ABSTRACT

In this paper, based on the theory of fractional-order calculus, we obtain some sufficient conditions for the uniform stability of fractional-order fuzzy BAM neural networks with delays in the leakage terms. Moreover, the existence, uniqueness and stability of its equilibrium point are also proved. A numerical example is presented to demonstrate the validity and feasibility of the proposed results.

Keywords:

Fractional-Order, Fuzzy BAM Neural Networks, Uniform Stability, Delay, Leakage Terms

1. Introduction

Fractional-order calculus is an area of mathematics that deals with extensions of derivatives and integrals to noninteger orders and represents a powerful tool in applied mathematics to study a myriad of problems from different fields [1] [2] [3] [4] . Analogously, starting with a linear difference equation, we are led to a definition of fractional difference of an arbitrary order [5] . Nowadays, studying on fractional-order calculus has become an active research field. In recent years, fractional operator is introduced into artificial neural networks, and the fractional-order formulation of artificial neural network models is also proposed in research results about biological neurons.

The analysis of fractional-order artificial neural networks has received some attention, and some important and interesting results have been obtained [6] - [11] . For instances, the stability and multi-stability (coexistence of several different stable states), bifurcations and chaos of fractional-order neural networks of Hopfield type were investigated in [6] . Finite-time stability in neural networks with delay has been discussed in [7] . Zhang and Yu [8] proposed fractional-order Hopfield neural networks with discontinuous activation functions and investigated its stability through the Lyapunov functionals. In [9] , authors considered the following fractional-order autonomous neural network:

$\begin{array}{l}{D}^{\alpha }{x}_i\left(t\right)=-c_i{x}_i\left(t\right)+{\displaystyle \underset{j=1}{\overset{n}{\sum }}{a}_{ij}}{f}_j\left(x_j\left(t\right)\right)+{\displaystyle \underset{j=1}{\overset{n}{\sum }}{b}_{ij}}{g}_j\left(x_j\left(t-\tau \right)\right)+I_i,\\ i\in N=1,2,\cdots ,n,t\in \left[0,T\right],T<+\infty \end{array}$

where $D^{\alpha }$ is the fractional derivative and $0<\alpha <1$ ; $x_i\left(t\right)$ corresponds to the state of the ith unit at time t; $f_j(\cdot )$ and $g_j(\cdot )$ denote the activation function of the jth neuron. A sufficient criterion ensuring the uniform stability of the system and the existence, uniqueness, and uniform stability of the equilibrium point is presented.

Recently, a typical time delay called leakage delay which is the time delay in the leakage term of the systems and a considerable factor affecting dynamics for the worse in the systems, has a great impact on the dynamical behavior of neural networks. Since leakage delays can have a destabilizing influence on the dynamical behaviors of neural networks, it is necessary to investigate leakage delay effects on the stability of neural networks (see [12] [13] [14] [15] [16] ). Fuzzy theory is considered as a more suitable method for the sake of taking vagueness into consideration; as a kind of important neural networks, studies have shown that the fuzzy neural networks are a very useful paradigm for image processing problems [17] [18] . Subsequently, various interesting results on the stability and other behaviors of delayed fuzzy BAM neural networks have been derived (see [19] [20] [21] [22] and references cited therein). However, to the best of our knowledge, there are few results on the uniform stability analysis of fractional-order fuzzy BAM neural networks with leakage delays.

Motivated by the above, in this paper, we are concerned with the following fractional-order fuzzy BAM neural network with delays in the leakage terms:

$\{\begin{array}{l}{}^c{D}^{\alpha }{x}_i\left(t\right)=-a_i{x}_i\left(t-\sigma \right)+{\displaystyle \underset{j=1}{\overset{m}{\sum }}{c}_{ji}}{f}_j\left(y_j\left(t-\tau \right)\right)+\underset{j=1}{\overset{m}{{\displaystyle \wedge }}}{\alpha }_{ji}{f}_j\left(y_j\left(t-\tau \right)\right)\\ +\underset{j=1}{\overset{m}{{\displaystyle \vee }}}{\beta }_{ji}{f}_j\left(y_j\left(t-\tau \right)\right)+\underset{j=1}{\overset{m}{{\displaystyle \wedge }}}{T}_{ji}{\mu }_j+\underset{j=1}{\overset{m}{{\displaystyle \vee }}}{H}_{ji}{\mu }_j+I_i,i=1,2,\cdots ,n,t\ge 0,\\ {}^c{D}^{\alpha }{y}_j\left(t\right)=-b_j{y}_j\left(t-\tau \right)+{\displaystyle \underset{i=1}{\overset{n}{\sum }}{d}_{ij}}{g}_i\left(x_i\left(t-\sigma \right)\right)+\underset{i=1}{\overset{n}{{\displaystyle \wedge }}}{p}_{ij}{g}_i\left(x_i\left(t-\sigma \right)\right)\\ +\underset{i=1}{\overset{n}{{\displaystyle \vee }}}{q}_{ij}{g}_i\left(x_i\left(t-\sigma \right)\right)+\underset{i=1}{\overset{n}{{\displaystyle \wedge }}}{K}_{ij}{\nu }_i+\underset{i=1}{\overset{n}{{\displaystyle \vee }}}{R}_{ij}{\nu }_i+J_j,j=1,2,\cdots ,m,t\ge 0,\end{array}$ (1)

where n and m correspond to the number of neurons in X-layer and Y-layer, respectively. ${}^{c}D{}^{\alpha }$ is the Caputo’s fractional derivative and $0<\alpha <1$ ; $x_i\left(t\right)$ and $y_j\left(t\right)$ are the activations of the ith neuron and the jth neuron, respectively; $a_i>0,b_j>0$ denote the rate with which the ith neurons and the jth neurons will reset its potential to the resting state in isolation when disconnected from the network and external inputs; $a_{ij},b_{ij},T_{ij}$ and $H_{ij}$ are the elements of fuzzy feedback MIN template and fuzzy feedback MAX template, fuzzy feed-forward MIN template and fuzzy feed-forward MAX template in X-layer, respectively; $p_{ij},q_{ij},K_{ij}$ and $R_{ij}$ are the elements of fuzzy feedback MIN template and fuzzy feedback MAX template, fuzzy feed-forward MIN template and fuzzy feed-forward MAX template in Y-layer, respectively; $\wedge$ and $\vee$ denote the fuzzy AND and fuzzy OR operation, respectively; ${\mu }_j$ and ${\nu }_i$ denote external input of the ith neurons in X-layer and external input of the jth neurons in Y-layer, respectively; $I_i$ and $J_j$ represent bias of the ith neurons in X-layer and bias of the jth neurons in Y-layer, respectively; $c_{ij}$ and $d_{ij}$ are the delayed feedback, $f_j(\cdot )$ and $g_j(\cdot )$ are the signal transmission functions, and $i=1,2,\cdots ,n;j=1,2,\cdots ,m$ .

Here, the initial conditions associated with system (1) are of the form

$\{\begin{array}{l}{x}_i\left(s\right)={\varphi }_i\left(s\right),{\varphi }_i\left(0\right)=0,s\in \left[-\sigma ,0\right],i=1,2,\cdots ,n,\\ y_j\left(s\right)={\phi }_j\left(s\right),{\phi }_j\left(0\right)=0,s\in \left[-\tau ,0\right],j=1,2,\cdots ,m,\end{array}$ (2)

where it is usually assumed that ${\varphi }_i(\cdot )$ and ${\phi }_j(\cdot )$ denote real-valued continuous functions defined on $\left[-\sigma ,\text{0}\right]$ and $\left[-\tau ,0\right]$ , respectively. If the initial value

$\psi \left(t\right)={\left({\varphi }_1\left(t\right),{\varphi }_2\left(t\right),\cdots ,{\varphi }_n\left(t\right),{\phi }_1\left(t\right),{\phi }_2\left(t\right),\cdots ,{\phi }_m\left(t\right)\right)}^{\text{T}},$

we denote the norm $\Vert \psi \Vert =\max \left\{{\Vert \varphi \Vert }_0,{\Vert \phi \Vert }_0\right\}$ , norm of ${\Vert \cdot \Vert }_0$ denoted by

${\Vert \varphi \Vert }_0={\displaystyle \underset{i=1}{\overset{n}{\sum }}\underset{t\in \left[-\sigma ,0\right]}{\sup }}\left\{{\text{e}}^{-t}\left|{\varphi }_i\left(t\right)\right|\right\},{\Vert \phi \Vert }_0={\displaystyle \underset{j=1}{\overset{m}{\sum }}\underset{t\in \left[-\tau ,0\right]}{\sup }}\left\{{\text{e}}^{-t}\left|{\phi }_j\left(t\right)\right|\right\}.$

The main purpose of this paper is to obtain some sufficient conditions for the uniform stability of the system. Then we study the existence, uniqueness, uniform stability of the equilibrium point.

This paper is organized as follows: In Section 2, we introduce some notations and definitions and state some preliminary results which are needed in later sections. In Section 3, we establish some sufficient conditions for the uniform stability of the system and the existence, uniqueness, and uniform stability of the equilibrium point. In Section 4, an example is given to illustrate that our results are feasible. The conclusion is made in Section 5.

2. Preliminaries

In this section, we shall recall some definitions and state some lemmas which will be used in the later section.

Definition 1. ( [1] , [2] ) The fractional integral (Riemann-Liouville integral) $I_{t_0,t}^{\alpha }$ with fractional order $\alpha \in R^{+}$ of function $x\left(t\right)$ is defined as

$I_{t_0,t}^{\alpha }x\left(t\right)=\frac{1}{\Gamma \left(\alpha \right)}{\displaystyle \underset{t_0}{\overset{t}{\int }}{\left(t-\tau \right)}^{\alpha -1}x\left(\tau \right)\text{d}\tau },$

where $\Gamma (\cdot )$ is the gamma function, $\Gamma \left(\alpha \right)={\displaystyle {\int }_0^{+\infty }{t}^{\alpha -1}{\text{e}}^{-t}\text{d}t}$ .

Definition 2. ( [1] , [2] ) The Riemann-Liouville derivative of fractional order $\alpha$ of function $x\left(t\right)$ is given as

$D_{t_0,t}^{\alpha }x\left(t\right)=\frac{{\text{d}}^n}{\text{d}{t}^n}\frac{1}{\Gamma \left(n-\alpha \right)}{\displaystyle \underset{t_0}{\overset{t}{\int }}{\left(t-\tau \right)}^{n-\alpha -1}x\left(\tau \right)\text{d}\tau },$

where $n-1<\alpha <n\in Z^{+}$ .

Definition 3. ( [1] , [2] ) The Caputo derivative of fractional order of function $x\left(t\right)$ is defined as follows

${}^c{D}_{t_0,t}^{\alpha }x\left(t\right)=\frac{1}{\Gamma \left(n-\alpha \right)}{\displaystyle \underset{t_0}{\overset{t}{\int }}\frac{x^{\left(n\right)}\left(\tau \right)}{{\left(t-\tau \right)}^{\alpha -n+1}}\text{d}\tau },$

where $n-1<\alpha <n\in Z^{+}$ .

Lemma 1. ( [1] , [2] ) If $x\left(t\right)\in C^n\left[0,+\infty \right)$ and $n-1<\alpha <n\in Z^{+}$ , then

1) $I^{\alpha }{I}^{\beta }x\left(t\right)=I^{\alpha +\beta }x\left(t\right),\alpha ,\beta \ge 0$ ;

2) ${}^c{D}^{\alpha }{I}^{\beta }x\left(t\right)=x\left(t\right),\alpha =\beta$ ;

3) $I^{\alpha }{}^{c}D{}^{\beta }x\left(t\right)=x\left(t\right)-{\displaystyle \underset{k=0}{\overset{n-1}{\sum }}\frac{t^k}{k!}x\left(t_0\right)},\alpha =\beta \ge 0$ .

Consider the initial value problem of the following fractional differential equation

$\{\begin{array}{l}{}^c{D}^{\alpha }x\left(t\right)=f\left(t,x\left(t\right)\right),t\in \left[0,+\infty \right),\\ x\left(0\right)=x_0,x_0\in R^m,\end{array}$ (3)

where $x={\left(x_1,x_2,\cdots ,x_m\right)}^{\text{T}}\in R^m$ , $0<\alpha <1$ , $f:\left[0,+\infty \right)\times R^m\to R^m$ is continuous in t and locally Lipschitz in x.

The equilibrium point of the Caputo’s fractional dynamic system has been defined in earlier work [23] [24] . We shall employ the following definitions of the equilibrium point and uniformly stable of the Caputo’s fractional dynamic system:

Definition 3. The constant $x^{\ast }$ is an equilibrium point of the Caputo’s fractional dynamic system (3) if and only if $f\left(t,x^{\ast }\right)=0$ for any $t\in \left[0,+\infty \right)$ .

Definition 4. The solution of system (1) is said to be stable if for any $\epsilon >0$ there exists $\delta \left(t_0,\epsilon \right)>0$ such that $t\ge t_0\ge 0,\Vert x_0-y_0\Vert <\delta$ imply $\Vert y\left(t,t_0,y_0\right)-x\left(t,t_0,x_0\right)\Vert <\epsilon$ for any two solutions $x\left(t,t_0,x_0\right)$ and $y\left(t,t_0,y_0\right)$ . It is uniformly stable if the above $\delta$ independent of $t_0$ .

Definition 5. The solution $z^{\ast }\left(t\right)={\left(x_1^{\ast }\left(t\right),x_2^{\ast }\left(t\right),\cdots ,x_n^{\ast }\left(t\right),y_1^{\ast }\left(t\right),y_2^{\ast }\left(t\right),\cdots ,y_m^{\ast }\left(t\right)\right)}^{\text{T}}$ with initial values ${\psi }^{\ast }\left(t\right)={\left({\varphi }_1^{\ast }\left(t\right),{\varphi }_2^{\ast }\left(t\right),\cdots ,{\varphi }_n^{\ast }\left(t\right),{\phi }_1^{\ast }\left(t\right),{\phi }_2^{\ast }\left(t\right),\cdots ,{\phi }_m^{\ast }\left(t\right)\right)}^{\text{T}}$ of system (1) is said to be uniformly stable if for any $\epsilon >0$ and $\delta \left(\epsilon \right)>0$ for any solution

$z\left(t\right)={\left(x_1\left(t\right),x_2\left(t\right),\cdots ,x_n\left(t\right),y_1\left(t\right),y_2\left(t\right),\cdots ,y_m\left(t\right)\right)}^{\text{T}}$

of (1) with initial value $\psi \left(t\right)={\left({\varphi }_1\left(t\right),{\varphi }_2\left(t\right),\cdots ,{\varphi }_n\left(t\right),{\phi }_1\left(t\right),{\phi }_2\left(t\right),\cdots ,{\phi }_m\left(t\right)\right)}^{\text{T}}$ , where

$\left({\varphi }_1,{\varphi }_2,\cdots ,{\varphi }_n\right)\in C\left(\left[-\sigma ,0\right],R^n\right),\left({\phi }_1,{\phi }_2,\cdots ,{\phi }_m\right),\in C\left(\left[-\tau ,0\right],R^m\right)$

and

${\varphi }_i\left(0\right)=0,{\phi }_j\left(0\right)=0,i=1,2,\cdots ,n;j=1,2,\cdots ,m,$

such that $\Vert {\psi }^{\ast }\left(t_0\right)-\psi \left(t_0\right)\Vert <\delta$ imply $\Vert z^{\ast }\left(t,t_0,{\psi }^{\ast }\left(t_0\right)\right)-z\left(t,t_0,\psi \left(t_0\right)\right)\Vert <\epsilon$ .

In order to obtain the main results, here, we make the following assumptions:

(H₁) The neuron activation functions $f_j,g_i\in C\left[R,R\right]$ $\left(i=1,2,\cdots ,n,j=1,2,\cdots ,m\right)$ satisfy the Lipschitz condition. That is, there exist positive constants $F_j,G_i$ such that

$\left|f_j\left(u\right)-f_j\left(v\right)\right|<F_j\left|u-v\right|,\left|g_i\left(u\right)-g_i\left(v\right)\right|<G_i\left|u-v\right|,\forall u,v\in R$

Lemma 2. ( [25] ) Suppose that $x,y\in R^n$ be the two states of the system (1). Then, one has

$\left|\underset{j=1}{\overset{n}{{\displaystyle \wedge }}}{\alpha }_{ij}{f}_j\left(x_j\right)-\underset{j=1}{\overset{n}{{\displaystyle \wedge }}}{\alpha }_{ij}{f}_j\left(y_j\right)\right|\le {\displaystyle \underset{j=1}{\overset{n}{\sum }}\left|{\alpha }_{ij}\right|\left|f_j\left(x_j\right)-f_j\left(y_j\right)\right|},$

$\left|\underset{j=1}{\overset{n}{{\displaystyle \vee }}}{\beta }_{ij}{f}_j\left(x_j\right)-\underset{j=1}{\overset{n}{{\displaystyle \vee }}}{\beta }_{ij}{f}_j\left(y_j\right)\right|\le {\displaystyle \underset{j=1}{\overset{n}{\sum }}\left|{\beta }_{ij}\right|\left|f_j\left(x_j\right)-f_j\left(y_j\right)\right|}.$

3. Uniform Stability of Fractional-Order Neural Networks

In this section, a sufficient condition for uniform stability of a class of fractional-order delayed neural networks on time scale, and the existence and uniqueness, uniform stability of equilibrium point are proposed, respectively.

Theorem 3. Let (H₁) holds. Suppose further that, for $i=1,2,\cdots ,n;j=1,2,\cdots ,m$ , (H₂) $a_i,b_j,c_{ji},{\alpha }_{ji},{\beta }_{ji},d_{ij},p_{ij},q_{ij},F_j$ and $G_i$ satisfy the following condition: $C>K$ , where

$w_{ji}=\left|c_{ji}\right|+\left|{\alpha }_{ji}\right|+\left|{\beta }_{ji}\right|,A={\displaystyle \underset{i=1}{\overset{n}{\sum }}{\omega }_i^{\ast }}={\displaystyle \underset{i=1}{\overset{n}{\sum }}\underset{\forall j}{\max }}\left\{w_{ji}{F}_j\right\},$

${\vartheta }_{ij}=\left|d_{ij}\right|+\left|p_{ij}\right|+\left|q_{ij}\right|,B={\displaystyle \underset{j=1}{\overset{m}{\sum }}{\vartheta }_j^{\ast }}={\displaystyle \underset{j=1}{\overset{m}{\sum }}\underset{\forall i}{\max }}\left\{{\vartheta }_{ij}{G}_i\right\},$

$\bar{a}=\underset{\forall i}{\max }\left\{a_i\right\},\bar{b}=\underset{\forall j}{\max }\left\{b_j\right\},\underset{\_}{a}=\underset{\forall i}{\max }\left\{a_i\right\},\underset{\_}{b}=\underset{\forall j}{\max }\left\{b_j\right\},$

$C=\min \left\{1-\bar{a},\underset{\_}{a},1-\bar{b},\underset{\_}{b}\right\},K=\max \left\{\left(\bar{b}+A\right){\text{e}}^{-\sigma },\left(\bar{a}+B\right){\text{e}}^{-\tau }\right\},$

then the system (1) is uniformly stable.

Proof: Assume that

$z\left(t\right)={\left(x_1\left(t\right),x_2\left(t\right),\cdots ,x_n\left(t\right),y_1\left(t\right),y_2\left(t\right),\cdots ,y_m\left(t\right)\right)}^{\text{T}}$

and

$z^{\ast }\left(t\right)={\left(x_1^{\ast }\left(t\right),x_2^{\ast }\left(t\right),\cdots ,x_n^{\ast }\left(t\right),y_1^{\ast }\left(t\right),y_2^{\ast }\left(t\right),\cdots ,y_m^{\ast }\left(t\right)\right)}^{\text{T}}$

are two solutions of system (1) with different initial values

$\psi \left(t\right)={\left({\varphi }_1\left(t\right),{\varphi }_2\left(t\right),\cdots ,{\varphi }_n\left(t\right),{\phi }_1\left(t\right),{\phi }_2\left(t\right),\cdots ,{\phi }_m\left(t\right)\right)}^{\text{T}},$

${\psi }^{\ast }\left(t\right)={\left({\varphi }_1^{\ast }\left(t\right),{\varphi }_2^{\ast }\left(t\right),\cdots ,{\varphi }_n^{\ast }\left(t\right),{\phi }_1^{\ast }\left(t\right),{\phi }_2^{\ast }\left(t\right),\cdots ,{\phi }_m^{\ast }\left(t\right)\right)}^{\text{T}},$

where

$\left({\varphi }_1,{\varphi }_2,\cdots ,{\varphi }_n\right),\left({\varphi }_1^{\ast },{\varphi }_2^{\ast },\cdots ,{\varphi }_n^{\ast }\right)\in C\left(\left[-\sigma ,0\right],R^n\right),$

$\left({\phi }_1,{\phi }_2,\cdots ,{\phi }_m\right),\left({\phi }_1^{\ast },{\phi }_2^{\ast },\cdots ,{\phi }_m^{\ast }\right)\in C\left(\left[-\tau ,0\right],R^m\right)$

and

${\varphi }_i\left(0\right)=0,{\varphi }_i^{\ast }\left(0\right)=0,i=1,2,\cdots ,n;{\phi }_j\left(0\right)=0,{\phi }_j^{\ast }\left(0\right)=0,j=1,2,\cdots ,m.$

If $z\left(t\right)={\left(x_1\left(t\right),x_2\left(t\right),\cdots ,x_n\left(t\right),y_1\left(t\right),y_2\left(t\right),\cdots ,y_m\left(t\right)\right)}^{\text{T}}$ , we use the following norm:

$\Vert z\Vert =\max \left\{{\Vert x\Vert }_0,{\Vert y\Vert }_0\right\},$

norm of ${\Vert .\Vert }_0$ denoted by

${\Vert x\Vert }_0={\displaystyle \underset{i=1}{\overset{n}{\sum }}\underset{t\in R}{\sup }}\left\{{\text{e}}^{-t}\left|x_i\left(t\right)\right|\right\},{\Vert y\Vert }_0={\displaystyle \underset{j=1}{\overset{m}{\sum }}\underset{t\in R}{\sup }}\left\{{\text{e}}^{-t}\left|y_j\left(t\right)\right|\right\}.$

Based on Lemma 2, one has

$\begin{array}{l}{\text{e}}^{-t}\left|x_i\left(t\right)-x_i^{\ast }\left(t\right)\right|\\ \le \frac{1}{\Gamma \left(\alpha \right)}{\text{e}}^{-t}{\displaystyle {\int }_0^t{\left(t-s\right)}^{\alpha -1}}\{a_i\left|x_i\left(s-\sigma \right)-x_i^{\ast }\left(s-\sigma \right)\right|\\ +{\displaystyle \underset{j=1}{\overset{m}{\sum }}\left(\left|c_{ji}\right|+\left|{\alpha }_{ji}\right|+\left|{\beta }_{ji}\right|\right)}\left|f_j\left(y_j\left(s-\tau \right)\right)-f_j\left(y_j^{\ast }\left(s-\tau \right)\right)\right|\}\text{d}s\\ \le a_i\frac{1}{\Gamma \left(\alpha \right)}{\displaystyle {\int }_0^{\sigma }{\left(t-s\right)}^{\alpha -1}{\text{e}}^{-\left(t-s\right)}{\text{e}}^{-s}\left|{\varphi }_i\left(s-\sigma \right)-{\varphi }_i^{\ast }\left(s-\sigma \right)\right|\text{d}s}\\ +a_i\frac{1}{\Gamma \left(\alpha \right)}{\displaystyle {\int }_{\sigma }^t{\left(t-s\right)}^{\alpha -1}{\text{e}}^{-\left(t-s\right)}{\text{e}}^{-s}\left|x_i\left(s-\sigma \right)-x_i^{\ast }\left(s-\sigma \right)\right|\text{d}s}\end{array}$

$\begin{array}{l}+{\displaystyle \underset{j=1}{\overset{m}{\sum }}{w}_{ji}}{F}_j\frac{1}{\Gamma \left(\alpha \right)}{\displaystyle {\int }_0^{\tau }{\left(t-s\right)}^{\alpha -1}{\text{e}}^{-\left(t-s\right)}{\text{e}}^{-s}\left|{\phi }_j\left(s-\tau \right)-{\phi }_j^{\ast }\left(s-\tau \right)\right|\text{d}s}\\ +{\displaystyle \underset{j=1}{\overset{m}{\sum }}{w}_{ji}}{F}_j\frac{1}{\Gamma \left(\alpha \right)}{\displaystyle {\int }_{\tau }^t{\left(t-s\right)}^{\alpha -1}{\text{e}}^{-\left(t-s\right)}{\text{e}}^{-s}\left|y_j\left(s-\tau \right)-y_j^{\ast }\left(s-\tau \right)\right|\text{d}s}\\ \le a_i\frac{1}{\Gamma \left(\alpha \right)}{\displaystyle {\int }_{-\sigma }^0{\left(t-\zeta -\sigma \right)}^{\alpha -1}{\text{e}}^{-\left(t-\zeta \right)}{\text{e}}^{-\zeta }\left|{\varphi }_i\left(\zeta \right)-{\varphi }_i^{\ast }\left(\zeta \right)\right|\text{d}\zeta }\\ +a_i\frac{1}{\Gamma \left(\alpha \right)}{\displaystyle {\int }_0^{t-\sigma }{\left(t-\zeta -\sigma \right)}^{\alpha -1}{\text{e}}^{-\left(t-\zeta \right)}{e}^{-\zeta }\left|x_i\left(\zeta \right)-x_i^{\ast }\left(\zeta \right)\right|\text{d}\zeta }\\ +w_i^{\ast }{\displaystyle \underset{j=1}{\overset{m}{\sum }}\frac{1}{\Gamma \left(\alpha \right)}}{\displaystyle {\int }_{-\tau }^0{\left(t-\gamma -\tau \right)}^{\alpha -1}{\text{e}}^{-\left(t-\gamma \right)}{\text{e}}^{-\gamma }\left|{\phi }_j\left(\gamma \right)-{\phi }_j^{\ast }\left(\gamma \right)\right|\text{d}\gamma }\end{array}$ (4)

$\begin{array}{l}+w_i^{\ast }{\displaystyle \underset{j=1}{\overset{m}{\sum }}\frac{1}{\Gamma \left(\alpha \right)}}{\displaystyle {\int }_0^{t-\tau }{\left(t-\gamma -\tau \right)}^{\alpha -1}{\text{e}}^{-\left(t-\gamma \right)}{\text{e}}^{-\gamma }\left|y_j\left(\gamma \right)-y_j^{\ast }\left(\gamma \right)\right|\text{d}\gamma }\\ \le a_i\underset{t\in ℝ}{\sup }\left\{{\text{e}}^{-t}\left|{\varphi }_i\left(t\right)-{\varphi }_i^{\ast }\left(t\right)\right|\right\}{\text{e}}^{-\sigma }\frac{1}{\Gamma \left(\alpha \right)}{\displaystyle {\int }_{t-\sigma }^t{\eta }^{\alpha -1}}{\text{e}}^{-\eta }\text{d}\eta \\ +a_i\underset{t\in ℝ}{\sup }\left\{{\text{e}}^{-t}\left|x_i\left(t\right)-x_i^{\ast }\left(t\right)\right|\right\}{\text{e}}^{-\sigma }\frac{1}{\Gamma \left(\alpha \right)}{\displaystyle {\int }_0^{t-\sigma }{\eta }^{\alpha -1}}{\text{e}}^{-\eta }\text{d}\eta \\ +w_i^{\ast }{\displaystyle \underset{j=1}{\overset{m}{\sum }}\underset{t\in ℝ}{\sup }}\left\{{\text{e}}^{-t}\left|{\phi }_j\left(t\right)-{\phi }_j^{\ast }\left(t\right)\right|\right\}{\text{e}}^{-\tau }\frac{1}{\Gamma \left(\alpha \right)}{\displaystyle {\int }_{t-\tau }^t{\theta }^{\alpha -1}}{\text{e}}^{-\theta }\text{d}\theta \\ +w_i^{\ast }{\displaystyle \underset{j=1}{\overset{n}{\sum }}\underset{t\in ℝ}{\sup }}\left\{{\text{e}}^{-t}\left|y_j\left(t\right)-y_j^{\ast }\left(t\right)\right|\right\}{\text{e}}^{-\tau }\frac{1}{\Gamma \left(\alpha \right)}{\displaystyle {\int }_0^{t-\tau }{\theta }^{\alpha -1}}{\text{e}}^{-\theta }\text{d}\theta \end{array}$

$\begin{array}{l}\le a_i\underset{t\in ℝ}{\sup }\left\{{\text{e}}^{-t}\left|{\varphi }_i\left(t\right)-{\varphi }_i^{\ast }\left(t\right)\right|\right\}{\text{e}}^{-\sigma }+a_i\underset{t\in ℝ}{\sup }\left\{{\text{e}}^{-t}\left|x_i\left(t\right)-x_i^{\ast }\left(t\right)\right|\right\}{\text{e}}^{-\sigma }\\ +w_i^{\ast }{\displaystyle \underset{j=1}{\overset{m}{\sum }}\underset{t\in ℝ}{\sup }}\left\{{\text{e}}^{-t}\left|{\phi }_j\left(t\right)-{\phi }_j^{\ast }\left(t\right)\right|\right\}{\text{e}}^{-\tau }+w_i^{\ast }{\displaystyle \underset{j=1}{\overset{m}{\sum }}\underset{t\in ℝ}{\sup }}\left\{{\text{e}}^{-t}\left|y_i\left(t\right)-y_i^{\ast }\left(t\right)\right|\right\}{\text{e}}^{-\tau }\\ \le a_i{e}^{-\sigma }{\Vert \varphi -{\varphi }^{\ast }\Vert }_0+a_i{\text{e}}^{-\sigma }{\Vert x-x^{\ast }\Vert }_0\\ +w_i^{\ast }{\text{e}}^{-\tau }{\Vert y-y^{\ast }\Vert }_0+w_i^{\ast }{\text{e}}^{-\tau }{\Vert \phi -{\phi }^{\ast }\Vert }_0,i=1,2,\cdots ,n.\end{array}$

From (4), one obtains

$\begin{array}{c}{\Vert x-x^{\ast }\Vert }_0={\displaystyle \underset{i=1}{\overset{n}{\sum }}\underset{t\in ℝ}{\sup }}\left\{{\text{e}}^{-t}\left|x_i\left(t\right)-x_i^{\ast }\left(t\right)\right|\right\}\\ \le \bar{a}{\text{e}}^{-\sigma }\left({\Vert \varphi -{\varphi }^{\ast }\Vert }_0+{\Vert x-x^{\ast }\Vert }_0\right)+A{\text{e}}^{-\tau }\left({\Vert y-y^{\ast }\Vert }_0+{\Vert \phi -{\phi }^{\ast }\Vert }_0\right),\end{array}$

which implies that

$\left(1-\bar{a}{\text{e}}^{-\sigma }\right){\Vert x-x^{\ast }\Vert }_0\le \bar{a}{\text{e}}^{-\sigma }{\Vert \varphi -{\varphi }^{\ast }\Vert }_0+A{\text{e}}^{-\tau }\left({\Vert y-y^{\ast }\Vert }_0+{\Vert \phi -{\phi }^{\ast }\Vert }_0\right).$ (5)

Similarly, we can also get

$\left(1-\bar{b}{\text{e}}^{-\tau }\right){\Vert y-y^{\ast }\Vert }_0\le \bar{b}{\text{e}}^{-\tau }{\Vert \phi -{\phi }^{\ast }\Vert }_0+B{\text{e}}^{-\sigma }\left({\Vert x-x^{\ast }\Vert }_0+{\Vert \varphi -{\varphi }^{\ast }\Vert }_0\right).$ (6)

In view of (5) and (6), one has

$\begin{array}{l}C\left({\Vert x-x^{\ast }\Vert }_0+{\Vert y-y^{\ast }\Vert }_0\right)\\ \le K\left({\Vert x-x^{\ast }\Vert }_0+{\Vert y-y^{\ast }\Vert }_0\right)+K\left({\Vert \varphi -{\varphi }^{\ast }\Vert }_0+{\Vert \phi -{\phi }^{\ast }\Vert }_0\right),\end{array}$

which implies that

${\Vert x-x^{\ast }\Vert }_0+{\Vert y-y^{\ast }\Vert }_0\le \frac{K}{C-K}\left({\Vert \varphi -{\varphi }^{\ast }\Vert }_0+{\Vert \phi -{\phi }^{\ast }\Vert }_0\right).$

Easily, we have

$\Vert z-z^{\ast }\Vert \le \frac{K}{C-K}\Vert \psi -{\psi }^{\ast }\Vert .$

Therefore, for $\forall \epsilon >0$ , there exist $\delta =\left(C-K\right)\epsilon /K>0$ such that $\Vert z-z^{\ast }\Vert <\epsilon$ when $\Vert \psi -{\psi }^{\ast }\Vert <\delta$ , which means that the solution $z\left(t\right)$ the system (1) is uniformly stable.

Theorem 4. Let (H₁), (H₂) hold, then there exists a unique equilibrium point in system (1), which is uniformly stable.

Proof: Let $a_i{x}_i^{\ast }=u_i^{\ast },b_j{y}_j^{\ast }=v_j^{\ast }$ , and constructing a mapping $\Theta :R^{n+m}\to R^{n+m}$ defined by

$\Theta \left(z\right)={\left({\Phi }_1{u}_1,\cdots ,{\Phi }_n{u}_n,{\Psi }_1{v}_1,\cdots ,{\Psi }_m{v}_m\right)}^{\text{T}},$

where

$z={\left(u_1,u_2,\cdots ,u_n,v_1,v_2,\cdots ,v_m\right)}^{\text{T}},$

$\begin{array}{l}{\Phi }_i{u}_i={\displaystyle \underset{j=1}{\overset{m}{\sum }}{c}_{ji}}{f}_j\left(\frac{v_j^{\ast }}{b_j}\right)+\underset{j=1}{\overset{m}{{\displaystyle \wedge }}}{\alpha }_{ji}{f}_j\left(\frac{v_j^{\ast }}{b_j}\right)+\underset{j=1}{\overset{m}{{\displaystyle \vee }}}{\beta }_{ji}{f}_j\left(\frac{v_j^{\ast }}{b_j}\right)\\ +\underset{j=1}{\overset{m}{{\displaystyle \wedge }}}{T}_{ji}{\mu }_j+\underset{j=1}{\overset{m}{{\displaystyle \vee }}}{H}_{ji}{\mu }_j+I_i,i=1,2,\cdots ,n,t\ge 0,\end{array}$

$\begin{array}{l}{\Psi }_j{v}_j={\displaystyle \underset{i=1}{\overset{n}{\sum }}{d}_{ij}}{g}_i\left(\frac{u_i^{\ast }}{c_i}\right)+\underset{i=1}{\overset{n}{{\displaystyle \wedge }}}{p}_{ij}{g}_i\left(\frac{u_i^{\ast }}{c_i}\right)+\underset{i=1}{\overset{n}{{\displaystyle \vee }}}{q}_{ij}{g}_i\left(\frac{u_i^{\ast }}{c_i}\right)\\ +\underset{i=1}{\overset{n}{{\displaystyle \wedge }}}{K}_{ij}{\nu }_i+\underset{i=1}{\overset{n}{{\displaystyle \vee }}}{R}_{ij}{\nu }_i+J_j,j=1,2,\cdots ,m,t\ge 0.\end{array}$

Now, we will show that $\Theta$ is a contraction mapping on $R^{n+m}$ endowed with the norm

$\Vert z\Vert =\max \left\{{\displaystyle \underset{i=1}{\overset{n}{\sum }}\left|u_i\right|},{\displaystyle \underset{j=1}{\overset{m}{\sum }}\left|v_j\right|}\right\}.$

In fact, for any two different points

$z={\left(u_1,u_2,\cdots ,u_n,v_1,v_2,\cdots ,v_m\right)}^{\text{T}}$

and

$\bar{z}={\left({\bar{u}}_1,{\bar{u}}_2,\cdots ,{\bar{u}}_n,{\bar{v}}_1,{\bar{v}}_2,\cdots ,{\bar{v}}_m\right)}^{\text{T}},$

we have

$\begin{array}{l}\Vert \Theta \left(z\right)-\Theta \left(\bar{z}\right)\Vert \\ =\max \left\{{\displaystyle \underset{i=1}{\overset{n}{\sum }}\left|{\Phi }_i{u}_i-{\Phi }_i\overline{u_i}\right|},{\displaystyle \underset{j=1}{\overset{m}{\sum }}\left|{\Psi }_j{v}_j-{\Psi }_j\overline{v_j}\right|}\right\}\\ \le \max \{{\displaystyle \underset{i=1}{\overset{n}{\sum }}(}{\displaystyle \underset{j=1}{\overset{m}{\sum }}\left|c_{ji}\right|\left|f_j\left(\frac{v_j}{b_j}\right)-f_j\left(\frac{\overline{v_j}}{b_j}\right)\right|}+{\displaystyle \underset{j=1}{\overset{m}{\sum }}\left|{\alpha }_{ji}\right|\left|f_j\left(\frac{v_j}{b_j}\right)-f_j\left(\frac{\overline{v_j}}{b_j}\right)\right|}\\ +{\displaystyle \underset{j=1}{\overset{m}{\sum }}\left|{\beta }_{ji}\right|\left|f_j\left(\frac{v_j}{b_j}\right)-f_j\left(\frac{\overline{v_j}}{b_j}\right)\right|}),{\displaystyle \underset{j=1}{\overset{m}{\sum }}({\displaystyle \underset{i=1}{\overset{n}{\sum }}\left|d_{ij}\right|\left|g_i\left(\frac{u_i}{c_i}\right)-\left(\frac{\overline{u_i}}{c_i}\right)\right|}}\\ +{\displaystyle \underset{i=1}{\overset{n}{\sum }}\left|p_{ij}\right|\left|g_i\left(\frac{u_i}{c_i}\right)-\left(\frac{\overline{u_i}}{c_i}\right)\right|}+{\displaystyle \underset{i=1}{\overset{n}{\sum }}\left|q_{ij}\right|\left|g_i\left(\frac{u_i}{c_i}\right)-\left(\frac{\overline{u_i}}{c_i}\right)\right|})\}\end{array}$

$\begin{array}{l}\le \max \left\{{\displaystyle \underset{i=1}{\overset{n}{\sum }}{\displaystyle \underset{j=1}{\overset{m}{\sum }}\frac{w_{ji}{F}_j}{b_j}}}\left|v_j-\overline{v_j}\right|,{\displaystyle \underset{j=1}{\overset{m}{\sum }}{\displaystyle \underset{i=1}{\overset{n}{\sum }}\frac{{\vartheta }_{ij}{G}_i}{a_i}}}\left|u_i-\overline{u_i}\right|\right\}\\ \le \max \left\{{\displaystyle \underset{i=1}{\overset{n}{\sum }}\frac{w_i^{\ast }}{\underset{\_}{b}}}{\displaystyle \underset{j=1}{\overset{m}{\sum }}\left|v_j\left(t\right)-\overline{v_j}\right|},{\displaystyle \underset{j=1}{\overset{m}{\sum }}\frac{{\vartheta }_i^{\ast }}{\underset{\_}{a}}}{\displaystyle \underset{i=1}{\overset{n}{\sum }}\left|u_i-\overline{u_i}\right|}\right\}\\ \le \max \left\{\frac{A}{C}{\displaystyle \underset{j=1}{\overset{m}{\sum }}\left|v_j-\overline{v_j}\right|},\frac{B}{C}{\displaystyle \underset{i=1}{\overset{n}{\sum }}\left|u_i-\overline{u_i}\right|}\right\}\\ \le \frac{K}{C}\max \left\{{\displaystyle \underset{j=1}{\overset{m}{\sum }}\left|v_j-\overline{v_j}\right|},{\displaystyle \underset{i=1}{\overset{n}{\sum }}\left|u_i-\overline{u_i}\right|}\right\}\\ \le \frac{K}{C}\Vert z-\bar{z}\Vert .\end{array}$

It follows from (H₂) that

$\Vert \Theta \left(z\right)-\Theta \left(\bar{z}\right)\Vert <\Vert z-\bar{z}\Vert ,$

which implies that $\Theta$ is a contraction mapping on $R^{n+m}$ . Hence, there exists a unique fixed point $u^{\ast }$ such that $\Theta \left(z^{\ast }\right)=z^{\ast }={\left(u_1^{\ast },u_2^{\ast },\cdots ,u_n^{\ast },v_1^{\ast },v_2^{\ast },\cdots ,v_m^{\ast }\right)}^{\text{T}}$ , i.e.

$\{\begin{array}{l}{u}_i^{\ast }={\displaystyle \underset{j=1}{\overset{m}{\sum }}{c}_{ji}}{f}_j\left(\frac{v_j^{\ast }}{b_j}\right)+\underset{j=1}{\overset{m}{{\displaystyle \wedge }}}{\alpha }_{ji}{f}_j\left(\frac{v_j^{\ast }}{b_j}\right)+\underset{j=1}{\overset{m}{{\displaystyle \vee }}}{\beta }_{ji}{f}_j\left(\frac{v_j^{\ast }}{b_j}\right)\\ +\underset{j=1}{\overset{m}{{\displaystyle \wedge }}}{T}_{ji}{\mu }_j+\underset{j=1}{\overset{m}{{\displaystyle \vee }}}{H}_{ji}{\mu }_j+I_i,i=1,2,\cdots ,n,t\ge 0,\\ v_j^{\ast }={\displaystyle \underset{i=1}{\overset{n}{\sum }}{d}_{ij}}{g}_i\left(\frac{u_i^{\ast }}{c_i}\right)+\underset{i=1}{\overset{n}{{\displaystyle \wedge }}}{p}_{ij}{g}_i\left(\frac{u_i^{\ast }}{c_i}\right)+\underset{i=1}{\overset{n}{{\displaystyle \vee }}}{q}_{ij}{g}_i\left(\frac{u_i^{\ast }}{c_i}\right)\\ +\underset{i=1}{\overset{n}{{\displaystyle \wedge }}}{K}_{ij}{\nu }_i+\underset{i=1}{\overset{n}{{\displaystyle \vee }}}{R}_{ij}{\nu }_i+J_j,j=1,2,\cdots ,m,t\ge 0.\end{array}$

That is

$\{\begin{array}{l}-a_i{x}_i^{\ast }+{\displaystyle \underset{j=1}{\overset{m}{\sum }}{c}_{ji}}{f}_j\left(y_j^{\ast }\right)+\underset{j=1}{\overset{m}{{\displaystyle \wedge }}}{\alpha }_{ji}{f}_j\left(y_j^{\ast }\right)+\underset{j=1}{\overset{m}{{\displaystyle \vee }}}{\beta }_{ji}{f}_j\left(y_j^{\ast }\right)\\ +\underset{j=1}{\overset{m}{{\displaystyle \wedge }}}{T}_{ji}{\mu }_j+\underset{j=1}{\overset{m}{{\displaystyle \vee }}}{H}_{ji}{\mu }_j+I_i=0,i=1,2,\cdots ,n,t\ge 0,\\ -b_j{y}_j^{\ast }+{\displaystyle \underset{i=1}{\overset{n}{\sum }}{d}_{ij}}{g}_i\left(x_i^{\ast }\right)+\underset{i=1}{\overset{n}{{\displaystyle \wedge }}}{p}_{ij}{g}_i\left(x_i^{\ast }\right)+\underset{i=1}{\overset{n}{{\displaystyle \vee }}}{q}_{ij}{g}_i\left(x_i^{\ast }\right)\\ +\underset{i=1}{\overset{n}{{\displaystyle \wedge }}}{K}_{ij}{\nu }_i+\underset{i=1}{\overset{n}{{\displaystyle \vee }}}{R}_{ij}{\nu }_i+J_j=0,j=1,2,\cdots ,m,t\ge 0,\end{array}$