Periodicity on Neutral-Type Inertial Neural Networks Incorporating Multiple Delays

Abstract

The classical Hopefield neural networks have obvious symmetry, thus the study related to its dynamic behaviors has been widely concerned. This research article is involved with the neutral-type inertial neural networks incorporating multiple delays. By making an appropriate Lyapunov functional, one novel sufficient stability criterion for the existence and global exponential stability of T-periodic solutions on the proposed system is obtained. In addition, an instructive numerical example is arranged to support the present approach. The obtained results broaden the application range of neutral-types inertial neural networks.

1. Introduction

The well-known inertial neural networks (INNs) were first introduced by Babcock and Westervelt [1,2], and can be expressed as the following functional differential equations:

$$s_j^{\prime \prime }\left(t\right)=-a_j{s}_j^{\prime }\left(t\right)-b_j{s}_j\left(t\right)+\sum _{k=1}^n{c}_{jk}{f}_k\left(s_k\left(t\right)\right)+\sum _{k=1}^n{d}_{jk}{f}_k\left(s_k\left(t-{\tau }_{jk}\right)\right)+I_j,t\ge 0,$$

(1)

with the initial value conditions:

$$s_j\left(v\right)={\phi }_j\left(v\right),s_j^{\prime }\left(0\right)={\psi }_j,-{\tau }_j^{+}\le vs.\le 0,{\phi }_j\in C([-{\tau }_j^{+},0],\mathbb{R}),{\psi }_j\in \mathbb{R},$$

(2)

where ${\tau }_j^{+}=\underset{1\le k\le n}{max}\left\{{\tau }_{jk}\right\}$, $s\left(t\right)=(s_1\left(t\right),s_2\left(t\right),\dots ,s_n\left(t\right))$ represents the state vector, $s_j^{\prime \prime }\left(t\right)$ is referred to an inertial item of the j-th neuron on (1), the parameters $a_j>0,b_j>0,c_{jk},d_{jk}$ and time delay ${\tau }_{jk}\ge 0$ are real numbers, $I_j$ is the external input, $f_k$ is a continuous activation function, $j,k\in \mathfrak{N}:=\{1,2,\dots ,n\}.$

During the past thirty decades, by utilizing the reduced-order transformation, numerous studies have been conducted on the stability and synchronization of system (1) and its generalizations, such as [3,4,5,6,7,8,9,10,11]. However, the reduced-order method will affect the dimensions of systems, thereby increasing a large amount of calculation, which will make it difficult to achieve in practice. For the sake of avoiding the traditional reduced-order method, the authors proposed several new criteria for the stability and synchronization of the system (1) in [12,13] through making a new Lyapunov functional. On this basis, references [14,15,16,17,18,19,20,21] extensively studied various dynamic behaviors of system (1) and its generalizations via applying the non-reduced order approach.

It is well known that the classical Hopefield neural networks have obvious symmetry, numerous researchers have carried out extensive research on its related dynamic behaviors [16,22,23,24]. In particular, the authors in [25] investigate the exponential stability and the almost sure exponential stability for a class of stochastic fuzzy Cohen-Grossberg neural networks by fabricating an appropriate Lyapunov functional. In practical application, owing to the finite switching and transmission speeds of signals in the networks, the existence of time delays is inevitable in the working networks. It should be pointed out that in addition to the state itself, there are also time delays in the derivatives of the state related to the networks. This kind of delay is deemed as neutral delay, which not only appears in the field of automatic control and population ecology [26,27], but also occurs in many physical systems, including transmission lines, Lotka–Volterra systems, chemical reactors, and others [23,24,28,29]. Particularly, if we use differential equations to model neural networks (NNs) for the realization of electronic circuits, the influence of neutral delay often exists. The authors in [24,29] investigated the effect of neutral delays on the partial element equivalent circuit. The circuit was represented to a neutral-type functional differential equation, and some new sufficient stability assertions were given by Lyapunov theory. Furthermore, the dynamic behaviors of neutral-type inertial neural networks (NTINNs) have been extensively studied by exploiting the reduced-order approach. For example, the authors in [30] used the finite-time stability theory, inequality techniques and analysis approaches to research the finite-time synchronization on fuzzy NTINNs. In [31], the stability of NTINNs is studied via utilizing the Lyapunov–Krasovskii functional approach and Linear Matrix Inequality (LMI) analysis.

On the other hand, periodic phenomena are widespread in biological systems. For instance, seasonal influences of weather and food supplies, electronic systems, NNs etc. Especially in the application of NNs, periodic phenomenon is one of the most important dynamic behaviors to describe the symmetry of the Hopefield neural networks model, and the existence and stability of periodic solutions will help us to understand the asymptotic behavior of mathematical biological systems. Therefore, it is a very meaningful thing to research the existence and stability of periodic solutions [24,32]. However, few researches have discussed the periodic problem of the following NTINNs involving multiple delays:

$$\begin{array}{ccc}\hfill s_j^{\prime \prime }\left(t\right)-\sum _{k=1}^n{e}_{jk}{s}_k^{\prime \prime }(t-{\xi }_{jk})& =& -a_j{s}_j^{\prime }\left(t\right)-b_j{s}_j\left(t\right)+\sum _{k=1}^n{c}_{jk}{f}_k\left(s_k\left(t\right)\right)\hfill \\ & & +\sum _{k=1}^n{d}_{jk}{f}_k\left(s_k(t-{\tau }_{jk})\right)+I_j\left(t\right),t\ge 0,\hfill \end{array}$$

(3)

with the initial conditions:

$$s_j\left(v\right)={\phi }_j\left(v\right),s_j^{\prime }\left(v\right)={\psi }_j\left(v\right),s_j^{\prime \prime }\left(v\right)={\varsigma }_j\left(v\right),-{\sigma }_j\le vs.\le 0,{\phi }_j,{\psi }_j,{\varsigma }_j\in C\left([-{\sigma }_j,0],\mathbb{R}\right),$$

(4)

where ${\xi }_j^{+}=\underset{1\le k\le n}{max}\left\{{\xi }_{jk}\right\},{\sigma }_j=max\{{\tau }_j^{+},{\xi }_j^{+}\},$$e_{jk}$ and the multiple neutral delays ${\xi }_{jk}\ge 0$ are constants, $I_j\left(t\right)$ is a continuous periodic function involving period $T>0$, and $j,k\in \mathfrak{N}$.

Enlightened by the above arguments, our major purpose in this article is to investigate the existence and stability of periodic solutions on NTINNs involving multiple delays through constructing a new and appropriate Lyapunov functional to replace the traditional reduced-order approach. Briefly speaking, the innovative contents of this article can be presented as below. (1) A class of NTINNs involving multiple delays is proposed; (2) Under certain assumptions, by exploiting the non-reduced order approach, one new sufficient stability criterion to guarantee the existence and stability of the T-periodic solutions on system (3) is gotten for the first time; (3) NTINNs here are second-order and involve multiple neutral delays, which are different from the traditional NNs [33,34,35,36,37,38,39,40] or INNs [3,4,5,6,7,8,9,11,12,13,14,15,17,18,19,20,21,30,31,32]. Compared with the results on exponential stability for the neutral-type neural networks (NTNNs) [26,29,39,41] and INNs [13,14,18,19], we give the exponential stability of the T-periodic solution for the NTINNs. (4) An instructive numerical simulation including comparisons is afforded to demonstrate the obtained theoretical results.

This article is systematized as below. In Section 2, a few indispensable lemmas, definitions and assumptions are given. In Section 3, the global exponential stability on the T-periodic solutions of the NTINNs (3) is proved. In Section 4, an instructive numerical simulation is afforded to evidence the validity and feasibility of the analytical results. A concise conclusion is offered in Section 5.

2. Preliminaries

Throughout this article, a few indispensable lemmas, definitions and assumptions are provided, which are useful in the following proving process.

Assumption 1.

There is a nonnegative real number $L_j$ obeying

$$\left|f_j\left(p\right)-f_j\left(q\right)\right|\le L_j|p-q|,\mathit{for}\mathit{all}p,q\in \mathbb{R}\mathit{and}j\in \mathfrak{N}.$$

Assumption 2.

For $j\in \mathfrak{N}$, there remain three real numbers ${\overline{\delta }}_j\ge 0,$ ${\overline{\beta }}_j\ge 0$ and ${\overline{\alpha }}_j>0$ agreeing with that

$${\mathcal{X}}_j<0,4{\mathcal{Y}}_j{\mathcal{X}}_j>{\mathcal{Z}}_j^2,$$

(5)

where

$$\begin{array}{cc}\hfill {\mathcal{X}}_j=& \sum _{k=1}^n\left({\overline{\delta }}_j+{\overline{\beta }}_j^2\right)|e_{jk}|+2{\overline{\alpha }}_j{\overline{\beta }}_j+\sum _{k=1}^n{\overline{\alpha }}_j{\overline{\beta }}_j|e_{jk}|-2a_j{\overline{\alpha }}_j^2+\sum _{k=1}^n{\overline{\alpha }}_j^2|c_{jk}|+\sum _{k=1}^n{\overline{\alpha }}_j^2|d_{jk}|\hfill \\ \hfill & +\sum _{k=1}^n{a}_j{\overline{\alpha }}_j^2|e_{jk}|+\sum _{k=1}^n{a}_j{\overline{\alpha }}_j{\overline{\beta }}_j|e_{jk}|+\sum _{k=1}^n{B}_{kj}^1+\sum _{k=1}^n\sum _{l=1}^n{\tilde{B}}_{kjl}^1,\hfill \\ \hfill {\mathcal{Y}}_j=& \sum _{k=1}^n({\overline{\delta }}_j+{\overline{\beta }}_j^2)|e_{jk}|+\sum _{k=1}^n{b}_j{\overline{\alpha }}_j^2|e_{jk}|-2b_j{\overline{\alpha }}_j{\overline{\beta }}_j+\sum _{k=1}^n{\overline{\alpha }}_j{\overline{\beta }}_j|c_{jk}|+\sum _{k=1}^n{\overline{\alpha }}_j{\overline{\beta }}_j|d_{jk}|\hfill \\ \hfill & +\sum _{k=1}^n{b}_j{\overline{\alpha }}_j{\overline{\beta }}_j|e_{jk}|+(\sum _{k=1}^n{\overline{\alpha }}_k^2|c_{kj}|+\sum _{k=1}^n\sum _{l=1}^n{\overline{\alpha }}_l^2|e_{lk}||e_{lj}|+\sum _{k=1}^n{\overline{\alpha }}_k{\overline{\beta }}_k|c_{kj}|\hfill \\ \hfill & +\sum _{k=1}^n\sum _{l=1}^n{\overline{\alpha }}_l{\overline{\beta }}_l|e_{lk}||c_{lj}|)L_j^2+\sum _{k=1}^n{A}_{kj}^1+\sum _{k=1}^n\sum _{l=1}^n{\tilde{A}}_{kjl}^1+(\sum _{k=1}^n{C}_{kj}+\sum _{k=1}^n\sum _{l=1}^n{\tilde{C}}_{kjl})L_j^2,\hfill \\ \hfill {\mathcal{Z}}_j=& 2({\overline{\delta }}_j+{\overline{\beta }}_j^2)-2b_j{\overline{\alpha }}_j^2-2a_j{\overline{\alpha }}_j{\overline{\beta }}_j,\hfill \end{array}$$

and

$$\begin{array}{cc}\hfill A_{kj}^1& =[{\overline{\delta }}_k+{\overline{\beta }}_k^2+(a_k+b_k){\overline{\alpha }}_k{\overline{\beta }}_k]|e_{kj}|,{\tilde{A}}_{kjl}^1={\overline{\alpha }}_k{\overline{\beta }}_k(|c_{kl}|+|d_{kl}|)|e_{kj}|,\hfill \\ \hfill B_{kj}^1& =\left({\overline{\delta }}_k+{\overline{\beta }}_k^2+2{\overline{\alpha }}_k{\overline{\beta }}_k+a_k{\overline{\alpha }}_k^2+b_k{\overline{\alpha }}_k^2\right)|e_{kj}|,\hfill \\ \hfill {\tilde{B}}_{kjl}^1& =(({\overline{\delta }}_k+{\overline{\beta }}_k^2+2{\overline{\alpha }}_k{\overline{\beta }}_k)|e_{kl}|+{\overline{\alpha }}_k^2(|c_{kl}|+|d_{kl}|))|e_{kj}|,\hfill \\ \hfill C_{kj}& =({\overline{\alpha }}_k^2+{\overline{\alpha }}_k{\overline{\beta }}_k)|d_{kj}|,{\tilde{C}}_{kjl}=({\overline{\alpha }}_k+{\overline{\beta }}_k){\overline{\alpha }}_k|e_{kl}||d_{kj}|,j,k,l\in \mathfrak{N}.\hfill \end{array}$$

Definition 1.

Given $x\left(t\right)= \left(x_1\left(t\right),x_2\left(t\right),\dots ,x_n\left(t\right)\right)$ and $y\left(t\right)=\left(y_1\left(t\right),y_2\left(t\right),\dots ,y_n\left(t\right)\right)$ as two solutions of the NTINNs (3) to satisfy

$$\begin{array}{cc}\hfill x_j\left(v\right)& ={\phi }_j^x\left(v\right),x_j^{\prime }\left(v\right)={\psi }_j^x\left(v\right),x_j^{\prime \prime }\left(v\right)={\varsigma }_j^x\left(v\right),\hfill \\ \hfill y_j\left(v\right)& ={\phi }_j^y\left(v\right),y_j^{\prime }\left(v\right)={\psi }_j^y\left(v\right),y_j^{\prime \prime }\left(v\right)={\varsigma }_j^y\left(v\right),j\in \mathfrak{N},\hfill \end{array}$$

(6)

where ${\phi }_j^x,{\psi }_j^x,{\varsigma }_j^x,{\phi }_j^y,{\psi }_j^y,{\varsigma }_j^y\in C([-{\sigma }_j,0],\mathbb{R}).$ The NTINNs (1.3) is said to have global exponential stability when there are constants $\lambda >0$ and $\Lambda =\Lambda \left({\phi }^x,{\psi }^x,{\varsigma }^x,{\phi }^y,{\psi }^y,{\varsigma }^y\right)>0$ obeying that

$$\left|x_j\left(t\right)-y_j\left(t\right)\right|\le \Lambda e^{-\lambda t},\left|x_j^{\prime }\left(t\right)-y_j^{\prime }\left(t\right)\right|\le \Lambda e^{-\lambda t},forallt\in [0,+\infty )\mathit{and}j\in \mathfrak{N}.$$

Lemma 1.

Under the Assumptions 1 and 2, every solution of NTINNs (3) incorporating initial values (4) exists and is unique on $[0,+\infty )$.

Proof. 

At first, set

$$\iota =\underset{1\le j\le n}{min}\left\{{\tau }_j^{-},{\xi }_j^{-}\right\},w_j\left(t\right)=s_j\left(t\right)-\sum _{k=1}^n{e}_{jk}{s}_k(t-{\xi }_{jk}),$$

where ${\tau }_j^{-}=\underset{1\le k\le n}{min}\left\{{\tau }_{jk}\right\},{\xi }_j^{-}=\underset{1\le k\le n}{min}\left\{{\xi }_{jk}\right\}\mathit{and}j\in \mathfrak{N}.$ Now, we prove that $s_j\left(t\right)$ exists and is unique on $[0,\iota ].$ Actually, for all $t\in [0,\iota ]$ and $j\in \mathfrak{N}$,

$$\begin{array}{cc}\hfill w_j^{\prime \prime }\left(t\right)=& s_j^{\prime \prime }\left(t\right)-\sum _{k=1}^n{e}_{jk}{s}_k^{\prime \prime }\left(t-{\xi }_{jk}\right)\hfill \\ \hfill =& -a_j{s}_j^{\prime }\left(t\right)-b_j{s}_j\left(t\right)+\sum _{k=1}^n{c}_{jk}{f}_k\left(s_k\left(t\right)\right)+\sum _{k=1}^n{d}_{jk}{f}_k\left(s_k\left(t-{\tau }_{jk}\right)\right)+I_j\left(t\right)\hfill \\ \hfill =& -a_j{w}_j^{\prime }\left(t\right)-b_j{w}_j\left(t\right)-a_j\sum _{k=1}^n{e}_{jk}{\psi }_k(t-{\xi }_{jk})-b_j\sum _{k=1}^n{e}_{jk}{\phi }_k(t-{\xi }_{jk})\hfill \\ \hfill & +\sum _{k=1}^n{c}_{jk}{f}_k(w_k\left(t\right)+\sum _{j=1}^n{e}_{kj}{\phi }_j(t-{\xi }_{kj}))+\sum _{k=1}^n{d}_{jk}{f}_k\left({\phi }_k\left(t-{\tau }_{jk}\right)\right)+I_j\left(t\right).\hfill \end{array}$$

(7)

From the Assumption 1, one can discover that the solution $w\left(t\right)$ of the second order ordinary differential Equation (7) with initial conditions $w\left(0\right)=\left\{{\phi }_j\left(0\right)-{\displaystyle \sum _{k=1}^n}{e}_{jk}{\phi }_k(-{\xi }_{jk})\right\}$ and $w^{\prime }\left(0\right)=\left\{{\psi }_j\left(0\right)-{\displaystyle \sum _{k=1}^n}{e}_{jk}{\psi }_k(-{\xi }_{jk})\right\}$ exists and is unique on $[0,\iota ]$. Consequently, $s\left(t\right)=w\left(t\right)+\left\{{\displaystyle \sum _{k=1}^n}{e}_{jk}{\phi }_k(t-{\xi }_{jk})\right\}$ exists and is unique on $[0,\iota ]$. Using the same method, one can discover $s\left(t\right)=w\left(t\right)+\left\{{\displaystyle \sum _{k=1}^n}{e}_{jk}{\phi }_k(t-{\xi }_{jk})\right\}$ exists and is unique on $[\iota ,2\iota ]$, $[2\iota ,3\iota ]$, ⋯. Hence, every solution of NTINNs (3) incorporating initial values (4) exists and is unique on $[0,+\infty )$. □

Lemma 2.

Under the Assumptions 1 and 2, NTINNs (3) possess global exponential stability.

Proof. 

Label $u_j\left(t\right)=x_j\left(t\right)-y_j\left(t\right)$, then

$$\begin{array}{cc}\hfill u_j^{\prime \prime }\left(t\right)-\sum _{k=1}^n{e}_{jk}{u}_k^{\prime \prime }(t-{\xi }_{jk})=& -a_j{u}_j^{\prime }\left(t\right)-b_j{u}_j\left(t\right)+\sum _{k=1}^n{c}_{jk}{\tilde{f}}_k\left(u_k\left(t\right)\right)\hfill \\ \hfill & +\sum _{k=1}^n{d}_{jk}{\tilde{f}}_k\left(u_k(t-{\tau }_{jk})\right),\hfill \end{array}$$

(8)

where

$${\tilde{f}}_k\left(u_k\left(t\right)\right)=f_k\left(x_k\left(t\right)\right)-f_k\left(y_k\left(t\right)\right),{\tilde{f}}_k\left(u_k(t-{\tau }_{jk})\right)=f_k\left(x_k(t-{\tau }_{jk})\right)-f_k\left(y_k(t-{\tau }_{jk})\right).$$

In view of the Assumption 2 and the boundedness of NTINNs (3), one can select a real number $\lambda >0$ such that

$${\mathcal{X}}_j^{\lambda }<0,4{\mathcal{Y}}_j^{\lambda }{\mathcal{X}}_j^{\lambda }>{\left({\mathcal{Z}}_j^{\lambda }\right)}^2,$$

(9)

where

$$\begin{array}{cc}\hfill {\mathcal{X}}_j^{\lambda }=& \sum _{k=1}^n\left({\overline{\delta }}_j+\lambda {\overline{\alpha }}_j{\overline{\beta }}_j+{\overline{\beta }}_j^2\right)|e_{jk}|+\lambda {\overline{\alpha }}_j^2+2{\overline{\alpha }}_j{\overline{\beta }}_j+\sum _{k=1}^n(\lambda {\overline{\alpha }}_j^2+{\overline{\alpha }}_j{\overline{\beta }}_j)|e_{jk}|\hfill \\ \hfill & -2a_j{\overline{\alpha }}_j^2+\sum _{k=1}^n{\overline{\alpha }}_j^2|c_{jk}|+\sum _{k=1}^n{\overline{\alpha }}_j^2|d_{jk}|+\sum _{k=1}^n{a}_j{\overline{\alpha }}_j^2|e_{jk}|+\sum _{k=1}^n{a}_j{\overline{\alpha }}_j{\overline{\beta }}_j|e_{jk}|\hfill \\ \hfill & +(\sum _{k=1}^n{B}_{kj}^2+\sum _{k=1}^n\sum _{l=1}^n{\tilde{B}}_{kjl}^2)e^{\lambda {\xi }_{kj}},\hfill \\ \hfill {\mathcal{Y}}_j^{\lambda }=& \lambda ({\overline{\delta }}_j+{\overline{\beta }}_j^2)+\lambda \sum _{k=1}^n({\overline{\delta }}_j+{\overline{\beta }}_j^2)|e_{jk}|+\sum _{k=1}^n({\overline{\delta }}_j+\lambda {\overline{\alpha }}_j{\overline{\beta }}_j+{\overline{\beta }}_j^2)|e_{jk}|\hfill \\ \hfill & +\sum _{k=1}^n{b}_j{\overline{\alpha }}_j^2|e_{jk}|-2b_j{\overline{\alpha }}_j{\overline{\beta }}_j+\sum _{k=1}^n{\overline{\alpha }}_j{\overline{\beta }}_j|c_{jk}|+\sum _{k=1}^n{\overline{\alpha }}_j{\overline{\beta }}_j|d_{jk}|+\sum _{k=1}^n{b}_j{\overline{\alpha }}_j{\overline{\beta }}_j|e_{jk}|\hfill \\ \hfill & +(\sum _{k=1}^n{\overline{\alpha }}_k^2|c_{kj}|+\sum _{k=1}^n\sum _{l=1}^n{\overline{\alpha }}_l^2|e_{lk}||e_{lj}|+\sum _{k=1}^n{\overline{\alpha }}_k{\overline{\beta }}_k|c_{kj}|+\sum _{k=1}^n\sum _{l=1}^n{\overline{\alpha }}_l{\overline{\beta }}_l|e_{lk}||c_{lj}|)L_j^2\hfill \\ \hfill & +(\sum _{k=1}^n{A}_{kj}^2+\sum _{k=1}^n\sum _{l=1}^n{\tilde{A}}_{kjl}^2)e^{\lambda {\xi }_{kj}}+(\sum _{k=1}^n{C}_{kj}+\sum _{k=1}^n\sum _{l=1}^n{\tilde{C}}_{kjl})L_j^2{e}^{\lambda {\tau }_{kj}},\hfill \\ \hfill {\mathcal{Z}}_j^{\lambda }=& 2({\overline{\delta }}_j+\lambda {\overline{\alpha }}_j{\overline{\beta }}_j+{\overline{\beta }}_j^2)-2b_j{\overline{\alpha }}_j^2-2a_j{\overline{\alpha }}_j{\overline{\beta }}_j\hfill \end{array}$$

and

$$\begin{array}{cc}\hfill A_{kj}^2& =\lambda ({\overline{\delta }}_k+{\overline{\beta }}_k^2)|e_{kj}|+({\overline{\delta }}_k+\lambda {\overline{\alpha }}_k{\overline{\beta }}_k+{\overline{\beta }}_k^2+(a_k+b_k){\overline{\alpha }}_k{\overline{\beta }}_k)|e_{kj}|,\hfill \\ \hfill {\tilde{A}}_{kjl}^2& =({\overline{\alpha }}_k{\overline{\beta }}_k(|c_{kl}|+|d_{kl}|)+\lambda ({\overline{\delta }}_k+{\overline{\beta }}_k^2)|e_{kl}|)|e_{kj}|,\hfill \\ \hfill B_{kj}^2& =\left({\overline{\delta }}_k+\lambda {\overline{\alpha }}_k{\overline{\beta }}_k+{\overline{\beta }}_k^2+\lambda {\overline{\alpha }}_k^2+2{\overline{\alpha }}_k{\overline{\beta }}_k+a_k{\overline{\alpha }}_k^2+b_k{\overline{\alpha }}_k^2\right)|e_{kj}|,\hfill \\ \hfill {\tilde{B}}_{kjl}^2& =(({\overline{\delta }}_k+\lambda {\overline{\alpha }}_k{\overline{\beta }}_k+{\overline{\beta }}_k^2+\lambda {\overline{\alpha }}_k^2+2{\overline{\alpha }}_k{\overline{\beta }}_k)|e_{kl}|+{\overline{\alpha }}_k^2(|c_{kl}|+|d_{kl}|))|e_{kj}|,j,k,l\in \mathfrak{N}.\hfill \end{array}$$

Set

$$z_j\left(t\right)=u_j\left(t\right)-\sum _{k=1}^n{e}_{jk}{u}_k(t-{\xi }_{jk}),$$

then

$$z_j^{\prime }\left(t\right)=u_j^{\prime }\left(t\right)-\sum _{k=1}^n{e}_{jk}{u}_k^{\prime }(t-{\xi }_{jk}),$$

and

$$z_j^{\prime \prime }\left(t\right)=u_j^{\prime \prime }\left(t\right)-\sum _{k=1}^n{e}_{jk}{u}_k^{\prime \prime }(t-{\xi }_{jk}),$$

which yield that

$$z_j^{\prime \prime }\left(t\right)=-a_j{u}_j^{\prime }\left(t\right)-b_j{u}_j\left(t\right)+\sum _{k=1}^n{c}_{jk}{\tilde{f}}_k\left(u_k\left(t\right)\right)+\sum _{k=1}^n{d}_{jk}{\tilde{f}}_k\left(u_k(t-{\tau }_{jk})\right).$$

(10)

Construct the Lyapunov functional:

$$W\left(t\right)=\sum _{j=1}^4{W}_j\left(t\right),$$

(11)

where

$$\begin{array}{cc}\hfill W_1\left(t\right)& =\sum _{j=1}^n{\overline{\delta }}_j{z}_j^2\left(t\right)e^{\lambda t}+\sum _{j=1}^n{({\overline{\alpha }}_j{z}_j^{\prime }\left(t\right)+{\overline{\beta }}_j{z}_j\left(t\right))}^2{e}^{\lambda t},\hfill \\ \hfill W_2\left(t\right)& =\sum _{j=1}^n\sum _{k=1}^n{\int }_{t-{\xi }_{kj}}^t{A}_{kj}^2{u}_j^2\left(v\right)e^{\lambda (v+{\xi }_{kj})}dv+\sum _{j=1}^n\sum _{k=1}^n\sum _{l=1}^n{\int }_{t-{\xi }_{kj}}^t{\tilde{A}}_{kjl}^2{u}_j^2\left(v\right)e^{\lambda (v+{\xi }_{kj})}dv,\hfill \\ \hfill W_3\left(t\right)& =\sum _{j=1}^n\sum _{k=1}^n{\int }_{t-{\xi }_{kj}}^t{B}_{kj}^2{\left(u_j^{\prime }\left(v\right)\right)}^2{e}^{\lambda (v+{\xi }_{kj})}dv+\sum _{j=1}^n\sum _{k=1}^n\sum _{l=1}^n{\int }_{t-{\xi }_{kj}}^t{\tilde{B}}_{kjl}^2{\left(u_j^{\prime }\left(v\right)\right)}^2{e}^{\lambda (v+{\xi }_{kj})}dv,\hfill \\ \hfill W_4\left(t\right)& =\sum _{j=1}^n\sum _{k=1}^n{\int }_{t-{\tau }_{kj}}^t{C}_{kj}{\tilde{f}}_j^2\left(u_j\left(v\right)\right)e^{\lambda (v+{\tau }_{kj})}dv+\sum _{j=1}^n\sum _{k=1}^n\sum _{l=1}^n{\int }_{t-{\tau }_{kj}}^t{\tilde{C}}_{kjl}{\tilde{f}}_j^2\left(u_j\left(v\right)\right)e^{\lambda (v+{\tau }_{kj})}dv.\hfill \end{array}$$

Firstly, the derivative of $W_1\left(t\right)$ along the trajectories of NTINNs (10) is obtained as follows:

$$\begin{array}{ccc}\hfill W_1^{\prime }\left(t\right)& =& \lambda \sum _{j=1}^n{\overline{\delta }}_j{z}_j^2\left(t\right)e^{\lambda t}+2\sum _{j=1}^n{\overline{\delta }}_j{z}_j\left(t\right)z_j^{\prime }\left(t\right)e^{\lambda t}+\lambda \sum _{j=1}^n{({\overline{\alpha }}_j{z}_j^{\prime }\left(t\right)+{\overline{\beta }}_j{z}_j\left(t\right))}^2{e}^{\lambda t}\hfill \\ & & +2\sum _{j=1}^n({\overline{\alpha }}_j{z}_j^{\prime }\left(t\right)+{\overline{\beta }}_j{z}_j\left(t\right))({\overline{\alpha }}_j{z}_j^{\prime \prime }\left(t\right)+{\overline{\beta }}_j{z}_j^{\prime }\left(t\right))e^{\lambda t}\hfill \\ & =& \lambda \sum _{j=1}^n\left({\overline{\delta }}_j+{\overline{\beta }}_j^2\right)z_j^2\left(t\right)e^{\lambda t}+2\sum _{j=1}^n\left({\overline{\delta }}_j+\lambda {\overline{\alpha }}_j{\overline{\beta }}_j+{\overline{\beta }}_j^2\right)z_j\left(t\right)z_j^{\prime }\left(t\right)e^{\lambda t}\hfill \\ & & +\sum _{j=1}^n(\lambda {\overline{\alpha }}_j^2+2{\overline{\alpha }}_j{\overline{\beta }}_j){\left(z_j^{\prime }\left(t\right)\right)}^2{e}^{\lambda t}+2\sum _{j=1}^n{\overline{\alpha }}_j\left({\overline{\alpha }}_j{z}_j^{\prime }\left(t\right)+{\overline{\beta }}_j{z}_j\left(t\right)\right)z_j^{\prime \prime }\left(t\right)e^{\lambda t}\hfill \\ & =& \lambda \sum _{j=1}^n\left({\overline{\delta }}_j+{\overline{\beta }}_j^2\right)u_j^2\left(t\right)e^{\lambda t}-2\lambda \sum _{j=1}^n({\overline{\delta }}_j+{\overline{\beta }}_j^2)u_j\left(t\right)\sum _{k=1}^n{e}_{jk}{u}_k(t-{\xi }_{jk})e^{\lambda t}\hfill \\ & & +\lambda \sum _{j=1}^n\left({\overline{\delta }}_j+{\overline{\beta }}_j^2\right)\sum _{k=1}^n{e}_{jk}{u}_k(t-{\xi }_{jk})\sum _{k=1}^n{e}_{jk}{u}_k(t-{\xi }_{jk})e^{\lambda t}\hfill \\ & & +2\sum _{j=1}^n\left({\overline{\delta }}_j+\lambda {\overline{\alpha }}_j{\overline{\beta }}_j+{\overline{\beta }}_j^2\right)(u_j\left(t\right)-\sum _{k=1}^n{e}_{jk}{u}_k\left(t-{\xi }_{jk}\right))\hfill \\ & & \times (u_j^{\prime }\left(t\right)-\sum _{k=1}^n{e}_{jk}{u}_k^{\prime }\left(t-{\xi }_{jk}\right))e^{\lambda t}\hfill \\ & & +\sum _{j=1}^n(\lambda {\overline{\alpha }}_j^2+2{\overline{\alpha }}_j{\overline{\beta }}_j){(u_j^{\prime }\left(t\right)-\sum _{k=1}^n{e}_{jk}{u}_k^{\prime }\left(t-{\xi }_{jk}\right))}^2{e}^{\lambda t}\hfill \\ & & +2\sum _{j=1}^n{\overline{\alpha }}_j[{\overline{\alpha }}_j(u_j^{\prime }\left(t\right)-\sum _{k=1}^n{e}_{jk}{u}_k^{\prime }\left(t-{\xi }_{jk}\right))+{\overline{\beta }}_j(u_j\left(t\right)-\sum _{k=1}^n{e}_{jk}{u}_k\left(t-{\xi }_{jk}\right))]\hfill \\ & & \times [-a_j{u}_j^{\prime }\left(t\right)-b_j{u}_j\left(t\right)+\sum _{k=1}^n{c}_{jk}{\tilde{f}}_k\left(u_k\left(t\right)\right)+\sum _{k=1}^n{d}_{jk}{\tilde{f}}_k\left(u_k\left(t-{\tau }_{jk}\right)\right)]e^{\lambda t}\hfill \\ \hfill & =& \lambda \sum _{j=1}^n({\overline{\delta }}_j+{\overline{\beta }}_j^2)u_j^2\left(t\right)e^{\lambda t}-2\lambda \sum _{j=1}^n\sum _{k=1}^n({\overline{\delta }}_j+{\overline{\beta }}_j^2)e_{jk}{u}_j\left(t\right)u_k(t-{\xi }_{jk})e^{\lambda t}\hfill \\ & & +\lambda \sum _{j=1}^n\sum _{k=1}^n({\overline{\delta }}_j+{\overline{\beta }}_j^2)e_{jk}{u}_k(t-{\xi }_{jk})\sum _{k=1}^n{e}_{jk}{u}_k(t-{\xi }_{jk})e^{\lambda t}\hfill \\ & & +2\sum _{j=1}^n\left({\overline{\delta }}_j+\lambda {\overline{\alpha }}_j{\overline{\beta }}_j+{\overline{\beta }}_j^2\right)u_j\left(t\right)u_j^{\prime }\left(t\right)e^{\lambda t}\hfill \\ & & -2\sum _{j=1}^n\sum _{k=1}^n\left({\overline{\delta }}_j+\lambda {\overline{\alpha }}_j{\overline{\beta }}_j+{\overline{\beta }}_j^2\right)e_{jk}{u}_j\left(t\right)u_k^{\prime }(t-{\xi }_{jk})e^{\lambda t}\hfill \\ & & -2\sum _{j=1}^n\sum _{k=1}^n\left({\overline{\delta }}_j+\lambda {\overline{\alpha }}_j{\overline{\beta }}_j+{\overline{\beta }}_j^2\right)e_{jk}{u}_k(t-{\xi }_{jk})u_j^{\prime }\left(t\right)e^{\lambda t}\hfill \\ & & +2\sum _{j=1}^n\sum _{k=1}^n\left({\overline{\delta }}_j+\lambda {\overline{\alpha }}_j{\overline{\beta }}_j+{\overline{\beta }}_j^2\right)e_{jk}{u}_k(t-{\xi }_{jk})\sum _{k=1}^n{e}_{jk}{u}_k^{\prime }(t-{\xi }_{jk})e^{\lambda t}\hfill \\ & & +\sum _{j=1}^n(\lambda {\overline{\alpha }}_j^2+2{\overline{\alpha }}_j{\overline{\beta }}_j){\left(u_j^{\prime }\left(t\right)\right)}^2{e}^{\lambda t}-2\sum _{j=1}^n\sum _{k=1}^n(\lambda {\overline{\alpha }}_j^2+2{\overline{\alpha }}_j{\overline{\beta }}_j)e_{jk}{u}_j^{\prime }\left(t\right)u_k^{\prime }(t-{\xi }_{jk})e^{\lambda t}\hfill \\ & & +\sum _{j=1}^n\sum _{k=1}^n(\lambda {\overline{\alpha }}_j^2+2{\overline{\alpha }}_j{\overline{\beta }}_j)e_{jk}{u}_k^{\prime }(t-{\xi }_{jk})\sum _{k=1}^n{e}_{jk}{u}_k^{\prime }(t-{\xi }_{jk})e^{\lambda t}\hfill \\ & & -2\sum _{j=1}^n{a}_j{\overline{\alpha }}_j^2{\left(u_j^{\prime }\left(t\right)\right)}^2{e}^{\lambda t}-2\sum _{j=1}^n{b}_j{\overline{\alpha }}_j^2{u}_j\left(t\right)u_j^{\prime }\left(t\right)e^{\lambda t}\hfill \\ & & +2\sum _{j=1}^n\sum _{k=1}^n{\overline{\alpha }}_j^2{c}_{jk}{u}_j^{\prime }\left(t\right){\tilde{f}}_k\left(u_k\left(t\right)\right)e^{\lambda t}+2\sum _{j=1}^n\sum _{k=1}^n{\overline{\alpha }}_j^2{d}_{jk}{u}_j^{\prime }\left(t\right){\tilde{f}}_k\left(u_k(t-{\tau }_{jk})\right)e^{\lambda t}\hfill \\ & & +2\sum _{j=1}^n\sum _{k=1}^n{a}_j{\overline{\alpha }}_j^2{e}_{jk}{u}_k^{\prime }(t-{\xi }_{jk})u_j^{\prime }\left(t\right)e^{\lambda t}+2\sum _{j=1}^n\sum _{k=1}^n{b}_j{\overline{\alpha }}_j^2{e}_{jk}{u}_k^{\prime }(t-{\xi }_{jk})u_j\left(t\right)e^{\lambda t}\hfill \\ & & -2\sum _{j=1}^n\sum _{k=1}^n{\overline{\alpha }}_j^2{e}_{jk}{u}_k^{\prime }(t-{\xi }_{jk})\sum _{k=1}^n{c}_{jk}{\tilde{f}}_k\left(u_k\left(t\right)\right)e^{\lambda t}\hfill \\ & & -2\sum _{j=1}^n\sum _{k=1}^n{\overline{\alpha }}_j^2{e}_{jk}{u}_k^{\prime }(t-{\xi }_{jk})\sum _{k=1}^n{d}_{jk}{\tilde{f}}_k\left(u_k(t-{\tau }_{jk})\right)e^{\lambda t}\hfill \\ & & -2\sum _{j=1}^n{a}_j{\overline{\alpha }}_j{\overline{\beta }}_j{u}_j\left(t\right)u_j^{\prime }\left(t\right)e^{\lambda t}-2\sum _{j=1}^n{b}_j{\overline{\alpha }}_j{\overline{\beta }}_j{u}_j^2\left(t\right)e^{\lambda t}\hfill \\ & & +2\sum _{j=1}^n\sum _{k=1}^n{\overline{\alpha }}_j{\overline{\beta }}_j{c}_{jk}{u}_j\left(t\right){\tilde{f}}_k\left(u_k\left(t\right)\right)e^{\lambda t}+2\sum _{j=1}^n\sum _{k=1}^n{\overline{\alpha }}_j{\overline{\beta }}_j{d}_{jk}{u}_j\left(t\right){\tilde{f}}_k\left(u_k(t-{\tau }_{jk})\right)e^{\lambda t}\hfill \\ & & +2\sum _{j=1}^n\sum _{k=1}^n{a}_j{\overline{\alpha }}_j{\overline{\beta }}_j{e}_{jk}{u}_k(t-{\xi }_{jk})u_j^{\prime }\left(t\right)e^{\lambda t}+2\sum _{j=1}^n\sum _{k=1}^n{b}_j{\overline{\alpha }}_j{\overline{\beta }}_j{e}_{jk}{u}_k(t-{\xi }_{jk})u_j\left(t\right)e^{\lambda t}\hfill \\ & & -2\sum _{j=1}^n\sum _{k=1}^n{\overline{\alpha }}_j{\overline{\beta }}_j{e}_{jk}{u}_k(t-{\xi }_{jk})\sum _{k=1}^n{c}_{jk}{\tilde{f}}_k\left(u_k\left(t\right)\right)e^{\lambda t}\hfill \\ & & +2\sum _{j=1}^n\sum _{k=1}^n{\overline{\alpha }}_j{\overline{\beta }}_j{e}_{jk}{u}_k(t-{\xi }_{jk})\sum _{k=1}^n{d}_{jk}{\tilde{f}}_k\left(u_k(t-{\tau }_{jk})\right)e^{\lambda t}.\hfill \end{array}$$

(12)

Moreover, one can give the following inequalities:

$$\begin{array}{ccc}\hfill & & \lambda \sum _{j=1}^n\left(\sum _{k=1}^n({\overline{\delta }}_j+{\overline{\beta }}_j^2)e_{jk}{u}_k(t-{\xi }_{jk})\right)(\sum _{k=1}^n{e}_{jk}{u}_k(t-{\xi }_{jk}))e^{\lambda t}\hfill \\ & =& \lambda \sum _{j=1}^n\left(\sum _{k=1}^n({\overline{\delta }}_j+{\overline{\beta }}_j^2)e_{jk}{u}_k(t-{\xi }_{jk})\right)(\sum _{l=1}^n{e}_{jl}{u}_l(t-{\xi }_{jl}))e^{\lambda t}\hfill \\ & =& \lambda \sum _{j=1}^n\sum _{k=1}^n\sum _{l=1}^n({\overline{\delta }}_j+{\overline{\beta }}_j^2)e_{jk}{e}_{jl}{u}_k(t-{\xi }_{jk})u_l(t-{\xi }_{jl})e^{\lambda t}\hfill \\ & \le & \lambda \sum _{j=1}^n\sum _{k=1}^n\sum _{l=1}^n({\overline{\delta }}_j+{\overline{\beta }}_j^2)|e_{jk}||e_{jl}||u_k(t-{\xi }_{jk})||u_l(t-{\xi }_{jl})|e^{\lambda t}\hfill \\ & \le & \frac{1}{2}\lambda \sum _{j=1}^n\sum _{k=1}^n\sum _{l=1}^n({\overline{\delta }}_j+{\overline{\beta }}_j^2)|e_{jk}||e_{jl}|u_k^2(t-{\xi }_{jk})e^{\lambda t}\hfill \\ & & +\frac{1}{2}\lambda \sum _{j=1}^n\sum _{k=1}^n\sum _{l=1}^n({\overline{\delta }}_j+{\overline{\beta }}_j^2)|e_{jk}||e_{jl}|u_l^2(t-{\xi }_{jl})e^{\lambda t}\hfill \\ & =& \frac{1}{2}\lambda \sum _{j=1}^n\sum _{k=1}^n\sum _{l=1}^n({\overline{\delta }}_k+{\overline{\beta }}_k^2)|e_{kj}||e_{kl}|u_j^2(t-{\xi }_{kj})e^{\lambda t}\hfill \\ & & +\frac{1}{2}\lambda \sum _{j=1}^n\sum _{k=1}^n\sum _{l=1}^n({\overline{\delta }}_k+{\overline{\beta }}_k^2)|e_{kl}||e_{kj}|u_j^2(t-{\xi }_{kj})e^{\lambda t}\hfill \\ & =& \lambda \sum _{j=1}^n\sum _{k=1}^n\sum _{l=1}^n({\overline{\delta }}_k+{\overline{\beta }}_k^2)|e_{kj}||e_{kl}|u_j^2(t-{\xi }_{kj})e^{\lambda t}.\hfill \end{array}$$

(13)

Similarly,

$$\begin{array}{ccc}& & 2\sum _{j=1}^n\left(\sum _{k=1}^n({\overline{\delta }}_j+\lambda {\overline{\alpha }}_j{\overline{\beta }}_j+{\overline{\beta }}_j^2)e_{jk}{u}_k(t-{\xi }_{jk})\right)(\sum _{k=1}^n{e}_{jk}{u}_k^{\prime }(t-{\xi }_{jk}))e^{\lambda t}\hfill \\ & \le & \sum _{j=1}^n\sum _{k=1}^n\sum _{l=1}^n({\overline{\delta }}_k+\lambda {\overline{\alpha }}_k{\overline{\beta }}_k+{\overline{\beta }}_k^2)|e_{kj}||e_{kl}|u_j^2(t-{\xi }_{kj})e^{\lambda t}\hfill \\ & & +\sum _{j=1}^n\sum _{k=1}^n\sum _{l=1}^n({\overline{\delta }}_k+\lambda {\overline{\alpha }}_k{\overline{\beta }}_k+{\overline{\beta }}_k^2)|e_{kl}||e_{kj}|(u_j^{\prime }(t-{\xi }_{kj}){)}^2{e}^{\lambda t},\hfill \end{array}$$

(14)

$$\begin{array}{ccc}& & \sum _{j=1}^n(\sum _{k=1}^n(\lambda {\overline{\alpha }}_j^2+2{\overline{\alpha }}_j{\overline{\beta }}_j)e_{jk}{u}_k^{\prime }(t-{\xi }_{jk}))\left(\sum _{k=1}^n{e}_{jk}{u}_k^{\prime }(t-{\xi }_{jk})\right)e^{\lambda t}\hfill \\ & \le & \sum _{j=1}^n\sum _{k=1}^n\sum _{l=1}^n(\lambda {\overline{\alpha }}_k^2+2{\overline{\alpha }}_k{\overline{\beta }}_k)|e_{kj}||e_{kl}|(u_j^{\prime }(t-{\xi }_{kj}){)}^2{e}^{\lambda t},\hfill \end{array}$$

(15)

$$\begin{array}{ccc}& & -2\sum _{j=1}^n\left(\sum _{k=1}^n{\overline{\alpha }}_j^2{e}_{jk}{u}_k^{\prime }(t-{\xi }_{jk})\right)\left(\sum _{k=1}^n{c}_{jk}{\tilde{f}}_k\left(u_k\left(t\right)\right)\right)e^{\lambda t}\hfill \\ & \le & \sum _{j=1}^n\sum _{k=1}^n\sum _{l=1}^n{\overline{\alpha }}_k^2|e_{kj}||c_{kl}|{\left(u_j^{\prime }(t-{\xi }_{kj})\right)}^2{e}^{\lambda t}\hfill \\ & & +\sum _{j=1}^n\sum _{k=1}^n\sum _{l=1}^n{\overline{\alpha }}_l^2|e_{lk}||c_{lj}|{\tilde{f}}_j^2\left(u_j\left(t\right)\right)e^{\lambda t},\hfill \end{array}$$

(16)

$$\begin{array}{ccc}& & -2\sum _{j=1}^n(\sum _{k=1}^n{\overline{\alpha }}_j^2{e}_{jk}{u}_k^{\prime }(t-{\xi }_{jk})\left)\right(\sum _{k=1}^n{d}_{jk}{\tilde{f}}_k\left(u_k(t-{\tau }_{jk})\right))e^{\lambda t}\hfill \\ & \le & \sum _{j=1}^n\sum _{k=1}^n\sum _{l=1}^n{\overline{\alpha }}_k^2|e_{kj}||d_{kl}|{\left(u_j^{\prime }(t-{\xi }_{kj})\right)}^2{e}^{\lambda t}\hfill \\ & & +\sum _{j=1}^n\sum _{k=1}^n\sum _{l=1}^n{\overline{\alpha }}_k^2|e_{kl}||d_{kj}|{\tilde{f}}_j^2\left(u_j(t-{\tau }_{kj})\right)e^{\lambda t},\hfill \end{array}$$

(17)

$$\begin{array}{ccc}& & -2\sum _{j=1}^n\sum _{k=1}^n{\overline{\alpha }}_j{\overline{\beta }}_j{e}_{jk}{u}_k(t-{\xi }_{jk})\sum _{k=1}^n{c}_{jk}{\tilde{f}}_k\left(u_k\left(t\right)\right)e^{\lambda t}\hfill \\ & \le & \sum _{j=1}^n\sum _{k=1}^n\sum _{l=1}^n{\overline{\alpha }}_k{\overline{\beta }}_k|e_{kj}||c_{kl}|u_j^2(t-{\xi }_{kj})e^{\lambda t}\hfill \\ & & +\sum _{j=1}^n\sum _{k=1}^n\sum _{l=1}^n{\overline{\alpha }}_l{\overline{\beta }}_l|e_{lk}||c_{lj}|{\tilde{f}}_j^2\left(u_j\left(t\right)\right)e^{\lambda t},\hfill \end{array}$$

(18)

$$\begin{array}{ccc}& & 2\sum _{j=1}^n\sum _{k=1}^n{\overline{\alpha }}_j{\overline{\beta }}_j{e}_{jk}{u}_k(t-{\xi }_{jk})\sum _{k=1}^n{d}_{jk}{\tilde{f}}_k\left(u_k(t-{\tau }_{jk})\right)e^{\lambda t}\hfill \\ & \le & \sum _{j=1}^n\sum _{k=1}^n\sum _{l=1}^n{\overline{\alpha }}_k{\overline{\beta }}_k|e_{kj}||d_{kl}|u_j^2(t-{\xi }_{kj})e^{\lambda t}\hfill \\ & & +\sum _{j=1}^n\sum _{k=1}^n\sum _{l=1}^n{\overline{\alpha }}_k{\overline{\beta }}_k|e_{kl}||d_{kj}|{\tilde{f}}_j^2\left(u_j(t-{\tau }_{kj})\right)e^{\lambda t},\hfill \end{array}$$

(19)

$$\begin{array}{ccc}& & -2\lambda \sum _{j=1}^n\sum _{k=1}^n\left({\overline{\delta }}_j+{\overline{\beta }}_j^2\right)e_{jk}{u}_j\left(t\right)u_k(t-{\xi }_{jk})e^{\lambda t}\hfill \\ & \le & \lambda \sum _{j=1}^n\sum _{k=1}^n\left({\overline{\delta }}_j+{\overline{\beta }}_j^2\right)|e_{jk}|u_j^2\left(t\right)e^{\lambda t}\hfill \\ & & +\lambda \sum _{j=1}^n\sum _{k=1}^n\left({\overline{\delta }}_k+{\overline{\beta }}_k^2\right)|e_{kj}|u_j^2(t-{\xi }_{kj})e^{\lambda t},\hfill \end{array}$$

(20)

$$\begin{array}{ccc}& & -2\sum _{j=1}^n\sum _{k=1}^n\left({\overline{\delta }}_j+\lambda {\overline{\alpha }}_j{\overline{\beta }}_j+{\overline{\beta }}_j^2\right)e_{jk}{u}_j\left(t\right)u_k^{\prime }(t-{\xi }_{jk})e^{\lambda t}\hfill \\ & \le & \sum _{j=1}^n\sum _{k=1}^n\left({\overline{\delta }}_j+\lambda {\overline{\alpha }}_j{\overline{\beta }}_j+{\overline{\beta }}_j^2\right)|e_{jk}|u_j^2\left(t\right)e^{\lambda t}\hfill \\ & & +\sum _{j=1}^n\sum _{k=1}^n\left({\overline{\delta }}_k+\lambda {\overline{\alpha }}_k{\overline{\beta }}_k+{\overline{\beta }}_k^2\right)|e_{kj}|{\left(u_j^{\prime }(t-{\xi }_{kj})\right)}^2{e}^{\lambda t},\hfill \end{array}$$

(21)

$$\begin{array}{ccc}& & -2\sum _{j=1}^n\sum _{k=1}^n\left({\overline{\delta }}_j+\lambda {\overline{\alpha }}_j{\overline{\beta }}_j+{\overline{\beta }}_j^2\right)e_{jk}{u}_k(t-{\xi }_{jk})u_j^{\prime }\left(t\right)e^{\lambda t}\hfill \\ & \le & \sum _{j=1}^n\sum _{k=1}^n\left({\overline{\delta }}_k+\lambda {\overline{\alpha }}_k{\overline{\beta }}_k+{\overline{\beta }}_k^2\right)|e_{kj}|u_j^2(t-{\xi }_{kj})e^{\lambda t}\hfill \\ & & +\sum _{j=1}^n\sum _{k=1}^n\left({\overline{\delta }}_j+\lambda {\overline{\alpha }}_j{\overline{\beta }}_j+{\overline{\beta }}_j^2\right)|e_{jk}|{\left(u_j^{\prime }\left(t\right)\right)}^2{e}^{\lambda t},\hfill \end{array}$$

(22)

$$\begin{array}{ccc}& & -2\sum _{j=1}^n\sum _{k=1}^n(\lambda {\overline{\alpha }}_j^2+2{\overline{\alpha }}_j{\overline{\beta }}_j)e_{jk}{u}_j^{\prime }\left(t\right)u_k^{\prime }(t-{\xi }_{jk})e^{\lambda t}\hfill \\ & \le & \sum _{j=1}^n\sum _{k=1}^n(\lambda {\overline{\alpha }}_j^2+2{\overline{\alpha }}_j{\overline{\beta }}_j)|e_{jk}|{\left(u_j^{\prime }\left(t\right)\right)}^2{e}^{\lambda t}\hfill \\ & & +\sum _{j=1}^n\sum _{k=1}^n(\lambda {\overline{\alpha }}_k^2+2{\overline{\alpha }}_k{\overline{\beta }}_k)|e_{kj}|{\left(u_j^{\prime }(t-{\xi }_{kj})\right)}^2{e}^{\lambda t},\hfill \end{array}$$

(23)

$$\begin{array}{ccc}& & 2\sum _{j=1}^n\sum _{k=1}^n{\overline{\alpha }}_j^2{c}_{jk}{u}_j^{\prime }\left(t\right){\tilde{f}}_k\left(u_k\left(t\right)\right)e^{\lambda t}\hfill \\ & \le & \sum _{j=1}^n\sum _{k=1}^n{\overline{\alpha }}_j^2|c_{jk}|{\left(u_j^{\prime }\left(t\right)\right)}^2{e}^{\lambda t}+\sum _{j=1}^n\sum _{k=1}^n{\overline{\alpha }}_k^2|c_{kj}|{\tilde{f}}_j^2\left(u_j\left(t\right)\right)e^{\lambda t},\hfill \end{array}$$

(24)

$$\begin{array}{ccc}& & 2\sum _{j=1}^n\sum _{k=1}^n{\overline{\alpha }}_j^2{d}_{jk}{u}_j^{\prime }\left(t\right){\tilde{f}}_k\left(u_k(t-{\tau }_{jk})\right)e^{\lambda t}\hfill \\ & \le & \sum _{j=1}^n\sum _{k=1}^n{\overline{\alpha }}_j^2|d_{jk}|{\left(u_j^{\prime }\left(t\right)\right)}^2{e}^{\lambda t}+\sum _{j=1}^n\sum _{k=1}^n{\overline{\alpha }}_k^2|d_{kj}|{\tilde{f}}_j^2\left(u_j(t-{\tau }_{kj})\right)e^{\lambda t},\hfill \end{array}$$

(25)

$$\begin{array}{ccc}& & 2\sum _{j=1}^n\sum _{k=1}^n{a}_j{\overline{\alpha }}_j^2{e}_{jk}{u}_k^{\prime }(t-{\xi }_{jk})u_j^{\prime }\left(t\right)e^{\lambda t}\hfill \\ & \le & \sum _{j=1}^n\sum _{k=1}^n{a}_k{\overline{\alpha }}_k^2|e_{kj}|{\left(u_j^{\prime }(t-{\xi }_{kj})\right)}^2{e}^{\lambda t}+\sum _{j=1}^n\sum _{k=1}^n{a}_j{\overline{\alpha }}_j^2|e_{jk}|{\left(u_j^{\prime }\left(t\right)\right)}^2{e}^{\lambda t},\hfill \end{array}$$

(26)

$$\begin{array}{ccc}& & 2\sum _{j=1}^n\sum _{k=1}^n{b}_j{\overline{\alpha }}_j^2{e}_{jk}{u}_k^{\prime }(t-{\xi }_{jk})u_j\left(t\right)e^{\lambda t}\hfill \\ & \le & \sum _{j=1}^n\sum _{k=1}^n{b}_k{\overline{\alpha }}_k^2|e_{kj}|{\left(u_j^{\prime }(t-{\xi }_{kj})\right)}^2{e}^{\lambda t}+\sum _{j=1}^n\sum _{k=1}^n{b}_j{\overline{\alpha }}_j^2|e_{jk}|u_j^2\left(t\right)e^{\lambda t},\hfill \end{array}$$

(27)

$$\begin{array}{ccc}& & 2\sum _{j=1}^n\sum _{k=1}^n{\overline{\alpha }}_j{\overline{\beta }}_j{c}_{jk}{u}_j\left(t\right){\tilde{f}}_k\left(u_k\left(t\right)\right)e^{\lambda t}\hfill \\ & \le & \sum _{j=1}^n\sum _{k=1}^n{\overline{\alpha }}_j{\overline{\beta }}_j|c_{jk}|u_j^2\left(t\right)e^{\lambda t}+\sum _{j=1}^n\sum _{k=1}^n{\overline{\alpha }}_k{\overline{\beta }}_k|c_{kj}|{\tilde{f}}_j^2\left(u_j\left(t\right)\right)e^{\lambda t},\hfill \end{array}$$

(28)

$$\begin{array}{ccc}& & 2\sum _{j=1}^n\sum _{k=1}^n{\overline{\alpha }}_j{\overline{\beta }}_j{d}_{jk}{u}_j\left(t\right){\tilde{f}}_k\left(u_k(t-{\tau }_{jk})\right)e^{\lambda t}\hfill \\ & \le & \sum _{j=1}^n\sum _{k=1}^n{\overline{\alpha }}_j{\overline{\beta }}_j|d_{jk}|u_j^2\left(t\right)e^{\lambda t}+\sum _{j=1}^n\sum _{k=1}^n{\overline{\alpha }}_k{\overline{\beta }}_k|d_{kj}|{\tilde{f}}_j^2\left(u_j(t-{\tau }_{kj})\right)e^{\lambda t},\hfill \end{array}$$

(29)

$$\begin{array}{ccc}& & 2\sum _{j=1}^n\sum _{k=1}^n{a}_j{\overline{\alpha }}_j{\overline{\beta }}_j{e}_{jk}{u}_k(t-{\xi }_{jk})u_j^{\prime }\left(t\right)e^{\lambda t}\hfill \\ & \le & \sum _{j=1}^n\sum _{k=1}^n{a}_k{\overline{\alpha }}_k{\overline{\beta }}_k|e_{kj}|u_j^2(t-{\xi }_{kj})e^{\lambda t}+\sum _{j=1}^n\sum _{k=1}^n{a}_j{\overline{\alpha }}_j{\overline{\beta }}_j|e_{jk}|{\left(u_j^{\prime }\left(t\right)\right)}^2{e}^{\lambda t},\hfill \end{array}$$

(30)

$$\begin{array}{ccc}& & 2\sum _{j=1}^n\sum _{k=1}^n{b}_j{\overline{\alpha }}_j{\overline{\beta }}_j{e}_{jk}{u}_k(t-{\xi }_{jk})u_j\left(t\right)e^{\lambda t}\hfill \\ & \le & \sum _{j=1}^n\sum _{k=1}^n{b}_k{\overline{\alpha }}_k{\overline{\beta }}_k|e_{kj}|u_j^2(t-{\xi }_{kj})e^{\lambda t}+\sum _{j=1}^n\sum _{k=1}^n{b}_j{\overline{\alpha }}_j{\overline{\beta }}_j|e_{jk}|u_j^2\left(t\right)e^{\lambda t}.\hfill \end{array}$$

(31)

Submitting (13)–(31) into (12), we obtain

$$\begin{array}{ccc}\hfill W_1^{\prime }\left(t\right)& \le & e^{\lambda t}\sum _{j=1}^n\left\{\right[\lambda ({\overline{\delta }}_j+{\overline{\beta }}_j^2)+\lambda \sum _{k=1}^n({\overline{\delta }}_j+{\overline{\beta }}_j^2)|e_{jk}|+\sum _{k=1}^n({\overline{\delta }}_j+\lambda {\overline{\alpha }}_j{\overline{\beta }}_j+{\overline{\beta }}_j^2)|e_{jk}|\hfill \\ & & +\sum _{k=1}^n{b}_j{\overline{\alpha }}_j^2|e_{jk}|-2b_j{\overline{\alpha }}_j{\overline{\beta }}_j+\sum _{k=1}^n{\overline{\alpha }}_j{\overline{\beta }}_j|c_{jk}|+\sum _{k=1}^n{\overline{\alpha }}_j{\overline{\beta }}_j|d_{jk}|+\sum _{k=1}^n{b}_j{\overline{\alpha }}_j{\overline{\beta }}_j|e_{jk}|]u_j^2\left(t\right)\hfill \\ & & +[2({\overline{\delta }}_j+\lambda {\overline{\alpha }}_j{\overline{\beta }}_j+{\overline{\beta }}_j^2)-2b_j{\overline{\alpha }}_j^2-2a_j{\overline{\alpha }}_j{\overline{\beta }}_j]u_j\left(t\right)u_j^{\prime }\left(t\right)\hfill \\ & & +[\sum _{k=1}^n\left({\overline{\delta }}_j+\lambda {\overline{\alpha }}_j{\overline{\beta }}_j+{\overline{\beta }}_j^2\right)|e_{jk}|+(\lambda {\overline{\alpha }}_j^2+2{\overline{\alpha }}_j{\overline{\beta }}_j)+\sum _{k=1}^n(\lambda {\overline{\alpha }}_j^2+{\overline{\alpha }}_j{\overline{\beta }}_j)|e_{jk}|\hfill \\ & & -2a_j{\overline{\alpha }}_j^2+\sum _{k=1}^n{\overline{\alpha }}_j^2|c_{jk}|+\sum _{k=1}^n{\overline{\alpha }}_j^2|d_{jk}|+\sum _{k=1}^n{a}_j{\overline{\alpha }}_j^2|e_{jk}|+\sum _{k=1}^n{a}_j{\overline{\alpha }}_j{\overline{\beta }}_j|e_{jk}|]{\left(u_j^{\prime }\left(t\right)\right)}^2\hfill \\ & & +[\sum _{k=1}^n{\overline{\alpha }}_k^2|c_{kj}|+\sum _{k=1}^n\sum _{l=1}^n{\overline{\alpha }}_l^2|e_{lk}||c_{lj}|+\sum _{k=1}^n{\overline{\alpha }}_k{\overline{\beta }}_k|c_{kj}|+\sum _{k=1}^n\sum _{l=1}^n{\overline{\alpha }}_l{\overline{\beta }}_l|e_{lk}||c_{lj}|]{\tilde{f}}_j^2\left(u_j\left(t\right)\right)\hfill \\ & & +[\sum _{k=1}^n{A}_{kj}^2+\sum _{k=1}^n\sum _{l=1}^n{\tilde{A}}_{kjl}^2]u_j^2(t-{\xi }_{kj})+[\sum _{k=1}^n{B}_{kj}^2+\sum _{k=1}^n\sum _{l=1}^n{\tilde{B}}_{kjl}^2]{\left(u_j^{\prime }(t-{\xi }_{kj})\right)}^2\hfill \\ & & +[\sum _{k=1}^n{C}_{kj}+\sum _{k=1}^n\sum _{l=1}^n{\tilde{C}}_{kjl}]{\tilde{f}}_j^2\left(u_j(t-{\tau }_{kj})\right)\}.\hfill \end{array}$$

(32)

In the following, by (10) and (11), we have

$$\begin{array}{ccc}\hfill W_2^{\prime }\left(t\right)& =& \sum _{j=1}^n\sum _{k=1}^n{A}_{kj}^2{u}_j^2\left(t\right)e^{\lambda (t+{\xi }_{kj})}-\sum _{j=1}^n\sum _{k=1}^n{A}_{kj}^2{u}_j^2(t-{\xi }_{kj})e^{\lambda t}\hfill \\ & & +\sum _{j=1}^n\sum _{k=1}^n\sum _{l=1}^n{\tilde{A}}_{kjl}^2{u}_j^2\left(t\right)e^{\lambda (t+{\xi }_{kj})}-\sum _{j=1}^n\sum _{k=1}^n\sum _{l=1}^n{\tilde{A}}_{kjl}^2{u}_j^2(t-{\xi }_{kj})e^{\lambda t},\hfill \end{array}$$

(33)

$$\begin{array}{ccc}\hfill W_3^{\prime }\left(t\right)& =& \sum _{j=1}^n\sum _{k=1}^n{B}_{kj}^2{\left(u_j^{\prime }\left(t\right)\right)}^2{e}^{\lambda (t+{\xi }_{kj})}-\sum _{j=1}^n\sum _{k=1}^n{B}_{kj}^2{\left(u_j^{\prime }(t-{\xi }_{kj})\right)}^2{e}^{\lambda t}\hfill \\ & & +\sum _{j=1}^n\sum _{k=1}^n\sum _{l=1}^n{\tilde{B}}_{kjl}^2{\left(u_j^{\prime }\left(t\right)\right)}^2{e}^{\lambda (t+{\xi }_{kj})}-\sum _{j=1}^n\sum _{k=1}^n\sum _{l=1}^n{\tilde{B}}_{kjl}^2{\left(u_j^{\prime }(t-{\xi }_{kj})\right)}^2{e}^{\lambda t},\hfill \end{array}$$

(34)

$$\begin{array}{ccc}\hfill W_4^{\prime }\left(t\right)& =& \sum _{j=1}^n\sum _{k=1}^n{C}_{kj}{\tilde{f}}_j^2\left(u_j\left(t\right)\right)e^{\lambda (t+{\tau }_{kj})}-\sum _{j=1}^n\sum _{k=1}^n{C}_{kj}{\tilde{f}}_j^2\left(u_j(t-{\tau }_{kj})\right)e^{\lambda t}\hfill \\ & & +\sum _{j=1}^n\sum _{k=1}^n\sum _{l=1}^n{\tilde{C}}_{kjl}{\tilde{f}}_j^2\left(u_j\left(t\right)\right)e^{\lambda (t+{\tau }_{kj})}-\sum _{j=1}^n\sum _{k=1}^n\sum _{l=1}^n{\tilde{C}}_{kjl}{\tilde{f}}_j^2\left(u_j(t-{\tau }_{kj})\right)e^{\lambda t}.\hfill \end{array}$$

(35)

From (32)–(35) and the Assumption 1, one can get

$$\begin{array}{ccc}\hfill W^{\prime }\left(t\right)& =& \sum _{j=1}^4{W}_j^{\prime }\left(t\right)\hfill \\ & \le & e^{\lambda t}\sum _{j=1}^n\left\{\right[\lambda ({\overline{\delta }}_j+{\overline{\beta }}_j^2)+\lambda \sum _{k=1}^n({\overline{\delta }}_j+{\overline{\beta }}_j^2)|e_{jk}|+\sum _{k=1}^n({\overline{\delta }}_j+\lambda {\overline{\alpha }}_j{\overline{\beta }}_j+{\overline{\beta }}_j^2)|e_{jk}|\hfill \\ & & +\sum _{k=1}^n{b}_j{\overline{\alpha }}_j^2|e_{jk}|-2b_j{\overline{\alpha }}_j{\overline{\beta }}_j+\sum _{k=1}^n{\overline{\alpha }}_j{\overline{\beta }}_j|c_{jk}|+\sum _{k=1}^n{\overline{\alpha }}_j{\overline{\beta }}_j|d_{jk}|+\sum _{k=1}^n{b}_j{\overline{\alpha }}_j{\overline{\beta }}_j|e_{jk}|\hfill \\ & & +(\sum _{k=1}^n{\overline{\alpha }}_k^2|c_{kj}|+\sum _{k=1}^n\sum _{l=1}^n{\overline{\alpha }}_l^2|e_{lk}||c_{lj}|+\sum _{k=1}^n{\overline{\alpha }}_k{\overline{\beta }}_k|c_{kj}|+\sum _{k=1}^n\sum _{l=1}^n{\overline{\alpha }}_l{\overline{\beta }}_l|e_{lk}||c_{lj}|)L_j^2\hfill \\ & & +(\sum _{k=1}^n{A}_{kj}^2+\sum _{k=1}^n\sum _{l=1}^n{\tilde{A}}_{kjl}^2)e^{\lambda {\xi }_{kj}}+(\sum _{k=1}^n{C}_{kj}+\sum _{k=1}^n\sum _{l=1}^n{\tilde{C}}_{kjl})L_j^2{e}^{\lambda {\tau }_{kj}}]u_j^2\left(t\right)\hfill \\ & & +[2({\overline{\delta }}_j+\lambda {\overline{\alpha }}_j{\overline{\beta }}_j+{\overline{\beta }}_j^2)-2b_j{\overline{\alpha }}_j^2-2a_j{\overline{\alpha }}_j{\overline{\beta }}_j]u_j\left(t\right)u_j^{\prime }\left(t\right)\hfill \\ & & +[\sum _{k=1}^n\left({\overline{\delta }}_j+\lambda {\overline{\alpha }}_j{\overline{\beta }}_j+{\overline{\beta }}_j^2\right)|e_{jk}|+\lambda {\overline{\alpha }}_j^2+2{\overline{\alpha }}_j{\overline{\beta }}_j+\sum _{k=1}^n(\lambda {\overline{\alpha }}_j^2+{\overline{\alpha }}_j{\overline{\beta }}_j)|e_{jk}|\hfill \\ & & -2a_j{\overline{\alpha }}_j^2+\sum _{k=1}^n{\overline{\alpha }}_j^2|c_{jk}|+\sum _{k=1}^n{\overline{\alpha }}_j^2|d_{jk}|+\sum _{k=1}^n{a}_j{\overline{\alpha }}_j^2|e_{jk}|+\sum _{k=1}^n{a}_j{\overline{\alpha }}_j{\overline{\beta }}_j|e_{jk}|\hfill \\ & & +(\sum _{k=1}^n{B}_{kj}^2+\sum _{k=1}^n\sum _{l=1}^n{\tilde{B}}_{kjl}^2)e^{\lambda {\xi }_{kj}}\left]{\left(u_j^{\prime }\left(t\right)\right)}^2\right\}\hfill \\ & =& e^{\lambda t}\sum _{j=1}^n\left({\mathcal{X}}_j^{\lambda }{\left(u_j^{\prime }\left(t\right)\right)}^2+{\mathcal{Z}}_j^{\lambda }{u}_j\left(t\right)u_j^{\prime }\left(t\right)+{\mathcal{Y}}_j^{\lambda }{u}_j^2\left(t\right)\right)\hfill \\ & =& e^{\lambda t}\sum _{j=1}^n{\mathcal{X}}_j^{\lambda }{\left(u_j^{\prime }\left(t\right)+\frac{{\mathcal{Z}}_j^{\lambda }}{2{\mathcal{X}}_j^{\lambda }}{u}_j\left(t\right)\right)}^2+\sum _{j=1}^n\left({\mathcal{Y}}_j^{\lambda }-\frac{{\left({\mathcal{Z}}_j^{\lambda }\right)}^2}{4{\mathcal{X}}_j^{\lambda }}\right)u_j^2\left(t\right)\hfill \\ & \le & 0.\hfill \end{array}$$

(36)

This implies that $W\left(t\right)\le W\left(0\right)$ on $[0,+\infty )$, and

$$\sum _{j=1}^n{\overline{\delta }}_j{z}_j^2\left(t\right)e^{\lambda t}+\sum _{j=1}^n{({\overline{\alpha }}_j{z}_j^{\prime }\left(t\right)+{\overline{\beta }}_j{z}_j\left(t\right))}^2{e}^{\lambda t}\le W\left(0\right).$$

Note that

$$\left|z_j^{\prime }\left(t\right)\right| \le \left|z_j^{\prime }\left(t\right)+z_j\left(t\right)\right| + \left|z_j\left(t\right)\right|,$$

we can easily see that there exists a constant $\Lambda >0$ obeying

$$\left|u_j^{\prime }\left(t\right)\right| \le \Lambda e^{-\lambda t},\left|u_j\left(t\right)\right| \le \Lambda e^{-\lambda t},\forall t\in [0,+\infty ),j\in \mathfrak{N}.$$

This completes the proof. □

Remark 1.

When $u_j\left(t\right)$ is a periodic solution of NTINNs (3), Lemma 2 shows that all solutions of NTINNs (3) and their derivatives are exponentially convergent to $u_j\left(t\right)$ and $u_j^{\prime }\left(t\right)$, respectively.

3. Periodicity of NTINNs

Theorem 1.

If the assumptions in Lemma 2 are satisfied, NTINNs (3) possess a globally exponentially stable T-periodic solution.

Proof. 

Denote ${\rho }_j\left(t\right)$ by setting

$$\begin{array}{ccc}\hfill {\rho }_j^{\prime \prime }\left(t\right)-\sum _{k=1}^n{e}_{jk}{\rho }_k^{\prime \prime }(t-{\xi }_{jk})& =& -a_j{\rho }_j^{\prime }\left(t\right)-b_j{\rho }_j\left(t\right)+\sum _{k=1}^n{c}_{jk}{f}_k\left({\rho }_k\left(t\right)\right)\hfill \\ & & +\sum _{k=1}^n{d}_{jk}{f}_k\left({\rho }_k(t-{\tau }_{jk})\right)+I_j\left(t\right),\hfill \end{array}$$

(37)

and

$${\rho }_j\left(v\right)={\phi }_j^{\rho }\left(v\right),{\rho }_j^{\prime }\left(v\right)={\psi }_j^{\rho }\left(v\right),{\rho }_j^{\prime \prime }\left(v\right)={\varsigma }_j^{\rho }\left(v\right),{\phi }_j^{\rho },{\psi }_j^{\rho },{\varsigma }_j^{\rho }\in C([-{\sigma }_j,0],\mathbb{R}),j\in \mathfrak{N}.$$

(38)

Hence, for any nonnegative integer $n,$

$$\begin{array}{ccc}\hfill & & {\rho }_j^{\prime \prime }(t+nT)-\sum _{k=1}^n{e}_{jk}{\rho }_k^{\prime \prime }\left(t+nT-{\xi }_{jk}\right)\hfill \\ & =& -a_j{\rho }_j^{\prime }(t+nT)-b_j{\rho }_j(t+nT)+\sum _{k=1}^n{c}_{jk}{f}_k\left({\rho }_k(t+nT)\right)\hfill \\ & & +\sum _{k=1}^n{d}_{jk}{f}_k\left({\rho }_k\left(t+nT-{\tau }_{jk}\right)\right)+I_j\left(t\right),\forall j\in \mathfrak{N},t+nT\ge 0,\hfill \end{array}$$

(39)

and $\mu \left(t\right)=\rho (t+T)$ is a solution of NTINNs (3), which satisfies

$${\phi }_j^{\mu }\left(v\right)={\rho }_j(v+T),{\psi }_j^{\mu }\left(v\right)={\rho }_j^{\prime }(v+T),{\varsigma }_j^{\mu }\left(v\right)={\rho }_j^{\prime \prime }(v+T),\forall j\in \mathfrak{N},v\in [-{\sigma }_j,0].$$

By using Lemma 2, one can select a constant $\Lambda =\Lambda ({\phi }^{\rho },{\psi }^{\rho },{\varsigma }^{\rho },{\phi }^{\mu },{\psi }^{\mu },{\varsigma }^{\mu })>0$ satisfying

$$|{\rho }_j\left(t\right)-{\mu }_j\left(t\right)|\le \Lambda e^{-\lambda t},|{\rho }_j^{\prime }\left(t\right)-{\mu }_j^{\prime }\left(t\right)|\le \Lambda e^{-\lambda t},\forall j\in \mathfrak{N},t\ge 0.$$

Therefore,

$$\begin{array}{cc}\hfill |{\rho }_j(t+mT)-{\rho }_j(t+(m+1)T)|=& |{\rho }_j(t+mT)-{\mu }_j(t+mT)|\hfill \\ \hfill \le & \Lambda e^{-\lambda (t+mT)},\forall j\in \mathfrak{N},t+mT\ge 0\hfill \end{array}$$

and

$$\begin{array}{cc}\hfill |{\rho }_j^{\prime }(t+mT)-{\rho }_j^{\prime }(t+(m+1)T)|=& |{\rho }_j^{\prime }(t+mT)-{\mu }_j^{\prime }(t+mT)|\hfill \\ \hfill \le & \Lambda e^{-\lambda (t+mT)},\forall j\in \mathfrak{N},t+mT\ge 0.\hfill \end{array}$$

Since

$${\rho }_j(t+nT)={\rho }_j\left(t\right)+\sum _{m=0}^{n-1}[{\rho }_j(t+(m+1)T)-{\rho }_j(t+mT)]$$

and

$${\rho }_j^{\prime }(t+nT)={\rho }_j^{\prime }\left(t\right)+\sum _{m=0}^{n-1}[{\rho }_j^{\prime }(t+(m+1)T)-{\rho }_j^{\prime }(t+mT)],j\in \mathfrak{N},$$

we can easily reveal that in any compact subset of $\mathbb{R}$, ${\left\{{\rho }_j(t+nT)\right\}}_{n\ge 1}$, ${\left\{{\rho }_j^{\prime }(t+nT)\right\}}_{n\ge 1}$ and ${\{{\rho }_j^{\prime }(t+(n+1)T)-{\displaystyle \sum _{k=1}^n}{e}_{jk}{\rho }_k^{\prime }\left(t+(n+1)T-{\xi }_{jk}\right)\}}_{n\ge 1}$ are uniformly convergent function sequences and there is a differentiable function $x\left(t\right)$ obeying

$$\underset{m\to +\infty }{lim}\rho (t+nT)=x\left(t\right),\underset{m\to +\infty }{lim}{\rho }^{\prime }(t+nT)=x^{\prime }\left(t\right).$$

Hence

$$x(t+T)=\underset{n\to +\infty }{lim}\rho (t+T+nT)=\underset{(n+1)\to +\infty }{lim}\rho (t+(n+1)T)=x\left(t\right),$$

which indicates that $x\left(t\right)$ is T-periodic on $\mathbb{R}$. In addition, from the Assumption 2 and the continuity of NTINNs (3), one can conclude that on any compact subset of $\mathbb{R}$, ${\{{\rho }_j^{\prime \prime }(t+(n+1)T)-{\displaystyle \sum _{k=1}^n}{e}_{jk}{\rho }_k^{\prime \prime }\left(t+(n+1)T-{\xi }_{jk}\right)\}}_{n\ge 1}$ is uniformly convergent. Setting $n\to +\infty$, it is easy to acquire that

$$\begin{array}{cc}\hfill x_j^{\prime \prime }\left(t\right)-\sum _{k=1}^n{e}_{jk}{x}_k^{\prime \prime }\left(t-{\xi }_{jk}\right)=& -a_j{x}_j^{\prime }\left(t\right)-b_j{x}_j\left(t\right)+\sum _{k=1}^n{c}_{jk}{f}_k\left(x_k\left(t\right)\right)\hfill \\ \hfill & +\sum _{k=1}^n{d}_{jk}{f}_k\left(x_k\left(t-{\tau }_{jk}\right)\right)+I_j\left(t\right),\hfill \end{array}$$

which reveals that $x\left(t\right)$ is a T-periodic solution of NTINNs (3). Finally, according to Lemma 2 and Remark 1, we obtain that $x\left(t\right)$ possesses global exponential stability. This ends the proof. □

Remark 2.

In recent years, the dynamic behaviors of NTNNs [26,29,39,41] and INNs [3,4,5,6,7,8,9,11,12,13,14,15,17,18,19,20,21,30,31,32]. have been widely studied. However, we note that the global exponential stability of T-periodic solutions on the NTINNs has not been studied, hence our research is novel and further promotes the previous research.

4. A Numerical Example

Example 1.

Label n = 2, and consider the following NTINNs involving multiple delays:

$$\left\{\begin{array}{c}{s}_1^{\prime \prime }\left(t\right)-0.2s_1^{\prime \prime }(t-1)+0.1s_2^{\prime \prime }(t-2)\hfill \\ =-6.8s_1^{\prime }\left(t\right)-8s_1\left(t\right)-0.4f_1\left(s_1\left(t\right)\right)+0.6f_2\left(s_2\left(t\right)\right)\hfill \\ +0.2f_1\left(s_1(t-0.4)\right)+0.3f_2\left(s_2(t-0.5)\right)+10sint,\hfill \\ s_2^{\prime \prime }\left(t\right)-0.1s_2^{\prime \prime }(t-1.2)+0.15s_2^{\prime \prime }(t-1.8)\hfill \\ =-10.2s_2^{\prime }\left(t\right)-11s_2\left(t\right)-0.2f_1\left(s_1\left(t\right)\right)+0.4f_2\left(s_2\left(t\right)\right)\hfill \\ +0.3f_1\left(s_1(t-0.2)\right)+0.4f_2\left(s_2(t-0.3)\right)+100cost,\hfill \end{array}\right.$$

(40)

where $f_j\left(u\right)=\frac{1}{8}(|u+1|-|u-1|),j=1,2$.

Take ${\delta }_1=34,{\delta }_2=66,{\overline{\beta }}_1=1,{\overline{\beta }}_2=1.2,{\overline{\alpha }}_1=1.8,{\overline{\alpha }}_2=2,L_j=\frac{1}{4},j=1,2,$ we get

$$\begin{array}{c}{\mathcal{X}}_1=-17.9544,{\mathcal{Y}}_1=-3.504,{\mathcal{Z}}_1=-6.32,\hfill \\ {\mathcal{X}}_2=-37.82,{\mathcal{Y}}_2=-15.22,{\mathcal{Z}}_2=-2.08.\hfill \end{array}$$

It is easy to see that

$$\begin{array}{c}\hfill {\mathcal{X}}_j<0,4{\mathcal{X}}_j{\mathcal{Y}}_j>{\left({\mathcal{Z}}_j\right)}^2,j=1,2.\end{array}$$

By utilizing Theorem 1, the NTINNs (40) possess a globally exponentially stable $2\pi$-periodic solution $x\left(t\right)$, and all solutions of (40) and their derivatives are exponentially convergent to $x\left(t\right)$ and $x^{\prime }\left(t\right)$, respectively. The simulation results of Figure 1 and Figure 2 show that the theoretical analysis is consistent with the numerical observation results.

Remark 3.

Since the global exponential stability of the T-periodic solutions on NTINNs involving multiple delays has never been studied, one can see that all the conclusions in references [42,43,44,45,46,47,48,49,50,51,52,53,54,55,56,57,58,59,60,61,62,63,64,65,66,67,68,69] cannot be directly employed to verify the global exponential stability of the $2\pi$-periodic solutions for NTINNs (40).

5. Conclusions

In this article, we researched the problem of the periodic solutions on NTINNs involving multiple delays. First, by exploring Lyapunov theory and inequality analysis, we establish the exponential attractivity of all solutions. Second, we obtained the existence of periodic solutions and their exponential stability. The effectiveness of the obtained results has been illustrated by an instructive numerical simulation. In addition, the method applied in this article offers a possible way to investigate the dynamic characteristics of other NTINNs, such as NTINNs involving D operators, fuzzy NTINNs, Cohen–Grossberg NTINNs and others.