑓. We define the Cauchy integral by ^𝑏_𝑎𝑓(𝑠)Δ𝑠=𝐹(𝑏)−𝐹(𝑎),∀𝑎,𝑏∈𝕋. (2.5) A function 𝐹∶𝕋→ℝ is called an antiderivative of 𝑓∶𝕋→ℝ provided that 𝐹^Δ(𝑡)=𝑓(𝑡),∀𝑡∈𝕋^𝑘. (2.6)

Lemma 2.6 (see [24]). If 𝑎,𝑏∈𝕋, 𝛼,𝛽∈ℝ, and 𝑓,𝑔∈𝐶(𝕋,ℝ), then (i)∫^𝑏_𝑎[𝛼𝑓(𝑡)+𝛽𝑔(𝑡)]Δ𝑡=𝛼∫^𝑏_𝑎𝑓(𝑡)Δ𝑡+𝛽∫^𝑏_𝑎𝑔(𝑡)Δ𝑡;(ii)if 𝑓(𝑡)≥0 for all 𝑎≤𝑡<𝑏, then ∫^𝑏_𝑎𝑓(𝑡)Δ𝑡≥0;(iii)if |𝑓(𝑡)|≤𝑔(𝑡) on [𝑎,𝑏)∶={𝑡∈𝕋∶𝑎≤𝑡<𝑏}, then |∫^𝑏_𝑎𝑓(𝑡)Δ𝑡|≤∫^𝑏_𝑎𝑔(𝑡)Δ𝑡.

A function 𝑝∶𝕋→ℝ is called regressive if 1+𝜇(𝑡)𝑝(𝑡)≠0 for all 𝑡∈𝕋^𝑘. The set of all regressive and rd-continuous functions 𝑓∶𝕋→ℝ will be denoted by ℛ=ℛ(𝕋)=ℛ(𝕋,ℝ). We define the set ℛ⁺ of all positively regressive elements of ℛ by ℛ⁺=ℛ⁺(𝕋,ℝ)={𝑝∈ℛ∶1+𝜇(𝑡)𝑝(𝑡)>0forall𝑡∈𝕋}. If 𝑝 is a regressive function, then the generalized exponential function 𝑒_𝑝 is defined by 𝑒_𝑝(𝑡,𝑠)=exp{∫^𝑡_𝑠𝜉_𝜇(𝜏)(𝑝(𝜏))Δ𝜏} for 𝑠,𝑡∈𝕋, with the cylinder transformation 𝜉_ℎ(𝑧)=Log(1+ℎ𝑧)ℎ,ifℎ≠0,𝑧,ifℎ=0. (2.7)

Let 𝑝,𝑞∶𝕋→ℝ be two regressive functions, then we define𝑝𝑝⊕𝑞=𝑝+𝑞+𝜇𝑝𝑞,⊖𝑝=−1+𝜇𝑝,𝑝⊖𝑞=𝑝⊕𝑝(⊖𝑞). (2.8)

The generalized exponential function has the following properties.

Lemma 2.7 (see [24]). Assume that 𝑝,𝑞∶𝕋→ℝ are two regressive functions, then, (i)𝑒_𝑝(𝜎(𝑡),𝑠)=(1+𝜇(𝑡)𝑝(𝑡))𝑒_𝑝(𝑡,𝑠);(ii)1/𝑒_𝑝(𝑡,𝑠)=𝑒_⊖𝑝(𝑡,𝑠);(iii)𝑒_𝑝(𝑡,𝑠)=1/𝑒_𝑝(𝑠,𝑡)=𝑒_⊖𝑝(𝑠,𝑡);(iv)𝑒_𝑝(𝑡,𝑠)𝑒_𝑝(𝑠,𝑟)=𝑒_𝑝(𝑡,𝑟);(v)[𝑒_𝑝(𝑡,𝑠)]^Δ=𝑝(𝑡)𝑒_𝑝(𝑡,𝑠);(vi)[𝑒_𝑝(𝑐,⋅)]^Δ=−𝑝[𝑒_𝑝(𝑐,⋅)]^𝜎 for 𝑐∈𝕋;(vii)𝑑/𝑑𝑧[𝑒_𝑧(𝑡,𝑠)]=(∫^𝑡_𝑠(1/(1+𝜇(𝜏)𝑧))Δ𝜏)𝑒_𝑧(𝑡,𝑠).

Lemma 2.8 (see [24]). Assume that 𝑓,𝑔∶𝕋→ℝ are delta differentiable at 𝑡∈𝕋^𝑘. Then, (𝑓𝑔)^Δ(𝑡)=𝑓^Δ(𝑡)𝑔(𝑡)+𝑓(𝜎(𝑡))𝑔^Δ(𝑡)=𝑔^Δ(𝑡)𝑓(𝑡)+𝑔(𝜎(𝑡))𝑓^Δ(𝑡). (2.9)

Next, let us introduce the Banach space which is suitable for (1.1)–(1.3) and some assumed conditions which are needed in this paper.

Let Ω={𝑥=(𝑥₁,𝑥₂,…,𝑥_𝑚)^𝑇∶|𝑥_𝑖|<𝑙_𝑖, 𝑖=1,2,…,𝑚} be an open bounded domain in ℝ^𝑚 with smooth boundary 𝜕Ω. Let 𝐶_𝑟𝑑(𝕋×Ω,ℝ^𝑛) be the set consisting of all the vector functions 𝑢(𝑡,𝑥) which are 𝑟𝑑-continuous with respect to 𝑡∈𝕋 and continuous with respect to 𝑥∈ℝ^𝑚, respectively. For every 𝑡∈𝕋 and 𝑥∈Ω, we define the set 𝐶_𝑡={𝑢(𝑡,⋅)∶𝑢∈𝐶(Ω,ℝ^𝑛)}. Then for every 𝑡∈𝕋, 𝐶_𝑡 is a Banach space with the norm ‖𝑢(𝑡,⋅)‖=(∑^𝑛_𝑖‖𝑢_𝑖(𝑡,⋅)‖²₂)^1/2, where 𝑢(𝑡,𝑥)=(𝑢₁(𝑡,𝑥),𝑢₂(𝑡,𝑥),…,𝑢_𝑛(𝑡,𝑥))^𝑇, ‖𝑢_𝑖(𝑡,⋅)‖₂=(∫_Ω|𝑢_𝑖(𝑡,𝑥)|²d𝑥)^1/2, 𝑖=1,2,…,𝑛. Obviously, 𝐶(Ω,ℝ^𝑛) is a Banach space equipped with the norm ‖𝜑‖₀=(∑^𝑛_𝑖‖𝜑_𝑖‖²_𝜏)^1/2, where 𝜑(𝑥)=(𝜑₁(𝑥),𝜑₂(𝑥),…,𝜑_𝑛(𝑥))^𝑇∈𝐶(Ω,ℝ^𝑛), ‖𝜑_𝑖‖_𝜏=(∫_Ω|𝜑_𝑖(𝑥)|²d𝑥)^1/2, 𝑖=1,2,…,𝑛.

Definition 2.9. A function 𝑢∶𝕋×Ω→ℝ^𝑛 is called a solution of (1.1)–(1.3) if and only if 𝑢 satisfies (1.1), initial value conditions (1.2) and Dirichlet boundary conditions (1.3).

Definition 2.10. A constant vector 𝑢^∗=(𝑢^∗₁,𝑢^∗₂,…,𝑢^∗_𝑛)^𝑇 is said to be an equilibrium solution to (1.1)–(1.3), if it satisfies −𝑏_𝑖𝑢^∗_𝑖+∑^𝑛_𝑗=1𝑐_𝑖𝑗𝑓_𝑗(𝑢^∗_𝑗)+𝐼_𝑖=0, 𝑖=1,2,…,𝑛.

Definition 2.11. The equilibrium solution 𝑢^∗=(𝑢^∗₁,𝑢^∗₂,…,𝑢^∗_𝑛)^𝑇 of recurrent neural network (1.1)–(1.3) is said to be globally exponentially stable if there exists a positive constant 𝛼∈ℛ⁺ and 𝑀≥1 such that every solution 𝑢(𝑡,𝑥)=(𝑢₁(𝑡,𝑥),𝑢₂(𝑡,𝑥),…,𝑢_𝑛(𝑡,𝑥))^𝑇 of (1.1)–(1.3) satisfies ‖𝑢(𝑡,⋅)−𝑢^∗‖≤𝑀𝑒_⊖𝛼(𝑡,0),𝑡∈𝕋⁺. (2.10)

Definition 2.12 (Lakshmikantham and Vatsala [25]). For each 𝑡∈𝕋, let 𝑁 be a neighborhood of 𝑡. Then, for 𝑉∈𝐶_𝑟𝑑[𝕋×ℝ^𝑛,ℝ⁺], define 𝐷⁺𝑉^Δ(𝑡,𝑥(𝑡)) to mean that, given 𝜀>0, there exists a right neighborhood 𝑁_𝜀∩𝑁 of 𝑡 such that 1𝜇(𝑡,𝑠)[𝑉(𝜎(𝑡),𝑥(𝜎(𝑡)))−𝑉(𝑠,𝑥(𝜎(𝑡)))−𝜇(𝑠,𝑡)𝑓(𝑡,𝑥(𝑡))]<𝐷⁺𝑉^Δ(𝑡,𝑥(𝑡))+𝜀 (2.11) for each 𝑠∈𝑁_𝜀,𝑠>𝑡, where 𝜇(𝑡,𝑠)≡𝜎(𝑡)−𝑠. If 𝑡 is rs and 𝑉(𝑡,𝑥(𝑡)) is continuous at 𝑡, this reduces to 𝐷⁺𝑉^Δ(𝑡,𝑥(𝑡))=𝑉(𝜎(𝑡),𝑥(𝜎(𝑡)))−𝑉(𝑡,𝑥(𝜎(𝑡))).𝜎(𝑡)−𝑡 (2.12)

3. Main Results

In this section, we will consider the existence, uniqueness, and global exponential stability of equilibrium of (1.1)–(1.3). To proceed, we need the following lemma.

Lemma 3.1 (see [14]). Let Ω be a cube |𝑥_𝑖|<𝑙_𝑖(𝑖=1,2,…,𝑚) and let ℎ(𝑥) be a real-valued function belonging to 𝐶¹(Ω) which vanish on the boundary 𝜕Ω of Ω, that is, ℎ(𝑥)|_𝜕Ω=0. Then, _Ωℎ²(𝑥)d𝑥≤𝑙²_𝑖Ω|𝜕ℎ𝜕𝑥_𝑖|²d𝑥. (3.1)

Throughout this paper, we always assume that (𝐻₁)𝑓_𝑗(⋅) is Lipschitz continuous, that is, there exists constant 𝐹_𝑗>0 such that |𝑓_𝑗(𝜉)−𝑓_𝑗(𝜂)|≤𝐹_𝑗|𝜉−𝜂|, for any 𝜉,𝜂∈ℝ, 𝑗=1,2,…,𝑛;(𝐻₂)𝑊=𝐵−𝐶⁺𝐹 is an M-matrix, where 𝐵=diag(𝑏₁,𝑏₂,…,𝑏_𝑛), 𝐶⁺=(|𝑐_𝑖𝑗|)_𝑛×𝑛, 𝐹=diag(𝐹₁,𝐹₂,…,𝐹_𝑛),(𝐻₃)−2𝑏_𝑖−∑^𝑚_𝑘=1(2𝑎_𝑖𝑘/𝑙²_𝑘)+∑^𝑛_𝑗=1(|𝑐_𝑖𝑗|𝐹_𝑗+|𝑐_𝑗𝑖|𝐹_𝑖)<0, 𝑖=1,2,…,𝑛.

Theorem 3.2. Assume that (𝐻₁) and (𝐻₂) hold, then (1.1)–(1.3) has at least one equilibrium solution 𝑢^∗=(𝑢^∗₁,𝑢^∗₂,…,𝑢^∗_𝑛)^𝑇.

Proof. By (𝐻₁), it follows that |𝑓_𝑗(𝑠)|≤𝐹_𝑗|𝑠|+|𝑓_𝑗(0)|,𝑗=1,2,…,𝑛,∀𝑠∈𝑅. (3.2) Let ℎ(𝑢,𝐼)=𝐵𝑢−𝐶𝑓(𝑢)−𝐼=0, (3.3) where 𝑓(𝑢)=(𝑓₁(𝑢₁),𝑓₂(𝑢₂),…,𝑓_𝑛(𝑢_𝑛))^𝑇. It is obvious that solutions to (3.3) are equilibria of (1.1)–(1.3). Let us define homotopic mapping 𝐻(𝑢,𝜆)=𝜆ℎ(𝑢,𝐼)+(1−𝜆)𝑢,𝜆∈𝐽=[0,1]. (3.4) We have |𝐻_𝑖(𝑢,𝜆)|=|𝜆[𝑏_𝑖𝑢_𝑖−(^𝑛_𝑗=1𝑐_𝑖𝑗𝑓_𝑗(𝑢_𝑗)+𝐼_𝑖)]+(1−𝜆)𝑢_𝑖|≥[1+𝜆(𝑏_𝑖−1)]|𝑢_𝑖|−𝜆^𝑛_𝑗=1|𝑐_𝑖𝑗|𝐹_𝑗|𝑢_𝑗|−𝜆[|𝐼_𝑖|+^𝑛_𝑗=1|𝑐_𝑖𝑗||𝑓_𝑗(0)|],𝑖=1,2,…,𝑛. (3.5) That is, 𝐻⁺≥[𝐸+𝜆(𝐵−𝐸),𝑢]⁺−𝜆𝐶⁺𝐹[𝑢]⁺−𝜆(𝐼⁺+𝐶⁺𝑓⁺(0))=(1−𝜆)[𝑢]⁺+𝜆(𝐵−𝐶⁺𝐹)[𝑢]⁺−𝜆(𝐼⁺+𝐶⁺𝑓⁺(0)), (3.6) where 𝐻⁺=(|𝐻₁|,|𝐻₂|,…,|𝐻_𝑛|)^𝑇, [𝑢]⁺=(|𝑢₁|,|𝑢₂|,…,|𝑢_𝑛|)^𝑇, 𝐼⁺=(|𝐼₁|,|𝐼₂|,…,|𝐼_𝑛|)^𝑇, 𝑓⁺(0)=(|𝑓₁(0)|,|𝑓₂(0)|,…,|𝑓_𝑛(0)|)^𝑇, and 𝐸 is an identity matrix.
Since 𝑊=𝐵−𝐶⁺𝐹 is an 𝑀-matrix, we have 𝑊⁻¹=(𝐵−𝐶⁺𝐹)⁻¹≥0 (nonnegative matrix) and there exists a 𝑄=(𝑄₁,𝑄₂,…,𝑄_𝑛)^𝑇>0(𝑄_𝑖>0,𝑖=1,2,…,𝑛) such that (𝐵−𝐶⁺𝐹)𝑄>0. Let𝑈(𝑅₀)={𝑢∶[𝑢]⁺≤𝑅₀=𝑄+(𝐵−𝐶⁺𝐹)⁻¹(𝐼⁺+𝐶⁺𝑓⁺(0))}. (3.7) Then, 𝑈(𝑅₀) is not empty and it follows from (3.7) that for any 𝑢∈𝜕𝑈(𝑅₀) (boundary of 𝑈(𝑅₀)), 𝐻⁺≥(1−𝜆)[𝑢]⁺+𝜆(𝐵−𝐶⁺𝐹)[[𝑢]⁺−(𝐵−𝐶⁺𝐹)⁻¹(𝐼⁺+𝐶⁺𝑓⁺(0))]=(1−𝜆)[𝑢]⁺𝜆(𝐵−𝐶⁺𝐹)𝑄>0,𝜆∈[0,1], (3.8) which implies that 𝐻(𝑢,𝜆)≠0, for 𝑢∈𝜕𝑈(𝑅₀), 𝜆∈[0,1]. So, from homotopy invariance theorem, we have deg(ℎ,𝑈(𝑅₀),0)=deg(𝐻(𝑢,1),𝑈(𝑅₀),0)=deg(𝐻(𝑢,0),𝑈(𝑅₀),0)=1, (3.9) where deg(ℎ,𝑈(𝑅₀),0) denotes topological degree. By topological degree theory, we can conclude that (3.3) has at least one solution in 𝑈(𝑅₀). That is, (1.1)–(1.3) has at least an equilibrium solution 𝑢^∗=(𝑢^∗₁,𝑢^∗₂,…,𝑢^∗_𝑛)^𝑇. This completes the proof.

Theorem 3.3. Assume that (𝐻₁)-(𝐻₃) hold, then the reaction-diffusion recurrent neural network (1.1)–(1.3) has a unique equilibrium solution 𝑢^∗=(𝑢^∗₁,𝑢^∗₂,…,𝑢^∗_𝑛)^𝑇 which is globally exponentially stable.

Proof. The existence of equilibrium solutions for (1.1)–(1.3) follows from Theorem 3.2. Now we only need to prove the uniqueness and global exponential stability of equilibrium solutions for (1.1)–(1.3).
Suppose that 𝑢(𝑡,𝑥) and 𝑣(𝑡,𝑥) are two arbitrary solutions of (1.1)–(1.3) with conditions 𝜙^𝑢(𝑥),𝜙^𝑣(𝑥)∈𝐶(Ω,ℝ^𝑛), and define 𝑧(𝑡,𝑥)=𝑢(𝑡,𝑥)−𝑣(𝑡,𝑥), 𝜙^𝑧(𝑥)=𝜙^𝑢(𝑥)−𝜙^𝑣(𝑥), then 𝑧(𝑡,𝑥) is governed by the following equations:𝑧^Δ_𝑖(𝑡,𝑥)=^𝑚_𝑘=1𝜕𝜕𝑥_𝑘(𝑎_𝑖𝑘𝜕𝑧_𝑖(𝑡,𝑥)𝜕𝑥_𝑘)−𝑏_𝑖𝑧_𝑖(𝑡,𝑥)+^𝑛_𝑗=1𝑐_𝑖𝑗(𝑓_𝑗(𝑢_𝑗(𝑡,𝑥))−𝑓_𝑗(𝑣_𝑗(𝑡,𝑥))), (3.10) where 𝑖=1,2,…,𝑛. Calculating the delta derivation of ‖𝑧_𝑖(𝑡,⋅)‖²₂ along the solution of (3.10), we have, for 𝑖=1,2,…,𝑛,(‖𝑧_𝑖(𝑡,⋅)‖²₂)^Δ=_Ω((𝑧_𝑖(𝑡,𝑥))²)^Δd𝑥=_Ω(𝑧_𝑖(𝑡,𝑥)+𝑧_𝑖(𝜎(𝑡),𝑥))(𝑧_𝑖(𝑡,𝑥))^Δ=d𝑥_Ω(2𝑧_𝑖(𝑡,𝑥)+𝜇(𝑡)(𝑧_𝑖(𝑡,𝑥))^Δ)(𝑧_i(𝑡,𝑥))^Δd𝑥=2_Ω𝑧_𝑖(𝑡,𝑥)(𝑧_𝑖(𝑡,𝑥))^Δd𝑥+𝜇(𝑡)_Ω((𝑧_𝑖(𝑡,𝑥))^Δ)²d𝑥=2^𝑚_𝑘=1Ω𝑧_𝑖𝜕(𝑡,𝑥)𝜕𝑥_𝑘(𝑎_𝑖𝑘𝜕𝑧_𝑖(𝑡,𝑥)𝜕𝑥_𝑘)d𝑥−2𝑏_𝑖Ω(𝑧_𝑖(𝑡,𝑥))²d𝑥+2^𝑛_𝑗=1𝑐_𝑖𝑗Ω𝑧_𝑖(𝑡,𝑥)(𝑓_𝑗(𝑢_𝑗(𝑡,𝑥))−𝑓_𝑗(𝑣_𝑗(𝑡,𝑥)))d𝑥+𝜇(𝑡)‖(𝑧_𝑖(𝑡,⋅))^Δ‖²₂. (3.11) From Green formula [26], Dirichlet boundary condition, and Lemma 3.1, we have, for 𝑖=1,2,…,𝑛,^𝑚_𝑘=1Ω𝑧_𝑖𝜕(𝑡,𝑥)𝜕𝑥_𝑘(𝑎_𝑖𝑘𝜕𝑧_𝑖(𝑡,𝑥)𝜕𝑥_𝑘)d𝑥=−^𝑚_𝑘=1Ω𝑎_𝑖𝑘(𝜕𝑧_𝑖(𝑡,𝑥)𝜕𝑥_𝑘)²d𝑥≤−^𝑚_𝑘=1Ω𝑎_𝑖𝑘𝑙²_𝑘(𝑧_𝑖(𝑡,𝑥))²d𝑥. (3.12) By (3.11), (3.12), condition (𝐻₁), and Holder inequality, we get (‖𝑧_𝑖(𝑡,⋅)‖²₂)^Δ≤−^𝑚_𝑘=12𝑎_𝑖𝑘𝑙²_𝑘‖𝑧_𝑖(𝑡,⋅)‖²₂−2𝑏_𝑖‖𝑧_𝑖(𝑡,⋅)‖²₂+2^𝑛_j=1|𝑐_𝑖𝑗|𝐹_𝑗‖𝑧_𝑖(𝑡,⋅)‖₂‖𝑧_𝑗(𝑡,⋅)‖₂+𝜇(𝑡)‖(𝑧_𝑖(𝑡,⋅))^Δ‖²₂=−^𝑚_𝑘=12𝑎_𝑖𝑘𝑙²_𝑘‖𝑧_𝑖(𝑡,⋅)‖²₂−2𝑏_𝑖‖𝑧_𝑖(𝑡,⋅)‖²₂+2^𝑛_𝑗=1|𝑐_𝑖𝑗|𝐹_𝑗‖𝑧_𝑖(𝑡,⋅)‖₂‖𝑧_𝑗(𝑡,⋅)‖₂+𝜇(𝑡)𝑞(𝑡)‖𝑧_𝑖(𝑡,⋅)‖²₂, (3.13) where ‖(𝑧_𝑖(𝑡,⋅))^Δ‖²₂=𝑞(𝑡)‖𝑧_𝑖(𝑡,⋅)‖²₂, 𝑞(𝑡)≥0,𝑖=1,2,…,𝑛.
If condition (𝐻₃) holds, we can always choose a positive number 𝜎>0 (may be very small) such that for 𝑖=1,2,…,𝑛,−2𝑏_𝑖−^𝑚_𝑘=12𝑎_𝑖𝑘𝑙²_𝑘+^𝑛_𝑗=1(|𝑐_𝑖𝑗|𝐹_𝑗+|𝑐_𝑗𝑖|𝐹_𝑖)+𝜎<0. (3.14)
Let us consider functions𝑞_𝑖(𝑦_𝑖)=𝑦_𝑖⊕𝑦_𝑖−2𝑏_𝑖−^𝑚_𝑘=12𝑎_𝑖𝑘𝑙²_𝑘+^𝑛_𝑗=1(|𝑐_𝑖𝑗|𝐹_𝑗+|𝑐_𝑗𝑖|𝐹_𝑖)+𝑤(𝑦_𝑖)𝜇(𝑡)𝑞(𝑡)max{𝑒_{𝑦𝑖⊕𝑦𝑖}(𝜎(𝑡),0),𝑒_{(𝑤(𝑦𝑖)−1)𝜇(𝑡)𝑞(𝑡)‖𝑧𝑖(𝑡,⋅)‖}²₂(𝑡,0)}𝑒_{𝑦𝑖⊕𝑦𝑖},(𝜎(𝑡),0) (3.15) where 𝑤(𝑦_𝑖)=∫^𝑦𝑖₀(𝑒^{𝑦𝑖−𝑠}/(𝑦_𝑖−𝑠)²)d𝑠,𝑖=1,2,…,𝑛. From (3.15), we obtain that 𝑞_𝑖(0)<−𝜎<0 and 𝑞_𝑖(𝑦_𝑖) is continuous for 𝑦_𝑖∈[0,+∞), furthermore, 𝑞_𝑖(𝑦_𝑖)→+∞ as 𝑦_𝑖→+∞, thus there exist constant 𝜀_𝑖∈(0,+∞) such that 𝑞_𝑖(𝜀^∗_𝑖)=0 and 𝑞_𝑖(𝜀_𝑖)<0, for 𝜀_𝑖∈(0,𝜀^∗_𝑖). By choosing 𝜀=min_{1≤𝑖≤𝑛}{𝜀_𝑖}, obviously 𝜀>0, we have, for 𝑖=1,2,…,𝑛,𝑞_𝑖(𝜀)=𝜀⊕𝜀−2𝑏_𝑖−^𝑚_𝑘=12𝑎_𝑖𝑘𝑙²_𝑘+^𝑛_𝑗=1(|𝑐_𝑖𝑗|𝐹_𝑗+|𝑐_𝑗𝑖|𝐹_𝑖)+𝑤(𝜀)𝜇(𝑡)𝑞(𝑡)max{𝑒_𝜀⊕𝜀(𝜎(𝑡),0),𝑒_{(𝑤(𝜀)−1)𝜇(𝑡)𝑞(𝑡)‖𝑧𝑖(𝑡,⋅)‖}²₂(𝑡,0)}𝑒_𝜀⊕𝜀(𝜎(𝑡),0)≤0. (3.16)
Now consider the Lyapunov functional𝑉(𝑡,𝑧(𝑡))=^𝑛_𝑖=1{𝑒_𝜀⊕𝜀(𝑡,0)‖𝑧_𝑖(𝑡,⋅)‖²₂+𝑒_{(𝑤(𝜀)−1)𝜇(𝑡)𝑞(𝑡)‖𝑧𝑖(𝑡,⋅)‖}²₂(𝑡,0)}. (3.17)
Calculating the delta derivatives of 𝑉(𝑡,𝑧(𝑡)) along the solution of (3.10) and noting that 𝑑/𝑑𝑧[𝑒_𝑧(𝑡,𝑠)]=(∫^𝑡_𝑠(1/(1+𝜇(𝜏)𝑧))Δ𝜏)𝑒_𝑧(𝑡,𝑠)>0 if and only if 𝑧∈ℛ⁺(i.e., 𝑒_𝑧(𝑡,𝑠) is increasing with respect to 𝑧 if and only if 𝑧∈ℛ⁺), we have𝐷⁺𝑉^Δ=(𝑡,𝑥(𝑡))_𝑛𝑖=1{(𝜀⊕𝜀)𝑒_𝜀⊕𝜀(𝑡,0)‖𝑧_𝑖(𝑡,⋅)‖²₂+𝑒_𝜀⊕𝜀(𝜎(𝑡),0)(‖𝑧_𝑖(𝑡,⋅)‖²₂)^Δ+(𝑤(𝜀)−1)𝜇(𝑡)𝑞(𝑡)‖𝑧_𝑖(𝑡,⋅)‖²₂𝑒_{(𝑤(𝜀)−1)𝜇(𝑡)𝑞(𝑡)‖𝑧𝑖(𝑡,⋅)‖}²₂≤(𝑡,0)}_𝑛𝑖=1{(𝜀⊕𝜀)𝑒_𝜀⊕𝜀(𝑡,0)‖𝑧_𝑖(𝑡,⋅)‖²₂+𝑒_𝜀⊕𝜀(𝜎(𝑡),0)×(−^𝑚_𝑘=12𝑎_𝑖𝑘𝑙²_𝑘‖𝑧_𝑖(𝑡,⋅)‖²₂−2𝑏_𝑖‖𝑧_𝑖(𝑡,⋅)‖²₂+2^𝑛_𝑗=1|𝑐_𝑖𝑗|𝐹_𝑗‖𝑧_𝑖(𝑡,⋅)‖₂‖𝑧_𝑗(𝑡,⋅)‖₂+𝜇(𝑡)𝑞(𝑡)‖𝑧_𝑖(𝑡,⋅)‖²₂)+(𝑤(𝜀)−1)𝜇(𝑡)𝑞(𝑡)‖𝑧_𝑖(𝑡,⋅)‖²₂𝑒_{(𝑤(𝜀)−1)𝜇(𝑡)𝑞(𝑡)‖𝑧𝑖(𝑡,⋅)‖}²₂(𝑡,0)}≤𝑒_𝜀⊕𝜀(𝜎(𝑡),0)^𝑛_𝑖=1{(𝜀⊕𝜀)‖𝑧_𝑖(𝑡,⋅)‖²₂−^𝑚_𝑘=12𝑎_𝑖𝑘𝑙²_𝑘‖𝑧_𝑖(𝑡,⋅)‖²₂−2𝑏_𝑖‖𝑧_𝑖(𝑡,⋅)‖²₂+^𝑛_𝑗=1|𝑐_𝑖𝑗|𝐹_𝑗(‖𝑧_𝑖(𝑡,⋅)‖²₂+‖𝑧_𝑗(𝑡,⋅)‖²₂+)max{𝑒_𝜀⊕𝜀(𝜎(𝑡),0),𝑒_{(𝑤(𝜀)−1)𝜇(𝑡)𝑞(𝑡)‖𝑧𝑖(𝑡,⋅)‖}²₂(𝑡,0)}𝑤(𝜀)𝜇(𝑡)𝑞(𝑡)𝑒_𝜀⊕𝜀(𝜎(𝑡),0)‖𝑧_𝑖(𝑡,⋅)‖²₂)}≤𝑒_𝜀⊕𝜀(𝜎(𝑡),0)^𝑛_𝑖=1‖𝑧_𝑖(𝑡,⋅)‖²₂{(𝜀⊕𝜀)−^𝑚_𝑘=12𝑎_𝑖𝑘𝑙²_𝑘+^𝑛_𝑗=1(|𝑐_𝑖𝑗|𝐹_𝑗+|𝑐_𝑗𝑖|𝐹_𝑖+)max{𝑒_𝜀⊕𝜀(𝜎(𝑡),0),𝑒_{(𝑤(𝜀)−1)𝜇(𝑡)𝑞(𝑡)‖𝑧𝑖(𝑡,⋅)‖}²₂(𝑡,0)}𝑤(𝜀)𝜇(𝑡)𝑞(𝑡)𝑒_𝜀⊕𝜀(𝜎(𝑡),0)}≤0. (3.18)(1)(2)(3)(4)(5)(6)(7)(8)(9)(10)
From (3.17) and (3.18), we have, for 𝑡>0,𝑒_𝜀⊕𝜀(𝑡,0)‖𝑧(𝑡,⋅)‖²=𝑒_𝜀⊕𝜀(𝑡,0)^𝑛_𝑖=1‖𝑧_𝑖(𝑡,⋅)‖²₂=≤𝑉(𝑡,𝑧(t))≤𝑉(0,𝑧(0))_𝑛𝑖=1{‖𝑧_𝑖(0,⋅)‖²₂+1}=^𝑛_𝑖=1{‖𝜙^𝑧_𝑖‖²_𝜏+1}=‖𝜙^𝑧‖²₀+𝑛, (3.19) which implies that ‖𝑧(𝑡,⋅)‖≤𝑀𝑒_⊖𝜀(𝑡,0), (3.20) where 𝑀=‖𝜙^𝑧‖²₀+𝑛>1.
Let 𝑢^∗=(𝑢^∗₁,𝑢^∗₂,…,𝑢^∗_𝑛)^𝑇 and 𝑢^∗∗=(𝑢₁^∗∗,𝑢₂^∗∗,…,𝑢_𝑛^∗∗)^𝑇 be two arbitrary equilibrium solutions of system (1.1)–(1.3). According to (3.20), we get ‖𝑢^∗−𝑢^∗∗‖≤𝑀𝑒_⊖𝜀(𝑡,0)→0(𝑡→+∞), here 𝑀=‖𝑢^∗−𝑢^∗∗‖²₀+𝑛>1. It follows that 𝑢^∗=𝑢^∗∗, that is, the equilibrium solution of (1.1)–(1.3) is unique.
Let 𝑢(𝑡,𝑥)=(𝑢₁(𝑡,𝑥),𝑢₂(𝑡,𝑥),…,𝑢_𝑛(𝑡,𝑥))^𝑇 and 𝑢^∗=(𝑢^∗₁,𝑢^∗₂,…,𝑢^∗_𝑛)^𝑇 be arbitrary solutions and an unique equilibrium solution of (1.1)–(1.3), respectively. In the light of (3.20), we obtain ‖𝑢(𝑡,𝑥)−𝑢^∗‖≤𝑀𝑒_⊖𝜀(𝑡,0), here 𝑀=‖𝜙−𝑢^∗‖²₀+𝑛>1. Thus, by Definition 2.11, we obtain the global exponential stability of unique equilibrium solution of (1.1)–(1.3). The proof is complete.

4. An Illustrative Example

Example 4.1. Consider the following reaction-diffusion recurrent neural network with Dirichlet boundary conditions on time scales: 𝑢^Δ_𝑖(𝑡,𝑥)=^𝑚_𝑘=1𝜕𝜕𝑥_𝑘(𝑎_𝑖𝑘𝜕𝑢_𝑖𝜕𝑥_𝑘)−𝑏_𝑖𝑢_𝑖(𝑡,𝑥)+^𝑛_𝑗=1𝑐_𝑖𝑗𝑓_𝑗(𝑢_𝑗(t,𝑥))+𝐼_𝑖𝑢,(𝑡,𝑥)∈𝕋×Ω,_𝑖(0,𝑥)=𝜙_𝑖𝑢(𝑥),𝑥∈Ω,_𝑖(𝑡,𝑥)=0,(𝑡,𝑥)∈[0,+∞)_𝕋×𝜕Ω, (4.1) where 𝕋⊂ℝ is a time scale and 𝕋∩[0,+∞)=[0,+∞)_𝕋 is unbounded, 𝑓₁(𝑣)=𝑓₂(𝑣)=(𝑒^𝑣−𝑒^−𝑣)/(𝑒^𝑣+𝑒^−𝑣), Ω={𝑥∶|𝑥_𝑖|<1, 𝑖=1,2}, and 𝐼=(𝐼₁,𝐼₂) is the constant input vector. Obviously, 𝑓_𝑗(𝑣) satisfies the Lipschitz condition with 𝐹_𝑗=1. Let 𝑎₁₁=0.5, 𝑎₁₂=0.5, 𝑎₂₁=0.3, 𝑎₂₂=0.7, 𝑏₁=1.5, 𝑏₂=0.4, 𝑐₁₁=0.5,𝑐₁₂=0.4, 𝑐₂₁=0.3, and 𝑐₂₂=0.2. By simple calculation, we have 𝐵−𝐶⁺𝐹=−=,1.5000.40.50.40.30.210011−0.4−0.30.2−2𝑏₁−²_𝑘=12𝑎_1𝑘𝑙²_𝑘+²_𝑗=1(|𝑐_1𝑗|𝐹_𝑗+|𝑐_𝑗1|𝐹₁)=−3.3<0,−2𝑏₂−²_𝑘=12𝑎_2𝑘𝑙²_𝑘+²_𝑗=1(|𝑐_2𝑗|𝐹_𝑗+|𝑐_𝑗2|𝐹₂)=−0.9<0, (4.2) that is, (𝐻₁), (𝐻₂), and (𝐻₃) hold. Hence, it follows from Theorem 3.3 that (4.1) has one unique equilibrium solution which is globally exponentially stable.

Acknowledgment

This work is supported by the National Natural Sciences Foundation of People’s Republic of China under Grant 10971183.