Almost periodic solutions for Hopfield neural networks with continuously distributed delays

Abstract

In this paper Hopfield neural networks with continuously distributed delays are considered. Without assuming the global Lipschitz conditions of activation functions, sufficient conditions for the existence and exponential stability of the almost periodic solutions are established by using the fixed point theorem and differential inequality techniques. The results of this paper are new and they complement previously known results.

Introduction

Consider the following models for Hopfield neural networks (HNNs) with continuously distributed delays

$$x_i^{\prime }(t)=-c_i(t)x_i(t)+\sum _{j=1}^n{b}_{ij}(t){\int }_0^{\infty }{K}_{ij}(u)g_j(x_j(t-u))\mathrm{d}u+I_i(t)\text{,}i=1\text{,}2\text{,}\dots \text{,}n\text{,}$$

in which n corresponds to the number of units in a neural network, $x_i(t)$ corresponds to the state vector of the ith unit at the time t, $c_i(t)>0$ 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 at the time t. $b_{ij}(t)\text{,}i\text{,}j=1\text{,}2\text{,}\dots \text{,}n$, are the connection weights at the time t, and $I_i(t)$ denote the external inputs at time t. $g_j$ ($j=1\text{,}2\text{,}\dots \text{,}n$) are activation functions of signal transmission.

It is well known that the HNNs have been successfully applied to signal and image processing, pattern recognition and optimization. Hence, they have been the object of intensive analysis by numerous authors in recent years. In particular, there have been extensive results on the problem of the existence and stability of periodic and almost periodic solutions of HNNs in the literature. We refer the reader to [1], [2], [3], [4], [5], [6], [8], [11], [12], [13], [14], [16], [17], [15] and the references cited therein. Moreover, in the above-mentioned literature, we observe that the following assumption

H₀

For each $j\in \{1\text{,}2\text{,}\dots \text{,}n\}$, $g_j:R\to R$ is global Lipschitz with Lipschitz constant $L_j$, i.e.,

$$|g_j(u_j)-g_j(v_j)|\le L_j|u_j-v_j|\text{,}\text{for all}{u}_j\text{,}{v}_j\in R\text{.}$$

has been considered as fundamental for the considered existence and stability of periodic and almost periodic solutions of HNNs. However, to the best of our knowledge, few authors have considered the problems of almost periodic solutions of HNNs without the assumptions H₀. Thus, it is worth while to continue to investigate the existence and stability of almost periodic solutions of HNNs.

The main purpose of this paper is to give the conditions for the existence and exponential stability of the almost periodic solutions for system (1.1). By applying fixed point theorem and differential inequality techniques, we derive some new sufficient conditions ensuring the existence, uniqueness and exponential stability of the almost periodic solution, which are new and they complement previously known results. In particular, we do not need the assumption H₀. Moreover, an example is also provided to illustrate the effectiveness of our results.

Throughout this paper, for $i\text{,}j=1\text{,}2\text{,}\dots \text{,}n$, it will be assumed that $c_i\text{,}{I}_i\text{,}{b}_{ij}:R\to R$ are almost periodic functions, and there exist constants ${\tilde{c}}_i\text{,}\overline{b_{ij}}$ and $\overline{I_i}$ such that

$$0<{\tilde{c}}_i=\underset{t\in R}{\inf }{c}_i(t)\text{,}\underset{t\in R}{\sup }\left|b_{ij}(t)\right|=\overline{b_{ij}}\text{,}\underset{t\in R}{\sup }\left|I_i(t)\right|=\overline{I_i}\text{,}i\text{,}j=1\text{,}2\text{,}\dots \text{,}n\text{.}$$

We also assume that the following conditions H₁, H₂, H₃ and H₄ hold.

H₁

For each $j\in \{1\text{,}2\text{,}\dots \text{,}n\}$, there exist $f_j\text{,}{h}_j\in C(R\text{,}R)$ and constants $L_j^f\text{,}{L}_j^h\text{,}{M}_j\in [0\text{,}+\infty )$ such that the following conditions are satisfied.

• $f_j(0)=0\text{,}{g}_j(u)=f_j(u)h_j(u)$, $\left|h_j(u)\right|\le M_j\text{,}\text{for all}u\in R$;

• $|f_j(u)-f_j(v)|\le L_j^f|u-v|\text{,}|h_j(u)-h_j(v)|\le L_j^h|u-v|\text{,}\text{for all}u\text{,}v\in R$.

H₂

For $i\text{,}j\in \{1\text{,}2\text{,}\dots \text{,}n\}$, the delay kernels $K_{ij}:[0\text{,}\infty )\to R$ are continuous, integrable and satisfy

$${\int }_0^{\infty }\left|K_{ij}(s)\right|\mathrm{d}s\le k_{ij}\text{.}$$

H₃

Assume that there exist nonnegative constants L, q and $\delta$ such that

$$L=\underset{1\le i\le n}{\max }\left\{\frac{\overline{I_i}}{{\tilde{c}}_i}\right\}\text{,}\delta =\underset{1\le i\le n}{\max }\left\{\sum _{j=1}^n{\tilde{c}}_i^{-1}\overline{b_{ij}}{k}_{ij}{L}_j^f{M}_j\right\}<1\text{,}q=\underset{1\le i\le n}{\max }\left\{\sum _{j=1}^n{\tilde{c}}_i^{-1}\overline{b_{ij}}{k}_{ij}{L}_j^f\left(L_j^h\frac{L}{1-\delta }+M_j\right)\right\}<1\text{.}$$

H₄

For $i\text{,}j\in \{1\text{,}2\text{,}\dots \text{,}n\}$, there exists a constant ${\lambda }_0>0$ such that

$${\int }_0^{\infty }\left|K_{ij}(s)\right|{\text{e}}^{{\lambda }_{0}s}\mathrm{d}s<+\infty \text{.}$$

For convenience, we introduce some notations. We will use $x={(x_1\text{,}{x}_2\text{,}\dots \text{,}{x}_n)}^{\text{T}}\in R^n$ to denote a column vector, in which the symbol $(\text{T})$ denotes the transpose of a vector. We let $\left|x\right|$ denote the absolute-value vector given by $\left|x\right|={(\left|x_1\right|\text{,}\left|x_2\right|\text{,}\dots \text{,}\left|x_n\right|)}^{\text{T}}$, and define $\Vert x\Vert ={\max }_{1\le i\le n}\left|x_i\right|$. A vector $x\ge 0$ means that all entries of x are greater than or equal to zero. $x>0$ is defined similarly. For vectors x and y, $x\ge y$ (resp. $x>y$) means that $x-y\ge 0$ (resp. $x-y>0$).

Throughout this paper, we set

$$\{x_j(t)\}={(x_1(t)\text{,}{x}_2(t)\text{,}\dots \text{,}{x}_n(t))}^{\text{T}}\text{,}\text{and}B=\{\varphi |\varphi =\{{\varphi }_j(t)\}={({\varphi }_1(t)\text{,}{\varphi }_2(t)\text{,}\dots \text{,}{\varphi }_n(t))}^{\text{T}}\}\text{,}$$

where $\varphi$ is an almost periodic function on R. For $\forall \varphi \in B$, we define induced module $\Vert \varphi {\Vert }_B={\sup }_{t\in R}\Vert \varphi (t)\Vert$, then B is a Banach space.

The initial conditions associated with system (1.1) are of the form

$$x_i(s)={\varphi }_i(s)\text{,}s\in (-\infty \text{,}0]\text{,}i=1\text{,}2\text{,}\dots \text{,}n\text{,}$$

where ${\varphi }_i(\cdot )$ denotes real-valued bounded continuous function defined on $(-\infty \text{,}0]$.

Definition 1

(see [7], [10]) Let $u(t):R\to R^n$ be continuous in t. $u(t)$ is said to be almost periodic on R if, for any $\epsilon >0$, the set $T(u\text{,}\epsilon )=\{\delta :|u(t+\delta )-u(t)|<\epsilon \text{,}\forall t\in R\}$ is relatively dense, i.e., for $\forall \epsilon >0$, it is possible to find a real number $l=l(\epsilon )>0$, for any interval with length $l(\epsilon )$, there exists a number $\delta =\delta (\epsilon )$ in this interval such that $|u(t+\delta )-u(t)|<\epsilon$, for $\forall t\in R$.

Definition 2

Let $Z^{\ast }(t)={(x_1^{\ast }(t)\text{,}{x}_2^{\ast }(t)\text{,}\dots \text{,}{x}_n^{\ast }(t))}^{\text{T}}$ be an almost periodic solution of system (1.1) with initial value ${\varphi }^{\ast }={({\varphi }_1^{\ast }(t)\text{,}{\varphi }_2^{\ast }(t)\text{,}\dots \text{,}{\varphi }_n^{\ast }(t))}^{\text{T}}$. If there exist constants $\lambda >0$ and $M\ge 1$ such that for every solution $Z(t)={(x_1(t)\text{,}{x}_2(t)\text{,}\dots \text{,}{x}_n(t))}^{\text{T}}$ of system (1.1) with any initial value $\varphi ={({\varphi }_1(t)\text{,}{\varphi }_2(t)\text{,}\dots \text{,}{\varphi }_n(t))}^{\text{T}}$,

$$|x_i(t)-x_i^{\ast }(t)|\le M\Vert \varphi -{\varphi }^{\ast }{\Vert }_1{\text{e}}^{-\lambda t}\text{,}\forall t>0\text{,}i=1\text{,}2\text{,}\dots \text{,}n\text{,}$$

where $\Vert \varphi -{\varphi }^{\ast }{\Vert }_1={\sup }_{-\infty \le s\le 0}{\max }_{1\le i\le n}|{\varphi }_i(s)-{\varphi }_i^{\ast }(s)|$. Then $Z^{\ast }(t)$ is said to be global exponential stable.

Definition 3

(see [7], [10]) Let $x\in R^n$ and $Q(t)$ be a $n\times n$ continuous matrix defined on R. The linear system

$$x^{\prime }(t)=Q(t)x(t)$$

is said to admit an exponential dichotomy on R if there exist positive constants $k\text{,}\alpha$, projection P and the fundamental solution matrix $X(t)$ of (1.5) satisfying

$$\begin{array}{c}\Vert X(t)P{X}^{-1}(s)\Vert \le k{\text{e}}^{-\alpha (t-s)}\text{,}\text{for}t\ge s\text{,}\hfill \\ \Vert X(t)(I-P)X^{-1}(s)\Vert \le k{\text{e}}^{-\alpha (s-t)}\text{,}\text{for}t\le s\text{.}\hfill \end{array}$$

Lemma 1.1

(see [7], [10]). If the linear system (1.5) admits an exponential dichotomy, then almost periodic system

$$x^{\prime }(t)=Q(t)x+g(t)$$

has a unique almost periodic solution $x(t)$, and

$$x(t)={\int }_{-\infty }^{t}X(t)P{X}^{-1}(s)g(s)\mathrm{d}s-{\int }_t^{+\infty }X(t)(I-P)X^{-1}(s)g(s)\mathrm{d}s\text{.}$$

Lemma 1.2

(see [7], [10]). Let $c_i(t)$ be an almost periodic function on R and

$$M[c_i]=\underset{T\to +\infty }{\lim }\frac{1}{T}{\int }_t^{t+T}{c}_i(s)\mathrm{d}s>0\text{,}i=1\text{,}2\text{,}\dots \text{,}n\text{.}$$

Then the linear system

$$x^{\prime }(t)=\text{diag}(-c_1(t)\text{,}-c_2(t)\text{,}\dots \text{,}-c_n(t))x(t)$$

admits an exponential dichotomy on R.

The remaining part of this paper is organized as follows. In Section 2, we shall derive new sufficient conditions for checking the existence of almost periodic solutions of (1.1). In Section 3, we present some new sufficient conditions for the uniqueness and exponential stability of the almost periodic solution of (1.1). In Section 4, we shall give some examples and remarks to illustrate our results obtained in the previous sections.