Fixed Point and Asymptotic Analysis of Cellular Neural Networks

Abstract

We firstly employ the fixed point theory to study the stability of cellular neural networks without delays and with time-varying delays. Some novel and concise sufficient conditions are given to ensure the existence and uniqueness of solution and the asymptotic stability of trivial equilibrium at the same time. Moreover, these conditions are easily checked and do not require the differentiability of delays.

1. Introduction

Cellular neural networks (CNNs) were firstly proposed by Chua and Yang in 1988 [1, 2] and have become a research focus for their numerous successful applications in various fields such as optimization, linear, and nonlinear programming, associative memory, pattern recognition, and computer vision. Owing to the finite switching speed of neurons and amplifiers in the implementation of neural networks, it turns out that the time delays are inevitable and therefore the model of delayed cellular neural networks (DCNNs) is of greater realistic significance. Research on the dynamic behaviors of CNNs and DCNNs has received much attention, and nowadays there have been a large number of achievements reported [3–5].

In fact, besides delay effects, stochastic and impulsive as well as diffusing effects are also likely to exist in the neural networks. As a result, they have formed complex CNNs including impulsive delayed reaction-diffusion CNNs, stochastic delayed reaction-diffusion CNNs, and so forth. One can refer to [6–11] for the relevant researches. Synthesizing the existing publications about complex CNNs, we find that Lyapunov method is the primary technique. However, we also notice that there exist lots of difficulties in the applications of corresponding results to practical problems and so it does seem that new methods are needed to address those difficulties.

Encouragingly, Burton and other authors have recently applied the fixed point theory to investigate the stability of deterministic systems and obtained more applicable results, for example, see the monograph [12] and the papers [13–24]. Furthermore, there has been found that the fixed point theory is also effective to the stability analysis of stochastic (delayed) differential equations, see [25–31]. Particularly, in [26–28], Luo used the fixed point theory to study the exponential stability of mild solutions of stochastic partial differential equations with bounded delays and with infinite delays. In [29, 30], Sakthivel and Luo used the fixed point theory to investigate the asymptotic stability in pth moment of mild solutions to nonlinear impulsive stochastic partial differential equations with bounded delays and with infinite delays. In [31], Luo used the fixed point theory to study the exponential stability of stochastic Volterra-Levin equations. With these motivations, we wonder if we can use the fixed point theory to study the stability of complex neural networks, thus obtaining more applicable results.

In the present paper, we aim to discuss the asymptotic stability of CNNs and DCNNs. Our method is based on the contraction mapping theory, which is different from the usual method of Lyapunov theory. Some new and easily checked algebraic criteria are presented ensuring the existence and uniqueness of solution and the asymptotic stability of trivial equilibrium at the same time. These sufficient conditions do not require even the differentiability of delays, let alone the monotone decreasing behavior of delays.

2. Preliminaries

Let R^𝑛 denote the n-dimensional Euclidean space and ‖⋅‖ represent the Euclidean norm. 𝒩≜{1,2,…,𝑛}. R₊=[0,∞). 𝐶(𝑋,𝑌) corresponds to the space of continuous mappings from the topological space 𝑋 to the topological space 𝑌.

In this paper, we consider the cellular neural network described by d𝑥_𝑖(𝑡)d𝑡=−𝑎^𝑖𝑥_𝑖(𝑡)+^𝑛_𝑗=1𝑥𝑏_𝑖𝑗𝑓_𝑗𝑥_𝑗(𝑡),𝑡≥0,(2.1)_𝑖(0)=𝑥_0𝑖,(2.2) and the following cellular neural network with time-varying delays as d𝑥_𝑖(𝑡)d𝑡=−𝑎^𝑖𝑥_𝑖(𝑡)+^𝑛_𝑗=1𝑏_𝑖𝑗𝑓_𝑗𝑥_𝑗(𝑡)+^𝑛_𝑗=1𝑥𝑐_𝑖𝑗𝑔_𝑗𝑥_𝑗𝑡−𝜏_𝑗(𝑡),𝑡≥0(2.3)_𝑖(𝑠)=𝜑_𝑖(𝑠),−𝜏≤𝑠≤0,(2.4) where 𝑖∈𝒩 and 𝑛 is the number of neurons in the neural network. 𝑥_𝑖(𝑡) corresponds to the state of the ith neuron at time 𝑡. 𝐱₀=(𝑥₀₁,…,𝑥_0𝑛)^T∈R^𝑛. 𝑓_𝑗(𝑥_𝑗(𝑡)) denotes the activation function of the jth neuron at time 𝑡 and 𝑔_𝑗(𝑥_𝑗(𝑡−𝜏_𝑗(𝑡))) is the activation function of the jth neuron at time 𝑡−𝜏_𝑗(𝑡). The constant 𝑏_𝑖𝑗 represents the connection weight of the jth neuron on the ith neuron at time 𝑡. The constant 𝑎_𝑖>0 represents the rate with which the ith neuron will reset its potential to the resting state when disconnected from the network and external inputs. The constant 𝑐_𝑖𝑗 represents the connection strength of the jth neuron on the ith neuron at time 𝑡−𝜏_𝑗(𝑡), where 𝜏_𝑗(𝑡) corresponds to the transmission delay along the axon of the jth neuron and satisfies 0≤𝜏_𝑗(𝑡)≤𝜏 (𝜏 is a constant). 𝑓_𝑗(⋅), 𝑔_𝑗(⋅)∈𝐶(R,R). 𝜑(𝑠)=(𝜑₁(𝑠),…,𝜑_𝑛(𝑠))^T∈R^𝑛 and 𝜑_𝑖(𝑠)∈𝐶([−𝜏,0],R). Denote |𝜑|=sup_{𝑠∈[−𝜏,0]}‖𝜑(𝑠)‖.

Throughout this paper, we always assume that 𝑓_𝑗(0)=𝑔_𝑗(0)=0 for 𝑗∈𝒩 and therefore (2.1) and (2.3) admit a trivial equilibrium 𝐱=0.

Denote by 𝐱(𝑡;0,𝐱₀)=(𝑥₁(𝑡;0,𝑥₀₁),…,𝑥_𝑛(𝑡;0,𝑥_0𝑛))^T∈R^𝑛 the solution of (2.1) with the initial condition (2.2) and denote by 𝐱(𝑡;𝑠,𝜑)=(𝑥₁(𝑡;𝑠,𝜑₁),…,𝑥_𝑛(𝑡;𝑠,𝜑_𝑛))^T∈R^𝑛 the solution of (2.3) with the initial condition (2.4).

Definition 2.1 (see [32]). The trivial equilibrium 𝐱=0 of (2.1) is said to be stable if for any 𝜀>0, there exists 𝛿>0 such that for any initial condition 𝐱₀ satisfying ‖𝐱₀‖<𝛿, ‖‖𝐱𝑡;0,𝐱₀‖‖<𝜀,𝑡≥0.(2.5)

Definition 2.2 (see [32]). The trivial equilibrium 𝐱=0 of (2.1) is said to be asymptotically stable if it is stable and for any 𝐱₀∈R^𝑛, lim_𝑡→∞‖‖𝐱𝑡;0,𝐱₀‖‖=0.(2.6)

Definition 2.3 (see [32]). The trivial equilibrium 𝐱=0 of (2.3) is said to be stable if for any 𝜀>0, there exists 𝛿>0 such that for any initial condition 𝜑(𝑠)∈𝐶([−𝜏,0],R^𝑛) satisfying |𝜑|<𝛿, ‖𝐱(𝑡;𝑠,𝜑)‖<𝜀,𝑡≥0.(2.7)

Definition 2.4 (see [32]). The trivial equilibrium 𝐱=0 of (2.3) is said to be asymptotically stable if it is stable and for any initial condition 𝜑(𝑠)∈𝐶([−𝜏,0],R^𝑛), lim_𝑡→∞‖𝐱(𝑡;𝑠,𝜑)‖=0.(2.8)

The consideration of this paper is based on the following fixed point theorem.

Lemma 2.5 (see [33]). Let Υ be a contraction operator on a complete metric space Θ, then there exists a unique point 𝜁∈Θ for which Υ(𝜁)=𝜁.

3. Asymptotic Stability of Cellular Neural Networks

In this section, we will simultaneously consider the existence and uniqueness of solution to (2.1)-(2.2) and the asymptotic stability of trivial equilibrium 𝐱=0 of (2.1) by means of the contraction mapping principle. Before proceeding, we firstly introduce the following assumption:(A1) There exist nonnegative constants 𝑙_𝑗 such that for 𝜂, 𝜐∈R, ||𝑓_𝑗(𝜂)−𝑓_𝑗(𝜐)||≤𝑙_𝑗||𝜂−𝜐||,𝑗∈𝒩.(3.1)

Let 𝒮=𝒮₁×𝒮₂×⋯×𝒮_𝑛, where 𝒮_𝑖 (𝑖∈𝒩) is the space consisting of continuous functions 𝜙_𝑖(𝑡)∶R₊→R such that 𝜙_𝑖(0)=𝑥_0𝑖 and 𝜙_𝑖(𝑡)→0 as 𝑡→∞, here 𝑥_0𝑖 is the same as defined in Section 2. Also 𝒮 is a complete metric space when it is equipped with a metric defined by 𝑑𝐪(𝑡),𝐡(𝑡)=sup_𝑛𝑡≥0_𝑖=1||𝑞_𝑖(𝑡)−ℎ_𝑖(𝑡)||,(3.2) where 𝐪(𝑡)=(𝑞₁(𝑡),…,𝑞_𝑛(𝑡))∈𝒮 and 𝐡(𝑡)=(ℎ₁(𝑡),…,ℎ_𝑛(𝑡))∈𝒮.

Theorem 3.1. Assume the condition (A1) holds. If the following inequalities hold ^𝑛_𝑖=1𝑎1_𝑖max_𝑗||𝑏_𝑖𝑗𝑙_𝑗||<1,max_𝑖∈𝒩𝜆_𝑖<1√𝑛,(3.3) where 𝜆_𝑖=(1/𝑎_𝑖)∑^𝑛𝑗=1|𝑏_𝑖𝑗𝑙_𝑗|, then the trivial equilibrium 𝐱=0 of (2.1) is asymptotically stable.

Proof. Multiplying both sides of (2.1) with e^𝑎𝑖𝑡 gives de^𝑎𝑖𝑡𝑥_𝑖(𝑡)=e^𝑎𝑖𝑡d𝑥_𝑖(𝑡)+𝑎_𝑖𝑥_𝑖(𝑡)e^𝑎𝑖𝑡d𝑡=e^{𝑎𝑖𝑡𝑛}_𝑗=1𝑏_𝑖𝑗𝑓_𝑗𝑥_𝑗(𝑡)d𝑡,𝑡≥0,𝑖∈𝒩,(3.4) which yields after integrating from 0 to 𝑡 as 𝑥_𝑖(𝑡)=𝑥_0𝑖e^{−𝑎𝑖𝑡}+e^{−𝑎𝑖𝑡}^𝑡0e^{𝑎𝑖𝑠𝑛}_𝑗=1𝑏_𝑖𝑗𝑓_𝑗𝑥_𝑗(𝑠)ds,𝑡≥0,𝑖∈𝒩.(3.5) Now, for any 𝐲(𝑡)=(𝑦₁(𝑡),…,𝑦_𝑛(𝑡))∈𝒮, we define the following operator Φ acting on 𝒮 as Φ𝐲(𝑡)=Φ𝑦₁(𝑡),…,Φ𝑦_𝑛(𝑡),𝑡≥0,(3.6) where Φ𝑦_𝑖(𝑡)=𝑥_0𝑖e^{−𝑎𝑖𝑡}+e^{−𝑎𝑖𝑡}^𝑡0e^{𝑎𝑖𝑠𝑛}_𝑗=1𝑏_𝑖𝑗𝑓_𝑗𝑦_𝑗(𝑠)ds,𝑖∈𝒩.(3.7)
The following proof is based on the contraction mapping principle, which can be divided into two steps as follows.
Step 1. We need to prove Φ(𝒮)⊂𝒮. Recalling the construction of 𝒮, we know that it is necessary to show the continuity of Φ on [0,∞) and Φ(𝑦_𝑖)(𝑡)|_𝑡=0=𝑥_0𝑖 as well as lim_𝑡→∞Φ(𝑦_𝑖)(𝑡)=0 for 𝑖∈𝒩.
From (3.7), it is easy to see Φ(𝑦_𝑖)(𝑡)|_𝑡=0=𝑥_0𝑖. Moreover, for a fixed time 𝑡₁≥0, we have Φ𝑦_𝑖𝑡₁+𝑟−Φ𝑦_𝑖𝑡₁=𝑥_0𝑖e^{−𝑎𝑖(𝑡1+𝑟)}−𝑥_0𝑖e^{−𝑎𝑖𝑡1}+e^{−𝑎𝑖(𝑡1+𝑟)}^𝑡10+𝑟e^{𝑎𝑖𝑠𝑛}_𝑗=1−e_{−𝑎𝑖𝑡1}𝑏_𝑖𝑗𝑓_𝑗𝑦_𝑗(𝑠)ds^𝑡1₀e^{𝑎𝑖𝑠𝑛}_𝑗=1𝑏_𝑖𝑗𝑓_𝑗𝑦_𝑗(𝑠)ds.(3.8) It is not difficult to see that Φ(𝑦_𝑖)(𝑡₁+𝑟)−Φ(𝑦_𝑖)(𝑡₁)→0 as 𝑟→0 which implies Φ is continuous on [0,∞).
Next we shall prove lim_𝑡→∞Φ(𝑦_𝑖)(𝑡)=0 for 𝑦_𝑖(𝑡)∈𝒮_𝑖. Since 𝑦_𝑗(𝑡)∈𝒮_𝑗, we get lim_𝑡→∞𝑦_𝑗(𝑡)=0. Then for any 𝜀>0, there exists a 𝑇_𝑗>0 such that 𝑠≥𝑇_𝑗 implies |𝑦_𝑗(𝑠)|<𝜀. Choose 𝑇^∗=max_𝑗∈𝒩{𝑇_𝑗}. It is then derived form (A1) that e^{−𝑎𝑖𝑡}^𝑡0e^{𝑎𝑖𝑠𝑛}_𝑗=1≤e_{−𝑎𝑖𝑡}𝑏_𝑖𝑗𝑓_𝑗𝑦_𝑗(𝑠)ds^𝑡0e^{𝑎𝑖𝑠𝑛}_𝑗=1||𝑏_𝑖𝑗=e_{−𝑎𝑖𝑡}𝑙_𝑗||||𝑦_𝑗(𝑠)||ds^𝑇∗₀e^{𝑎𝑖𝑠𝑛}_𝑗=1||𝑏_𝑖𝑗𝑙_𝑗||||𝑦_𝑗(𝑠)||ds+e^{−𝑎𝑖𝑡}^𝑡𝑇_∗e^{𝑎𝑖𝑠𝑛}_𝑗=1||𝑏_𝑖𝑗≤e_{−𝑎𝑖𝑡𝑛}𝑙_𝑗||||𝑦_𝑗(𝑠)||ds_𝑗=1||𝑏^𝑖𝑗𝑙_𝑗||sup_{𝑠∈0,𝑇∗}||𝑦_𝑗(𝑠)||^𝑇∗₀e^𝑎𝑖𝑠ds+𝜀e^{−𝑎𝑖𝑡𝑛}_𝑗=1||𝑏_𝑖𝑗≤e_{−𝑎𝑖𝑡𝑛}𝑙_𝑗||^𝑡𝑇_∗e^𝑎𝑖𝑠ds_𝑗=1||𝑏^𝑖𝑗𝑙_𝑗||sup_{𝑠∈0,𝑇∗}||𝑦_𝑗(𝑠)||^𝑇∗₀e^𝑎𝑖𝑠ds+𝜀𝑎_{𝑖𝑛𝑗=1}||𝑏_𝑖𝑗𝑙_𝑗||.(3.9) As 𝑎_𝑖>0, we obtain e^{−𝑎𝑖𝑡}∫^𝑡0e^𝑎𝑖𝑠∑^𝑛𝑗=1𝑏_𝑖𝑗𝑓_𝑗(𝑦_𝑗(𝑠))ds→0 as 𝑡→∞. So lim_𝑡→∞Φ(𝑦_𝑖)(𝑡)=0 for 𝑖∈𝒩. We therefore conclude that Φ(𝒮)⊂𝒮.
Step 2. We need to prove Φ is contractive. For any 𝐲=(𝑦₁(𝑡),…,𝑦_𝑛(𝑡))∈𝒮 and 𝐳=(𝑧₁(𝑡),…,𝑧_𝑛(𝑡))∈𝒮, we compute sup_{[]𝑛𝑡∈0,𝑇}_𝑖=1≤su||pΦ_{[]𝑛𝑡∈0,𝑇}𝑦_𝑖(𝑡)−Φ𝑧_𝑖(𝑡)||_𝑖=1e^{−𝑎𝑖𝑡}^𝑡0e^{𝑎𝑖𝑠𝑛}_𝑗=1||𝑏_𝑖𝑗≤sup_{[]𝑛𝑡∈0,𝑇}||||𝑓_𝑗𝑦_𝑗(𝑠)−𝑓_𝑗𝑧_𝑗(𝑠)||ds_𝑖=1e^{−𝑎𝑖𝑡}^𝑡0e^{𝑎𝑖𝑠𝑛}_𝑗=1||𝑏_𝑖𝑗≤sup_{[]𝑛𝑡∈0,𝑇}𝑙_𝑗||||𝑦_𝑗(𝑠)−𝑧_𝑗(𝑠)||ds_𝑖=1max^𝑗∈𝒩||𝑏_𝑖𝑗𝑙_𝑗||sup_{[]𝑠∈0,𝑇}^𝑛_𝑗=1||𝑦_𝑗(𝑠)−𝑧_𝑗(𝑠)||e^{−𝑎𝑖𝑡}^𝑡0≤e^𝑎𝑖𝑠ds_𝑛_𝑖=1𝑎1_𝑖max_𝑗∈𝒩||𝑏_𝑖𝑗𝑙_𝑗||sup_{[]𝑠∈0,𝑇}^𝑛_𝑗=1||𝑦_𝑗(𝑠)−𝑧_𝑗(𝑠)||.(3.10) As ∑^𝑛𝑖=1{(1/𝑎_𝑖)max_𝑗∈𝒩{|𝑏_𝑖𝑗𝑙_𝑗|}}<1, Φ is a contraction mapping.
Therefore, by the contraction mapping principle, we see there must exist a unique fixed point 𝐮(⋅) of Φ in 𝒮 which means 𝐮^T(⋅) is the solution of (2.1)-(2.2) and ‖𝐮^T(⋅)‖→0 as 𝑡→∞.
To obtain the asymptotic stability, we still need to prove that the trivial equilibrium 𝐱=0 of (2.1) is stable. For any 𝜀>0, from the conditions of Theorem 3.1, we can find 𝛿 satisfying 0<𝛿<𝜀 such that 𝛿+max_𝑖∈𝒩{𝜆_𝑖}𝜀≤𝜀/√𝑛.
Let ‖𝐱₀‖<𝛿. According to what have been discussed above, we know that there must exist a unique solution 𝐱(𝑡;0,𝐱₀)=(𝑥₁(𝑡;0,𝑥₀₁),…,𝑥_𝑛(𝑡;0,𝑥_0𝑛))^T∈R^𝑛 to (2.1)-(2.2), and 𝑥_𝑖(𝑡)=Φ𝑥_𝑖(𝑡)=𝐽₁(𝑡)+𝐽₂(𝑡),𝑡≥0,(3.11) where 𝐽₁(𝑡)=𝑥_0𝑖e^{−𝑎𝑖𝑡}, 𝐽₂(𝑡)=e^{−𝑎𝑖𝑡}∫^𝑡0e^𝑎𝑖𝑠∑^𝑛𝑗=1𝑏_𝑖𝑗𝑓_𝑗(𝑥_𝑗(𝑠))ds.
Suppose there exists 𝑡^∗>0 such that ‖𝐱(𝑡^∗;0,𝐱₀)‖=𝜀 and ‖𝐱(𝑡;0,𝐱₀)‖<𝜀 as 0≤𝑡<𝑡^∗. It follows from (3.11) that |𝑥_𝑖(𝑡^∗)|≤|𝐽₁(𝑡^∗)|+|𝐽₂(𝑡^∗)|.
As |𝐽₁(𝑡^∗)|=|𝑥_0𝑖e^{−𝑎𝑖𝑡∗}|≤𝛿 and |𝐽₂(𝑡^∗)|≤e^{−𝑎𝑖𝑡∗}∫^𝑡∗₀e^𝑎𝑖𝑠∑^𝑛𝑗=1|𝑏_𝑖𝑗𝑙_𝑗𝑥_𝑗(𝑠)|ds<(𝜀/𝑎_𝑖)∑^𝑛𝑗=1|𝑏_𝑖𝑗𝑙_𝑗|, we obtain |𝑥_𝑖(𝑡^∗)|<𝛿+𝜆_𝑖𝜀. Hence ‖‖𝐱𝑡^∗;0,𝐱₀‖‖²=^𝑛_𝑖=1||𝑥^𝑖𝑡^∗||²<^𝑛_𝑖=1||𝛿+𝜆^𝑖𝜀||²≤𝑛|||𝛿+max_𝑖∈𝒩𝜆_𝑖𝜀|||²≤𝜀².(3.12)
This contradicts to the assumption of ‖𝐱(𝑡^∗;0,𝐱₀)‖=𝜀. Therefore, ‖𝐱(𝑡;0,𝐱₀)‖<𝜀 holds for all 𝑡≥0. This completes the proof.

4. Asymptotic Stability of Delayed Cellular Neural Networks

In this section, we will simultaneously consider the existence and uniqueness of solution to (2.3)-(2.4) and the asymptotic stability of trivial equilibrium 𝐱=0 of (2.3) by means of the contraction mapping principle. Before proceeding, we give the assumption as follows.(A2) There exist nonnegative constants 𝑘_𝑗 such that for 𝜂, 𝜐∈R, ||𝑔_𝑗(𝜂)−𝑔_𝑗(𝜐)||≤𝑘_𝑗||𝜂−𝜐||,𝑗∈𝒩.(4.1)

Let ℋ=ℋ₁×⋯×ℋ_𝑛, where ℋ_𝑖 (𝑖∈𝒩) is the space consisting of continuous functions 𝜙_𝑖(𝑡)∶[−𝜏,∞)→R such that 𝜙_𝑖(𝑠)=𝜑_𝑖(𝑠) on 𝑠∈[−𝜏,0] and 𝜙_𝑖(𝑡)→0 as 𝑡→∞, here 𝜑_𝑖(𝑠) is the same as defined in Section 2. Also ℋ is a complete metric space when it is equipped with a metric defined by 𝑑𝐪(𝑡),𝐡(𝑡)=sup_{𝑛𝑡≥−𝜏}_𝑖=1||𝑞_𝑖(𝑡)−ℎ_𝑖(𝑡)||,(4.2) where 𝐪(𝑡)=(𝑞₁(𝑡),…,𝑞_𝑛(𝑡))∈ℋ and 𝐡(𝑡)=(ℎ₁(𝑡),…,ℎ_𝑛(𝑡))∈ℋ.

Theorem 4.1. Assume the conditions (A1)-(A2) hold. If the following inequalities hold ^𝑛_𝑖=1𝑎1_𝑖max_𝑗∈𝒩||𝑏_𝑖𝑗𝑙_𝑗||+max_𝑗∈𝒩||𝑐_𝑖𝑗𝑘_𝑗||<1,max_𝑖∈𝒩𝜆_𝑖∗<1√𝑛(4.3) where 𝜆_𝑖∗=(1/𝑎_𝑖)∑^𝑛𝑗=1|𝑏_𝑖𝑗𝑙_𝑗|+(1/𝑎_𝑖)∑^𝑛𝑗=1|𝑐_𝑖𝑗𝑘_𝑗|, then the trivial equilibrium 𝐱=0 of (2.3) is asymptotically stable.

Proof. Multiplying both sides of (2.3) with e^𝑎𝑖𝑡 gives de^𝑎𝑖𝑡𝑥_𝑖(𝑡)==ee^𝑎𝑖𝑡_𝑎𝑖𝑡d𝑥_𝑖(𝑡)+𝑎_𝑖𝑥_𝑖(𝑡)e^𝑎𝑖𝑡d𝑡^𝑛_𝑗=1𝑏_𝑖𝑗𝑓_𝑗𝑥_𝑗(𝑡)+^𝑛_𝑗=1𝑐_𝑖𝑗𝑔_𝑗𝑥_𝑗𝑡−𝜏_𝑗(𝑡)d𝑡,(4.4) which yields after integrating from 0 to 𝑡 as 𝑥_𝑖(𝑡)=𝜑_𝑖(0)e^{−𝑎𝑖𝑡}+e^{−𝑎𝑖𝑡}^𝑡0e^𝑎𝑖𝑠^𝑛_𝑗=1𝑏_𝑖𝑗𝑓_𝑗𝑥_𝑗(𝑠)+^𝑛_𝑗=1𝑐_𝑖𝑗𝑔_𝑗𝑥_𝑗𝑠−𝜏_𝑗(𝑠)ds.(4.5)
Now for any 𝐲(𝑡)=(𝑦₁(𝑡),…,𝑦_𝑛(𝑡))∈ℋ, we define the following operator 𝜋 acting on ℋ𝜋𝐲(𝑡)=𝜋𝑦₁(𝑡),…,𝜋𝑦_𝑛(𝑡),(4.6) where 𝜋𝑦_𝑖(𝑡)=𝜑_𝑖(0)e^{−𝑎𝑖𝑡}+e^{−𝑎𝑖𝑡}^𝑡0e^𝑎𝑖𝑠^𝑛_𝑗=1𝑏_𝑖𝑗𝑓_𝑗𝑦_𝑗(𝑠)+^𝑛_𝑗=1𝑐_𝑖𝑗𝑔_𝑗𝑦_𝑗𝑠−𝜏_𝑗(𝑠)ds,𝑡≥0,(4.7) and 𝜋(𝑦_𝑖)(𝑠)=𝜑_𝑖(𝑠) on 𝑠∈[−𝜏,0] for 𝑖∈𝒩.
Similar to the proof of Theorem 3.1, we shall apply the contraction mapping principle to prove Theorem 4.1. The subsequent proof can be divided into two steps.
Step 1. We need prove 𝜋(ℋ)⊂ℋ. To prove 𝜋(ℋ)⊂ℋ, it is necessary to show the continuity of 𝜋 on [−𝜏,∞) and lim_𝑡→∞𝜋(𝑦_𝑖)(𝑡)=0 for 𝑦_𝑖(𝑡)∈ℋ_𝑖 and 𝑖∈𝒩. In light of (4.7), we have, for a fixed time 𝑡₁≥0, 𝜋𝑦_𝑖𝑡₁+𝑟−𝜋𝑦_𝑖𝑡₁=𝐼₁+𝐼₂+𝐼_3,(4.8) where 𝐼𝐼₁₂=𝜑_𝑖(0)e^{−𝑎𝑖(𝑡1+𝑟)}−𝜑_𝑖(0)e^{−𝑎𝑖𝑡1},=e^{−𝑎𝑖(𝑡1+𝑟)}^𝑡10+𝑟e^{𝑎𝑖𝑠𝑛}_𝑗=1𝑏_𝑖𝑗𝑓_𝑗𝑦_𝑗(𝑠)ds−e^{−𝑎𝑖𝑡1}^𝑡1₀e^{𝑎𝑖𝑠𝑛}_𝑗=1𝐼𝑏_𝑖𝑗𝑓_𝑗𝑦_𝑗(𝑠)ds,₃=e^{−𝑎𝑖(𝑡1+𝑟)}^𝑡10+𝑟e^{𝑎𝑖𝑠𝑛}_𝑗=1𝑐_𝑖𝑗𝑔_𝑗𝑦_𝑗𝑠−𝜏_𝑗(𝑠)ds−e^{−𝑎𝑖𝑡1}^𝑡1₀e^{𝑎𝑖𝑠𝑛}_𝑗=1𝑐_𝑖𝑗𝑔_𝑗𝑦_𝑗𝑠−𝜏_𝑗(𝑠)ds.(4.9) It is easy to see that lim_𝑟→0{𝜋(𝑦_𝑖)(𝑡₁+𝑟)−𝜋(𝑦_𝑖)(𝑡₁)}=0. Thus, 𝜋 is continuous on [0,∞). Noting 𝜑_𝑖(𝑠)∈𝐶([−𝜏,0],R) and 𝜋(𝑦_𝑖)(0)=𝜑_𝑖(0), we obtain 𝜋 is indeed continuous on [−𝜏,∞).
Next, we will prove lim_𝑡→∞𝜋(𝑦_𝑖)(𝑡)=0 for 𝑦_𝑖(𝑡)∈ℋ_𝑖. As we did in Section 3, we know lim_𝑡→∞e^{−𝑎𝑖𝑡}=0 and e^{−𝑎𝑖𝑡}∫^𝑡0e^𝑎𝑖𝑠∑^𝑛𝑗=1𝑏_𝑖𝑗𝑓_𝑗(𝑦_𝑗(𝑠))ds→0 as 𝑡→∞. In what follows, we will show e^{−𝑎𝑖𝑡}∫^𝑡0e^𝑎𝑖𝑠∑^𝑛𝑗=1𝑐_𝑖𝑗𝑔_𝑗(𝑦_𝑗(𝑠−𝜏_𝑗(𝑠)))ds→0 as 𝑡→∞. In fact, since 𝑦_𝑗(𝑡)∈ℋ_𝑗, we have lim_𝑡→∞𝑦_𝑗(𝑡)=0. Then for any 𝜀>0, there exists a 𝑇_𝑗>0 such that 𝑠≥𝑇_𝑗−𝜏 implies |𝑦_𝑗(𝑠)|<𝜀. Select 𝑇=max_𝑗∈𝒩{𝑇_𝑗}. It is then derived from (A2) that e^{−𝑎𝑖𝑡}^𝑡0e^{𝑎𝑖𝑠𝑛}_𝑗=1≤e_{−𝑎𝑖𝑡}𝑐_𝑖𝑗𝑔_𝑗𝑦_𝑗𝑠−𝜏_𝑗(𝑠)ds^𝑡0e^{𝑎𝑖𝑠𝑛}_𝑗=1||𝑐_𝑖𝑗=e_{−𝑎𝑖𝑡}𝑘_𝑗||||𝑦_𝑗𝑠−𝜏_𝑗(𝑠)||ds𝑇₀e^{𝑎𝑖𝑠𝑛}_𝑗=1||𝑐_𝑖𝑗+e_{−𝑎𝑖𝑡}𝑘_𝑗||||𝑦_𝑗𝑠−𝜏_𝑗(𝑠)||ds^𝑡𝑇e^{𝑎𝑖𝑠𝑛}_𝑗=1||𝑐_𝑖𝑗≤e_{−𝑎𝑖𝑡𝑛}𝑘_𝑗||||𝑦_𝑗𝑠−𝜏_𝑗(𝑠)||ds_𝑗=1⎧⎪⎨⎪⎩||𝑐_𝑖𝑗𝑘_𝑗||sup_{𝑠∈−𝜏,}𝑇_||𝑦_𝑗(𝑠)||⎫⎪⎬⎪⎭⎧⎪⎨⎪⎩₀𝑇e^aisds⎫⎪⎬⎪⎭+𝜀e^{−𝑎𝑖𝑡𝑛}_𝑗=1||𝑐_𝑖𝑗𝑘_𝑗||^𝑡𝑇e^𝑎𝑖𝑠≤e_{−𝑎𝑖𝑡𝑛}ds_𝑗=1⎧⎪⎨⎪⎩||𝑐_𝑖𝑗𝑘_𝑗||sup_{𝑠∈−𝜏,}𝑇_||𝑦_𝑗(𝑠)||⎫⎪⎬⎪⎭⎧⎪⎨⎪⎩₀𝑇e^𝑎𝑖𝑠ds⎫⎪⎬⎪⎭+𝜀𝑎_{𝑖𝑛𝑗=1}||𝑐_𝑖𝑗𝑘_𝑗||.(4.10) As lim_𝑡→∞e^{−𝑎𝑖𝑡}=0, we obtain e^{−𝑎𝑖𝑡}∫^𝑡0e^𝑎𝑖𝑠∑^𝑛𝑗=1𝑐_𝑖𝑗𝑔_𝑗(𝑦_𝑗(𝑠−𝜏_𝑗(𝑠)))ds→0 as 𝑡→∞, which leads to lim_𝑡→∞𝜋(𝑦_𝑖)(𝑡)=0 for 𝑦_𝑖(𝑡)∈ℋ_𝑖 and 𝑖∈𝒩. We therefore conclude 𝜋(ℋ)⊂ℋ.
Step 2. We need to prove 𝜋 is contractive. For any 𝐲=(𝑦₁(𝑡),…,𝑦_𝑛(𝑡))∈ℋ and 𝐳=(𝑧₁(𝑡),…,𝑧_𝑛(𝑡))∈ℋ, we estimate ^𝑛_𝑖=1||≤𝜋𝑦_𝑖(𝑡)−𝜋𝑧_𝑖(𝑡)_𝑛||_𝑖=1e^{−𝑎𝑖𝑡}^𝑡0e^{𝑎𝑖𝑠𝑛}_𝑗=1||𝑏_𝑖𝑗+||||𝑓_𝑗𝑦_𝑗(𝑠)−𝑓_𝑗𝑧_𝑗(𝑠)||ds_𝑛_𝑖=1e^{−𝑎𝑖𝑡}^𝑡0e^{𝑎𝑖𝑠𝑛}_𝑗=1||𝑐_𝑖𝑗≤||||𝑔_𝑗𝑦_𝑗𝑠−𝜏_𝑗(𝑠)−𝑔_𝑗𝑧_𝑗𝑠−𝜏_𝑗(𝑠)||ds_𝑛_𝑖=1e^{−𝑎𝑖𝑡}^𝑡0e^{𝑎𝑖𝑠𝑛}_𝑗=1||𝑏_𝑖𝑗+𝑙_𝑗||||𝑦_𝑗(𝑠)−𝑧_𝑗(𝑠)||ds_𝑛_𝑖=1e^{−𝑎𝑖𝑡}^𝑡0e^{𝑎𝑖𝑠𝑛}_𝑗=1||𝑐_𝑖𝑗≤𝑘_𝑗||||𝑦_𝑗𝑠−𝜏_𝑗(𝑠)−𝑧_𝑗𝑠−𝜏_𝑗(𝑠)||ds_𝑛_𝑖=1max^𝑗∈𝒩||𝑏_𝑖𝑗𝑙_𝑗||sup_{[]𝑠∈0,𝑡}^𝑛_𝑗=1||𝑦_𝑗(𝑠)−𝑧_𝑗(𝑠)||𝑒^{−𝑎𝑖𝑡}^𝑡0+𝑒^𝑎𝑖𝑠ds_𝑛_𝑖=1max^𝑗∈𝒩||𝑐_𝑖𝑗𝑘_𝑗||sup_{[]𝑠∈−𝜏,𝑡}^𝑛_𝑗=1||𝑦_𝑗(𝑠)−𝑧_𝑗(𝑠)||e^{−𝑎𝑖𝑡}^𝑡0≤e^𝑎𝑖𝑠ds_𝑛_𝑖=1𝑎1_𝑖max_𝑗∈𝒩||𝑏_𝑖𝑗𝑙_𝑗||sup_{[]𝑠∈0,𝑡}^𝑛_𝑗=1+||𝑦_𝑗(𝑠)−𝑧_𝑗(𝑠)_𝑛||_𝑖=1𝑎1_𝑖max_𝑗∈𝒩||𝑐_𝑖𝑗𝑘_𝑗||sup_{[]𝑠∈−𝜏,𝑡}^𝑛_𝑗=1||𝑦_𝑗(𝑠)−𝑧_𝑗(𝑠)||.(4.11)
Hence, sup_{[]𝑛𝑡∈−𝜏,𝑇}_𝑖=1≤||𝜋𝑦_𝑖(𝑡)−𝜋𝑧_𝑖(𝑡)_𝑛||_𝑖=1𝑎1_𝑖max_𝑗∈𝒩||𝑏_𝑖𝑗𝑙_𝑗||+^𝑛_𝑖=1𝑎1_𝑖max_𝑗∈𝒩||𝑐_𝑖𝑗𝑘_𝑗||sup_{[]𝑠∈−𝜏,𝑇}^𝑛_𝑗=1||𝑦_𝑗(𝑠)−𝑧_𝑗(𝑠)||.(4.12)
As ∑^𝑛𝑖=1{(1/𝑎_𝑖)(max_𝑗∈𝒩|𝑏_𝑖𝑗𝑙_𝑗|+max_𝑗∈𝒩|𝑐_𝑖𝑗𝑘_𝑗|)}<1, 𝜋 is a contraction mapping and hence there exists a unique fixed point 𝐮(⋅) of 𝜋 in ℋ which means 𝐮^T(⋅) is the solution of (2.3)-(2.4) and ‖𝐮^T(⋅)‖→0 as 𝑡→∞.
To obtain the asymptotic stability, we still need to prove that the trivial equilibrium of (2.3) is stable. For any 𝜀>0, from the conditions of Theorem 4.1, we can find 𝛿 satisfying 0<𝛿<𝜀 such that 𝛿+max_𝑖∈𝒩{𝜆^∗𝑖}𝜀≤𝜀/√𝑛.
Let |𝜑|<𝛿. According to what have been discussed above, we know that there exists a unique solution 𝐱(𝑡;𝑠,𝜑)=(𝑥₁(𝑡;𝑠,𝜑₁),…,𝑥_𝑛(𝑡;𝑠,𝜑_𝑛))^T∈R^𝑛 to (2.3)-(2.4), and 𝑥_𝑖(𝑡)=𝜋𝑥_𝑖(𝑡)=𝐽₁(𝑡)+𝐽₂(𝑡)+𝐽₃(𝑡),𝑡≥0,(4.13) where 𝐽₁(𝑡)=𝑥_0𝑖e^{−𝑎𝑖𝑡},𝐽₂(𝑡)=e^{−𝑎𝑖𝑡}^𝑡0e^{𝑎𝑖𝑠𝑛}_𝑗=1𝐽𝑏_𝑖𝑗𝑓_𝑗𝑥_𝑗(𝑠)ds,₃=e^{−𝑎𝑖𝑡}^𝑡0e^{𝑎𝑖𝑠𝑛}_𝑗=1𝑐_𝑖𝑗𝑔_𝑗𝑥_𝑗𝑠−𝜏_𝑗(𝑠)ds.(4.14)
Suppose there exists 𝑡^∗>0 such that ‖𝐱(𝑡^∗;𝑠,𝜑)‖=𝜀 and ‖𝐱(𝑡;𝑠,𝜑)‖<𝜀 as 0≤𝑡<𝑡^∗. It follows from (4.13) that |𝑥_𝑖(𝑡^∗)|≤|𝐽₁(𝑡^∗)|+|𝐽₂(𝑡^∗)|+|𝐽₃(𝑡^∗)|.
As |𝐽₁(𝑡^∗)|=|𝑥_0𝑖e^{−𝑎𝑖𝑡∗}|≤𝛿, |𝐽₂(𝑡^∗)|<(𝜀/𝑎_𝑖)∑^𝑛𝑗=1|𝑏_𝑖𝑗𝑙_𝑗| and ||𝐽₃𝑡^∗||≤e^{−𝑎𝑖𝑡∗}^𝑡∗₀e^{𝑎𝑖𝑠𝑛}_𝑗=1||𝑐_𝑖𝑗𝑘_𝑗𝑥_𝑗𝑠−𝜏_𝑗(𝑠)||ds<𝜀𝑎_{𝑖𝑛𝑗=1}||𝑐_𝑖𝑗𝑘_𝑗||,(4.15) we obtain |𝑥_𝑖(𝑡^∗)|<𝛿+𝜆_𝑖∗𝜀. Hence ‖‖𝐱𝑡^∗;𝑠,𝜑‖‖²=^𝑛_𝑖=1||𝑥^𝑖𝑡^∗||²<^𝑛_𝑖=1||𝛿+𝜆^∗𝑖𝜀||²≤𝑛|||𝛿+max_𝑖∈𝒩𝜆^∗𝑖𝜀|||²≤𝜀².(4.16)
This contradicts to the assumption of ‖𝐱(𝑡^∗;𝑠,𝜑)‖=𝜀. Therefore, ‖𝐱(𝑡;𝑠,𝜑)‖<𝜀 holds for all 𝑡≥0. This completes the proof.

Remark 4.2. In Theorems 3.1 and 4.1, we use the contraction mapping principle to study the existence and uniqueness of solution and the asymptotic stability of trivial equilibrium at the same time, while Lyapunov method fails to do this.

Remark 4.3. The provided sufficient conditions in Theorem 4.1 do not require even the differentiability of delays, let alone the monotone decreasing behavior of delays which is necessary in some relevant works.

5. Example

Consider the following two-dimensional cellular neural network with time-varying delays d𝑥_𝑖(𝑡)d𝑡=−𝑎^𝑖𝑥_𝑖(𝑡)+²_𝑗=1𝑏_𝑖𝑗𝑓_𝑗𝑥_𝑗(𝑡)+²_𝑗=1𝑐_𝑖𝑗𝑔_𝑗𝑥_𝑗𝑡−𝜏_𝑗(𝑡),(5.1) with the initial conditions 𝑥₁(𝑠)=cos(𝑠), 𝑥₂(𝑠)=sin(𝑠) on −𝜏≤𝑠≤0, where 𝑎₁=𝑎₂=3,𝑏₁₁=0,𝑏₁₂=1/7,𝑏₂₁=−1/7,𝑏₂₂=−1/7,𝑐₁₁=3/7,𝑐₁₂=2/7,𝑐₂₁=0,𝑐₂₂=1/7, 𝑓_𝑗(𝑠)=𝑔_𝑗(𝑠)=(|𝑠+1|−|𝑠−1|)/2, 𝜏_𝑗(𝑡) is bounded by 𝜏.

It is easily to know that 𝑙_𝑗=𝑘_𝑗=1 for 𝑗=1,2. Compute ²_𝑖=1𝑎1_𝑖max_𝑗=1,2||𝑏_𝑖𝑗𝑙_𝑗||+max_𝑗=1,2||𝑐_𝑖𝑗𝑘_𝑗||<1,max_𝑖∈𝒩𝑎1_{𝑖𝑛𝑗=1}||𝑏_𝑖𝑗𝑙_𝑗||+1𝑎_{𝑖𝑛𝑗=1}||𝑐_𝑖𝑗𝑘_𝑗||<1√2.(5.2) From Theorem 4.1, we conclude that the trivial equilibrium 𝐱=0 of this two-dimensional cellular neural network is asymptotically stable.

Acknowledgments

This work is supported by National Natural Science Foundation of China under Grant 60904028 and 71171116.