Piecewise pseudo-almost periodic solution for impulsive non-autonomous high-order Hopfield neural networks with variable delays

Abstract

In this paper, we provide a new method to study non-autonomous dynamic systems with variable pseudo-almost periodic coefficients. By using some fixed point theorems in Banach space and inequality technique, some completely new sufficient conditions of the existence and exponential stability of piecewise pseudo-almost periodic solutions are established for impulsive non-autonomous high-order Hopfield neural networks with variable coefficients and delays. Finally, a numerical example and simulations are given to illustrate that the obtained results are feasible and effective. It is the first time that the existence and stability of the pseudo-almost periodic solutions for impulsive nonautonomous neural networks are obtained.

Introduction

Recently, neural networks have been studied extensively by many authors and found many applications in different areas such as in engineering, artificial intelligence, parallel computation and so on (see Refs.[1], [2], [3], [4], [5], [6], [7], [8], [9], [10], [11], [12], [13], [14], [15], [16], [17], [18], [19]). It is known to all that studies on neural dynamic systems not only involve a discussion of stability properties, but also involve many dynamic behaviors such as periodic oscillatory behavior, almost periodic oscillatory properties, chaos and bifurcation.

Furthermore, in the aspect of studying the almost periodic problems for dynamic systems and its related topics, the existence of almost periodic, asymptotically almost periodic, pseudo-almost periodic solutions become the most attractive hot issues in qualitative theory of differential equations due to their applications, especially in biology, economics and physics (see Refs. [20], [21]). The concept of pseudo-almost periodicity, which is the central subject in this paper, was introduced by Zhang (see Refs. [22], [23]) in the early nineties. Since then, such a notion became of great interest to the classical almost periodicity in the sense of Bohr and Bochner. Thus such a concept is welcome for implementing another existing generalization of almost periodicity, the so-called asymptotically almost periodicity due to Frechet; see e.g. [23]. For more on the concepts of almost periodicity and pseudo-almost periodicity and related issues, we refer the reader to Refs. [21], [22], [23] (for both the almost periodicity and asymptotic almost periodicity) and to Refs. [24], [25] (for the pseudo-almost periodicity).

In view of the effects of the environmental factors, we can assume that the parameters of the system are almost periodic function, or more general, pseudo-almost periodic function, which are the natural generalization of the concept of almost periodicity. It is worth noting that the applications of the pseudo-almost periodic theory are involved in various research fields, some results of applications to neural networks in this field have been published, which show that such a theory is a powerful tool in investigation of neural networks and more general than almost periodic theory in real world applications (see Refs. [26], [27], [28], [29]). Since the exponential convergent rate can be unveiled, the global exponential stability plays a key role in characterizing the behavior of dynamical system. Thus, it is worthwhile to investigate the existence and global exponential stability of pseudo-almost periodic solutions of neural networks. For instance, in Ref. [26], F. Chérif studied the existence and global exponential stability of pseudo-almost periodic solution for shunting inhibitory cellular neural networks with the following form:

$${\dot{x}}_{\mathit{ij}}\left(t\right)=-a_{\mathit{ij}}{x}_{\mathit{ij}}\left(t\right)-\sum _{B_{\mathit{ij}}^{\mathit{kl}}\in N_r(i,j)}{B}_{\mathit{ij}}^{\mathit{kl}}\left(t\right)g\left(x_{\mathit{kl}}(t-\tau )\right)x_{\mathit{ij}}\left(t\right)-\sum _{C_{\mathit{ij}}^{\mathit{kl}}\in N_r(i,j)}{C}_{\mathit{ij}}^{\mathit{kl}}\left(t\right)\left({\int }_0^{\infty }{K}_{\mathit{ij}}\left(u\right)f\left(x_{\mathit{kl}}(t-u\right)\mathrm{d}u)\right)x_{\mathit{ij}}\left(t\right)+L_{\mathit{ij}}\left(t\right),$$

all coefficients of system (1.1) are pseudo-almost periodic functions. Moreover, in Ref. [28], B. Ammar and F. Chérif were concerned with the following RNNs with time-varying coefficients and mixed delays:

$${\dot{x}}_i\left(t\right)=-a_i{x}_i\left(t\right)+\sum _{j=1}^n\left(c_{\mathit{ij}}\left(t\right)f_j\left(x_j\left(t\right)\right)+d_{\mathit{ij}}\left(t\right)g_j\left(x_j(t-\tau )\right)+p_{\mathit{ij}}\left(t\right){\int }_{-\infty }^t{k}_{\mathit{ij}}(t-s)h_j\left(x_j\left(s\right)\right)\mathrm{d}s\right)+J_i\left(t\right),1\le i\le n.$$

After that, the pseudo-almost periodicity for some kinds of neural networks were investigated and some new effective results which are more general than almost periodicity were obtained (see Refs. [27], [29]). In paper [30], by using the exponential dichotomy theory and contraction mapping fixed point theorem, the author established the existence and uniqueness of pseudo-almost periodic solution for the following cellular neural networks:

$$x_i^{\prime }\left(t\right)=-c_i\left(t\right){\int }_0^{\infty }{h}_i\left(s\right)x_i(t-s)\mathrm{d}s+\sum _{j=1}^n{a}_{\mathit{ij}}\left(t\right)f_j\left(x_j(t-{\tau }_{\mathit{ij}}(t\left)\right)\right)+\sum _{j=1}^n{b}_{\mathit{ij}}\left(t\right){\int }_0^{\infty }{k}_{\mathit{ij}}\left(u\right)g_j\left(x_j(t-u)\right)\mathrm{d}u+I_i\left(t\right),i=1,2,\dots ,n.$$

On the other hand, the theory of impulsive differential equations is now being recognized to be not only richer than the corresponding theory of differential equations without impulses, but also represents a more natural framework for mathematical modeling of many real-world phenomena, such as population dynamic systems and the neural networks. In recent years, the impulsive differential equations have been extensively studied (see Refs. [31], [32], [33], [34], [35], [36], [37], [38], [39]).

However, to the best of our knowledge, there is no published papers considering the piecewise pseudo-almost periodic solutions for non-autonomous high-order Hopfield-type neural networks with variable pseudo-almost periodic coefficients. Motivated by the above, in this paper, for the first time, we study the impulsive non-autonomous high-order Hopfield-type neural networks with pseudo-almost periodic coefficients as follows:

$$\{\begin{array}{c}{x}_i^{\prime }\left(t\right)=\sum _{j=1}^n{c}_{\mathit{ij}}\left(t\right)x_j\left(t\right)+\sum _{j=1}^n{a}_{\mathit{ij}}\left(t\right)f_j\left(t,x_j(t-{\gamma }_{\mathit{ij}}(t\left)\right)\right)\hfill \\ +\sum _{j=1}^n\sum _{l=1}^n{b}_{\mathit{ijl}}\left(t\right)g_j\left(t,x_j(t-{\sigma }_{\mathit{ijl}}(t\left)\right)\right)g_l\left(t,x_l(t-v_{\mathit{ijl}}(t\left)\right)\right)+{\gamma }_i\left(t\right),\hfill \\ t\ne t_k,i=1,2,\dots ,n,k\in \mathbb{Z},\hfill \\ \mathrm{\Delta }x\left(t_k\right)={\alpha }_{k}x\left(t_k\right)+I_k\left(x\left(t_k\right)\right)+c_k,t=t_k,i=1,2,\dots ,n,k\in \mathbb{Z},\hfill \end{array}$$

where 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_ij(t) 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, a_ij(t) and b_ijl(t) are the first- and second-order connection weights of the neural network, ${\gamma }_{\mathit{ij}}\ge 0,{\sigma }_{\mathit{ijl}}\left(t\right)\ge 0$ and $v_{\mathit{ijl}}\left(t\right)\ge 0$ correspond to the transmission delays at time t. ${\gamma }_i\left(t\right)$ denote the external inputs at time t, f_j and g_j are the activation functions of signal transmission; $\mathrm{\Delta }{x}_i\left(t_k\right)=x_i\left(t_k^{+}\right)-x_i\left(t_k^{-}\right)$ are impulses at moments t_k and $t_1<t_2<\cdots$ is a strictly increasing sequence such that ${\lim }_{k\to \infty }{t}_k=+\infty$, ${\alpha }_k\in {\mathbb{R}}^{n\times n},I_k\in C(\Omega ,{\mathbb{R}}^n),\Omega \subset {\mathbb{R}}^n,c_k\in {\mathbb{R}}^n,k\in \mathbb{Z}$.

Denote

$$C\left(t\right)={\left(c_{\mathit{ij}}\left(t\right)\right)}_{n\times n},\gamma \left(t\right)={\left({\gamma }_1\left(t\right),\dots ,{\gamma }_n\left(t\right)\right)}^T,c_k={(c_1^k,\dots ,c_n^k)}^T,A\left(t\right)={\left(a_{\mathit{ij}}\left(t\right)\right)}_{n\times n},\varphi \left(t\right)={\left({\varphi }_1\left(t\right),\dots ,{\varphi }_n\left(t\right)\right)}^T,I_k\left(\varphi \left(t_k\right)\right)={\left(I_1^k\left({\varphi }_1\left(t_k\right)\right),\dots ,I_n^k\left({\varphi }_n\left(t_k\right)\right)\right)}^T,{\alpha }_k={({\alpha }_{\mathit{ij}}^k)}_{n\times n},$$

and $\mathit{PC}(J,{\mathbb{R}}^n),J\subset {\mathbb{R}}^n$, the space of all piecewise continuous functions $x:J\to {\mathbb{R}}^n$ with points of discontinuity of the first kind $t_k,k=\pm 1,\pm 2,\dots$ and which are continuous from the left, i.e., $x\left(t_k^{-}\right)=x\left(t_k\right)$.

Let $t_0\in \mathbb{R}$, introduce the notation: $\mathit{PC}\left(t_0\right)$ is the space of all functions $\varphi :[t_0-\tilde{\tau },t_0]\to {\Omega }_0\subset {\mathbb{R}}^n$ with points of discontinuity of the first kind ${\theta }_1,{\theta }_2,\dots ,{\theta }_s\in (t_0-\tilde{\tau },t_0)$, where $\tilde{\tau }={\max }_{i,j,l}{\sup }_{t\in \mathbb{R}}\left\{{\gamma }_{\mathit{ij}}\left(t\right),{\sigma }_{\mathit{ijl}}\left(t\right),v_{\mathit{ijl}}\left(t\right)\right\}$.

Let ϕ₀ be an element of $\mathit{PC}\left(t_0\right)$. Denote by

$$x\left(t\right)=x(t;t_0,{\varphi }_0)={\left(x_1\left(t\right),x_2\left(t\right),\dots ,x_n\left(t\right)\right)}^T,{\varphi }_0=\left({\varphi }_{01}\left(t\right),{\varphi }_{02}\left(t\right),\dots ,{\varphi }_{0n}\left(t\right)\right)\in \mathit{PC}\left(t_0\right),$$

the solution of system (1.2), satisfying the initial conditions:

$$\{\begin{array}{c}x(t;t_0,{\varphi }_0)={\varphi }_0\left(t\right),t\in (t_0-\tilde{\tau },t_0),\hfill \\ x(t_0^{+};t_0,{\varphi }_0)={\varphi }_0\left(t_0\right).\hfill \end{array}$$

Remark 1.1

We say x(t) is the solution for the system (1.2) with the initial condition (1.3) if x(t) satisfies (1.2), (1.3). ${\theta }_1,{\theta }_2,\dots ,{\theta }_s$ are the first kind discontinuous points located in the interval $(t_0-\tilde{\tau },t_0)$, which is dependent on the initial point t₀ and the distribution of the discontinuous points.

The rest of this paper is organized as follows. In Section 2, we will establish some useful and completely new lemmas for a general impulsive non-autonomous dynamic system with pseudo-almost periodic coefficients, which will be used to obtain our main results. Section 3 is devoted to establishing some criteria for the existence and exponential stability of piecewise pseudo-almost periodic solution for the system (1.2) by the new method in Section 2. In Section 4, a numerical example and simulations are given to illustrate the feasibility and effectiveness of the obtained results.