Exponential Stability of Hopfield Neural Network Model with Non-Instantaneous Impulsive Effects

Abstract

We introduce a non-instantaneous impulsive Hopfield neural network model in this paper. Firstly, we prove the existence and uniqueness of an almost periodic solution of this model. Secondly, we prove that the solution of this model is exponentially stable. Finally, we give an example of this model.

1. Introduction

It is well known that neural network models have many applications in the area of parallel computing, associative memory, pattern recognition, computer vision etc. [1,2,3,4,5]. Therefore, more and more experts and scholars pay attention to neural network models.

The studies on neurocomputing have been improved very fast after the work of McCulloch et al. [6]. One of the neural networks model was given by Hopfield [7,8]. In the actual situation, system can be affected by short-term fluctuations in the environment. Impulses are commonly used to describe this phenomenon. For instance, according to Arbib [9] and Haykin [10], when a stimulus from the body or the external environment is received by receptors, the electrical impulses will be conveyed to the neural net and impulsive effects arise naturally in the net. Stamova and Stamov [11] proposed a Hopfield neural network with impulsive effects at fixed moments as follows

$$\begin{array}{c}\hfill \left\{\begin{array}{c}{\dot{w}}_i\left(\iota \right)={\displaystyle \sum _{j=1}^n}{a}_{ij}\left(\iota \right)w_j\left(\iota \right)+{\displaystyle \sum _{j=1}^n}{b}_{ij}\left(\iota \right)f_j\left(w_j\left(\iota \right)\right)+g_i\left(\iota \right),\iota \ne {\iota }_k,\hfill \\ \Delta w\left({\iota }_k\right)=B_{k}w\left({\iota }_k\right)+C_k\left(w\left({\iota }_k\right)\right)+h_k,k\in {\mathbb{N}}_{+},\hfill \end{array}\right.\end{array}$$

(1)

where $\iota \in \mathbb{J}:=\left\{0\right\}\bigcup {\mathbb{R}}_{+}$, ${\mathbb{R}}_{+}:=\left\{b\right|b$ is a positive real number}, ${\iota }_i$$(0\le {\iota }_1<{\iota }_2<\cdots )$ stand for the times which are impulses, $a_{ij},b_{ij},g_i\in C(\mathbb{J},\mathbb{R})$, $f_j\in C(\mathbb{J},\mathbb{R})$, $i=1,2,\cdots ,n,j=1,2,\cdots ,n$, $C(\mathbb{J},\mathbb{R})$ is the space of all the continuous functions from $\mathbb{J}$ to $\mathbb{R}$, $w\left(\iota \right)=col(w_1\left(\iota \right),w_2\left(\iota \right),\cdots ,w_n\left(\iota \right))$, $\Delta w\left({\iota }_k\right)=w\left({\iota }_k^{+}\right)-w\left({\iota }_k^{-}\right)$, $w\left({\iota }_k^{+}\right)$ is the right limits of $w\left({\iota }_k\right)$ and $w\left({\iota }_k^{-}\right)$ is the left limits of $w\left({\iota }_k\right)$, $B_k\in {\mathbb{R}}^{n\times n}$, $C_k\in C({\mathbb{R}}_{+}^n,{\mathbb{R}}^n)$, ${\mathbb{R}}_{+}^n=\{x=(x_1,x_2,\cdots ,x_n)\in {\mathbb{R}}^n|x_i>0,i=1,2,\cdots ,n\}$, where ${\mathbb{R}}^n,n\in \mathbb{N}$ is n-dimensional Euclidean space, $h_k\in {\mathbb{R}}^n,k\in {\mathbb{N}}_{+}$, $\mathbb{N}:=\{0,1,2,3,\cdots \}$ and ${\mathbb{N}}_{+}:=\mathbb{N}/\left\{0\right\}$.

However, most systems do not return to normal immediately after the impulse [12]. The system stays active for a limited period of time. Therefore, Hernández et al. [13] firstly introduced the theory of non-instantaneous impulses and established the existence of solutions for a class of impulsive differential equations. After that, Wang et al. [14,15,16] generalized this model and carried out more in-depth research on non-instantaneous impulsive differential equations. In general, there are no impulses that happen instantaneously, that is to say, it is non-instantaneous, even if the event occurs over a short period of time. Non-instantaneous impulsive effects exist in Hopfield neural network. For instance, in implementation of electronic networks, the state of the network is often subject to non-instantaneous perturbations, which may be caused by noise instances. Moreover, many evolutionary processes, particularly some biological systems, such as biological neural networks and bursting rhythm models in pathology, might exhibit non-instantaneous impulsive effects as well. Therefore, it is beneficial to study a class of differential equations with non-instantaneous impulses.

Then, we consider the case of the model (1) with non-instantaneous impulses as follows

$$\begin{array}{c}\hfill \left\{\begin{array}{c}{\dot{w}}_i\left(\iota \right)={\displaystyle \sum _{j=1}^n}{a}_{ij}\left(\iota \right)w_j\left(\iota \right)+{\displaystyle \sum _{j=1}^n}{b}_{ij}\left(\iota \right)f_j\left(w_j\left(\iota \right)\right)+g_i\left(\iota \right),\iota \in (l_k,m_{k+1}],k\in \mathbb{N},\hfill \\ w\left(m_k^{+}\right)=B_{k}w\left(m_k^{-}\right)+C_k\left(w\left(m_k^{-}\right)\right)+h_k,k\in {\mathbb{N}}_{+},\hfill \\ w\left(\iota \right)=B_{k}w\left(m_k^{-}\right)+C_k\left(w\left(m_k^{-}\right)\right)+h_k,\iota \in (m_k,l_k],k\in {\mathbb{N}}_{+},\hfill \\ w\left(l_k^{+}\right)=w\left(l_k^{-}\right),k\in {\mathbb{N}}_{+},\hfill \end{array}\right.\end{array}$$

(2)

where $0=l_0<m_1<l_1<m_2<l_2<\cdots <m_k<l_k<m_{k+1}<\cdots$. The solution $w\left(\iota \right)=w(\iota ;{\iota }_0,w_0)$ of model (2) with the initial condition $w\left({\iota }_0^{+}\right)=w_0\in {\mathbb{R}}_{+}^n,{\iota }_0\in \mathbb{J}$ is a piecewise continuous function with points of discontinuity of the first kind at the moments $m_k,k\in {\mathbb{N}}_{+}$, at which it is continuous from the left.

Periodic phenomenon is one of the phenomena widely existing in nature [17]. But many motion processes in the present world are approximate to periodic instead of strictly periodicity. Therefore, Danish mathematicians Bohr [18] first proposed the concept of almost periodic (AP), which is a significant generalization for practical application. Many scholars have demonstrated that it is more realistic to adopt an AP hypothesis in the process of AP study, when taking into account the impact of environmental factors, and this has certain ergodicity [19,20,21,22,23,24,25].

The rest of this paper is arranged as follows. In Section 2, we provide some of the necessary preliminaries for this paper. In Section 3, we prove the existence, uniqueness and exponential stability of the AP solution to (2). In Section 4, we present an Example to support our theoretical results.

2. Preliminaries

For the sequences $\left\{m_k\right\}$ and $\left\{l_k\right\}$, $k\in {\mathbb{N}}_{+}$, assume that $\underset{k\to +\infty }{lim}{m}_k=+\infty$, $\underset{k\to +\infty }{lim}{l}_k=+\infty$. Let the norm $\parallel j\left(\iota \right)\parallel =max\left\{\right|j_1\left(\iota \right)|,|j_2\left(\iota \right)|,\cdots ,|j_n\left(\iota \right)\left|\right\}$ for $j\left(\iota \right)={(j_1\left(\iota \right),j_2\left(\iota \right),\cdots ,j_n\left(\iota \right))}^{\top }$. The space $PC([0,\infty ),{\mathbb{R}}^n):=\{w:[0,\infty )\to {\mathbb{R}}^n:$ $w\in C((m_i,m_{i+1}]$, ${\mathbb{R}}^n)$, $w\left(m_i^{-}\right)=w\left(m_i\right)$, $w\left(m_i^{+}\right)$ exist for any $i\in \mathbb{N}\}$ endowed with norm ${\parallel w\parallel }_{PC}=\underset{\iota \in [0,\infty )}{sup}\parallel w\left(\iota \right)\parallel$, where $C((m_i,m_{i+1}],{\mathbb{R}}^n)$ represents the space which is made up of all the continuous functions from $(m_i,m_{i+1}]$ to ${\mathbb{R}}^n$. It is obvious that $(PC([0,\infty ),{\mathbb{R}}^n),\parallel ·{\parallel }_{PC})$ is a Banach space.

Definition 1

(see [26]). For the sequences ${\left\{M_i\right\}}_{i\in {\mathbb{N}}_{+}}$, $M_i\in {\mathbb{R}}^n$, if for any $i\in {\mathbb{N}}_{+}$ there exist $\epsilon >0$ and integer p such that the following inequality hold

$$\parallel M_{i+p}-M_i\parallel <\epsilon ,$$

(3)

then p is called to be ε-AP of the ${\left\{M_i\right\}}_{i\in {\mathbb{N}}_{+}}$, $M_i\in {\mathbb{R}}^n$.

Definition 2

(see [27]). ${\left\{M_i\right\}}_{i\in {\mathbb{N}}_{+}}$, $M_i\in {\mathbb{R}}^n$ are called to be AP sequences if for any $\epsilon >0$, there exists a relatively dense set of its ε-AP.

Definition 3

(see [26]). The $w\in PC([0,\infty ),{\mathbb{R}}^n)$ is called an AP function if all of the conditions are satisfied as follows

(i)

$\{m_i^j\}$, $i,j\in {\mathbb{N}}_{+}$ are uniformly AP sequences, where $m_i^j=m_{i+j}-m_i$.

(ii)

For any $\epsilon >0$, there exists a number $\delta =\delta \left(\epsilon \right)$ which is positive, such that if ${\iota }_1$ and ${\iota }_2$ are the points in the same continuous interval and $|{\iota }_1-{\iota }_2|<\delta$, then $\parallel w\left({\iota }_1\right)-w\left({\iota }_2\right)\parallel <\epsilon$.

(iii)

For any $\epsilon >0$, there exists a relatively dense set Γ of $\epsilon -$AP, such that if $\vartheta \in \mathsf{\Gamma }$, then $\parallel w(\iota +\vartheta )-w\left(\iota \right)\parallel <\epsilon$ for all $\iota \in [0,\infty )$ satisfying the condition $|\iota -m_i|>\epsilon ,i\in {\mathbb{N}}_{+}$.

Together with model (2), we shall consider the linear model

$$\begin{array}{c}\hfill \left\{\begin{array}{c}\dot{w}\left(\iota \right)=A\left(\iota \right)w\left(\iota \right),\iota \in (l_k,m_{k+1}],k\in \mathbb{N},\hfill \\ w\left(m_k^{+}\right)=B_{k}w\left(m_k^{-}\right),k\in {\mathbb{N}}_{+},\hfill \\ w\left(\iota \right)=B_{k}w\left(m_k^{-}\right),\iota \in (m_k,l_k],k\in {\mathbb{N}}_{+},\hfill \\ w\left(l_k^{+}\right)=w\left(l_k^{-}\right),k\in {\mathbb{N}}_{+},\hfill \end{array}\right.\end{array}$$

(4)

where $A\left(\iota \right)=\left(a_{ij}\right),i=1,2,3,\cdots ,n,j=1,2,\cdots ,n$.

Let $w\left(\iota \right)=\mathcal{W}(\iota ,{\iota }_0)w_{{\iota }_0}$, $0\le {\iota }_0\le \iota$ represents the solution of (4) with $w\left({\iota }_0\right)=w_{{\iota }_0}$, where $\mathcal{W}(\iota ,{\iota }_0)$ is the Cauchy matrix of model (4) which can be looked up on [28].

We propose some assumptions as follows.

(H₁)

The sequences $\left\{l_k^{\tau }\right\}$, $l_k^{\tau }=l_{k+\tau }-l_k$ and $\left\{m_k^{\tau }\right\}$, $m_k^{\tau }=m_{k+\tau }-m_k$, $k,\tau \in {\mathbb{N}}_{+}$ are uniformly AP and $0<l_k-m_k\le \theta <+\infty$, $0<\varsigma \le m_{k+1}-l_k\le \overline{\theta }<+\infty$, $k\in {\mathbb{N}}_{+}$.

(H₂)

The matrix function $A\in C(\mathbb{J},{\mathbb{R}}^{n\times n})$ is AP in the sense of Bohr.

(H₃)

The sequence $\left\{B_k\right\},k\in {\mathbb{N}}_{+}$ is AP.

(H₄)

The functions $f_j\left(\iota \right)$ are AP in the sense of Bohr, and

$$0<\underset{\iota \in \mathbb{J}}{sup}\left|f_j\left(\iota \right)\right|<\infty ,f_j\left(0\right)=0,$$

and there exists an $L_1>0$ such that for $\iota ,s\in \mathbb{R}$,

$$\underset{j=1,2,3,\cdots ,n}{max}|f_j\left(\iota \right)-f_j\left(s\right)|<L_1|\iota -s|.$$

(H₅)

The functions $b_{ij}\left(\iota \right)$ are AP in the sense of Bohr, and

$$0<\underset{\iota \in \mathbb{J}}{sup}\left|b_{ij}\left(\iota \right)\right|={\overline{b}}_{ij}<\infty .$$

(H₆)

The functions $g_i\left(\iota \right),i=1,2,3,\cdots ,n$, are AP in the sense of Bohr, the sequences $\left\{h_k\right\},k\in {\mathbb{N}}_{+}$ are AP and there exists a $\mathbf{C}>0$ such that

$$max\left\{{\parallel g\parallel }_{PC},\underset{k\in {\mathbb{N}}_{+}}{sup}\parallel h_k\parallel \right\}\le \mathbf{C},$$

where $g\left(\iota \right)=(g_1\left(\iota \right),g_2\left(\iota \right),\cdots ,g_n\left(\iota \right))$.

(H₇)

The sequence of functions $\left\{C_k\left(x\right)\right\},k\in {\mathbb{N}}_{+}$ is AP uniformly with respect to $x\in {\mathbb{R}}_{+}^n$, and there exists an $L_2>0$ such that

$$\parallel C_k\left(x\right)-C_k\left(y\right)\parallel \le L_2\parallel x-y\parallel ,$$

for $k\in {\mathbb{N}}_{+},x,y\in {\mathbb{R}}^n$. $C_k\left(x\right)=x$ if and only if $x=(0,0,\cdots ,0)$.

Now, we need the following Lemmas.

Lemma 4

(see [28]). Assume that $\left(H_1\right)$–$\left(H_3\right)$ hold. Then, for the Cauchy matrix $\mathcal{W}(\iota ,{\iota }_0)$ of model (4) there exist positive constants $\mathbf{K}\ge 0$ and ${\rm Y}>0$ such that

$$\parallel \mathcal{W}(\iota ,{\iota }_0)\parallel \le \mathbf{K}{e}^{-{\rm Y}(\iota -{\iota }_0)},0\le {\iota }_0\le \iota .$$

Lemma 5

(see [28]). For any $\epsilon >0$, $0\le {\iota }_0<\iota$, $|\iota -m_i|>\epsilon$, $|\iota -l_i|>\epsilon$, $|{\iota }_0-m_i|>\epsilon$ and $|{\iota }_0-l_i|>\epsilon$, $i\in {\mathbb{N}}_{+}$, there exist a constant $K>0$ and a relatively dense set of Γ of ε-AP such that

$$\parallel \mathcal{W}(\iota +r,{\iota }_0+r)-\mathcal{W}(\iota ,{\iota }_0)\parallel \le \epsilon K{e}^{-\frac{1}{2}{\rm Y}(\iota -{\iota }_0)},r\in \mathsf{\Gamma }.$$

Lemma 6

(see [11]). Let conditions $\left(H_1\right)$–$\left(H_6\right)$ hold. Then for each $\epsilon >0$, there exist ${\epsilon }_1$, $0<{\epsilon }_1<\epsilon$, a relatively dense set Γ of real numbers and a set Q of integers such that the following relations are fulfilled.

(a)

$\parallel A(\iota +r)-A\left(\iota \right)\parallel <\epsilon ,\iota \in \mathbb{J},r\in \mathsf{\Gamma }$;

(b)

$|b_{ij}(\iota +r)-b_{ij}\left(\iota \right)|<\epsilon ,\iota \in \mathbb{J},r\in \mathsf{\Gamma },i,j=1,2,3,\cdots ,n$;

(c)

$|f_j(\iota +r)-f_j\left(\iota \right)|<\epsilon ,\iota \in \mathbb{J},r\in \mathsf{\Gamma },j=1,2,3,\cdots ,n$;

(d)

$|g_j(\iota +r)-g_j\left(\iota \right)|<\epsilon ,\iota \in \mathbb{J},r\in \mathsf{\Gamma },j=1,2,3,\cdots ,n$;

(e)

$\parallel B_{k+q}-B_k\parallel <\epsilon ,q\in Q,k\in {\mathbb{R}}_{+}$;

(f)

$|h_{k+q}-h_k|<\epsilon ,q\in Q,k\in {\mathbb{R}}_{+}$;

(g)

$|l_k^q-r|<{\epsilon }_1$, $|m_k^q-r|<{\epsilon }_1$, $q\in Q$, $r\in \mathsf{\Gamma }$, $k\in {\mathbb{N}}_{+}$.

Lemma 7

(see [26]). If the sequences $\{m_i^j\}$, $i,j\in \mathbb{N}$ are uniformly AP, then we can get

(i)

There exists a constant $\rho >0$ such that $\underset{t\to +\infty }{sup}\frac{\mu (\iota +t,\iota )}{t}=\rho$ which is uniformly with respect to $\iota >0$.

(ii)

For any $p>0$, there exists N which is a positive integer such that the number of elements in the sequences $\left\{m_i\right\}$ on each interval of length p does not exceed N. We can choose $N\ge \rho$.

3. Main Results

Theorem 8.

Assume that conditions $\left(H_1\right)$–$\left(H_7\right)$ are satisfied, model (2) has a unique positive AP solution if

$$\mathbf{K}\left\{{\displaystyle \frac{L_1}{{\rm Y}}}\underset{i=1,2,\cdots ,n}{max}\sum _{j=1}^n{\overline{b}}_{ij}+L_2{N}_1\right\}<1.$$

Proof. 

Let $N_1=\underset{\iota \in \mathbb{J}}{sup}{\displaystyle \sum _{k=1}^{\mu (\iota ,0)}}{e}^{-{\rm Y}(\iota -m_k^{+})}$, $N_2=\underset{\iota \in \mathbb{J}}{sup}{\displaystyle \sum _{k=1}^{\mu (\iota ,0)}}{e}^{-\frac{1}{2}{\rm Y}(\iota -m_k^{+})}$, $\mathsf{\Omega }:=\{w\in PC(\mathbb{J},{\mathbb{R}}_{+}^n),w$ is AP $(\parallel w(·+r)-w(·)\parallel <\epsilon ,r\in \mathsf{\Gamma })$ and ${\parallel w\parallel }_{PC}\le \aleph \}$, where $\mathsf{\Gamma }$ is mentioned in Lemma 5.

For $l_k<\iota <m_{k+1},k\in \mathbb{N}$, let

$$\begin{array}{ccc}\hfill \phi & =& \sum _{k=0}^{\mu (\iota ,0)-1}{\int }_{l_k}^{m_{k+1}}\mathcal{W}(\iota ,u)g\left(u\right)du+{\int }_{l_{\mu (\iota ,0)}}^{\iota }\mathcal{W}(\iota ,u)g\left(u\right)du\hfill \\ & & +\sum _{k=1}^{\mu (\iota ,0)}\mathcal{W}(\iota ,m_k^{+})h_k.\hfill \end{array}$$

Then,

$$\begin{array}{ccc}\hfill {\parallel \phi \parallel }_{PC}& \le & \underset{\iota \in \mathbb{J}}{sup}\{\sum _{k=0}^{\mu (\iota ,0)-1}{\int }_{l_k}^{m_{k+1}}\parallel \mathcal{W}(\iota ,u)\parallel \parallel g\left(u\right)\parallel du+{\int }_{l_{\mu (\iota ,0)}}^{\iota }\parallel \mathcal{W}(\iota ,u)\parallel \parallel g\left(u\right)\parallel du\hfill \\ & & +\sum _{k=1}^{\mu (\iota ,0)}\parallel \mathcal{W}(\iota ,m_k^{+})\parallel \parallel h_k\parallel \}\hfill \\ & \le & \underset{\iota \in \mathbb{J}}{sup}\left\{{\int }_0^{\iota }\parallel \mathcal{W}(\iota ,u)\parallel \parallel g\left(u\right)\parallel du+\sum _{k=1}^{\mu (\iota ,0)}\parallel \mathcal{W}(\iota ,m_k^{+})\parallel \parallel h_k\parallel \right\}\hfill \\ & \le & \underset{\iota \in \mathbb{J}}{sup}\left\{\underset{i=1,2,\cdots ,n}{max}{\int }_0^{\iota }\parallel \mathcal{W}(\iota ,u)\parallel |g_i\left(u\right)|du+\sum _{k=1}^{\mu (\iota ,0)}\parallel \mathcal{W}(\iota ,m_k^{+})\parallel \parallel h_k\parallel \right\}\hfill \\ & \le & \underset{\iota \in \mathbb{J}}{sup}\{{\int }_0^{\iota }\mathbf{K}{e}^{-{\rm Y}(\iota -u)}\mathbf{C}du+\sum _{k=1}^{\mu (\iota ,0)}\mathbf{K}{e}^{-{\rm Y}(\iota -m_k^{+})}\mathbf{C}\}\hfill \\ & \le & {\displaystyle \frac{\mathbf{K}\mathbf{C}}{{\rm Y}}}+\mathbf{K}{N}_1\mathbf{C}\hfill \\ & \le & \mathbf{K}\mathbf{C}\left({\displaystyle \frac{1}{{\rm Y}}}+N_1\right)=\aleph .\hfill \end{array}$$

(5)

Let $r\in \mathsf{\Gamma }$, $q\in Q$, where the sets $\mathsf{\Gamma }$ and Q are determined in Lemma 6. Then,

$$\begin{array}{ccc}\hfill \underset{\iota \in \mathbb{J}}{sup}\parallel \phi (\iota +r)-\phi \left(\iota \right)\parallel & \le & \underset{\iota \in \mathbb{J}}{sup}\{\sum _{k=0}^{\mu (\iota ,0)-1}{\int }_{l_k}^{m_{k+1}}\parallel \mathcal{W}(\iota +r,u+r)-\mathcal{W}(\iota ,u)\parallel \parallel g(u+r)\parallel du\hfill \\ & & +\sum _{k=0}^{\mu (\iota ,0)-1}{\int }_{l_k}^{m_{k+1}}\parallel \mathcal{W}(\iota ,u)\parallel \parallel g(u+r)-g\left(u\right)\parallel du\hfill \\ & & +{\int }_{l_{\mu (\iota ,0)}}^{\iota }\parallel \mathcal{W}(\iota +r,u+r)-\mathcal{W}(\iota ,u)\parallel \parallel g(u+r)\parallel du\hfill \\ & & +{\int }_{l_{\mu (\iota ,0)}}^{\iota }\parallel \mathcal{W}(\iota ,u)\parallel \parallel g(u+r)-g\left(u\right)\parallel du\hfill \\ & & +\sum _{k=1}^{\mu (\iota ,0)}\parallel \mathcal{W}(\iota +r,m_{k+q}^{+})-\mathcal{W}(\iota ,m_k^{+})\parallel \parallel h_{k+q}\parallel \hfill \\ & & +\sum _{k=1}^{\mu (\iota ,0)}\parallel \mathcal{W}(\iota ,m_k^{+})\parallel \parallel h_{k+q}-h_k\parallel \}\hfill \\ & \le & \underset{\iota \in \mathbb{J}}{sup}\{{\int }_0^{\iota }\parallel \mathcal{W}(\iota +r,u+r)-\mathcal{W}(\iota ,u)\parallel \parallel g(u+r)\parallel du\hfill \\ & & +{\int }_0^{\iota }\parallel \mathcal{W}(\iota ,u)\parallel \parallel g(u+r)-g\left(u\right)\parallel du\hfill \\ & & +\sum _{k=1}^{\mu (\iota ,0)}\parallel \mathcal{W}(\iota +r,m_{k+q}^{+})-\mathcal{W}(\iota ,m_k^{+})\parallel \parallel h_{k+q}\parallel \hfill \\ & & +\sum _{k=1}^{\mu (\iota ,0)}\parallel \mathcal{W}(\iota ,m_k^{+})\parallel \parallel h_{k+q}-h_k\parallel \}\hfill \\ & \le & \underset{\iota \in \mathbb{J}}{sup}\{{\int }_0^{\iota }\epsilon K{e}^{-\frac{1}{2}{\rm Y}(\iota -u)}\mathbf{C}du+{\int }_0^{\iota }\mathbf{K}{e}^{-{\rm Y}(\iota -u)}\epsilon du\hfill \\ & & +\sum _{k=1}^{\mu (\iota ,0)}\epsilon K{e}^{-\frac{1}{2}{\rm Y}(\iota -m_k^{+})}\mathbf{C}+\sum _{k=1}^{\mu (\iota ,0)}\mathbf{K}{e}^{-{\rm Y}(\iota -m_k^{+})}\epsilon \}\hfill \\ & \le & \epsilon K\mathbf{C}{\displaystyle \frac{2}{{\rm Y}}}+\mathbf{K}\epsilon {\displaystyle \frac{1}{{\rm Y}}}+\epsilon K{N}_2\mathbf{C}+\mathbf{K}\epsilon N_1\hfill \\ & \le & \epsilon \left(K\mathbf{C}{\displaystyle \frac{2}{{\rm Y}}}+\mathbf{K}{\displaystyle \frac{1}{{\rm Y}}}+K{N}_2\mathbf{C}+\mathbf{K}{N}_1\right).\hfill \end{array}$$

Set

$$F(\iota ,w)=col\{F_1(\iota ,w),F_2(\iota ,w),\cdots ,F_n(\iota ,w)\},$$

where

$$F_i(\iota ,w)=\sum _{j=1}^n{b}_{ij}\left(\iota \right)f_j\left(w_j\left(\iota \right)\right),i=1,2,3,\cdots ,n.$$

We define in $\mathsf{\Omega }$ an operator T,

$$\begin{array}{ccc}\hfill T_w& =& \sum _{k=0}^{\mu (\iota ,0)-1}{\int }_{l_k}^{m_{k+1}}\mathcal{W}(\iota ,u)(F(u,w\left(u\right))+g\left(u\right))du\hfill \\ & & +{\int }_{l_{\mu (\iota ,0)}}^{\iota }\mathcal{W}(\iota ,u)(F(u,w\left(u\right))+g\left(u\right))du\hfill \\ & & +\sum _{k=1}^{\mu (\iota ,0)}\mathcal{W}(\iota ,m_k^{+})(C_k\left(w\left(m_k^{-}\right)\right)+h_k)\hfill \end{array}$$

(6)

and consider a subset $\widehat{\mathsf{\Omega }}\subset \mathsf{\Omega }$, where

$$\widehat{\mathsf{\Omega }}=\left\{{w\in \mathsf{\Omega }:\parallel w-\phi \parallel }_{PC}\le {\displaystyle \frac{R\aleph }{1-R}}\right\}.$$

Consequently, for an arbitrary $w\in \widehat{\mathsf{\Omega }}$ from (5) and (6) it follows that

$${\parallel w\parallel }_{PC}\le {\parallel w-\phi \parallel }_{PC}+{\parallel \phi \parallel }_{PC}\le {\displaystyle \frac{R\aleph }{1-R}}+\aleph ={\displaystyle \frac{\aleph }{1-R}}.$$

Now, we prove that T is a self-mapping from $\widehat{\mathsf{\Omega }}$ to $\widehat{\mathsf{\Omega }}$.

For $w\in \widehat{\mathsf{\Omega }}$ we have

$$\begin{array}{ccc}\hfill & & \parallel T_w{-\phi \parallel }_{PC}\hfill \\ \hfill & \le & \underset{\iota \in \mathbb{J}}{sup}\{\sum _{k=0}^{\mu (\iota ,0)-1}{\int }_{l_k}^{m_{k+1}}\parallel \mathcal{W}(\iota ,u)\parallel \parallel F(u,w\left(u\right))\parallel du\hfill \\ \hfill & & +{\int }_{l_{\mu (\iota ,0)}}^{\iota }\parallel \mathcal{W}(\iota ,u)\parallel \parallel F(u,w\left(u\right))\parallel du\hfill \\ \hfill & & +\sum _{k=1}^{\mu (\iota ,0)}\parallel \mathcal{W}(\iota ,m_k^{+})\parallel \parallel C_k\left(w\left(m_k^{-}\right)\right)\parallel \}\hfill \\ \hfill & \le & \underset{\iota \in \mathbb{J}}{sup}\{{\int }_0^{\iota }\parallel \mathcal{W}(\iota ,u)\parallel \parallel F(u,w\left(u\right))\parallel du\hfill \\ \hfill & & +\sum _{k=1}^{\mu (\iota ,0)}\parallel \mathcal{W}(\iota ,m_k^{+})\parallel \parallel C_k\left(w\left(m_k^{-}\right)\right)\parallel \}\hfill \\ \hfill & \le & \underset{\iota \in \mathbb{J}}{sup}\{\underset{i=1,2,\cdots ,n}{max}{\int }_0^{\iota }\parallel \mathcal{W}(\iota ,u)\parallel \sum _{j=1}^n\parallel b_{ij}\left(\iota \right)\parallel \parallel f_j(w_j\left(u\right)\parallel du\hfill \\ \hfill & & +\sum _{k=1}^{\mu (\iota ,0)}\parallel \mathcal{W}(\iota ,m_k^{+})\parallel \parallel C_k\left(w\left(m_k^{-}\right)\right)\parallel \}\hfill \\ \hfill & \le & \underset{\iota \in \mathbb{J}}{sup}\{\underset{i=1,2,\cdots ,n}{max}{\int }_0^{\iota }\parallel \mathcal{W}(\iota ,u)\parallel \sum _{j=1}^n{\overline{b}}_{ij}{L}_1\parallel w\left(u\right)\parallel du\hfill \\ \hfill & & +\sum _{k=1}^{\mu (\iota ,0)}\parallel \mathcal{W}(\iota ,m_k^{+})\parallel L_2\parallel w\left(m_k^{-}\right)\parallel \}\hfill \\ \hfill & \le & \{\underset{i=1,2,\cdots ,n}{max}{\int }_0^{\iota }\mathbf{K}{e}^{-{\rm Y}(\iota -u)}{L}_1\sum _{j=1}^n{\overline{b}}_{ij}du\hfill \\ \hfill & & +\sum _{k=1}^{\mu (\iota ,0)}\mathbf{K}{e}^{-{\rm Y}(\iota -m_k^{+})}{L}_2\}{\parallel w\parallel }_{PC}\hfill \\ \hfill & \le & \mathbf{K}\left\{\underset{i=1,2,\cdots ,n}{max}{\displaystyle \frac{L_1}{{\rm Y}}}\sum _{j=1}^n{\overline{b}}_{ij}+N_1{L}_2\right\}{\parallel w\parallel }_{PC}\hfill \\ & =& {R\parallel w\parallel }_{PC}\le {\displaystyle \frac{R\aleph }{1-R}}.\hfill \end{array}$$

(7)

Let $r\in \mathsf{\Gamma }$, $q\in Q$, where the sets $\mathsf{\Gamma }$ and Q are determined in Lemma 6. Then

$$\begin{array}{ccc}\hfill & & \parallel T_w(\iota +r)-T_w\left(\iota \right)\parallel \hfill \\ \hfill & \le & \underset{\iota \in \mathbb{J}}{sup}\parallel (T_w(\iota +r)-T_w\left(\iota \right))-(\phi (\iota +r)-\phi \left(\iota \right))\parallel +\underset{\iota \in \mathbb{J}}{sup}\parallel \phi (\iota +r)-\phi \left(\iota \right)\parallel \hfill \\ \hfill & \le & \underset{\iota \in \mathbb{J}}{sup}\parallel (T_w(\iota +r)-\phi (\iota +r))-(T_w\left(\iota \right)-\phi \left(\iota \right))\parallel +\underset{\iota \in \mathbb{J}}{sup}\parallel \phi (\iota +r)-\phi \left(\iota \right)\parallel \hfill \\ \hfill & \le & \underset{\iota \in \mathbb{J}}{sup}\{\sum _{k=0}^{\mu (\iota ,0)-1}{\int }_{l_k}^{m_{k+1}}\parallel \mathcal{W}(\iota +r,u+r)-\mathcal{W}(\iota ,u)\parallel \parallel F(u+r,w(u+r))\parallel du\hfill \\ \hfill & & +\sum _{k=0}^{\mu (\iota ,0)-1}{\int }_{l_k}^{m_{k+1}}\parallel \mathcal{W}(\iota ,u)\parallel \parallel F(u+r,w(u+r))-F(u,w\left(u\right))\parallel du\hfill \\ \hfill & & +{\int }_{l_{\mu (\iota ,0)}}^{\iota }\parallel \mathcal{W}(\iota +r,u+r)-\mathcal{W}(\iota ,u)\parallel \parallel F(u+r,w(u+r))\parallel du\hfill \\ \hfill & & +{\int }_{l_{\mu (\iota ,0)}}^{\iota }\parallel \mathcal{W}(\iota ,u)\parallel \parallel F(u+r,w(u+r))-F(u,w\left(u\right))\parallel du\hfill \\ \hfill & & +\sum _{k=1}^{\mu (\iota ,0)}\parallel \mathcal{W}(\iota +r,m_{k+q}^{+})-\mathcal{W}(\iota ,m_k^{+})\parallel \parallel C_{k+q}(w(m_{k+q}^{-}))\parallel \hfill \\ \hfill & & +\sum _{k=1}^{\mu (\iota ,0)}\parallel \mathcal{W}(\iota ,m_k^{+})\parallel \parallel C_{k+q}(w(m_{k+q}^{-}))-C_k\left(w\left(m_k^{-}\right)\right)\parallel \}\hfill \\ \hfill & & +\underset{\iota \in \mathbb{J}}{sup}\parallel \phi (\iota +r)-\phi \left(\iota \right)\parallel \hfill \\ \hfill & \le & \underset{\iota \in \mathbb{J}}{sup}\{{\int }_0^{\iota }\parallel \mathcal{W}(\iota +r,u+r)-\mathcal{W}(\iota ,u)\parallel \parallel F(u+r,w(u+r))\parallel du\hfill \\ \hfill & & +{\int }_0^{\iota }\parallel \mathcal{W}(\iota ,u)\parallel \parallel F(u+r,w(u+r))-F(u,w\left(u\right))\parallel du\hfill \\ \hfill & & +\sum _{k=1}^{\mu (\iota ,0)}\parallel \mathcal{W}(\iota +r,m_{k+q}^{+})-\mathcal{W}(\iota ,m_k^{+})\parallel \parallel C_{k+q}(w(m_{k+q}^{-}))\parallel \hfill \\ \hfill & & +\sum _{k=1}^{\mu (\iota ,0)}\parallel \mathcal{W}(\iota ,m_k^{+})\parallel \parallel C_{k+q}(w(m_{k+q}^{-}))-C_k\left(w\left(m_k^{-}\right)\right)\parallel \}\hfill \\ \hfill & & +\underset{\iota \in \mathbb{J}}{sup}\parallel \phi (\iota +r)-\phi \left(\iota \right)\parallel \hfill \\ \hfill & \le & \underset{\iota \in \mathbb{J}}{sup}\left\{\underset{i=1,2,\cdots ,n}{max}\right({\int }_0^{\iota }\parallel \mathcal{W}(\iota +r,u+r)-\mathcal{W}(\iota ,u)\parallel \left|\sum _{j=1}^n{b}_{ij}(u+r)f_j\left(w_j(u+r)\right)\right|du\hfill \\ \hfill & & +{\int }_0^{\iota }\parallel \mathcal{W}(\iota ,u)\parallel |\sum _{j=1}^n{b}_{ij}(u+r)f_j\left(w_j(u+r)\right)-\sum _{j=1}^n{b}_{ij}\left(u\right)f_j\left(w_j\left(u\right)\right)|du)\hfill \\ \hfill & & +\sum _{k=1}^{\mu (\iota ,0)}\parallel \mathcal{W}(\iota +r,m_{k+q}^{+})-\mathcal{W}(\iota ,m_k^{+})\parallel \parallel C_{k+q}(w(m_{k+q}^{-}))\parallel \hfill \\ \hfill & & +\sum _{k=1}^{\mu (\iota ,0)}\parallel \mathcal{W}(\iota ,m_k^{+})\parallel \parallel C_{k+q}(w(m_{k+q}^{-}))-C_k\left(w\left(m_k^{-}\right)\right)\parallel \}\hfill \\ \hfill & & +\underset{\iota \in \mathbb{J}}{sup}\parallel \phi (\iota +r)-\phi \left(\iota \right)\parallel \hfill \end{array}$$

$$\begin{array}{ccc}\hfill & \le & \underset{\iota \in \mathbb{J}}{sup}\left\{\underset{i=1,2,\cdots ,n}{max}\right({\int }_0^{\iota }\parallel \mathcal{W}(\iota +r,u+r)-\mathcal{W}(\iota ,u)\parallel \left|\sum _{j=1}^n{b}_{ij}(u+r)f_j\left(w_j(u+r)\right)\right|du\hfill \\ \hfill & & +{\int }_0^{\iota }\parallel \mathcal{W}(\iota ,u)\parallel |\sum _{j=1}^n(b_{ij}(u+r)-b_{ij}\left(u\right))f_j\left(w_j(u+r)\right)\hfill \\ \hfill & & +\sum _{j=1}^n{b}_{ij}\left(u\right)(f_j\left(w_j(u+r)\right)-f_j\left(w_j\left(u\right)\right))\left|du\right)\hfill \\ \hfill & & +\sum _{k=1}^{\mu (\iota ,0)}\parallel \mathcal{W}(\iota +r,m_{k+q}^{+})-\mathcal{W}(\iota ,m_k^{+})\parallel \parallel C_{k+q}(w(m_{k+q}^{-}))\parallel \hfill \\ \hfill & & +\sum _{k=1}^{\mu (\iota ,0)}\parallel \mathcal{W}(\iota ,m_k^{+})\parallel \parallel C_{k+q}(w(m_{k+q}^{-}))-C_k\left(w\left(m_k^{-}\right)\right)\parallel \}\hfill \\ \hfill & & +\underset{\iota \in \mathbb{J}}{sup}\parallel \phi (\iota +r)-\phi \left(\iota \right)\parallel \hfill \\ \hfill & \le & \underset{\iota \in \mathbb{J}}{sup}\left\{\underset{i=1,2,\cdots ,n}{max}\right({\int }_0^{\iota }\epsilon K{e}^{-\frac{1}{2}{\rm Y}(\iota -u)}\sum _{j=1}^n{\overline{b}}_{ij}{L}_1|w_j(u+r)\left)\right|du\hfill \\ \hfill & & +{\int }_0^{\iota }\mathbf{K}{e}^{-{\rm Y}(\iota -u)}\left(\sum _{j=1}^n\epsilon L_1|w_j(u+r)|+\sum _{j=1}^n{\overline{b}}_{ij}{L}_1|w_j(u+r)-w_j\left(u\right)|\right)du)\hfill \\ \hfill & & +\sum _{k=1}^{\mu (\iota ,0)}\epsilon K{e}^{-\frac{1}{2}{\rm Y}(\iota -m_k^{+})}{L}_2\parallel w(m_{k+q}^{-})\parallel \hfill \\ \hfill & & +\sum _{k=1}^{\mu (\iota ,0)}\mathbf{K}{e}^{-{\rm Y}(\iota -m_k^{-})}{L}_2\parallel w(m_{k+q}^{-})-w(m_k^{-})\parallel \}\hfill \\ \hfill & & +\underset{\iota \in \mathbb{J}}{sup}\parallel \phi (\iota +r)-\phi \left(\iota \right)\parallel \hfill \\ \hfill & \le & \underset{\iota \in \mathbb{J}}{sup}\{\underset{i=1,2,\cdots ,n}{max}\left({\displaystyle \frac{\epsilon 2K}{{\rm Y}}}\sum _{j=1}^n{\overline{b}}_{ij}{L}_1{\displaystyle \frac{\aleph }{1-R}}+{\displaystyle \frac{\mathbf{K}}{{\rm Y}}}\left(\epsilon L_1{\displaystyle \frac{\aleph }{1-R}}+\sum _{j=1}^n{\overline{b}}_{ij}{L}_1\epsilon \right)\right)\hfill \\ \hfill & & +\sum _{k=1}^{\mu (\iota ,0)}\epsilon K{e}^{-\frac{1}{2}{\rm Y}(\iota -m_k^{+})}{L}_2{\displaystyle \frac{\aleph }{1-R}}+\sum _{k=1}^{\mu (\iota ,0)}\mathbf{K}{e}^{-{\rm Y}(\iota -m_k^{-})}{L}_2\epsilon \}\hfill \\ \hfill & & +\underset{\iota \in \mathbb{J}}{sup}\parallel \phi (\iota +r)-\phi \left(\iota \right)\parallel \hfill \\ \hfill & \le & \epsilon \{{\displaystyle \frac{L_1}{{\rm Y}}}\left(\underset{i=1,2,\cdots ,n}{max}\left(\sum _{j=1}^{n}2K{\overline{b}}_{ij}{\displaystyle \frac{\aleph }{1-R}}+\mathbf{K}{\displaystyle \frac{\aleph }{1-R}}+\mathbf{K}\sum _{j=1}^n{\overline{b}}_{ij}\right)+K{L}_2{N}_2{\displaystyle \frac{\aleph }{1-R}}+\mathbf{K}{L}_2{N}_1\right\}\hfill \\ \hfill & & +\epsilon \left(K\mathbf{C}{\displaystyle \frac{2}{{\rm Y}}}+\mathbf{K}{\displaystyle \frac{1}{{\rm Y}}}+K{N}_2\mathbf{C}+\mathbf{K}{N}_1\right)\hfill \\ \hfill & \le & \epsilon \left\{{\displaystyle \frac{L_1}{{\rm Y}}}\right(\underset{i=1,2,\cdots ,n}{max}\left(\sum _{j=1}^{n}2K{\overline{b}}_{ij}{\displaystyle \frac{\aleph }{1-R}}+\mathbf{K}{\displaystyle \frac{\aleph }{1-R}}+\mathbf{K}\sum _{j=1}^n{\overline{b}}_{ij}\right)+K{L}_2{N}_2{\displaystyle \frac{\aleph }{1-R}}+\mathbf{K}{L}_2{N}_1\hfill \\ & & +K\mathbf{C}{\displaystyle \frac{2}{{\rm Y}}}+\mathbf{K}{\displaystyle \frac{1}{{\rm Y}}}+K{N}_2\mathbf{C}+\mathbf{K}{N}_1\}.\hfill \end{array}$$

(8)

Consequently, after (7) and (8), we obtain that $T_w\in \widehat{\mathsf{\Omega }}$.

Let $\varphi \in \widehat{\mathsf{\Omega }}$, $\xi \in \widehat{\mathsf{\Omega }}$. Then,

$$\begin{array}{ccc}\hfill & & \parallel T_{\varphi }-T_{\xi }{\parallel }_{PC}\hfill \\ \hfill & \le & \underset{\iota \in \mathbb{J}}{sup}\{\sum _{k=0}^{\mu (\iota ,0)-1}{\int }_{l_k}^{m_{k+1}}\parallel \mathcal{W}(\iota ,u)\parallel \parallel F(u,\varphi \left(u\right))-F(u,\xi \left(u\right))\parallel du\hfill \\ \hfill & & +{\int }_{l_{\mu (\iota ,0)}}^{\iota }\parallel \mathcal{W}(\iota ,u)\parallel \parallel F(u,\varphi \left(u\right))-F(u,\xi \left(u\right))\parallel du\hfill \\ \hfill & & +\sum _{k=1}^{\mu (\iota ,0)}\parallel \mathcal{W}(\iota ,m_k^{+})\parallel \parallel C_k\left(\varphi \left(m_k^{-}\right)\right)-C_k\left(\xi \left(m_k^{-}\right)\right)\parallel \hfill \\ \hfill & \le & \underset{\iota \in \mathbb{J}}{sup}\{{\int }_0^{\iota }\parallel \mathcal{W}(\iota ,u)\parallel \parallel F(u,\varphi \left(u\right))-F(u,\xi \left(u\right))\parallel du\hfill \\ \hfill & & +\sum _{k=1}^{\mu (\iota ,0)}\parallel \mathcal{W}(\iota ,m_k^{+})\parallel \parallel C_k\left(\varphi \left(m_k^{-}\right)\right)-C_k\left(\xi \left(m_k^{-}\right)\right)\parallel \}\hfill \\ \hfill & \le & \underset{\iota \in \mathbb{J}}{sup}\{{\int }_0^{\iota }\underset{i=1,2,\cdots ,n}{max}\mathbf{K}{e}^{-{\rm Y}(\iota -u)}\sum _{j=1}^n{\overline{b}}_{ij}{L}_{1}du\hfill \\ \hfill & & +\sum _{k=1}^{\mu (\iota ,0)}\mathbf{K}{e}^{-{\rm Y}(\iota -m_k^{+})}{L}_2\}{\parallel \varphi -\xi \parallel }_{PC}\hfill \\ & \le & \mathbf{K}\left\{{\displaystyle \frac{L_1}{{\rm Y}}}\underset{i=1,2,\cdots ,n}{max}\sum _{j=1}^n{\overline{b}}_{ij}+L_2{N}_1\right\}{\parallel \varphi -\xi \parallel }_{PC}.\hfill \end{array}$$

(9)

Then from (9) it follows that T is a contracting operator in $\widehat{\mathsf{\Omega }}$, and there exists a unique AP solution of (2). □

Theorem 9.

Assume that all conditions in Theorem 8 and

$$\mathbf{K}{L}_1\underset{i=1,2,\cdots ,n}{max}\sum _{j=1}^n{\overline{b}}_{ij}+Nln(1+\mathbf{K}{L}_2)<{\rm Y}$$

hold. Then, the solution of (2) is globally exponentially stable.

Proof. 

Let now $x\left(\iota \right)$ be an arbitrary solution of (2). Then, we obtain

$$\begin{array}{ccc}\hfill \parallel w\left(\iota \right)-x\left(\iota \right)\parallel & \le & \mathbf{K}{e}^{-{\rm Y}(\iota -{\iota }_0)}\parallel w\left({\iota }_0\right)-x\left({\iota }_0\right)\parallel \hfill \\ & & +{\int }_{{\iota }_0}^{\iota }\underset{i=1,2,\cdots ,n}{max}\mathbf{K}{e}^{-{\rm Y}(\iota -u)}\sum _{j=1}^n{\overline{b}}_{ij}{L}_1\parallel w\left(u\right)-x\left(u\right)\parallel du\hfill \\ & & +\sum _{k=\mu ({\iota }_0,0)+1}^{\mu (\iota ,0)}\mathbf{K}{e}^{-{\rm Y}(\iota -m_k^{+})}{L}_2\parallel w\left(m_k^{-}\right)-x\left(m_k^{-}\right)\parallel .\hfill \end{array}$$

Set $v\left(\iota \right)=\parallel w\left(\iota \right)-x\left(\iota \right)\parallel e^{{\rm Y}\iota }$, then by means of Gronwall-Bellman’s inequality, it follows that

$$\begin{array}{ccc}\hfill \parallel w\left(\iota \right)-x\left(\iota \right)\parallel & \le & \mathbf{K}{e}^{-{\rm Y}(\iota -{\iota }_0)}\parallel w\left({\iota }_0\right)-x\left({\iota }_0\right)\parallel \prod _{k=\mu ({\iota }_0,0)+1}^{\mu (\iota ,0)}(1+\mathbf{K}{L}_2{e}^{-{\rm Y}(\iota -m_k^{+})})\hfill \\ & & e^{{\int }_{{\iota }_0}^{\iota }\underset{i=1,2,\cdots ,n}{max}\mathbf{K}{e}^{-{\rm Y}(\iota -u)}{\displaystyle \sum _{j=1}^n}{\overline{b}}_{ij}{L}_1}\hfill \\ & \le & \mathbf{K}\parallel w\left({\iota }_0\right)-x\left({\iota }_0\right)\parallel {(1+\mathbf{K}{L}_2)}^{\mu (\iota ,{\iota }_0)}{e}^{-{\rm Y}(\iota -{\iota }_0)}{e}^{\mathbf{K}{L}_1\underset{i=1,2,\cdots ,n}{max}{\displaystyle \sum _{j=1}^n}{\overline{b}}_{ij}(\iota -{\iota }_0)}\hfill \\ & \le & \mathbf{K}\parallel w\left({\iota }_0\right)-x\left({\iota }_0\right)\parallel {(1+\mathbf{K}{L}_2)}^{\mu (\iota ,{\iota }_0)}{e}^{\left(-{\rm Y}+\mathbf{K}{L}_1\underset{i=1,2,\cdots ,n}{max}{\displaystyle \sum _{j=1}^n}{\overline{b}}_{ij}\right)(\iota -{\iota }_0)}\hfill \\ & \le & \mathbf{K}\parallel w\left({\iota }_0\right)-x\left({\iota }_0\right)\parallel e^{N(\iota -{\iota }_0)ln(1+\mathbf{K}{L}_2)}{e}^{\left(-{\rm Y}+\mathbf{K}{L}_1\underset{i=1,2,\cdots ,n}{max}{\displaystyle \sum _{j=1}^n}{\overline{b}}_{ij}\right)(\iota -{\iota }_0)}\hfill \\ & \le & \mathbf{K}\parallel w\left({\iota }_0\right)-x\left({\iota }_0\right)\parallel e^{(-{\rm Y}+\mathbf{K}{L}_1\underset{i=1,2,\cdots ,n}{max}{\displaystyle \sum _{j=1}^n}{\overline{b}}_{ij}+Nln(1+\mathbf{K}{L}_2))(\iota -{\iota }_0)}.\hfill \end{array}$$

Obviously, if there exists $\mathbf{K}{L}_1\underset{i=1,2,\cdots ,n}{max}{\displaystyle \sum _{j=1}^n}{\overline{b}}_{ij}+Nln(1+\mathbf{K}{L}_2)<{\rm Y}$, then the solution of (2) is exponentially. □

4. Example

Example 10.

We shall consider the classical model of Hopfield neural networks

$$\begin{array}{c}\hfill \left\{\begin{array}{c}{\dot{w}}_i\left(\iota \right)=-{\displaystyle \frac{1}{R_i}}{w}_i\left(\iota \right)+{\displaystyle \sum _{j=1}^n}{b}_{ij}{f}_j\left(w_j\left(\iota \right)\right)+g_i\left(\iota \right),\iota \in (l_k,m_{k+1}],k\in \mathbb{N},\hfill \\ w\left(m_k^{+}\right)=Bw\left(m_k^{-}\right)+C_k\left(w\left(m_k^{-}\right)\right)+h_k,k\in {\mathbb{N}}_{+},\hfill \\ w\left(\iota \right)=Bw\left(m_k^{-}\right)+C_k\left(w\left(m_k^{-}\right)\right)+h_k,\iota \in (m_k,l_k],k\in {\mathbb{N}}_{+},\hfill \\ w\left(l_k^{+}\right)=w\left(l_k^{-}\right),k\in {\mathbb{N}}_{+},\hfill \end{array}\right.\end{array}$$

(10)

where $\iota \in \mathbb{J}$, $R_i>0$, $b_{ij}\in \mathbb{R},{\gamma }_i\in C(\mathbb{J},\mathbb{R}),f_j\in C({\mathbb{R}}_{+},\mathbb{R}),i=1,2,\cdots ,n,j=1,2,\cdots ,n$, $x\left(\iota \right)=col(x_1\left(\iota \right),x_2\left(\iota \right),\cdots ,x_n\left(\iota \right))$, $B=diag\left[b_i\right],b_i\in \mathbb{R},i=1,2,\cdots ,n,C_k\in C({\mathbb{R}}_{+}^n,\mathbb{R}),h_k\in {\mathbb{R}}^n$.

Let

$$\begin{array}{c}\hfill \left\{\begin{array}{c}{\dot{w}}_i\left(\iota \right)=-{\displaystyle \frac{1}{R_i}}{w}_i\left(\iota \right),\iota \in (l_k,m_{k+1}],k\in \mathbb{N},\hfill \\ w\left(m_k^{+}\right)=Bw\left(m_k^{-}\right),k\in {\mathbb{N}}_{+},\hfill \\ w\left(\iota \right)=Bw\left(m_k^{-}\right),\iota \in (m_k,l_k],k\in {\mathbb{N}}_{+},\hfill \\ w\left(l_k^{+}\right)=w\left(l_k^{-}\right),k\in {\mathbb{N}}_{+},\hfill \end{array}\right.\end{array}$$

be the linear part of (10).

The Cauchy matrix $\mathcal{W}(\iota ,{\iota }_0)$ of (10) is in the form

$$\begin{array}{c}\hfill \mathcal{W}(\iota ,{\iota }_0)=\left\{\begin{array}{c}{B}^{\mu (\iota ,{\iota }_0)+1}{e}^{A(\iota -l_{\mu (\iota ,0)})}{\displaystyle \prod _{k=\mu (\iota ,0)}^{\mu ({\iota }_0,0)+2}}{e}^{A(m_k-l_{k-1})}{e}^{A(m_{\mu ({\iota }_0,0)+1}-{\iota }_0)},\hfill \\ {\iota }_0<m_{\mu ({\iota }_0,0)+1}<\cdots <l_{\mu (\iota ,0)}<\iota ,\hfill \\ B^{\mu (\iota ,{\iota }_0)+1}{\displaystyle \prod _{k=\mu (\iota ,0)}^{\mu ({\iota }_0,0)+2}}{e}^{A(m_k-l_{k-1})}{e}^{A(m_{\mu ({\iota }_0,0)+1}-{\iota }_0)},\hfill \\ {\iota }_0<m_{\mu ({\iota }_0,0)+1}<\cdots <m_{\mu (\iota ,0)}<\iota ,\hfill \\ B^{\mu (\iota ,{\iota }_0)+1}{e}^{A(\iota -l_{\mu (\iota ,0)})}{\displaystyle \prod _{k=\mu (\iota ,0)}^{\mu ({\iota }_0,0)+1}}{e}^{A(m_k-l_{k-1})},\hfill \\ {\iota }_0<l_{\mu ({\iota }_0,0)}<\cdots <l_{\mu (\iota ,0)}<\iota ,\hfill \\ B^{\mu (\iota ,{\iota }_0)+1}{\displaystyle \prod _{k=\mu (\iota ,0)}^{\mu ({\iota }_0,0)+1}}{e}^{A(m_k-l_{k-1})},\hfill \\ {\iota }_0<l_{\mu ({\iota }_0,0)}<\cdots <m_{\mu (\iota ,0)}<\iota .\hfill \end{array}\right.\end{array}$$

Then,

$$\begin{array}{ccc}\hfill \parallel \mathcal{W}(\iota ,{\iota }_0)\parallel & \le & ∥B^{\mu (\iota ,{\iota }_0)+1}{e}^{A(\iota -{\iota }_0)}∥\hfill \\ & \le & ∥e^{ln{B}^{\mu (\iota ,{\iota }_0)+1}}{e}^{A(\iota -{\iota }_0)}∥\hfill \\ & \le & ∥e^{(\mu (\iota ,{\iota }_0)+1)lnB}{e}^{A(\iota -{\iota }_0)}∥\hfill \\ & \le & ∥e^{lnB}{e}^{N(\iota -{\iota }_0)lnB}{e}^{A(\iota -{\iota }_0)}∥\hfill \\ & \le & ∥e^{lnB}{e}^{(A+NlnB)(\iota -{\iota }_0)}∥\hfill \\ & \le & e^{\underset{i=1,2,\cdots ,n}{max}ln{b}_i}{e}^{\left(\underset{i=1,2,\cdots ,n}{max}-\frac{1}{R_i}+\underset{i=1,2,\cdots ,n}{max}Nln{b}_i\right)(\iota -{\iota }_0)}\hfill \\ & \le & e^{\underset{i=1,2,\cdots ,n}{max}ln{b}_i}{e}^{-\left(\underset{i=1,2,\cdots ,n}{min}\frac{1}{R_i}-\underset{i=1,2,\cdots ,n}{max}Nln{b}_i\right)(\iota -{\iota }_0)}.\hfill \end{array}$$

Let $\mathbf{K}=exp\left(\underset{i=1,2,\cdots ,n}{max}ln{b}_i\right)$, ${\rm Y}=\underset{i=1,2,\cdots ,n}{min}\frac{1}{R_i}-\underset{i=1,2,\cdots ,n}{max}Nln{b}_i$, then we can obtain $\parallel \mathcal{W}(\iota ,{\iota }_0)\parallel \le \mathbf{K}{e}^{-{\rm Y}(\iota -{\iota }_0)}$.

According to the Theorems 8 and 9, assume that $\left(H_1\right)$–$\left(H_7\right)$ are met and the following inequalities hold

$$\begin{array}{ccc}& & \mathbf{K}=exp\left(\underset{i=1,2,\cdots ,n}{max}ln{b}_i\right),\hfill \\ & & {\rm Y}=\underset{i=1,2,\cdots ,n}{min}\frac{1}{R_i}-\underset{i=1,2,\cdots ,n}{max}Nln{b}_i,\hfill \\ & & \mathbf{K}\left\{{\displaystyle \frac{L_1}{{\rm Y}}}\underset{i=1,2,\cdots ,n}{max}\sum _{j=1}^n{\overline{b}}_{ij}+L_2{N}_1\right\}<1.\hfill \end{array}$$

Then, there exists a unique AP solution $w\left(\iota \right)$ of (10).

In addition, if the following inequalities hold

$$\mathbf{K}{L}_1\underset{i=1,2,\cdots ,n}{max}\sum _{j=1}^n{\overline{b}}_{ij}+Nln(1+\mathbf{K}{L}_2)<{\rm Y},$$

then the solution $w\left(\iota \right)$ is globally exponentially stable.

5. Conclusions

Neural network models with impulses can study many phenomena in life. We note that Stamova and Stamov [11] proposed a Hopfield neural network with impulsive effects at fixed moments. We are very interested in this work. After careful reading, we introduced the non-instantaneous impulse factor into this model and proposed a Hopfield neural network non-instantaneous impulsive model. Then, we provided conditions for the existence of a unique AP solution and the exponential stability of the solution for this model.

There are many limitations to our work. It is known that asymptotic stability of solutions to impulsive systems can be treated in both weak (convergence towards the solution depends only on the elapsed time) and strong (convergence depends on the elapsed on the elapsed time and the number of impulses) flavors [29,30]. We deal with the classical weak stability in this paper. Then, we will gradually consider the case of strong stability for the model (2) with non-instantaneous impulses in future.