Skip to main content

Anti-periodicity on high-order inertial Hopfield neural networks involving mixed delays

Abstract

This paper deals with a class of high-order inertial Hopfield neural networks involving mixed delays. Utilizing differential inequality techniques and the Lyapunov function method, we obtain a sufficient assertion to ensure the existence and global exponential stability of anti-periodic solutions of the proposed networks. Moreover, an example with a numerical simulation is furnished to illustrate the effectiveness and feasibility of the theoretical results.

1 Introduction

The inertial neural networks model, which was first proposed by Babcock and Westerwelt [1, 2], is one of the other popular artificial neural network models used in a variety of application areas. This type of neural networks has received much attention by many researchers. In particular, numerous works have been devoted to study the dynamic behaviors on inertial neural networks with time-varying delays and some excellent results are reported, for example, stability [3–5], Hopf bifurcation [6–11], and synchronization [12–14]. To the best of our knowledge, the dynamics analysis on inertial neural networks is usually to convert them into a first-order differential system by reducing order variable substitution under the assumption that the activation functions are bounded [15–17]. However, the authors in [12, 18–21] pointed out that the above method not only raises the dimension in the inertial neural networks system, but also increases huge amount of computation which makes it difficult to realize in practice. For the above reasons, the authors of [19, 20] and [21], respectively, developed some non-reduced order techniques to investigate the stability and synchronization of inertial neural networks with different types of time delays.

On the other hand, in neural networks dynamics involving fields such as communication, economics, biology or ecology, the relevant state variables are usually considered as proteins and molecules, light intensity levels or electric charges, which are naturally anti-periodic [22–24]. Considering this factor, many recurrent neural network models involving time-varying delays and anti-periodic environments have been widely investigated in [16, 17, 24–26]. It is worth noting that the high-order Hopfield neural networks have the advantages of faster convergence speed, larger storage capacity and stronger fault tolerance than lower-order neural networks [27–29]. Consequently, Yao [30] studied the existence and global exponential stability of anti-periodic solutions for a class of proportional delayed high-order inertial Hopfield neural networks with time-varying delays.

In recent years, the authors in [21] have mentioned that many parallel routes with a series of different axon sizes and lengths appear in neural networks, and it is desired to explain the dynamics behaviors of neural networks by involving continuously distributed delays. Furthermore, the dynamic behaviors of many recurrent neural networks with continuously distributed delays have been revealed in [27, 31–35]. However, few articles have considered the anti-periodic problem for the following high-order inertial Hopfield neural networks (HIHNNs) involving time-varying delays and continuously distributed delays:

$$ \begin{aligned}[b] x_{i}''(t)=&-a_{i}(t)x_{i}'(t)-b_{i}(t)x_{i}(t)+ \sum_{j=1}^{n} c_{ij}(t)A_{j} \bigl(x_{j}(t)\bigr) \\ &\quad {}+\sum_{j=1}^{n} d_{ij}(t)B_{j}\bigl(x_{j}\bigl(t- q_{ij}( t)\bigr) \bigr) \\ &\quad {}+ \sum_{j=1}^{n}\sum _{l=1}^{n}\theta _{ijl}(t)Q_{j} \bigl(x_{j}\bigl( t-\eta _{ijl} (t) \bigr) \bigr)Q_{l}\bigl(x_{l}\bigl( t-\xi _{ijl}( t) \bigr)\bigr) \\ &\quad {}+ \sum_{j=1}^{n}h_{ij} (t) \int _{0}^{+\infty }\sigma _{ij}(u) K_{j}\bigl(x_{j}(t-u)\bigr)\,du \\ &\quad {}+ \sum_{j=1}^{n}\sum _{l=1}^{n}p_{ijl}(t) \int _{0}^{+ \infty }\hat{\sigma }_{ijl}(u) R_{j}\bigl(x_{j}(t-u)\bigr)\,du \\ &\quad {} \times \int _{0}^{+\infty } \bar{\sigma } _{ijl}(u) R_{l}\bigl(x_{l}(t-u)\bigr)\,du+J_{i}(t), \end{aligned} $$
(1.1)

and the initial value conditions:

$$ x_{i}(s)=\varphi _{i}(s),\qquad x_{i}'(0)= \psi _{i},\quad -\infty \leq s \leq 0, \varphi _{i} \in BC\bigl((- \infty , 0], \mathbb{R}\bigr), \psi _{i}\in \mathbb{R}, $$
(1.2)

where \(BC((-\infty ,0], \mathbb{R})\) is the set of all continuous and bounded functions from \((-\infty ,0]\) to \(\mathbb{R}\), \(J_{i}, c_{ij}, d_{ij}, \theta _{ijl}, h_{ij}, p_{ijl} :\mathbb{R} \rightarrow \mathbb{R}\), \(a_{i}, b_{i} : \mathbb{R}\rightarrow (0, +\infty )\) and \(q_{ij} , \eta _{ijl} , \xi _{ijl} :\mathbb{R}\rightarrow \mathbb{R}^{+}\) are bounded and continuous functions, \(a_{i}\), \(b_{i}\), \(q_{ij}\), \(\eta _{ijl}\), \(\xi _{ijl}\) are periodic functions with period \(T>0\), the input term \(J_{i}\) is T-anti-periodic (i.e. \(J_{i}(t+T)=-J_{i}(t)\) for all \(t\in \mathbb{R}\)), and \(i,j,l\in D= \{1,2,\ldots ,n\}\).

Motivated by the previous discussions, in this paper, without adopting the reduced order method, we shall install new results concerning the anti-periodic dynamics for HIHNNs with time-varying delays and continuously distributed delays. Some sufficient conditions ensuring the existence and global exponential stability on the anti-periodic solution of system (1.1) are established by using differential inequalities and the Lyapunov function method, which improve and complement some earlier publications [16, 17, 36–40].

We organize the paper as follows. In Sect. 2, some assumptions and an important lemmas are listed. Section 3 presents the main results and their detailed proof. Section 4 gives a numerical example to demonstrate the feasibility of the main results. Conclusions are drawn in Sect. 5.

2 Preliminary results

In this section, some assumptions and a key lemma are provided.

Assumptions

\((G_{1})\) :

There are nonnegative constants \(L^{A}_{j}\), \(L^{B}_{j}\), \(L^{Q}_{j}\), \(L^{K}_{j}\), \(L^{R}_{j}\), \(M^{Q}_{j}\) and \(M^{R}_{j}\) such that

$$\begin{aligned}& \textstyle\begin{cases} \vert A_{j}(u )-A_{j}(v ) \vert \leq L^{A} _{j} \vert u -v \vert , \qquad \vert B_{j}(u )-B_{j}(v ) \vert \leq L^{B}_{j} \vert u -v \vert , \\ \vert Q_{j}(u )-Q_{j}(v ) \vert \leq L^{Q}_{j} \vert u -v \vert , \qquad \vert K_{j}(u )-K_{j}(v ) \vert \leq L^{K} _{j} \vert u -v \vert , \\ \vert R_{j}(u )-R_{j}(v ) \vert \leq L^{R} _{j} \vert u -v \vert , \qquad \vert Q_{j}(u) \vert \leq M_{j}^{Q},\qquad \vert R_{j}(u) \vert \leq M_{j}^{R} , \end{cases}\displaystyle \\& \textstyle\begin{cases} c_{ij}(t+ T )A_{j}(u)=-c_{ij}(t)A_{j}(-u), \qquad d_{ij}(t+T)B_{j}(u)=-d_{ij}(t)B_{j}(-u), \\ \theta _{ijl}(t+T)Q_{j}(u)Q_{l}(v)=-\theta _{ijl}(t)Q_{j}(-u)Q_{l}(-v), \\ h_{ij}(t+T)\int _{0}^{+\infty }\sigma _{ij}(s)K_{j}(u)\,ds=- h_{ij}(t) \int _{0}^{+\infty }\sigma _{ij}(s)K_{j}(-u)\,ds, \end{cases}\displaystyle \end{aligned}$$

and

$$\begin{aligned}& p_{ijl}(t+T) \int _{0}^{+\infty }\hat{\sigma }_{ijl}(s)R_{j}(u) \,ds \int _{0}^{+ \infty }\bar{\sigma }_{ijl}(s)R_{l}(u) \,ds \\& \quad = -p_{ijl}(t) \int _{0}^{+\infty }\hat{\sigma }_{ijl}(s)R_{j}(-u) \,ds \int _{0}^{+\infty }\bar{\sigma }_{ijl}(s)R_{l}(-u) \,ds, \end{aligned}$$

for all \(u , v \in \mathbb{R} \), \(i, j, l \in D \).

\((G_{2})\) :

For \(i, j, l \in D \), \(|\sigma _{ij}(t)|e^{\mu t}\), \(|\hat{\sigma }_{ijl}(t)|e^{\mu t}\), \(|\bar{\sigma }_{ijl}(t)|e^{\mu t}\) are integrable on \([0, +\infty )\) for a positive constant μ.

\((G_{3})\) :

There are constants \(\beta _{i}>0\) and \(\alpha _{i}\geq 0\), \(\gamma _{i}\geq 0\) obeying

$$ C_{i} (t)< 0,\qquad 4C_{i}(t)D_{i }(t)> E_{i}^{2}(t),\quad \forall t \in \mathbb{R}, i\in D, $$
(2.1)

where

$$\begin{aligned}& \begin{aligned} C_{i}(t) ={}&\alpha _{i}\gamma _{i}-a_{i}(t) \alpha _{i}^{2} + \frac{1}{2}\alpha _{i}^{2}\sum_{j=1}^{n} \bigl( \bigl\vert c_{ij}(t) \bigr\vert L^{A}_{j} + \bigl\vert d_{ij}(t) \bigr\vert L^{B}_{j} \bigr) \\ &{} +\frac{1}{2} \alpha _{i}^{2}\sum _{j=1}^{n}\sum_{l=1}^{n} \bigl\vert \theta _{ijl}(t) \bigr\vert \bigl(M^{Q}_{j}L^{Q}_{l}+L^{Q}_{j}M^{Q}_{l} \bigr) \\ &{} +\frac{1}{2}\alpha _{i}^{2}\sum _{j=1}^{n} \bigl\vert h_{ij}(t) \bigr\vert L_{j}^{K} \int _{0}^{+\infty } \bigl\vert \sigma _{ij}(u) \bigr\vert \,du \\ &{} +\frac{1}{2}\alpha _{i}^{2}\sum _{j=1}^{n}\sum_{l=1}^{n} \bigl\vert p_{ijl}(t) \bigr\vert M_{j}^{R}L_{l}^{R} \int _{0}^{+\infty } \bigl\vert \hat{\sigma }_{ijl}(u) \bigr\vert \,du \int _{0}^{+\infty } \bigl\vert \bar{\sigma }_{ijl}(u) \bigr\vert \,du \\ &{} +\frac{1}{2}\alpha _{i}^{2}\sum _{j=1}^{n}\sum_{l=1}^{n} \bigl\vert p_{ijl}(t) \bigr\vert M_{l}^{R}L_{j}^{R} \int _{0}^{+\infty } \bigl\vert \bar{\sigma }_{ijl}(u) \bigr\vert \,du \int _{0}^{+\infty } \bigl\vert \hat{\sigma }_{ijl}(u) \bigr\vert \,du, \end{aligned} \\& \begin{aligned} D_{i}(t) ={}& {-}b_{i}(t)\alpha _{i}\gamma _{i} +\frac{1}{2}\sum _{j=1}^{n}\bigl( \bigl\vert c_{ij}(t) \bigr\vert L^{A}_{j}+ \bigl\vert d_{ij}(t) \bigr\vert L^{B}_{j} \bigr) \vert \alpha _{i}\gamma _{i} \vert \\ &{}+\frac{1}{2}\sum_{j=1}^{n} \alpha _{j}^{2}\biggl( \bigl\vert c_{ji}(t) \bigr\vert L^{A}_{i} + d_{ji}^{+} L^{B}_{i}\frac{1}{1-\dot{q} _{ji} ^{+}}e^{2\lambda q_{ji}^{+}} \biggr) \\ &{}+\frac{1}{2}\sum_{j=1}^{n} \biggl( \bigl\vert c_{ji}(t) \bigr\vert L^{A}_{i} + d_{ji}^{+} L^{B}_{i} \frac{1}{1-\dot{q} _{ji} ^{+}}e^{2\lambda q_{ji}^{+}}\biggr) \vert \alpha _{j} \gamma _{j} \vert \\ &{}+\frac{1}{2} \sum_{j=1}^{n} \sum_{l=1}^{n} \vert \alpha _{i}\gamma _{i} \vert \bigl\vert \theta _{ijl}(t) \bigr\vert \bigl(M^{Q}_{j}L^{Q}_{l}+L^{Q}_{j}M^{Q}_{l} \bigr) \\ &{}+\frac{1}{2} \sum_{l=1}^{n} \sum_{j=1}^{n}\bigl( \alpha _{l}^{2}+ \vert \alpha _{l}\gamma _{l} \vert \bigr) \theta _{lji} ^{+} M^{Q}_{j}L^{Q}_{i} e^{2\lambda \xi _{lji} ^{+}} \frac{1}{1-\dot{\xi } _{lji} ^{+}} \\ &{}+\frac{1}{2} \sum_{j=1}^{n} \sum_{l=1}^{n}\bigl(\alpha _{j}^{2}+ \vert \alpha _{j}\gamma _{j} \vert \bigr) \theta _{jil} ^{+}L^{Q}_{i}M^{Q}_{l} e^{2 \lambda \eta _{jil} ^{+}} \frac{1}{1-\dot{\eta } _{jil} ^{+}} \\ &{}+\frac{1}{2}\sum_{j=1}^{n} \bigl(\alpha _{j}^{2} h_{ji}^{+}+ \vert \alpha _{j}\gamma _{j} \vert h_{ji}^{+}\bigr)L_{i}^{K} \int _{0}^{+\infty } \bigl\vert \sigma _{ji}(u) \bigr\vert e^{2\lambda u}\,du \\ &{}+\frac{1}{2}\sum_{l=1}^{n} \sum_{j=1}^{n}\bigl(\alpha _{l}^{2} + \vert \alpha _{l}\gamma _{l} \vert \bigr)p_{lji}^{+} \int _{0}^{+\infty } \bigl\vert \hat{\sigma }_{lji}(u) \bigr\vert \,du \int _{0}^{+\infty } \bigl\vert \bar{\sigma }_{lji}(u) \bigr\vert e^{2 \lambda u}\,duM_{j}^{R}L_{i}^{R} \\ &{}+\frac{1}{2}\sum_{j=1}^{n} \sum_{l=1}^{n}\bigl(\alpha _{j}^{2} + \vert \alpha _{j}\gamma _{j} \vert \bigr)p_{jil}^{+} \int _{0}^{+\infty } \bigl\vert \bar{\sigma }_{jil}(u) \bigr\vert \,du \int _{0}^{+\infty } \bigl\vert \hat{\sigma }_{jil}(u) \bigr\vert e^{2 \lambda u}\,duM_{l}^{R}L_{i}^{R} \\ &{}+\frac{1}{2}\sum_{j=1}^{n} \vert \alpha _{i}\gamma _{i} \vert \bigl\vert h_{ij}(t) \bigr\vert L_{j}^{K} \int _{0}^{+\infty } \bigl\vert \sigma _{ij}(u) \bigr\vert \,du \\ &{}+\frac{1}{2}\sum_{j=1}^{n} \sum_{l=1}^{n} \vert \alpha _{i} \gamma _{i} \vert \bigl\vert p_{ijl}(t) \bigr\vert M_{j}^{R}L_{l}^{R} \int _{0}^{+\infty } \bigl\vert \hat{\sigma }_{ijl}(u) \bigr\vert \,du \int _{0}^{+\infty } \bigl\vert \bar{\sigma }_{ijl}(u) \bigr\vert \,du \\ &{}+\frac{1}{2}\sum_{j=1}^{n} \sum_{l=1}^{n} \vert \alpha _{i} \gamma _{i} \vert \bigl\vert p_{ijl}(t) \bigr\vert M_{l}^{R}L_{j}^{R} \int _{0}^{+\infty } \bigl\vert \bar{\sigma }_{ijl}(u) \bigr\vert \,du \int _{0}^{+\infty } \bigl\vert \hat{\sigma }_{ijl}(u) \bigr\vert \,du , \end{aligned} \\& E_{i}(t) = \beta _{i}+\gamma _{i}^{2} -a_{i}(t)\alpha _{i}\gamma _{i}-b_{i}(t) \alpha _{i}^{2}, \qquad \dot{q}_{ij}^{+}= \max_{t\in [0, T]}q'_{ij}(t), \\& \dot{\eta }_{ijl}^{+}=\max_{t\in [0, T]}\eta '_{ijl}(t),\qquad \dot{\xi }_{ijl}^{+}= \max_{t\in [0, T]}\xi '_{ijl}(t), \\& q_{ij}^{+}=\max_{t\in [0, T]}q_{ij}(t),\qquad \eta _{ijl}^{+}= \max_{t\in [0, T]}\eta _{ijl}(t), \end{aligned}$$

and

$$\begin{aligned}& \xi _{ijl}^{+}=\max_{t\in [0, T]}\xi _{ijl}(t), \qquad c_{ij}^{+}=\sup _{t\in \mathbb{R}} \bigl\vert c_{ij}(t) \bigr\vert ,\qquad d_{ij}^{+}=\sup_{t\in \mathbb{R}} \bigl\vert d_{ij}(t) \bigr\vert , \\& \theta _{ijl}^{+}= \sup_{t\in \mathbb{R}} \bigl\vert \theta _{ijl}(t) \bigr\vert ,\quad i,j, l\in D . \end{aligned}$$
\((G_{4})\) :

For \(i, j, l\in D\), \(q_{ij}\), \(\eta _{ijl}\) and \(\xi _{ijl}\) are continuously differentiable, \(q_{ij}'(t)=\dot{q_{ij}} (t)< 1\), \(\eta _{ijl}'(t)=\dot{\eta _{ijl} }(t)< 1\) and \(\xi _{ijl}'(t)=\dot{\xi _{ijl}} (t)< 1\), for all \(t\in \mathbb{R}\).

Remark 2.1

According to \((G_{1})\) and the basic theory on functional differential equation with infinite delay in [41], one can show that all solutions of (1.1) and (1.2) exist in \([0, +\infty )\).

Lemma 2.1

Assume that\((G_{1})\), \((G_{2})\), \((G_{3})\)and\((G_{4})\)hold. Let\(x (t)=(x_{1}(t), x_{2}(t), \ldots , x_{n}(t))\)and\(y (t)=(y_{1}(t), y_{2}(t),\ldots , y_{n}(t))\)be two solutions of system (1.1) satisfying

$$ x_{i}(s)=\varphi ^{x}_{i}(s),\qquad x_{i}'(0) = \psi ^{x}_{i} ,\qquad y_{i}(s)= \varphi ^{y}_{i}(s),\qquad y_{i}'(0) = \psi ^{y}_{i}, $$
(2.2)

where\(i\in D\), \(\varphi ^{x}_{i}, \varphi ^{y}_{i} \in BC((-\infty , 0], \mathbb{R})\), \(\psi ^{x}_{i}, \psi ^{y}_{i}\in \mathbb{R} \). Then there are two positive constantsλand\(M=M(\varphi ^{x}, \psi ^{x},\varphi ^{y}, \psi ^{y})\)such that

$$ \bigl\vert x_{i}(t)- y_{i}(t) \bigr\vert \leq M e^{ -\lambda t},\qquad \bigl\vert x_{i}'(t)- y_{i}'(t) \bigr\vert \leq M e^{- \lambda t},\quad \textit{for all } t\geq 0, i\in D. $$
(2.3)

Proof

Denote \(z_{i}(t) =y_{i}(t)-x_{i}(t) \), then

$$\begin{aligned} &z_{i}''(t) \\ &\quad =-a_{i}(t)z_{i}'(t)-b_{i}(t)z_{i}(t) \\ &\qquad {}+\sum_{j=1}^{n} c_{ij}(t) \widetilde{A}_{j}\bigl(z_{j}(t)\bigr) +\sum _{j=1}^{n} d_{ij}(t) \widetilde{B}_{j}\bigl(z_{j}\bigl(t- q_{ij}( t)\bigr) \bigr) + \sum_{j=1}^{n}\sum _{l=1}^{n}\theta _{ijl}(t) \\ &\qquad {} \times \bigl[Q_{j}\bigl(y_{j}\bigl( t-\eta _{ijl} (t) \bigr)\bigr)Q_{l}\bigl(y_{l} \bigl( t-\xi _{ijl}( t) \bigr)\bigr)-Q_{j} \bigl(y_{j}\bigl( t-\eta _{ijl} (t) \bigr) \bigr)Q_{l}\bigl(x_{l}\bigl( t-\xi _{ijl}( t) \bigr)\bigr) \\ &\qquad {} + Q_{j}\bigl(y_{j}\bigl(t-\eta _{ijl} (t) \bigr)\bigr)Q_{l}\bigl(x_{l}\bigl( t-\xi _{ijl}( t) \bigr)\bigr)- Q_{j}\bigl(x_{j} \bigl(t- \eta _{ijl} (t) \bigr)\bigr)Q_{l} \bigl(x_{l}\bigl( t-\xi _{ijl}( t) \bigr)\bigr)\bigr] \\ &\qquad {} + \sum_{j=1}^{n}h_{ij} (t) \int _{0}^{+\infty }\sigma _{ij}(u) \widetilde{K}_{j}\bigl(z_{j}(t-u)\bigr)\,du + \sum _{j=1}^{n}\sum _{l=1}^{n}p_{ijl}(t) \\ &\qquad {} \times \biggl[ \int _{0}^{+\infty }\hat{\sigma }_{ijl}(u) R_{j}\bigl(y_{j}(t-u)\bigr)\,du \int _{0}^{+\infty } \bar{\sigma } _{ijl}(u) R_{l}\bigl(y_{l}(t-u)\bigr)\,du \\ &\qquad {} - \int _{0}^{+\infty }\hat{\sigma }_{ijl}(u) R_{j}\bigl(y_{j}(t-u)\bigr)\,du \int _{0}^{+ \infty } \bar{\sigma } _{ijl}(u) R_{l}\bigl(x_{l}(t-u)\bigr)\,du \\ &\qquad {} + \int _{0}^{+\infty }\hat{\sigma }_{ijl}(u) R_{j}\bigl(y_{j}(t-u)\bigr)\,du \int _{0}^{+ \infty } \bar{\sigma } _{ijl}(u) R_{l}\bigl(x_{l}(t-u)\bigr)\,du \\ &\qquad {} - \int _{0}^{+\infty }\hat{\sigma }_{ijl}(u) R_{j}\bigl(x_{j}(t-u)\bigr)\,du \int _{0}^{+ \infty } \bar{\sigma } _{ijl}(u) R_{l}\bigl(x_{l}(t-u)\bigr)\,du\biggr], \end{aligned}$$

where

$$ \textstyle\begin{cases} \widetilde{A}_{j}(z_{j}(t)) = A_{j}(y_{j}(t))- A_{j}(x_{j}(t)), \\ \widetilde{B}_{j}(z_{j}(t-q_{ij}(t)))= B_{j}(y_{j}(t-q_{ij}(t))) - B_{j}(x_{j}(t-q_{ij}(t))), \\ \widetilde{K}_{j}(z_{j}(t-u))=K_{j}(y_{j}(t-u))-K_{j}(x_{j}(t-u)), \end{cases} $$

and \(i, j\in D\).

According to \((G_{2})\) and the boundedness of (1.1), one can select a constant \(\lambda >0\) such that

$$ C_{i}^{\lambda }(t) < 0,\qquad 4C_{i}^{\lambda }(t)D_{i }^{\lambda }(t)> \bigl(E_{i}^{\lambda }(t)\bigr)^{2}, \quad \forall t\in \mathbb{R}, $$
(2.4)

where

$$\begin{aligned}& \begin{aligned} C_{i} ^{\lambda }(t) ={}& \lambda \alpha _{i}^{2}+ \alpha _{i}\gamma _{i}-a_{i}(t) \alpha _{i}^{2} +\frac{1}{2}\alpha _{i}^{2}\sum_{j=1}^{n} \bigl( \bigl\vert c_{ij}(t) \bigr\vert L^{A}_{j} + \bigl\vert d_{ij}(t) \bigr\vert L^{B}_{j} \bigr) \\ &{}+\frac{1}{2} \alpha _{i}^{2}\sum _{j=1}^{n}\sum_{l=1}^{n} \bigl\vert \theta _{ijl}(t) \bigr\vert \bigl(M^{Q}_{j}L^{Q}_{l}+L^{Q}_{j}M^{Q}_{l} \bigr) \\ &{}+\frac{1}{2}\alpha _{i}^{2}\sum _{j=1}^{n} \bigl\vert h_{ij}(t) \bigr\vert L_{j}^{K} \int _{0}^{+\infty } \bigl\vert \sigma _{ij}(u) \bigr\vert \,du \\ &{}+\frac{1}{2}\alpha _{i}^{2}\sum _{j=1}^{n}\sum_{l=1}^{n} \bigl\vert p_{ijl}(t) \bigr\vert M_{j}^{R}L_{l}^{R} \int _{0}^{+\infty } \bigl\vert \hat{\sigma }_{ijl}(u) \bigr\vert \,du \int _{0}^{+\infty } \bigl\vert \bar{\sigma }_{ijl}(u) \bigr\vert \,du \\ &{}+\frac{1}{2}\alpha _{i}^{2}\sum _{j=1}^{n}\sum_{l=1}^{n} \bigl\vert p_{ijl}(t) \bigr\vert M_{l}^{R}L_{j}^{R} \int _{0}^{+\infty } \bigl\vert \bar{\sigma }_{ijl}(u) \bigr\vert \,du \int _{0}^{+\infty } \bigl\vert \hat{\sigma }_{ijl}(u) \bigr\vert \,du, \end{aligned} \\ & D_{i} ^{\lambda }(t) \\ & \quad =-b_{i}(t)\alpha _{i}\gamma _{i}+ \lambda \beta _{i}+\lambda \gamma _{i}^{2} +\frac{1}{2}\sum_{j=1}^{n} \bigl( \bigl\vert c_{ij}(t) \bigr\vert L^{A}_{j}+ \bigl\vert d_{ij}(t) \bigr\vert L^{B}_{j} \bigr) \vert \alpha _{i}\gamma _{i} \vert \\ & \qquad {}+\frac{1}{2}\sum_{j=1}^{n} \alpha _{j}^{2}\biggl( \bigl\vert c_{ji}(t) \bigr\vert L^{A}_{i} + d_{ji}^{+} L^{B}_{i}\frac{1}{1-\dot{q} _{ji} ^{+}}e^{2\lambda q_{ji}^{+}} \biggr) \\ & \qquad {}+\frac{1}{2}\sum_{j=1}^{n} \biggl( \bigl\vert c_{ji}(t) \bigr\vert L^{A}_{i} + d_{ji}^{+} L^{B}_{i} \frac{1}{1-\dot{q} _{ji} ^{+}}e^{2\lambda q_{ji}^{+}}\biggr) \vert \alpha _{j} \gamma _{j} \vert \\ & \qquad {}+\frac{1}{2} \sum_{j=1}^{n} \sum_{l=1}^{n} \vert \alpha _{i} \gamma _{i} \vert \bigl\vert \theta _{ijl}(t) \bigr\vert \bigl(M^{Q}_{j}L^{Q}_{l}+L^{Q}_{j}M^{Q}_{l} \bigr) \\ & \qquad {}+\frac{1}{2} \sum_{l=1}^{n} \sum_{j=1}^{n}\bigl(\alpha _{l}^{2}+ \vert \alpha _{l}\gamma _{l} \vert \bigr) \theta _{lji} ^{+} M^{Q}_{j}L^{Q}_{i} e^{2 \lambda \xi _{lji} ^{+}} \frac{1}{1-\dot{\xi } _{lji} ^{+}} \\ & \qquad {}+\frac{1}{2} \sum_{j=1}^{n} \sum_{l=1}^{n}\bigl(\alpha _{j}^{2}+ \vert \alpha _{j}\gamma _{j} \vert \bigr) \theta _{jil} ^{+}L^{Q}_{i}M^{Q}_{l} e^{2 \lambda \eta _{jil} ^{+}} \frac{1}{1-\dot{\eta } _{jil} ^{+}} \\ & \qquad {} +\frac{1}{2}\sum_{j=1}^{n} \bigl(\alpha _{j}^{2} h_{ji}^{+}+ \vert \alpha _{j}\gamma _{j} \vert h_{ji}^{+}\bigr)L_{i}^{K} \int _{0}^{+\infty } \bigl\vert \sigma _{ji}(u) \bigr\vert e^{2\lambda u}\,du \\ & \qquad {} +\frac{1}{2}\sum_{l=1}^{n} \sum_{j=1}^{n}\bigl(\alpha _{l}^{2} + \vert \alpha _{l}\gamma _{l} \vert \bigr)p_{lji}^{+} \int _{0}^{+\infty } \bigl\vert \hat{\sigma }_{lji}(u) \bigr\vert \,du \int _{0}^{+\infty } \bigl\vert \bar{\sigma }_{lji}(u) \bigr\vert e^{2 \lambda u}\,duM_{j}^{R}L_{i}^{R} \\ & \qquad {} +\frac{1}{2}\sum_{j=1}^{n} \sum_{l=1}^{n}\bigl(\alpha _{j}^{2} + \vert \alpha _{j}\gamma _{j} \vert \bigr)p_{jil}^{+} \int _{0}^{+\infty } \bigl\vert \bar{\sigma }_{jil}(u) \bigr\vert \,du \int _{0}^{+\infty } \bigl\vert \hat{\sigma }_{jil}(u) \bigr\vert e^{2 \lambda u}\,duM_{l}^{R}L_{i}^{R} \\ & \qquad {}+\frac{1}{2}\sum_{j=1}^{n} \vert \alpha _{i}\gamma _{i} \vert \bigl\vert h_{ij}(t) \bigr\vert L_{j}^{K} \int _{0}^{+\infty } \bigl\vert \sigma _{ij}(u) \bigr\vert \,du \\ & \qquad {}+\frac{1}{2}\sum_{j=1}^{n} \sum_{l=1}^{n} \vert \alpha _{i} \gamma _{i} \vert \bigl\vert p_{ijl}(t) \bigr\vert M_{j}^{R}L_{l}^{R} \int _{0}^{+\infty } \bigl\vert \hat{\sigma }_{ijl}(u) \bigr\vert \,du \int _{0}^{+\infty } \bigl\vert \bar{\sigma }_{ijl}(u) \bigr\vert \,du \\ & \qquad {}+\frac{1}{2}\sum_{j=1}^{n} \sum_{l=1}^{n} \vert \alpha _{i} \gamma _{i} \vert \bigl\vert p_{ijl}(t) \bigr\vert M_{l}^{R}L_{j}^{R} \int _{0}^{+\infty } \bigl\vert \bar{\sigma }_{ijl}(u) \bigr\vert \,du \int _{0}^{+\infty } \bigl\vert \hat{\sigma }_{ijl}(u) \bigr\vert \,du, \end{aligned}$$

and

$$ E_{i}^{\lambda }(t) = \beta _{i}+\gamma _{i}^{2}+2\lambda \alpha _{i} \gamma _{i} -a_{i}(t)\alpha _{i}\gamma _{i}-b_{i}(t)\alpha _{i}^{2},\quad i \in D . $$

Set

$$\begin{aligned} &W(t) \\ &\quad = \frac{1}{2}\sum_{i=1}^{n} \beta _{i}z_{i}^{2}(t)e^{2 \lambda t} + \frac{1}{2}\sum_{i=1}^{n}\bigl( \alpha _{i} z _{i}'(t)+ \gamma _{i}z_{i}(t)\bigr)^{2}e^{2\lambda t} \\ &\qquad {} +\frac{1}{2}\sum_{i=1}^{n} \sum_{j=1}^{n}\bigl(\alpha _{i}^{2} d_{ij} ^{+}+ \vert \alpha _{i}\gamma _{i} \vert d_{ij} ^{+}\bigr)e^{ 2\lambda q_{ij}^{+} }L^{B}_{j} \int _{t-q _{ij}(t)}^{t}z_{j}^{2}(s) \frac{1}{1-\dot{q} _{ij} ^{+}}e^{2\lambda s}\,ds \\ &\qquad {} +\frac{1}{2}\sum_{i=1}^{n} \sum_{l=1}^{n}\sum _{j=1}^{n}\bigl(\alpha _{l}^{2}+ \vert \alpha _{l}\gamma _{l} \vert \bigr) \theta _{lji} ^{+} M^{Q}_{j}L^{Q}_{i} e^{2\lambda \xi _{lji} ^{+}} \int _{t-\xi _{lji}( t) }^{t}z_{i}^{2}(s) \frac{1}{1-\dot{\xi } _{lji} ^{+}}e^{2\lambda s}\,ds \\ &\qquad {} +\frac{1}{2}\sum_{i=1}^{n} \sum_{j=1}^{n}\sum _{l=1}^{n}\bigl(\alpha _{j}^{2}+ \vert \alpha _{j}\gamma _{j} \vert \bigr) \theta _{jil} ^{+}L^{Q}_{i}M^{Q}_{l} e^{2\lambda \eta _{jil} ^{+}} \int _{t-\eta _{jil}( t) }^{t}z_{i}^{2}(s) \frac{1}{1-\dot{\eta } _{jil} ^{+}}e^{2\lambda s}\,ds \\ &\qquad {}+\frac{1}{2}\sum_{i=1}^{n} \sum_{j=1}^{n}\bigl(\alpha _{i}^{2} h_{ij}^{+}+ \vert \alpha _{i}\gamma _{i} \vert h_{ij}^{+} \bigr)L_{j}^{K} \int _{0}^{+ \infty } \bigl\vert \sigma _{ij}(u) \bigr\vert \int _{t-u}^{t}z_{j}^{2}(s)e^{2\lambda (u+s)} \,ds\,du \\ &\qquad {} +\frac{1}{2}\sum_{i=1}^{n} \sum_{l=1}^{n}\sum _{j=1}^{n}\bigl(\alpha _{l}^{2}+ \vert \alpha _{l}\gamma _{l} \vert \bigr)p_{lji}^{+} M_{j}^{R}L_{i}^{R} \int _{0}^{+\infty } \bigl\vert \hat{\sigma }_{lji}(u) \bigr\vert \,du \\ &\qquad {} \times \int _{0}^{+\infty } \bigl\vert \bar{\sigma }_{lji}(u) \bigr\vert \int _{t-u}^{t}z_{i}^{2}(s)e^{2 \lambda (u+s)} \,ds\,du \\ &\qquad {} +\frac{1}{2}\sum_{i=1}^{n} \sum_{j=1}^{n}\sum _{l=1}^{n}\bigl(\alpha _{j}^{2}+ \vert \alpha _{j}\gamma _{j} \vert \bigr)p_{jil}^{+} M_{l}^{R}L_{i}^{R} \int _{0}^{+\infty } \bigl\vert \bar{\sigma }_{jil}(u) \bigr\vert \,du \\ &\qquad {} \times \int _{0}^{+\infty } \bigl\vert \hat{\sigma }_{jil}(u) \bigr\vert \int _{t-u}^{t}z_{i}^{2}(s)e^{2 \lambda (u+s)} \,ds\,du. \end{aligned}$$

A straightforward computation yields

$$\begin{aligned}& W'(t) \\ & \quad = 2\lambda \Biggl[\frac{1}{2}\sum_{i=1}^{n} \beta _{i}z_{i}^{2}(t)e^{2 \lambda t}+ \frac{1}{2}\sum_{i=1}^{n}\bigl( \alpha _{i}z '_{i}(t)+ \gamma _{i}z_{i}(t)\bigr)^{2}e^{2\lambda t} \Biggr] \\ & \qquad {} +\sum_{i=1}^{n}\bigl(\beta _{i}+\gamma _{i}^{2}\bigr)z_{i}(t)z '_{i}(t)e^{2 \lambda t}+\sum _{i=1}^{n}\alpha _{i}\bigl(\alpha _{i}z '_{i}(t)+ \gamma _{i}z_{i}(t)\bigr)e^{2\lambda t} \\ & \qquad {} \times \Biggl[-a_{i}(t)z '_{i}(t)-b_{i}(t)z_{i}(t)+ \sum_{j=1}^{n}c_{ij}(t) \widetilde{A}_{j}\bigl(z_{j}(t)\bigr) +\sum _{j=1}^{n}d_{ij}(t) \widetilde{B}_{j}\bigl(z_{j}\bigl(t-q_{ij}(t) \bigr)\bigr) \\ & \qquad {} +\sum_{j=1}^{n}\sum _{l=1}^{n}\theta _{ijl}(t) \bigl(Q_{j}\bigl(y_{j}\bigl(t- \eta _{ijl} (t) \bigr)\bigr)Q_{l}\bigl(y_{l}\bigl( t-\xi _{ijl} (t) \bigr)\bigr) \\ & \qquad {} -Q_{j}\bigl(y_{j}\bigl(t-\eta _{ijl} (t) \bigr)\bigr)Q_{l}\bigl(x_{l}\bigl( t-\xi _{ijl} (t) \bigr)\bigr)+Q_{j}\bigl(y_{j} \bigl( t-\eta _{ijl}( t) \bigr)\bigr)Q_{l} \bigl(x_{l}\bigl( t-\xi _{ijl} (t) \bigr)\bigr) \\ & \qquad {} -Q_{j}\bigl(x_{j}\bigl(t-\eta _{ijl}( t )\bigr)\bigr)Q_{l}\bigl(x_{l}\bigl(t- \xi _{ijl} (t )\bigr)\bigr)\bigr) \\ & \qquad {} + \sum_{j=1}^{n}h_{ij} (t) \int _{0}^{+\infty }\sigma _{ij}(u) \widetilde{K}_{j}\bigl(z_{j}(t-u)\bigr)\,du + \sum _{j=1}^{n}\sum _{l=1}^{n}p_{ijl}(t) \\ & \qquad {} \times \biggl( \int _{0}^{+\infty }\hat{\sigma }_{ijl}(u) R_{j}\bigl(y_{j}(t-u)\bigr)\,du \int _{0}^{+\infty } \bar{\sigma } _{ijl}(u) R_{l}\bigl(y_{l}(t-u)\bigr)\,du \\ & \qquad {} - \int _{0}^{+\infty }\hat{\sigma }_{ijl}(u) R_{j}\bigl(y_{j}(t-u)\bigr)\,du \int _{0}^{+ \infty } \bar{\sigma } _{ijl}(u) R_{l}\bigl(x_{l}(t-u)\bigr)\,du \\ & \qquad {} + \int _{0}^{+\infty }\hat{\sigma }_{ijl}(u) R_{j}\bigl(y_{j}(t-u)\bigr)\,du \int _{0}^{+ \infty } \bar{\sigma } _{ijl}(u) R_{l}\bigl(x_{l}(t-u)\bigr)\,du \\ & \qquad {} - \int _{0}^{+\infty }\hat{\sigma }_{ijl}(u) R_{j}\bigl(x_{j}(t-u)\bigr)\,du \int _{0}^{+ \infty } \bar{\sigma } _{ijl}(u) R_{l}\bigl(x_{l}(t-u)\bigr)\,du\biggr)\Biggr] \\ & \qquad {} +\sum_{i=1}^{n}\alpha _{i}\gamma _{i}\bigl(z '_{i} (t)\bigr)^{2}e^{2 \lambda t} +\frac{1}{2}\sum _{i=1}^{n}\sum_{j=1}^{n} \bigl( \alpha _{i}^{2}d_{ij} ^{+}+ \vert \alpha _{i}\gamma _{i} \vert d_{ij} ^{+}\bigr)e^{ 2 \lambda q_{ij}^{+} }L^{B}_{j} \\& \qquad {} \times \biggl[z_{j}^{2}(t)\frac{1}{1-\dot{q} _{ij} ^{+}}e^{2\lambda t}-z_{j}^{2} \bigl(t-q_{ij}(t)\bigr)e^{2 \lambda (t-q_{ij}(t))}\frac{ 1- q'_{ij}(t)}{1-\dot{q} _{ij} ^{+}} \biggr] \\& \qquad {} +\frac{1}{2}\sum_{i=1}^{n} \sum_{l=1}^{n}\sum _{j=1}^{n}\bigl(\alpha _{l}^{2}+ \vert \alpha _{l}\gamma _{l} \vert \bigr) \theta _{lji} ^{+} M^{Q}_{j}L^{Q}_{i} e^{2\lambda \xi _{lji} ^{+}} \\& \qquad {} \times \biggl[z_{i}^{2}(t)\frac{1}{1-\dot{\xi } _{lji} ^{+}} e^{2\lambda t}- z_{i}^{2}\bigl(t-\xi _{lji}( t)\bigr)e^{2\lambda (t-\xi _{lji}( t))} \frac{1- \xi _{lji}'( t) }{1-\dot{\xi } _{lji} ^{+}}\biggr] \\& \qquad {} +\frac{1}{2}\sum_{i=1}^{n} \sum_{j=1}^{n}\sum _{l=1}^{n}\bigl(\alpha _{j}^{2}+ \vert \alpha _{j}\gamma _{j} \vert \bigr) \theta _{jil} ^{+}L^{Q}_{i}M^{Q}_{l} e^{2\lambda \eta _{jil} ^{+}} \\& \qquad {} \times \biggl[z_{i}^{2}(t) \frac{1}{1-\dot{\eta } _{jil} ^{+}}e^{2 \lambda t}-z_{i}^{2} \bigl(t-\eta _{jil}( t)\bigr)e^{2\lambda (t-\eta _{jil}( t))} \frac{1-\eta _{jil}'( t)}{1-\dot{\eta } _{jil} ^{+}} \biggr] \\& \qquad {} +\frac{1}{2}\sum_{i=1}^{n} \sum_{j=1}^{n}\bigl(\alpha _{i}^{2} h_{ij}^{+}+ \vert \alpha _{i}\gamma _{i} \vert h_{ij}^{+} \bigr)L_{j}^{K} \\& \qquad {} \times \biggl[ \int _{0}^{+\infty } \bigl\vert \sigma _{ij}(u) \bigr\vert e^{2\lambda u}\,duz_{j}^{2}(t)e^{2 \lambda t}- \int _{0}^{+\infty } \bigl\vert \sigma _{ij}(u) \bigr\vert z_{j}^{2}(t-u) \,due^{2 \lambda t}\biggr] \\& \qquad {} +\frac{1}{2}\sum_{i=1}^{n} \sum_{l=1}^{n}\sum _{j=1}^{n}\bigl(\alpha _{l}^{2} + \vert \alpha _{l}\gamma _{l} \vert \bigr)p_{lji}^{+} M_{j}^{R}L_{i}^{R} \int _{0}^{+\infty } \bigl\vert \hat{\sigma }_{lji}(u) \bigr\vert \,du \\& \qquad {} \times \biggl[ \int _{0}^{+\infty } \bigl\vert \bar{\sigma }_{lji}(u) \bigr\vert e^{2\lambda u}\,duz_{i}^{2}(t)e^{2 \lambda t} - \int _{0}^{+\infty } \bigl\vert \bar{\sigma }_{lji}(u) \bigr\vert z_{i}^{2}(t-u) \,due^{2 \lambda t}\biggr] \\& \qquad {} +\frac{1}{2}\sum_{i=1}^{n} \sum_{j=1}^{n}\sum _{l=1}^{n}\bigl(\alpha _{j}^{2} + \vert \alpha _{j}\gamma _{j} \vert \bigr)p_{jil}^{+} M_{l}^{R}L_{i}^{R} \int _{0}^{+\infty } \bigl\vert \bar{\sigma }_{jil}(u) \bigr\vert \,du \\& \qquad {} \times \biggl[ \int _{0}^{+\infty } \bigl\vert \hat{\sigma }_{jil}(u) \bigr\vert e^{2\lambda u}\,duz_{i}^{2}(t)e^{2 \lambda t} - \int _{0}^{+\infty } \bigl\vert \hat{\sigma }_{jil}(u) \bigr\vert z_{i}^{2}(t-u) \,due^{2 \lambda t}\biggr] \\& \quad \leq e^{2\lambda t}\Biggl\{ \sum_{i=1}^{n} \bigl(\beta _{i}+\gamma _{i}^{2}+2 \lambda \alpha _{i}\gamma _{i} -a_{i}(t)\alpha _{i}\gamma _{i}-b_{i}(t) \alpha _{i}^{2}\bigr)z_{i}(t)z '_{i}(t) \end{aligned}$$
$$\begin{aligned}& \qquad {} +\sum_{i=1}^{n}\bigl(\lambda \alpha _{i}^{2}+\alpha _{i} \gamma _{i}-a_{i}(t)\alpha _{i}^{2} \bigr) \bigl(z '_{i}(t)\bigr)^{2}-\sum _{i=1}^{n}\bigl(b_{i}(t) \alpha _{i}\gamma _{i}-\lambda \beta _{i}- \lambda \gamma _{i}^{2}\bigr)z_{i}^{2}(t) \\& \qquad {} +\frac{1}{2}\sum_{i=1}^{n} \sum_{j=1}^{n}\bigl(\alpha _{i}^{2}d_{ij} ^{+}+ \vert \alpha _{i}\gamma _{i} \vert d_{ij} ^{+}\bigr)L^{B}_{j}e^{2 \lambda q_{ij}^{+} }z_{j}^{2}(t) \frac{1}{1-\dot{q} _{ij}^{+}} \\& \qquad {} +\frac{1}{2}\sum_{i=1}^{n} \sum_{l=1}^{n}\sum _{j=1}^{n}\bigl(\alpha _{l}^{2}+ \vert \alpha _{l}\gamma _{l} \vert \bigr) \theta _{lji} ^{+} M^{Q}_{j}L^{Q}_{i} e^{2\lambda \xi _{lji} ^{+}} z_{i}^{2}(t) \frac{1}{1-\dot{\xi } _{lji} ^{+}} \\& \qquad {} +\frac{1}{2}\sum_{i=1}^{n} \sum_{j=1}^{n}\sum _{l=1}^{n}\bigl(\alpha _{j}^{2}+ \vert \alpha _{j}\gamma _{j} \vert \bigr) \theta _{jil} ^{+}L^{Q}_{i}M^{Q}_{l} e^{2\lambda \eta _{jil} ^{+}} z_{i}^{2}(t) \frac{1}{1-\dot{\eta } _{jil} ^{+}} \\& \qquad {}-\frac{1}{2}\sum_{i=1}^{n} \sum_{j=1}^{n}\bigl(\alpha _{i}^{2}d_{ij} ^{+}+ \vert \alpha _{i}\gamma _{i} \vert d_{ij} ^{+}\bigr) L^{B}_{j} z_{j}^{2} \bigl(t-q_{ij}(t)\bigr) \\& \qquad {} -\frac{1}{2}\sum_{i=1}^{n} \sum_{l=1}^{n}\sum _{j=1}^{n}\bigl(\alpha _{l}^{2}+ \vert \alpha _{l}\gamma _{l} \vert \bigr) \theta _{lji} ^{+} M^{Q}_{j}L^{Q}_{i} z_{i}^{2}\bigl(t-\xi _{lji}( t)\bigr) \\& \qquad {}- \frac{1}{2}\sum_{i=1}^{n} \sum_{j=1}^{n}\sum _{l=1}^{n}\bigl(\alpha _{j}^{2}+ \vert \alpha _{j}\gamma _{j} \vert \bigr) \theta _{jil} ^{+}L^{Q}_{i}M^{Q}_{l} z_{i}^{2}\bigl(t-\eta _{jil}( t)\bigr) \\& \qquad {} +\sum_{i=1}^{n}\sum _{j=1}^{n}\bigl(\alpha _{i}^{2} \bigl\vert z '_{i}(t) \bigr\vert + \vert \alpha _{i}\gamma _{i} \vert \bigl\vert z_{i}(t) \bigr\vert \bigr) \bigl\vert c_{ij}(t) \bigr\vert \bigl\vert \widetilde{A}_{j}\bigl(z_{j}(t) \bigr) \bigr\vert \\& \qquad {} +\sum_{i=1}^{n}\sum _{j=1}^{n}\bigl(\alpha _{i}^{2} \bigl\vert z '_{i}(t) \bigr\vert + \vert \alpha _{i}\gamma _{i} \vert \bigl\vert z_{i}(t) \bigr\vert \bigr) \bigl\vert d_{ij}(t) \bigr\vert \bigl\vert \widetilde{B}_{j}\bigl(z_{j} \bigl(t-q_{ij}(t)\bigr)\bigr) \bigr\vert \\& \qquad {} +\sum_{i=1}^{n}\bigl(\alpha _{i}^{2} \bigl\vert z '_{i}(t) \bigr\vert + \vert \alpha _{i} \gamma _{i} \vert \bigl\vert z_{i}(t) \bigr\vert \bigr) \\& \qquad {}\times \sum_{j=1}^{n}\sum _{l=1}^{n} \bigl\vert \theta _{ijl}(t) \bigr\vert \bigl(M^{Q}_{j}L^{Q}_{l} \bigl\vert z_{l}\bigl(t- \xi _{ijl} (t) \bigr) \bigr\vert +M^{Q}_{l}L^{Q}_{j} \bigl\vert z_{j}\bigl( t-\eta _{ijl} (t) \bigr) \bigr\vert \bigr) \\& \qquad {} +\sum_{i=1}^{n}\sum _{j=1}^{n}\bigl(\alpha _{i}^{2} \bigl\vert z'_{i}(t) \bigr\vert + \vert \alpha _{i}\gamma _{i} \vert \bigl\vert z_{i}(t) \bigr\vert \bigr) \bigl\vert h_{ij}(t) \bigr\vert \int _{0}^{+\infty } \bigl\vert \sigma _{ij}(u) \bigr\vert \bigl\vert \widetilde{K}_{j} \bigl(z_{j}(t-u)\bigr) \bigr\vert \,du \\& \qquad {} +\sum_{i=1}^{n}\sum _{j=1}^{n}\sum_{l=1}^{n} \bigl( \alpha _{i}^{2} \bigl\vert z'_{i}(t) \bigr\vert + \vert \alpha _{i}\gamma _{i} \vert \bigl\vert z_{i}(t) \bigr\vert \bigr) \bigl\vert p_{ijl}(t) \bigr\vert \int _{0}^{+\infty } \bigl\vert \hat{\sigma }_{ijl}(u) \bigr\vert \,du \\& \qquad {} \times \int _{0}^{+\infty } \bigl\vert \bar{\sigma }_{ijl}(u) \bigr\vert \bigl\vert z_{l}(t-u) \bigr\vert \,du M_{j}^{R}L_{l}^{R} \\& \qquad {} +\sum_{i=1}^{n}\sum _{j=1}^{n}\sum_{l=1}^{n} \bigl( \alpha _{i}^{2} \bigl\vert z'_{i}(t) \bigr\vert + \vert \alpha _{i}\gamma _{i} \vert \bigl\vert z_{i}(t) \bigr\vert \bigr) \bigl\vert p_{ijl}(t) \bigr\vert \int _{0}^{+\infty } \bigl\vert \bar{\sigma }_{ijl}(u) \bigr\vert \,du \\& \qquad {} \times \int _{0}^{+\infty } \bigl\vert \hat{\sigma }_{ijl}(u) \bigr\vert \bigl\vert z_{j}(t-u) \bigr\vert \,du M_{l}^{R}L_{j}^{R} \\& \qquad {} +\frac{1}{2}\sum_{i=1}^{n} \sum_{j=1}^{n}\bigl(\alpha _{i}^{2} h_{ij}^{+}+ \vert \alpha _{i}\gamma _{i} \vert h_{ij}^{+} \bigr)L_{j}^{K} \int _{0}^{+ \infty } \bigl\vert \sigma _{ij}(u) \bigr\vert e^{2\lambda u}\,du z_{j}^{2}(t) \\& \qquad {} -\frac{1}{2}\sum_{i=1}^{n} \sum_{j=1}^{n}\bigl(\alpha _{i}^{2} h_{ij}^{+}+ \vert \alpha _{i}\gamma _{i} \vert h_{ij}^{+} \bigr)L_{j}^{K} \int _{0}^{+ \infty } \bigl\vert \sigma _{ij}(u) \bigr\vert z_{j}^{2}(t-u)\,du \end{aligned}$$
$$\begin{aligned}& \qquad {} +\frac{1}{2}\sum_{i=1}^{n} \sum_{l=1}^{n}\sum _{j=1}^{n}\bigl(\alpha _{l}^{2} + \vert \alpha _{l}\gamma _{l} \vert \bigr)p_{lji}^{+} \int _{0}^{+\infty } \bigl\vert \hat{\sigma }_{lji}(u) \bigr\vert \,du \int _{0}^{+\infty } \bigl\vert \bar{\sigma }_{lji}(u) \bigr\vert e^{2\lambda u}\,duz_{i}^{2}(t)M_{j}^{R}L_{i}^{R} \\& \qquad {} -\frac{1}{2}\sum_{i=1}^{n} \sum_{l=1}^{n}\sum _{j=1}^{n}\bigl(\alpha _{l}^{2} + \vert \alpha _{l}\gamma _{l} \vert \bigr)p_{lji}^{+} \int _{0}^{+\infty } \bigl\vert \hat{\sigma }_{lji}(u) \bigr\vert \,du \int _{0}^{+\infty } \bigl\vert \bar{\sigma }_{lji}(u) \bigr\vert z_{i}^{2}(t-u) \,duM_{j}^{R}L_{i}^{R} \\& \qquad {} +\frac{1}{2}\sum_{i=1}^{n} \sum_{j=1}^{n}\sum _{l=1}^{n}\bigl(\alpha _{j}^{2} + \vert \alpha _{j}\gamma _{j} \vert \bigr)p_{jil}^{+} \int _{0}^{+\infty } \bigl\vert \bar{\sigma }_{jil}(u) \bigr\vert \,du \int _{0}^{+\infty } \bigl\vert \hat{\sigma }_{jil}(u) \bigr\vert e^{2\lambda u}\,duz_{i}^{2}(t)M_{l}^{R}L_{i}^{R} \\& \qquad {} -\frac{1}{2}\sum_{i=1}^{n} \sum_{j=1}^{n}\sum _{l=1}^{n}\bigl(\alpha _{j}^{2} + \vert \alpha _{j}\gamma _{j} \vert \bigr)p_{jil}^{+} \int _{0}^{+\infty } \bigl\vert \bar{\sigma }_{jil}(u) \bigr\vert \,du \int _{0}^{+\infty } \bigl\vert \hat{\sigma }_{jil}(u) \bigr\vert z_{i}^{2}(t-u) \,duM_{l}^{R}L_{i}^{R} \Biggr\} \\& \quad = e^{2\lambda t}\Biggl\{ \sum_{i=1}^{n} \bigl(\beta _{i}+\gamma _{i}^{2}+2 \lambda \alpha _{i}\gamma _{i} -a_{i}(t)\alpha _{i}\gamma _{i}-b_{i}(t) \alpha _{i}^{2}\bigr)z_{i}(t) z '_{i}(t) \\& \qquad {}+\sum_{i=1}^{n}\bigl(\lambda \alpha _{i}^{2}+\alpha _{i}\gamma _{i}-a_{i}(t) \alpha _{i}^{2} \bigr) \bigl(z '_{i}(t)\bigr)^{2} \\& \qquad {} +\sum_{i=1}^{n}\Biggl[-b_{i}(t) \alpha _{i}\gamma _{i}+\lambda \beta _{i}+ \lambda \gamma _{i}^{2} +\frac{1}{2}\sum _{j=1}^{n}\bigl( \alpha _{j}^{2} d_{ji} ^{+}+ \vert \alpha _{j}\gamma _{j} \vert d_{ji} ^{+}\bigr)L^{B}_{i}e^{ 2\lambda q_{ji}^{+} } \frac{1}{1-\dot{q} _{ji} ^{+}} \\& \qquad {} +\frac{1}{2} \sum_{l=1}^{n} \sum_{j=1}^{n}\bigl( \alpha _{l}^{2}+ \vert \alpha _{l}\gamma _{l} \vert \bigr) \theta _{lji} ^{+} M^{Q}_{j}L^{Q}_{i} e^{2\lambda \xi _{lji} ^{+}} \frac{1}{1-\dot{\xi } _{lji} ^{+}} \\& \qquad {} +\frac{1}{2} \sum_{j=1}^{n} \sum_{l=1}^{n}\bigl( \alpha _{j}^{2}+ \vert \alpha _{j}\gamma _{j} \vert \bigr) \theta _{jil} ^{+}L^{Q}_{i}M^{Q}_{l} e^{2\lambda \eta _{jil} ^{+}} \frac{1}{1-\dot{\eta } _{jil} ^{+}} \\& \qquad {} +\frac{1}{2}\sum_{j=1}^{n} \bigl(\alpha _{j}^{2} h_{ji}^{+}+ \vert \alpha _{j}\gamma _{j} \vert h_{ji}^{+}\bigr)L_{i}^{K} \int _{0}^{+\infty } \bigl\vert \sigma _{ji}(u) \bigr\vert e^{2\lambda u}\,du \\& \qquad {} +\frac{1}{2}\sum_{l=1}^{n} \sum_{j=1}^{n}\bigl(\alpha _{l}^{2} + \vert \alpha _{l}\gamma _{l} \vert \bigr)p_{lji}^{+} \int _{0}^{+\infty } \bigl\vert \hat{\sigma }_{lji}(u) \bigr\vert \,du \int _{0}^{+\infty } \bigl\vert \bar{\sigma }_{lji}(u) \bigr\vert e^{2 \lambda u}\,duM_{j}^{R}L_{i}^{R} \\& \qquad {} +\frac{1}{2}\sum_{j=1}^{n} \sum_{l=1}^{n}\bigl(\alpha _{j}^{2} + \vert \alpha _{j}\gamma _{j} \vert \bigr)p_{jil}^{+} \int _{0}^{+\infty } \bigl\vert \bar{\sigma }_{jil}(u) \bigr\vert \,du \int _{0}^{+\infty } \bigl\vert \hat{\sigma }_{jil}(u) \bigr\vert e^{2 \lambda u}\,duM_{l}^{R}L_{i}^{R} \Biggr]z_{i}^{2}(t) \\& \qquad {} -\frac{1}{2}\sum_{i=1}^{n} \sum_{j=1}^{n}\bigl(\alpha _{i}^{2}d_{ij} ^{+}+ \vert \alpha _{i}\gamma _{i} \vert d_{ij} ^{+}\bigr) L^{B}_{j} z_{j}^{2} \bigl(t-q_{ij}(t)\bigr) \\& \qquad {} -\frac{1}{2}\sum_{i=1}^{n} \sum_{l=1}^{n}\sum _{j=1}^{n}\bigl(\alpha _{l}^{2}+ \vert \alpha _{l}\gamma _{l} \vert \bigr) \theta _{lji} ^{+} M^{Q}_{j}L^{Q}_{i} z_{i}^{2}\bigl(t-\xi _{lji}( t)\bigr) \\& \qquad {}-\frac{1}{2}\sum_{i=1}^{n} \sum_{j=1}^{n}\sum _{l=1}^{n}\bigl(\alpha _{j}^{2}+ \vert \alpha _{j}\gamma _{j} \vert \bigr) \theta _{jil} ^{+}L^{Q}_{i}M^{Q}_{l} z_{i}^{2}\bigl(t-\eta _{jil}( t)\bigr) \\& \qquad {} -\frac{1}{2}\sum_{i=1}^{n} \sum_{j=1}^{n}\bigl(\alpha _{i}^{2} h_{ij}^{+}+ \vert \alpha _{i}\gamma _{i} \vert h_{ij}^{+} \bigr)L_{j}^{K} \int _{0}^{+ \infty } \bigl\vert \sigma _{ij}(u) \bigr\vert z_{j}^{2}(t-u)\,du \\& \qquad {} -\frac{1}{2}\sum_{i=1}^{n} \sum_{l=1}^{n}\sum _{j=1}^{n}\bigl(\alpha _{l}^{2} + \vert \alpha _{l}\gamma _{l} \vert \bigr)p_{lji}^{+} \int _{0}^{+\infty } \bigl\vert \hat{\sigma }_{lji}(u) \bigr\vert \,du \int _{0}^{+\infty } \bigl\vert \bar{\sigma }_{lji}(u) \bigr\vert z_{i}^{2}(t-u) \,duM_{j}^{R}L_{i}^{R} \end{aligned}$$
$$\begin{aligned}& \qquad {} -\frac{1}{2}\sum_{i=1}^{n} \sum_{j=1}^{n}\sum _{l=1}^{n}\bigl(\alpha _{j}^{2} + \vert \alpha _{j}\gamma _{j} \vert \bigr)p_{jil}^{+} \int _{0}^{+\infty } \bigl\vert \bar{\sigma }_{jil}(u) \bigr\vert \,du \int _{0}^{+\infty } \bigl\vert \hat{\sigma }_{jil}(u) \bigr\vert z_{i}^{2}(t-u) \,duM_{l}^{R}L_{i}^{R} \\& \qquad {} +\sum_{i=1}^{n}\sum _{j=1}^{n}\bigl(\alpha _{i}^{2} \bigl\vert z '_{i}(t) \bigr\vert + \vert \alpha _{i}\gamma _{i} \vert \bigl\vert z_{i}(t) \bigr\vert \bigr) \bigl\vert c_{ij}(t) \bigr\vert \bigl\vert \widetilde{A}_{j}\bigl(z_{j}(t) \bigr) \bigr\vert \\& \qquad {} +\sum_{i=1}^{n}\sum _{j=1}^{n}\bigl(\alpha _{i}^{2} \bigl\vert z '_{i}(t) \bigr\vert + \vert \alpha _{i}\gamma _{i} \vert \bigl\vert z_{i}(t) \bigr\vert \bigr) \bigl\vert d_{ij}(t) \bigr\vert \bigl\vert \widetilde{B}_{j}\bigl(z_{j} \bigl(t-q_{ij}(t)\bigr)\bigr) \bigr\vert \\& \qquad {} +\sum_{i=1}^{n}\bigl(\alpha _{i}^{2} \bigl\vert z '_{i}(t) \bigr\vert + \vert \alpha _{i} \gamma _{i} \vert \bigl\vert z_{i}(t) \bigr\vert \bigr) \\& \qquad {}\times \sum_{j=1}^{n}\sum _{l=1}^{n} \bigl\vert \theta _{ijl}(t) \bigr\vert \bigl(M^{Q}_{j}L^{Q}_{l} \bigl\vert z_{l}\bigl(t- \xi _{ijl} (t) \bigr) \bigr\vert +M^{Q}_{l}L^{Q}_{j} \bigl\vert z_{j}\bigl( t-\eta _{ijl} (t) \bigr) \bigr\vert \bigr) \\& \qquad {} +\sum_{i=1}^{n}\sum _{j=1}^{n}\bigl(\alpha _{i}^{2} \bigl\vert z'_{i}(t) \bigr\vert + \vert \alpha _{i}\gamma _{i} \vert \bigl\vert z_{i}(t) \bigr\vert \bigr) \bigl\vert h_{ij}(t) \bigr\vert \int _{0}^{+\infty } \bigl\vert \sigma _{ij}(u) \bigr\vert \bigl\vert \widetilde{K} _{j} \bigl(z_{j}(t-u)\bigr) \bigr\vert \,du \\& \qquad {} +\sum_{i=1}^{n}\sum _{j=1}^{n}\sum_{l=1}^{n} \bigl( \alpha _{i}^{2} \bigl\vert z'_{i}(t) \bigr\vert + \vert \alpha _{i}\gamma _{i} \vert \bigl\vert z_{i}(t) \bigr\vert \bigr) \bigl\vert p_{ijl}(t) \bigr\vert \\& \qquad {} \times \int _{0}^{+\infty } \bigl\vert \hat{\sigma }_{ijl}(u) \bigr\vert \,du \int _{0}^{+ \infty } \bigl\vert \bar{\sigma }_{ijl}(u) \bigr\vert \bigl\vert z_{l}(t-u) \bigr\vert \,du M_{j}^{R}L_{l}^{R} \\& \qquad {} +\sum_{i=1}^{n}\sum _{j=1}^{n}\sum_{l=1}^{n} \bigl( \alpha _{i}^{2} \bigl\vert z'_{i}(t) \bigr\vert + \vert \alpha _{i}\gamma _{i} \vert \bigl\vert z_{i}(t) \bigr\vert \bigr) \bigl\vert p_{ijl}(t) \bigr\vert \\& \qquad {} \times \int _{0}^{+\infty } \bigl\vert \bar{\sigma }_{ijl}(u) \bigr\vert \,du \int _{0}^{+ \infty } \bigl\vert \hat{\sigma }_{ijl}(u) \bigr\vert \bigl\vert z_{j}(t-u) \bigr\vert \,du M_{l}^{R}L_{j}^{R} \Biggr\} ,\quad \forall t\in [0,+\infty ). \end{aligned}$$
(2.5)

It follows from \((G_{1})\) and \(PQ \leq \frac{1}{2} (P^{2} + Q^{2})\) (\(P, Q\in \mathbb{R}\)) that

$$\begin{aligned}& \sum_{i=1}^{n}\sum _{j=1}^{n}\bigl(\alpha _{i}^{2} \bigl\vert z'_{i}(t) \bigr\vert + \vert \alpha _{i}\gamma _{i} \vert \bigl\vert z_{i}(t) \bigr\vert \bigr) \bigl\vert c_{ij}(t) \bigr\vert \bigl\vert \widetilde{A}_{j}\bigl(z_{j}(t )\bigr) \bigr\vert \\& \quad \leq \frac{1}{2}\sum_{i=1}^{n} \sum_{j=1}^{n} \alpha _{i}^{2} \bigl\vert c_{ij}(t) \bigr\vert L^{A}_{j}\bigl( \bigl(z '_{i}(t) \bigr)^{2}+z_{j}^{2}(t)\bigr) \\& \qquad {} +\frac{1}{2}\sum_{i=1}^{n} \sum_{j=1}^{n} \vert \alpha _{i} \gamma _{i} \vert \bigl\vert c_{ij}(t) \bigr\vert L^{A}_{j} \bigl(z_{i}^{2}(t)+z_{j}^{2}(t) \bigr) \\& \quad = \frac{1}{2}\sum_{i=1}^{n} \sum_{j=1}^{n}\alpha _{i}^{2} \bigl\vert c_{ij}(t) \bigr\vert L^{A}_{j} \bigl(z'_{i}(t)\bigr)^{2} \\& \qquad {} +\frac{1}{2}\sum_{i=1}^{n} \sum_{j=1}^{n}\bigl( \vert \alpha _{i} \gamma _{i} \vert \bigl\vert c_{ij}(t) \bigr\vert L^{A}_{j} +\alpha _{j}^{2} \bigl\vert c_{ji}(t) \bigr\vert L^{A}_{i}+ \vert \alpha _{j}\gamma _{j} \vert \bigl\vert c_{ji}(t) \bigr\vert L^{A}_{i}\bigr)z_{i}^{2}(t), \\& \sum_{i=1}^{n}\sum _{j=1}^{n}\bigl(\alpha _{i}^{2} \bigl\vert z'_{i}(t) \bigr\vert + \vert \alpha _{i}\gamma _{i} \vert \bigl\vert z_{i}(t) \bigr\vert \bigr) \bigl\vert d_{ij}(t) \bigr\vert \bigl\vert \widetilde{B}_{j}\bigl(z_{j} \bigl(t-q_{ij}(t)\bigr)\bigr) \bigr\vert \\& \quad \leq \frac{1}{2}\sum_{i=1}^{n} \sum_{j=1}^{n} \alpha _{i}^{2} \bigl\vert d_{ij}(t) \bigr\vert L^{B}_{j}\bigl( \bigl(z' _{i}(t)\bigr)^{2}+z_{j}^{2} \bigl(t-q_{ij}(t)\bigr)\bigr) \\& \qquad {} +\frac{1}{2}\sum_{i=1}^{n} \sum_{j=1}^{n} \vert \alpha _{i} \gamma _{i} \vert \bigl\vert d_{ij}(t) \bigr\vert L^{B}_{j} \bigl(z_{i}^{2}(t)+z_{j}^{2} \bigl(t-q_{ij}(t)\bigr)\bigr) \\& \quad = \frac{1}{2}\sum_{i=1}^{n} \sum_{j=1}^{n}\alpha _{i}^{2} \bigl\vert d_{ij}(t) \bigr\vert L^{B}_{j} \bigl(z '_{i}(t)\bigr)^{2} + \frac{1}{2}\sum_{i=1}^{n}\sum _{j=1}^{n} \vert \alpha _{i}\gamma _{i} \vert \bigl\vert d_{ij}(t) \bigr\vert L^{B}_{j}z_{i}^{2}(t) \\& \qquad {} +\frac{1}{2}\sum_{i=1}^{n} \sum_{j=1}^{n}\bigl(\alpha _{i}^{2} \bigl\vert d_{ij}(t) \bigr\vert L^{B}_{j}+ \vert \alpha _{i}\gamma _{i} \vert \bigl\vert d_{ij}(t) \bigr\vert L^{B}_{j}\bigr)z_{j}^{2} \bigl(t-q_{ij}(t)\bigr), \\& \sum_{i=1}^{n}\bigl(\alpha _{i}^{2} \bigl\vert z '_{i}(t) \bigr\vert + \vert \alpha _{i} \gamma _{i} \vert \bigl\vert z_{i}(t) \bigr\vert \bigr) \\& \qquad {} \times \sum_{j=1}^{n}\sum _{l=1}^{n} \bigl\vert \theta _{ijl}(t) \bigr\vert \bigl(M^{Q}_{j}L^{Q}_{l} \bigl\vert z_{l}\bigl( t-\xi _{ijl} (t) \bigr) \bigr\vert +L^{Q}_{j} \bigl\vert z_{j} \bigl(t- \eta _{ijl} (t) \bigr) \bigr\vert M^{Q}_{l} \bigr) \\& \quad \leq \frac{1}{2}\sum_{i=1}^{n} \sum_{j=1}^{n}\sum _{l=1}^{n}\alpha _{i}^{2} \bigl\vert \theta _{ijl}(t) \bigr\vert M^{Q}_{j}L^{Q}_{l} \bigl( \bigl(z' _{i}(t)\bigr)^{2}+z_{l}^{2} \bigl(t-\xi _{ijl} (t) \bigr)\bigr) \\& \qquad {} +\frac{1}{2}\sum_{i=1}^{n} \sum_{j=1}^{n}\sum _{l=1}^{n} \vert \alpha _{i}\gamma _{i} \vert \bigl\vert \theta _{ijl}(t) \bigr\vert M^{Q}_{j}L^{Q}_{l} \bigl( \bigl(z _{i}(t)\bigr)^{2}+z_{l}^{2} \bigl(t-\xi _{ijl} (t) \bigr)\bigr) \\& \qquad {} + \frac{1}{2}\sum_{i=1}^{n} \sum_{j=1}^{n}\sum _{l=1}^{n}\alpha _{i}^{2} \bigl\vert \theta _{ijl}(t) \bigr\vert L^{Q}_{j}M^{Q}_{l} \bigl( \bigl(z' _{i}(t)\bigr)^{2}+z_{j}^{2} \bigl(t-\eta _{ijl}( t) \bigr)\bigr) \\& \qquad {} +\frac{1}{2}\sum_{i=1}^{n} \sum_{j=1}^{n}\sum _{l=1}^{n} \vert \alpha _{i}\gamma _{i} \vert \bigl\vert \theta _{ijl}(t) \bigr\vert L^{Q}_{j}M^{Q}_{l} \bigl( \bigl(z _{i}(t)\bigr)^{2}+z_{j}^{2} \bigl(t-\eta _{ijl} (t) \bigr)\bigr) \\& \quad = \frac{1}{2}\sum_{i=1}^{n} \sum_{j=1}^{n}\sum _{l=1}^{n}\alpha _{i}^{2} \bigl\vert \theta _{ijl}(t) \bigr\vert \bigl(M^{Q}_{j}L^{Q}_{l}+L^{Q}_{j}M^{Q}_{l} \bigr) \bigl(z' _{i}(t)\bigr)^{2} \\& \qquad {} +\frac{1}{2}\sum_{i=1}^{n} \sum_{j=1}^{n}\sum _{l=1}^{n} \vert \alpha _{i}\gamma _{i} \vert \bigl\vert \theta _{ijl}(t) \bigr\vert \bigl(M^{Q}_{j}L^{Q}_{l}+L^{Q}_{j}M^{Q}_{l} \bigr) \bigl(z _{i}(t)\bigr)^{2} \\& \qquad {} +\frac{1}{2}\sum_{i=1}^{n} \sum_{j=1}^{n}\sum _{l=1}^{n}\bigl(\alpha _{i}^{2}+ \vert \alpha _{i}\gamma _{i} \vert \bigr) \bigl\vert \theta _{ijl}(t) \bigr\vert M^{Q}_{j}L^{Q}_{l} z_{l}^{2}\bigl(t-\xi _{ijl} (t) \bigr) \\& \qquad {} +\frac{1}{2}\sum_{i=1}^{n} \sum_{j=1}^{n}\sum _{l=1}^{n}\bigl(\alpha _{i}^{2}+ \vert \alpha _{i}\gamma _{i} \vert \bigr) \bigl\vert \theta _{ijl}(t) \bigr\vert L^{Q}_{j}M^{Q}_{l} z_{j}^{2}\bigl(t-\eta _{ijl} (t) \bigr) \\& \quad = \frac{1}{2}\sum_{i=1}^{n} \sum_{j=1}^{n}\sum _{l=1}^{n}\alpha _{i}^{2} \bigl\vert \theta _{ijl}(t) \bigr\vert \bigl(M^{Q}_{j}L^{Q}_{l}+L^{Q}_{j}M^{Q}_{l} \bigr) \bigl(z' _{i}(t)\bigr)^{2} \\& \qquad {} +\frac{1}{2}\sum_{i=1}^{n} \sum_{j=1}^{n}\sum _{l=1}^{n} \vert \alpha _{i}\gamma _{i} \vert \bigl\vert \theta _{ijl}(t) \bigr\vert \bigl(M^{Q}_{j}L^{Q}_{l}+L^{Q}_{j}M^{Q}_{l} \bigr) \bigl(z _{i}(t)\bigr)^{2} \\& \qquad {} +\frac{1}{2}\sum_{l=1}^{n} \sum_{j=1}^{n}\sum _{i=1}^{n}\bigl(\alpha _{l}^{2}+ \vert \alpha _{l}\gamma _{l} \vert \bigr) \bigl\vert \theta _{lji}(t) \bigr\vert M^{Q}_{j}L^{Q}_{i} z_{i}^{2}\bigl(t-\xi _{lji} (t) \bigr) \\& \qquad {} +\frac{1}{2}\sum_{j=1}^{n} \sum_{i=1}^{n}\sum _{l=1}^{n}\bigl(\alpha _{j}^{2}+ \vert \alpha _{j}\gamma _{j} \vert \bigr) \bigl\vert \theta _{jil}(t) \bigr\vert L^{Q}_{i}M^{Q}_{l} z_{i}^{2}\bigl(t-\eta _{jil} (t) \bigr) , \\& \sum_{i=1}^{n}\sum _{j=1}^{n}\bigl(\alpha _{i}^{2} \bigl\vert z'_{i}(t) \bigr\vert + \vert \alpha _{i}\gamma _{i} \vert \bigl\vert z_{i}(t) \bigr\vert \bigr) \bigl\vert h_{ij}(t) \bigr\vert \int _{0}^{+\infty } \bigl\vert \sigma _{ij}(u) \bigr\vert \bigl\vert \widetilde{K}_{j} \bigl(z_{j}(t-u)\bigr) \bigr\vert \,du \\& \quad \leq \frac{1}{2}\sum_{i=1}^{n} \sum_{j=1}^{n} \alpha _{i}^{2} \bigl\vert h_{ij}(t) \bigr\vert L_{j}^{K} \int _{0}^{+\infty } \bigl\vert \sigma _{ij}(u) \bigr\vert \,du\bigl(z'_{i}(t) \bigr)^{2} \\& \qquad {} +\frac{1}{2}\sum_{i=1}^{n} \sum_{j=1}^{n} \vert \alpha _{i} \gamma _{i} \vert \bigl\vert h_{ij}(t) \bigr\vert L_{j}^{K} \int _{0}^{+\infty } \bigl\vert \sigma _{ij}(u) \bigr\vert \,duz_{i}^{2}(t) \\& \qquad {} +\frac{1}{2}\sum_{i=1}^{n} \sum_{j=1}^{n}\bigl(\alpha _{i}^{2} \bigl\vert h_{ij}(t) \bigr\vert L_{j}^{K}+ \vert \alpha _{i}\gamma _{i} \vert \bigl\vert h_{ij}(t) \bigr\vert L_{j}^{K}\bigr) \int _{0}^{+\infty } \bigl\vert \sigma _{ij}(u) \bigr\vert z_{j}^{2}(t-u)\,du, \\& \sum_{i=1}^{n}\sum _{j=1}^{n}\sum_{l=1}^{n} \bigl( \alpha _{i}^{2} \bigl\vert z'_{i}(t) \bigr\vert + \vert \alpha _{i}\gamma _{i} \vert \bigl\vert z_{i}(t) \bigr\vert \bigr) \bigl\vert p_{ijl}(t) \bigr\vert \\& \qquad {} \times \int _{0}^{+\infty } \bigl\vert \hat{\sigma }_{ijl}(u) \bigr\vert \,du \int _{0}^{+ \infty } \bigl\vert \bar{\sigma }_{ijl}(u) \bigr\vert \bigl\vert z_{l}(t-u) \bigr\vert \,du M_{j}^{R}L_{l}^{R} \\& \quad \leq \frac{1}{2}\sum_{i=1}^{n} \sum_{j=1}^{n} \sum _{l=1}^{n}\alpha _{i}^{2} \bigl\vert p_{ijl}(t) \bigr\vert M_{j}^{R}L_{l}^{R} \int _{0}^{+\infty } \bigl\vert \hat{\sigma }_{ijl}(u) \bigr\vert \,du \int _{0}^{+\infty } \bigl\vert \bar{\sigma }_{ijl}(u) \bigr\vert \,du\bigl(z'_{i}(t) \bigr)^{2} \\& \qquad {} +\frac{1}{2}\sum_{i=1}^{n} \sum_{j=1}^{n}\sum _{l=1}^{n} \vert \alpha _{i}\gamma _{i} \vert \bigl\vert p_{ijl}(t) \bigr\vert M_{j}^{R}L_{l}^{R} \int _{0}^{+\infty } \bigl\vert \hat{\sigma }_{ijl}(u) \bigr\vert \,du \int _{0}^{+\infty } \bigl\vert \bar{\sigma }_{ijl}(u) \bigr\vert \,duz^{2}_{i}(t) \\& \qquad {} +\frac{1}{2}\sum_{l=1}^{n} \sum_{j=1}^{n}\sum _{i=1}^{n}\bigl(\alpha _{l}^{2}+ \vert \alpha _{l}\gamma _{l} \vert \bigr) \bigl\vert p_{lji}(t) \bigr\vert M_{j}^{R}L_{i}^{R} \int _{0}^{+\infty } \bigl\vert \hat{\sigma }_{lji}(u) \bigr\vert \,du \\& \qquad {} \times \int _{0}^{+\infty } \bigl\vert \bar{\sigma }_{lji}(u) \bigr\vert z^{2}_{i}(t-u)\,du, \end{aligned}$$

and

$$\begin{aligned} &\sum_{i=1}^{n}\sum _{j=1}^{n}\sum_{l=1}^{n} \bigl( \alpha _{i}^{2} \bigl\vert z'_{i}(t) \bigr\vert + \vert \alpha _{i}\gamma _{i} \vert \bigl\vert z_{i}(t) \bigr\vert \bigr) \bigl\vert p_{ijl}(t) \bigr\vert \\ &\qquad {} \times \int _{0}^{+\infty } \bigl\vert \bar{\sigma }_{ijl}(u) \bigr\vert \,du \int _{0}^{+ \infty } \bigl\vert \hat{\sigma }_{ijl}(u) \bigr\vert \bigl\vert z_{j}(t-u) \bigr\vert \,du M_{l}^{R}L_{j}^{R} \\ &\quad \leq \frac{1}{2}\sum_{i=1}^{n} \sum_{j=1}^{n} \sum _{l=1}^{n}\alpha _{i}^{2} \bigl\vert p_{ijl}(t) \bigr\vert M_{l}^{R}L_{j}^{R} \int _{0}^{+\infty } \bigl\vert \bar{\sigma }_{ijl}(u) \bigr\vert \,du \int _{0}^{+\infty } \bigl\vert \hat{\sigma }_{ijl}(u) \bigr\vert \,du\bigl(z'_{i}(t) \bigr)^{2} \\ &\qquad {} +\frac{1}{2}\sum_{i=1}^{n} \sum_{j=1}^{n}\sum _{l=1}^{n} \vert \alpha _{i}\gamma _{i} \vert \bigl\vert p_{ijl}(t) \bigr\vert M_{l}^{R}L_{j}^{R} \int _{0}^{+\infty } \bigl\vert \bar{\sigma }_{ijl}(u) \bigr\vert \,du \int _{0}^{+\infty } \bigl\vert \hat{\sigma }_{ijl}(u) \bigr\vert \,duz^{2}_{i}(t) \\ &\qquad {} +\frac{1}{2}\sum_{j=1}^{n} \sum_{i=1}^{n}\sum _{l=1}^{n}\bigl(\alpha _{j}^{2}+ \vert \alpha _{j}\gamma _{j} \vert \bigr) \bigl\vert p_{jil}(t) \bigr\vert M_{l}^{R}L_{i}^{R} \int _{0}^{+\infty } \bigl\vert \bar{\sigma }_{jil}(u) \bigr\vert \,du \\ &\qquad {} \times \int _{0}^{+\infty } \bigl\vert \hat{\sigma }_{jil}(u) \bigr\vert z^{2}_{i}(t-u)\,du, \end{aligned}$$

which, together with (2.4) and (2.5), entails

$$\begin{aligned} W'(t) \leq & e^{2\lambda t}\Biggl\{ \sum _{i=1}^{n}\bigl(\beta _{i}+ \gamma _{i}^{2}+2\lambda \alpha _{i}\gamma _{i} -a_{i}(t)\alpha _{i} \gamma _{i}-b_{i}(t)\alpha _{i}^{2} \bigr)z_{i}(t) z'_{i}(t) \\ &{}+\sum_{i=1}^{n}\Biggl[\lambda \alpha _{i}^{2}+\alpha _{i}\gamma _{i}-a_{i}(t) \alpha _{i}^{2} + \frac{1}{2}\alpha _{i}^{2}\sum _{j=1}^{n}\bigl( \bigl\vert c_{ij}(t) \bigr\vert L^{A}_{j} + \bigl\vert d_{ij}(t) \bigr\vert L^{B}_{j}\bigr) \\ &{} +\frac{1}{2} \alpha _{i}^{2}\sum _{j=1}^{n}\sum_{l=1}^{n} \bigl\vert \theta _{ijl}(t) \bigr\vert \bigl(M^{Q}_{j}L^{Q}_{l}+L^{Q}_{j}M^{Q}_{l} \bigr) \\ &{} +\frac{1}{2}\alpha _{i}^{2}\sum _{j=1}^{n} \bigl\vert h_{ij}(t) \bigr\vert L_{j}^{K} \int _{0}^{+\infty } \bigl\vert \sigma _{ij}(u) \bigr\vert \,du \\ &{} +\frac{1}{2}\alpha _{i}^{2}\sum _{j=1}^{n}\sum_{l=1}^{n} \bigl\vert p_{ijl}(t) \bigr\vert M_{j}^{R}L_{l}^{R} \int _{0}^{+\infty } \bigl\vert \hat{\sigma }_{ijl}(u) \bigr\vert \,du \int _{0}^{+\infty } \bigl\vert \bar{\sigma }_{ijl}(u) \bigr\vert \,du \\ &{} +\frac{1}{2}\alpha _{i}^{2}\sum _{j=1}^{n}\sum_{l=1}^{n} \bigl\vert p_{ijl}(t) \bigr\vert M_{l}^{R}L_{j}^{R} \int _{0}^{+\infty } \bigl\vert \bar{\sigma }_{ijl}(u) \bigr\vert \,du \int _{0}^{+\infty } \bigl\vert \hat{\sigma }_{ijl}(u) \bigr\vert \,du\Biggr]\bigl(z'_{i}(t) \bigr)^{2} \\ &{}+\sum_{i=1}^{n}\Biggl[-b_{i}(t) \alpha _{i}\gamma _{i}+\lambda \beta _{i}+ \lambda \gamma _{i}^{2} +\frac{1}{2}\sum _{j=1}^{n}\bigl( \bigl\vert c_{ij}(t) \bigr\vert L^{A}_{j}+ \bigl\vert d_{ij}(t) \bigr\vert L^{B}_{j} \bigr) \vert \alpha _{i}\gamma _{i} \vert \\ &{}+\frac{1}{2}\sum_{j=1}^{n} \alpha _{j}^{2}\biggl( \bigl\vert c_{ji}(t) \bigr\vert L^{A}_{i} + d_{ji}^{+} L^{B}_{i}\frac{1}{1-\dot{q} _{ji} ^{+}}e^{2\lambda q_{ji}^{+}} \biggr) \\ &{}+\frac{1}{2}\sum_{j=1}^{n} \biggl( \bigl\vert c_{ji}(t) \bigr\vert L^{A}_{i} + d_{ji}^{+} L^{B}_{i} \frac{1}{1-\dot{q} _{ji} ^{+}}e^{2\lambda q_{ji}^{+}}\biggr) \vert \alpha _{j} \gamma _{j} \vert \\ &{} +\frac{1}{2} \sum_{j=1}^{n} \sum_{l=1}^{n} \vert \alpha _{i} \gamma _{i} \vert \bigl\vert \theta _{ijl}(t) \bigr\vert \bigl(M^{Q}_{j}L^{Q}_{l}+L^{Q}_{j}M^{Q}_{l} \bigr) \\ &{} +\frac{1}{2} \sum_{l=1}^{n} \sum_{j=1}^{n}\bigl(\alpha _{l}^{2}+ \vert \alpha _{l}\gamma _{l} \vert \bigr) \theta _{lji} ^{+} M^{Q}_{j}L^{Q}_{i} e^{2 \lambda \xi _{lji} ^{+}} \frac{1}{1-\dot{\xi } _{lji} ^{+}} \\ &{} +\frac{1}{2} \sum_{j=1}^{n} \sum_{l=1}^{n}\bigl(\alpha _{j}^{2}+ \vert \alpha _{j}\gamma _{j} \vert \bigr) \theta _{jil} ^{+}L^{Q}_{i}M^{Q}_{l} e^{2 \lambda \eta _{jil} ^{+}} \frac{1}{1-\dot{\eta } _{jil} ^{+}} \\ &{} +\frac{1}{2}\sum_{j=1}^{n} \bigl(\alpha _{j}^{2} h_{ji}^{+}+ \vert \alpha _{j}\gamma _{j} \vert h_{ji}^{+}\bigr)L_{i}^{K} \int _{0}^{+\infty } \bigl\vert \sigma _{ji}(u) \bigr\vert e^{2\lambda u}\,du \\ &{} +\frac{1}{2}\sum_{l=1}^{n} \sum_{j=1}^{n}\bigl(\alpha _{l}^{2} + \vert \alpha _{l}\gamma _{l} \vert \bigr)p_{lji}^{+} \int _{0}^{+\infty } \bigl\vert \hat{\sigma }_{lji}(u) \bigr\vert \,du \int _{0}^{+\infty } \bigl\vert \bar{\sigma }_{lji}(u) \bigr\vert e^{2 \lambda u}\,duM_{j}^{R}L_{i}^{R} \\ &{} +\frac{1}{2}\sum_{j=1}^{n} \sum_{l=1}^{n}\bigl(\alpha _{j}^{2} + \vert \alpha _{j}\gamma _{j} \vert \bigr)p_{jil}^{+} \int _{0}^{+\infty } \bigl\vert \bar{\sigma }_{jil}(u) \bigr\vert \,du \int _{0}^{+\infty } \bigl\vert \hat{\sigma }_{jil}(u) \bigr\vert e^{2 \lambda u}\,duM_{l}^{R}L_{i}^{R} \\ &{} +\frac{1}{2}\sum_{j=1}^{n} \vert \alpha _{i}\gamma _{i} \vert \bigl\vert h_{ij}(t) \bigr\vert L_{j}^{K} \int _{0}^{+\infty } \bigl\vert \sigma _{ij}(u) \bigr\vert \,du \\ &{} +\frac{1}{2}\sum_{j=1}^{n} \sum_{l=1}^{n} \vert \alpha _{i} \gamma _{i} \vert \bigl\vert p_{ijl}(t) \bigr\vert M_{j}^{R}L_{l}^{R} \int _{0}^{+\infty } \bigl\vert \hat{\sigma }_{ijl}(u) \bigr\vert \,du \int _{0}^{+\infty } \bigl\vert \bar{\sigma }_{ijl}(u) \bigr\vert \,du \\ &{} +\frac{1}{2}\sum_{j=1}^{n} \sum_{l=1}^{n} \vert \alpha _{i} \gamma _{i} \vert \bigl\vert p_{ijl}(t) \bigr\vert M_{l}^{R}L_{j}^{R} \int _{0}^{+\infty } \bigl\vert \bar{\sigma }_{ijl}(u) \bigr\vert \,du \int _{0}^{+\infty } \bigl\vert \hat{\sigma }_{ijl}(u) \bigr\vert \,du\Biggr]z_{i}^{2}(t) \Biggr\} \\ = & e^{2\lambda t}\Biggl\{ \sum_{i=1}^{n} \bigl(C^{\lambda }_{i} (t) \bigl(z'_{i}(t) \bigr)^{2}+D^{ \lambda }_{i}(t)z_{i}^{2}(t)+E^{\lambda }_{i}(t)z_{i}(t) z'_{i}(t)\bigr)\Biggr\} \\ = &e^{2\lambda t}\Biggl\{ \sum_{i=1}^{n}C_{i}^{\lambda }(t) \biggl( z '_{i}(t)+ \frac{E_{i}^{\lambda }(t)}{2C_{i}^{\lambda }(t)}z_{i}(t) \biggr)^{2} +\sum_{i=1}^{n} \biggl(D_{i}^{\lambda }(t)- \frac{(E^{\lambda }_{i}(t)) ^{2}}{4C_{i}^{\lambda }(t)}\biggr) z_{i}^{2}(t)\Biggr\} \\ \leq & 0,\quad \forall t \in [0, +\infty ). \end{aligned}$$

This indicates that \(W (t)\leq W (0)\) for all \(t \in [0, +\infty )\), and

$$ \frac{1}{2}\sum_{i=1}^{n} \beta _{i}z_{i}^{2}(t)e^{2\lambda t}+ \frac{1}{2}\sum_{i=1}^{n}\bigl( \alpha _{i} z '_{i}(t)+\gamma _{i}z_{i}(t)\bigr)^{2}e^{2 \lambda t} \leq W (0),\quad t \in [0, +\infty ). $$

Manifestly,

$$ \bigl(\alpha _{i} z'_{i}(t) e^{ \lambda t}+\gamma _{i}z_{i}(t)e^{ \lambda t} \bigr)^{2}=\bigl( \alpha _{i}z'_{i}(t)+ \gamma _{i}z_{i}(t)\bigr)^{2}e^{2\lambda t} $$

and

$$ \alpha _{i} \bigl\vert z'_{i}(t) \bigr\vert e^{ \lambda t}\leq \bigl\vert \alpha _{i} z'_{i}(t) e^{ \lambda t}+\gamma _{i}z_{i}(t)e^{ \lambda t} \bigr\vert + \bigl\vert \gamma _{i}z_{i}(t)e^{ \lambda t} \bigr\vert . $$

Combining with the Cauchy–Schwarz inequality, one can pick a constant \(M >0\) such that

$$ \bigl\vert z'_{i}(t) \bigr\vert \leq M e^{ -\lambda t},\qquad \bigl\vert z_{i}(t) \bigr\vert \leq M e^{- \lambda t},\quad t\geq 0, i\in D , $$

which proves Lemma 2.1. □

Remark 2.2

More precisely, according to Lemma 2.1, we know that, if \(y(t)\) is an equilibrium point or a T-anti-periodic solution of (1.1), then all solutions of the system (1.1) and their derivatives are exponentially convergent to \(y(t)\) and \(y'(t)\), respectively. Referring to the definition of stability adopted in [5, 18–21, 40, 42–45], this indicates that \(y(t)\) is globally exponentially stable.

3 Anti-periodicity of HIHNNs (1.1)

Now, we set out to present the main result of this paper as follows.

Theorem 3.1

Under conditions\((G_{1})\)–\((G_{4})\), system (1.1) has a globally exponentially stableT-anti-periodic solution.

Proof

Let \(v(t)=(v_{1}(t), v_{2}(t),\ldots , v_{n}(t))\) be a solution of system (1.1) with initial conditions:

$$ v_{i}(s)=\varphi ^{v}_{i}(s),\qquad v_{i}'(0)=\psi ^{v}_{i},\quad \varphi ^{v}_{i} \in BC\bigl( (-\infty , 0], \mathbb{R} \bigr), \psi ^{v}_{i}\in \mathbb{R}, i\in D. $$
(3.1)

Clearly, for any nonnegative integer m,

$$\begin{aligned} & \bigl((-1)^{m+1}v_{i} \bigl(t + (m+1) T\bigr) \bigr)'' \\ &\quad = -{a}_{i}(t) \bigl((-1)^{m+1}v_{i} \bigl(t + (m+1) T\bigr)\bigr)'-{b}_{i}(t) \bigl((-1)^{m+1}v_{i}\bigl(t+ (m+1) T\bigr)\bigr) \\ &\qquad {}+\sum_{j=1}^{n}{c}_{ij}(t)A_{j} \bigl((-1)^{m+1}v_{j}\bigl(t+ (m+1)T \bigr)\bigr) \\ &\qquad {} +\sum_{j=1}^{n}{d}_{ij}(t)B_{j} \bigl((-1)^{m+1}v_{j}\bigl( t+ (m+1)T-q_{ij}(t )\bigr) \bigr)+ \sum_{j=1}^{n}\sum _{l=1}^{n}\theta _{ijl}(t) \\ &\qquad {} \times Q_{j}\bigl((-1)^{m+1}v_{j}\bigl(t+ (m+1)T- \eta _{ijl} (t ) \bigr)\bigr) \\ &\qquad {} \times Q_{l}\bigl((-1)^{m+1}v_{l}\bigl( t+ (m+1)T-\xi _{ijl} (t ) \bigr)\bigr) \\ &\qquad {}+ \sum_{j=1}^{n}h_{ij} (t) \int _{0}^{+\infty }\sigma _{ij}(u) K_{j}\bigl((-1)^{m+1}v_{j}(t-u)\bigr)\,du \\ &\qquad {}+ \sum_{j=1}^{n}\sum _{l=1}^{n}p_{ijl}(t) \int _{0}^{+ \infty }\hat{\sigma }_{ijl}(u) R_{j}\bigl((-1)^{m+1}v_{j}(t-u)\bigr)\,du \\ &\qquad {} \times \int _{0}^{+\infty } \bar{\sigma } _{ijl}(u) R_{l}\bigl((-1)^{m+1}v_{l}(t-u)\bigr) \,du+J_{i}(t) , \end{aligned}$$
(3.2)

for all \(i\in D\), \(t+(m+1)T\geq 0 \). It is easy to see that \((-1)^{m+1} v (t + (m+1)T) \) is a solution of (1.1), and \(u(t)= -v (t + T)\) satisfies system (1.1) involving initial values:

$$ \varphi ^{u}_{i}(s)=-v_{i}(s+T),\qquad \psi ^{u}_{i} =-v_{i}'(T),\quad \mbox{for all } s\in (-\infty , 0], i\in D. $$

According to Lemma 2.1, we can take a constant \(N=N(\varphi ^{v}, \psi ^{v},\varphi ^{u}, \psi ^{u})\) satisfying

$$ \bigl\vert v_{i}(t)- u_{i}(t) \bigr\vert \leq N e^{-\lambda t} ,\qquad \bigl\vert v_{i}'(t)- u_{i}'(t) \bigr\vert \leq N e^{-\lambda t} ,\quad \mbox{for all } t\geq 0, i\in D. $$

Thus,

$$ \left . \textstyle\begin{array}{@{}l@{}} \vert (-1)^{p } v_{i}(t+pT)- (-1)^{p+1}v _{i}(t+(p+1)T) \vert \\ \quad = \vert v_{i}(t+pT )- u_{i}(t+pT) \vert \leq N e^{-\lambda (t+pT)} , \\ \vert ((-1)^{p }v_{i}(t+pT))'- ((-1)^{p+1}v_{i}(t+(p+1)T))' \vert \\ \quad = \vert v_{i}'(t+pT)- u_{i}'(t+pT) \vert \leq N e^{-\lambda (t+pT)} , \end{array}\displaystyle \right \}\quad \forall i\in D, t+pT\geq 0. $$

Thus, together with the facts that

$$\begin{aligned}& (-1)^{m+1}v_{i}\bigl(t+(m+1)T\bigr) \\ & \quad = v_{i} (t ) +\sum_{p=0}^{m }\bigl[ (-1)^{p+1}v _{i}\bigl(t + (p+1)T\bigr)- (-1)^{p }v_{i} (t + pT)\bigr]\quad (i\in D) \end{aligned}$$

and

$$\begin{aligned}& \bigl((-1)^{m+1}v_{i}\bigl(t+(m+1)T\bigr) \bigr)' \\ & \quad = v_{i} '(t ) +\sum_{p=0}^{m } \bigl[ \bigl((-1)^{p+1}v _{i}\bigl(t + (p+1)T\bigr) \bigr)'- \bigl((-1)^{p+1}v_{i} (t + pT) \bigr)'\bigr]\quad (i\in D), \end{aligned}$$

then, we can show that there exists a continuous differentiable function \(\kappa (t)=(\kappa _{1} (t), \kappa _{2} (t), \ldots ,\kappa _{n} (t)) \) such that \(\{ (-1)^{m }v (t + mT)\}_{m\geq 1}\) and \(\{ ((-1)^{m }v (t + mT))'\}_{m\geq 1}\) are uniformly convergent to \(\kappa (t) \) and \(\kappa '(t) \) on any compact set of \(\mathbb{R}\), respectively. Moreover,

$$\begin{aligned} \kappa (t+T)&=\lim_{m\rightarrow +\infty }(-1)^{m }v (t +T+ m T) \\ &= - \lim_{(m+1)\rightarrow +\infty }(-1)^{m +1} v \bigl(t + (m +1)T\bigr)=- \kappa (t ) \end{aligned}$$

involves that \(\kappa (t)\) is T-anti-periodic on \(\mathbb{R}\). It follows from \((G_{1})\)–\((G_{4})\) and the continuity on (3.2) that \(\{ v'' (t + (m+1)T) \}_{m\geq 1}\) uniformly converges to a continuous function on any compact set of \(\mathbb{R}\). Furthermore, for any compact set of \(\mathbb{R}\), setting \(m \longrightarrow +\infty \), we obtain

$$\begin{aligned} \kappa _{i}''(t)={}&{-}a_{i}(t) \kappa _{i}'(t)-b_{i}(t)\kappa _{i}(t)+ \sum_{j=1}^{n} c_{ij}(t)A_{j}\bigl(\kappa _{j}(t)\bigr) \\ &{}+\sum_{j=1}^{n} d_{ij}(t)B_{j} \bigl(\kappa _{j}\bigl(t- q_{ij}( t)\bigr) \bigr) \\ &{}+ \sum_{j=1}^{n}\sum _{l=1}^{n}\theta _{ijl}(t)Q_{j} \bigl( \kappa _{j}\bigl( t-\eta _{ijl} (t) \bigr) \bigr)Q_{l}\bigl(\kappa _{l}\bigl( t-\xi _{ijl}( t) \bigr)\bigr) \\ &{}+ \sum_{j=1}^{n}h_{ij} (t) \int _{0}^{+\infty }\sigma _{ij}(u) K_{j}\bigl( \kappa _{j}(t-u)\bigr)\,du \\ &{}+ \sum_{j=1}^{n}\sum _{l=1}^{n}p_{ijl}(t) \int _{0}^{+ \infty }\hat{\sigma }_{ijl}(u) R_{j}\bigl(\kappa _{j}(t-u)\bigr)\,du \\ &{} \times \int _{0}^{+\infty } \bar{\sigma } _{ijl}(u) R_{l}\bigl(\kappa _{l}(t-u)\bigr)\,du+J_{i}(t),\quad i\in D, \end{aligned}$$

which involves the fact that \(\kappa (t)\) is a T-anti-periodic solution of (1.1). With the aid of Lemma 2.1 and Remark 2.2, we find that \(\kappa (t)\) is globally exponentially stable. This completes the proof of Theorem 3.1. □

Remark 3.1

In this present paper, the exponential convergence on every solution and its derivative in system (1.1) is established. In particular, one can see that the results in [19–21, 30, 34] are a special case of Theorem 3.1. This indicates that our results generalize and improve the previous references.

4 A numerical example

In this section, we give an example with a simulation to demonstrate the feasibility and the validity of our theoretical results.

Example 4.1

Let \(n=2\), and let us regard the following high-order inertial Hopfield neural networks with mixed delays

$$ \textstyle\begin{cases} x_{1}''(t) = -(14.92+0.1 \vert \sin t \vert ) x_{1}'(t) -(27.89+0.2 \vert \sin t \vert )x_{1}(t) \\ \hphantom{x_{1}''(t) ={}}{}+ 2.28(\sin t) A_{1}(x_{1}(t))+ 2.19 (\cos t) A_{2}(x_{2}(t)) \\ \hphantom{x_{1}''(t) ={}}{}-0.84 (\cos 2t) B_{1}(x_{1}( t-0.2 \sin ^{2} t ))+2.41 (\cos 2t )B_{2}(x_{2}( t-0.3 \sin ^{2} t )) \\ \hphantom{x_{1}''(t) ={}}{}+4 (\sin t)Q_{1}(x_{1}(t-0.4 \sin ^{2} t)) Q_{2}(x_{2}(t-0.5 \sin ^{2} t)) \\ \hphantom{x_{1}''(t) ={}}{}-0.95(\cos 2t)\int _{0}^{+\infty }\sin (4u)e^{-0.5u}K_{1}(x_{1}(t-u))\,du \\ \hphantom{x_{1}''(t) ={}}{}+2.52(\sin 2t)\int _{0}^{+\infty }\sin (3u)e^{-u}K_{2}(x_{2}(t-u))\,du \\ \hphantom{x_{1}''(t) ={}}{}+3.8(\cos t)\int _{0}^{+\infty }\cos (2u)e^{-0.5u}R_{1}(x_{1}(t-u))\,du \\ \hphantom{x_{1}''(t) ={}}{}\times \int _{0}^{+\infty }\cos (2u)e^{-u}R_{2}(x_{2}(t-u))\,du +55\sin t , \\ x_{2}''(t) = -(15.11+0.1 \vert \cos t \vert ) x_{2}'(t) -(31.05- 0.1 \vert \sin t \vert )) x_{2}(t) \\ \hphantom{x_{2}''(t) ={}}{}-1.88 (\sin t) A_{1}(x_{1}(t)) - 2.33 (\cos t) A_{2}(x_{2}(t)) \\ \hphantom{x_{2}''(t) ={}}{}-2.18 (\sin 2t) B_{1}(x_{1}( t-0.2 \cos ^{2} t ))+3.18( \sin 2t) B_{2}(x_{2}( t-0.3 \cos ^{2} t )) \\ \hphantom{x_{2}''(t) ={}}{}+3.8 (\sin t)Q_{1}(x_{1}( t-0.4 \cos ^{2} t)) Q_{2}(x_{2}( t-0.5 \cos ^{2} t)) \\ \hphantom{x_{2}''(t) ={}}{}-2.28(\cos 2t)\int _{0}^{+\infty }\sin (2u)e^{-0.5u}K_{1}(x_{1}(t-u))\,du \\ \hphantom{x_{2}''(t) ={}}{}+3.28(\sin 2t)\int _{0}^{+\infty }\sin (5u)e^{-u}K_{2}(x_{2}(t-u))\,du \\ \hphantom{x_{2}''(t) ={}}{}+4(\cos t)\int _{0}^{+\infty }\cos (4u)e^{-0.5u}R_{1}(x_{1}(t-u))\,du \\ \hphantom{x_{2}''(t) ={}}{}\times \int _{0}^{+\infty }\cos (4u)e^{-u}R_{2}(x_{2}(t-u))\,du+48 \sin t, \end{cases} $$
(4.1)

where \(A_{1}(u) = A_{2}(u) =\frac{1}{35} |u | \), \(B_{1}(u) = B_{2}(u) = \frac{1}{48} u\), \(Q_{1}(u) = Q_{2}(u) = \frac{1}{110} (|u+1| -|u-1|)\), \(K_{1}(u)=K_{2}(u)=\frac{1}{4\pi }\arctan u\), \(R_{1}(u)=R_{2}(u)= \frac{1}{2\pi }|\arctan u| \).

Obviously, one can take \(\lambda =0.01\), \(\alpha _{i}=\gamma _{i}=1.1\), \(\beta _{i}=5\), \(L_{i}^{A}=\frac{1}{35}\), \(L_{i}^{B}=\frac{1}{48}\), \(L_{i}^{Q}= \frac{1}{55}\), \(L_{i}^{K}= L_{i}^{R}=\frac{1}{2}\), \(M_{i}^{Q}=\frac{1}{55}\), \(M_{i}^{R}= \frac{1}{4}\), \(i=1,2\), such that (4.1) satisfies (2.4) and all the conditions assumed in Sect. 2. By Theorem 3.1, we know that system (4.1) has a globally exponentially stable π-anti-periodic solution \(x^{*}(t)\), and every solution of (4.1) and its derivative are exponentially convergent to \(x^{*}(t)\) and \((x^{*}(t))'\), respectively. Simulations in Figs. 1 and 2 reflect that the theoretical periodicity is in agreement with the numerically observed behavior.

Figure 1
figure 1

Numerical solutions \(x(t)\) to system (4.1) with initial values: \((\varphi _{1}(s),\varphi _{2}(s),\psi _{1}(s),\psi _{2}(s))\equiv ( 2 \sin t+1, -2\cos t-3 , 2, 0), (2\cos t+2, 3\sin t-1, 0, 3), (-3 \sin t-2, -4\sin t+3, -3, -4)\)

Figure 2
figure 2

Numerical solutions \(x'(t)\) to system (4.1) with initial value \((\varphi _{1}(s),\varphi _{2}(s), \psi _{1}(s),\psi _{2}(s))\equiv ( 2\sin t+1, -2\cos t-3, 2, 0), (2\cos t+2, 3\sin t-1, 0, 3), (-3 \sin t-2, -4\sin t+3, -3, -4)\)

Remark 4.1

By the way, there are many excellent results on inertial Hopfield neural networks with time-varying delays [18–21, 27, 29, 30]. However, the anti-periodicity on high-order inertial Hopfield neural networks with bounded time-varying delays and unbounded continuously distributed delays have never been touched upon by using the non-reduced order method. In addition, the corresponding results of [17–21, 27, 29, 30] and [46–95] cannot be used to reveal the convergence of the anti-periodic solution of the system (4.1).

5 Conclusions

In this paper, abandoning the traditional reduced order method, we explore the global convergence dynamics on a class of anti-periodic high-order inertial Hopfield neural networks with bounded time-varying delays and unbounded continuously distributed delays. Some sufficient conditions have been obtained to guarantee that every solution and its derivative of the addressed model is exponentially convergent to a anti-periodic solution and its derivative by combining differential inequality techniques with the Lyapunov function method. It should be mentioned that the results obtained in this manuscript are novel, and the method adopted provides a possible effective approach for studying other types high-order inertial neural networks with mixed delays.

References

  1. Babcock, K., Westervelt, R.: Stability and dynamics of simple electronic neural networks with added inertia. Physica D 23, 464–469 (1986)

    Google Scholar 

  2. Babcock, K., Westervelt, R.: Dynamics of simple electronic neural networks. Physica D 28, 305–316 (1987)

    MathSciNet  Google Scholar 

  3. Yu, S., Zhang, Z., Quan, Z.: New global exponential stability conditions for inertial Cohen–Grossberg neural networks with time delays. Neurocomputing 151, 1446–1454 (2015)

    Google Scholar 

  4. Cai, Z., Huang, J., Huang, L.: Periodic orbit analysis for the delayed Filippov system. Proc. Am. Math. Soc. 146, 4667–4682 (2018)

    MathSciNet  MATH  Google Scholar 

  5. Huang, C.: Exponential stability of inertial neural networks involving proportional delays and non-reduced order method. J. Exp. Theor. Artif. Intell. 32(1), 133–146 (2020)

    Google Scholar 

  6. Li, C., Chen, G., Liao, L., Yu, J.: Hopf bifurcation and chaos in a single inertial neuron model with time delay. Eur. Phys. J. B 41, 337–343 (2004)

    Google Scholar 

  7. Zhao, H., Yu, X., Wang, L.: Bifurcation and control in an inertial two-neuron system with time delays. Int. J. Bifurc. Chaos 22(2), 1250036 (2012)

    MathSciNet  MATH  Google Scholar 

  8. Wang, J., Tian, L.: Global Lagrange stability for inertial neural networks with mixed time varying delays. Neurocomputing 235, 140–146 (2017)

    Google Scholar 

  9. Ge, J., Xu, J.: Weak resonant double Hopf bifurcations in an inertial four-neuron model with time delay. Int. J. Neural Syst. 22, 63–75 (2012)

    Google Scholar 

  10. Zhang, H., Cao, Q., Yang, H.: Asymptotically almost periodic dynamics on delayed Nicholson-type system involving patch structure. J. Inequal. Appl. 2020, 102 (2020)

    MathSciNet  Google Scholar 

  11. Chen, T., Huang, L., Yu, P., Huang, W.: Bifurcation of limit cycles at infinity in piecewise polynomial systems. Nonlinear Anal., Real World Appl. 41, 82–106 (2018)

    MathSciNet  MATH  Google Scholar 

  12. Li, X., Li, X., Hu, C.: Some new results on stability and synchronization for delayed inertial neural networks based on non-reduced order method. Neural Netw. 96, 91–100 (2017)

    MATH  Google Scholar 

  13. Yang, X., Wen, S., Liu, Z., Li, C., Huang, C.: Dynamic properties of foreign exchange complex network. Mathematics 7, 832 (2019). https://doi.org/10.3390/math7090832

    Article  Google Scholar 

  14. Huang, C., Yang, L., Cao, J.: Asymptotic behavior for a class of population dynamics. AIMS Math. 54(4), 3378–3390 (2020). https://doi.org/10.3934/math.2020218

    Article  Google Scholar 

  15. Ke, Y., Miao, C.: Stability and existence of periodic solutions in inertial BAM neural networks with time delay. Neural Comput. Appl. 23, 1089–1099 (2013)

    Google Scholar 

  16. Ke, Y., Miao, C.: Anti-periodic solutions of inertial neural networks with time delays. Neural Process. Lett. 45, 523–538 (2017)

    Google Scholar 

  17. Xu, C., Zhang, Q.: Existence and global exponential stability of anti-periodic solutions for BAM neural networks with inertial term and delay. Neurocomputing 153, 108–116 (2015)

    Google Scholar 

  18. Huang, C., Liu, B.: New studies on dynamic analysis of inertial neural networks involving non-reduced order method. Neurocomputing 325, 283–287 (2019)

    Google Scholar 

  19. Huang, C., Yang, L., Liu, B.: New results on periodicity of non-autonomous inertial neural networks involving non-reduced order method. Neural Process. Lett. 50, 595–606 (2019)

    Google Scholar 

  20. Huang, C., Zhang, H.: Periodicity of non-autonomous inertial neural networks involving proportional delays and non-reduced order method. Int. J. Biomath. 12(2), 1–13 (2019)

    MathSciNet  MATH  Google Scholar 

  21. Zhang, J., Huang, C.: Dynamics analysis on a class of delayed neural networks involving inertial terms. Adv. Differ. Equ. 2020, 120 (2020). https://doi.org/10.1186/s13662-020-02566-4

    Article  MathSciNet  Google Scholar 

  22. Belley, J.-M., Bondo, E.: Anti-periodic solutions of Liénard equations with state dependent impulses. J. Differ. Equ. 261(7), 4164–4187 (2016)

    MATH  Google Scholar 

  23. Peng, J., Zhang, Y.: Heron triangles with figurate number sides. Acta Math. Hung. 157(2), 478–488 (2019)

    MathSciNet  MATH  Google Scholar 

  24. Huang, C., Wang, J., Huang, L.: Asymptotically almost periodicity of delayed Nicholson-type system involving patch structure. Electron. J. Differ. Equ. 2020, 61 (2020)

    Google Scholar 

  25. Huang, C., Wen, S., Huang, L.: Dynamics of anti-periodic solutions on shunting inhibitory cellular neural networks with multi-proportional delays. Neurocomputing 357, 47–52 (2019)

    Google Scholar 

  26. Huang, C., Long, X., Cao, J.: Stability of antiperiodic recurrent neural networks with multiproportional delays. Math. Methods Appl. Sci. 43(9), 6093–6102 (2020). https://doi.org/10.1002/mma.6350

    Article  Google Scholar 

  27. Li, Y., Meng, X., Xiong, L.: Pseudo almost periodic solutions for neutral type high-order Hopfield neural networks with mixed time-varying delays and leakage delays on time scales. Int. J. Mach. Learn. Cybern. 8(6), 1–13 (2017)

    Google Scholar 

  28. Li, J., Ying, J., Xie, D.: On the analysis and application of an ion size-modified Poisson–Boltzmann equation. Nonlinear Anal., Real World Appl. 47, 188–203 (2019)

    MathSciNet  MATH  Google Scholar 

  29. Xiao, B., Meng, H.: Existence and exponential stability of positive almost periodic solutions for high-order Hopfield neural networks. Appl. Math. Mech. 33, 532–542 (2009)

    MathSciNet  MATH  Google Scholar 

  30. Yao, L.: Global exponential stability on anti-periodic solutions in proportional delayed HIHNNs. J. Exp. Theor. Artif. Intell. (2020). https://doi.org/10.1080/0952813X.2020.1721571

    Article  Google Scholar 

  31. Xu, Y.: Exponential stability of weighted pseudo almost periodic solutions for HCNNs with mixed delays. Neural Process. Lett. 46(2), 507–519 (2017)

    Google Scholar 

  32. Huang, C., Liu, B., Tian, X.: Global convergence on asymptotically almost periodic SICNNs with nonlinear decay functions. Neural Process. Lett. 49, 625–641 (2019)

    Google Scholar 

  33. Huang, C., Kuang, H., Chen, X., Wen, F.: An LMI approach for dynamics of switched cellular neural networks with mixed delays. Abstr. Appl. Anal. 2013, 870486 (2013). https://doi.org/10.1155/2013/870486

    Article  MathSciNet  MATH  Google Scholar 

  34. Xu, Y.: Convergence on non-autonomous inertial neural networks with unbounded distributed delays. J. Exp. Theor. Artif. Intell. 32(3), 503–513 (2020). https://doi.org/10.1080/0952813X.2019.1652941

    Article  Google Scholar 

  35. Zhao, C., Wang, Z.: Exponential convergence of a SICNN with leakage delays and continuously distributed delays of neutral type. Neural Process. Lett. 41, 239–247 (2015)

    Google Scholar 

  36. Li, W., Huang, L., Ji, J.: Periodic solution and its stability of a delayed Beddington–Deangelis type predator–prey system with discontinuous control strategy. Math. Methods Appl. Sci. 42(13), 4498–4515 (2019)

    MathSciNet  MATH  Google Scholar 

  37. Iswarya, M., Raja, R., Rajchakit, G., Cao, J., Alzabut, J., Huang, C.: Existence, uniqueness and exponential stability of periodic solution for discrete-time delayed BAM neural networks based on coincidence degree theory and graph theoretic method. Mathematics 7(11), 1055 (2019). https://doi.org/10.3390/math7111055

    Article  Google Scholar 

  38. Zhao, J., Liu, J., Fang, L.: Anti-periodic boundary value problems of second-order functional differential equations. Bull. Malays. Math. Sci. Soc. 37(2), 311–320 (2014)

    MathSciNet  MATH  Google Scholar 

  39. Xu, Y., Cao, Q., Guo, X.: Stability on a patch structure Nicholson’s blowflies system involving distinctive delays. Appl. Math. Lett. 105, 106340 (2020). https://doi.org/10.1016/j.aml.2020.106340

    Article  MathSciNet  MATH  Google Scholar 

  40. Ke, Y., Miao, C.: Stability analysis of inertial Cohen–Grossberg-type neural networks with time delays. Neurocomputing 117, 196–205 (2013)

    Google Scholar 

  41. Hino, Y., Murakami, S., Naito, T.: Functional Differential Equations with Infinite Delay. Lecture in Mathematics, vol. 1473. Springer, Berlin (1991)

    MATH  Google Scholar 

  42. Duan, L., Huang, C.: Existence and global attractivity of almost periodic solutions for a delayed differential neoclassical growth model. Math. Methods Appl. Sci. 40(3), 814–822 (2017)

    MathSciNet  MATH  Google Scholar 

  43. Huang, C., Cao, J., Cao, J.: Stability analysis of switched cellular neural networks: a mode-dependent average dwell time approach. Neural Netw. 82, 84–99 (2016). https://doi.org/10.1016/j.neunet.2016.07.009

    Article  MATH  Google Scholar 

  44. Liu, B.: Finite-time stability of CNNs with neutral proportional delays and time-varying leakage delays. Math. Methods Appl. Sci. 40, 167–174 (2017)

    MathSciNet  MATH  Google Scholar 

  45. Huang, C., Yang, H., Cao, J.: Weighted pseudo almost periodicity of multi-proportional delayed shunting inhibitory cellular neural networks with D operator. Discrete Contin. Dyn. Syst., Ser. S (2020). https://doi.org/10.3934/dcdss.2020372

    Article  Google Scholar 

  46. Huang, C., Yang, Z., Yi, T., Zou, X.: On the basins of attraction for a class of delay differential equations with non-monotone bistable nonlinearities. J. Differ. Equ. 256(7), 2101–2114 (2014). https://doi.org/10.1016/j.jde.2013.12.015

    Article  MathSciNet  MATH  Google Scholar 

  47. Huang, C., Long, X., Huang, L., Fu, S.: Stability of almost periodic Nicholson’s blowflies model involving patch structure and mortality terms. Canad. Math. Bull., 63(2), 405–422 (2020). https://doi.org/10.4153/S0008439519000511

    Article  MathSciNet  MATH  Google Scholar 

  48. Huang, C., Zhang, H., Huang, L.: Almost periodicity analysis for a delayed Nicholson’s blowflies model with nonlinear density-dependent mortality term. Commun. Pure Appl. Anal. 18(6), 3337–3349 (2019)

    MathSciNet  Google Scholar 

  49. Hu, H., Yi, T., Zou, X.: On spatial-temporal dynamics of a Fisher–KPP equation with a shifting environment. Proc. Am. Math. Soc. 148, 213–221 (2020)

    MathSciNet  MATH  Google Scholar 

  50. Hu, H., Yuan, X., Huang, L., Huang, C.: Global dynamics of an SIRS model with demographics and transfer from infectious to susceptible on heterogeneous networks. Math. Biosci. Eng. 16(5), 5729–5749 (2019)

    MathSciNet  Google Scholar 

  51. Hu, H., Zou, X.: Existence of an extinction wave in the Fisher equation with a shifting habitat. Proc. Am. Math. Soc. 145(11), 4763–4771 (2017)

    MathSciNet  MATH  Google Scholar 

  52. Wang, J., Huang, C., Huang, L.: Discontinuity-induced limit cycles in a general planar piecewise linear system of saddle-focus type. Nonlinear Anal. Hybrid Syst. 33, 162–178 (2019)

    MathSciNet  MATH  Google Scholar 

  53. Wang, J., Chen, X., Huang, L.: The number and stability of limit cycles for planar piecewise linear systems of node-saddle type. J. Math. Anal. Appl. 469(1), 405–427 (2019)

    MathSciNet  MATH  Google Scholar 

  54. Zhang, H., Qian, C.: Convergence analysis on inertial proportional delayed neural networks. Adv. Differ. Equ. (2020). https://doi.org/10.1186/s13662-020-02737-3

    Article  MathSciNet  Google Scholar 

  55. Qian, C.: New periodic stability for a class of Nicholson’s blowflies models with multiple different delays. Int. J. Control (2020). https://doi.org/10.1080/00207179.2020.1766118

    Article  Google Scholar 

  56. Huang, C., Su, R., Cao, J., Xiao, S.: Asymptotically stable high-order neutral cellular neural networks with proportional delays and D operators. Math. Comput. Simul. 107, 127–135 (2020). https://doi.org/10.1016/j.matcom.2019.06.001

    Article  MathSciNet  Google Scholar 

  57. Zhang, X., Hu, H.: Convergence in a system of critical neutral functional differential equations. Appl. Math. Lett. 107, 106385 (2020). https://doi.org/10.1016/j.aml.2020.106385

    Article  MathSciNet  MATH  Google Scholar 

  58. Tan, Y., Huang, C., Sun, B., Wang, T.: Dynamics of a class of delayed reaction–diffusion systems with Neumann boundary condition. J. Math. Anal. Appl. 458(2), 1115–1130 (2018)

    MathSciNet  MATH  Google Scholar 

  59. Huang, C., Yang, X., Cao, J.: Stability analysis of Nicholson’s blowflies equation with two different delays. Math. Comput. Simul. 171, 201–206 (2020). https://doi.org/10.1016/j.matcom.2019.09.023

    Article  MathSciNet  Google Scholar 

  60. Zhang, Y.: Some observations on the Diophantine equation \(f(x)f(y) = f(z)^{2}\). Colloq. Math. 142(2), 275–283 (2016)

    MathSciNet  MATH  Google Scholar 

  61. Li, L., Jin, Q., Yao, B.: Regularity of fuzzy convergence spaces. Open Math. 16, 1455–1465 (2018)

    MathSciNet  MATH  Google Scholar 

  62. Lv, B., Huang, L., Wang, K.: Endomorphisms of twisted Grassmann graphs. Graphs Comb. 33(1), 157–169 (2017)

    MathSciNet  MATH  Google Scholar 

  63. Huang, C., Peng, C., Chen, X., Wen, F.: Dynamics analysis of a class of delayed economic model. Abstr. Appl. Anal. 2013, 962738 (2013). https://doi.org/10.1155/2013/962738

    Article  MathSciNet  MATH  Google Scholar 

  64. Zhang, Y.: Right triangle and parallelogram pairs with a common area and a common perimeter. J. Number Theory 164, 179–190 (2016)

    MathSciNet  MATH  Google Scholar 

  65. Huang, L., Su, H., Tang, G., Wang, J.: Bilinear forms graphs over residue class rings. Linear Algebra Appl. 523, 13–32 (2017)

    MathSciNet  MATH  Google Scholar 

  66. Cao, Q., Guo, X.: Anti-periodic dynamics on high-order inertial Hopfield neural networks involving time-varying delays. AIMS Math. 5(6), 5402–5421 (2020). https://doi.org/10.3934/math.2020347

    Article  Google Scholar 

  67. Li, X., Liu, Y., Wu, J.: Flocking and pattern motion in a modified Cucker–Smale model. Bull. Korean Math. Soc. 53(5), 1327–1339 (2016)

    MathSciNet  MATH  Google Scholar 

  68. Xie, Y., Li, Q., Zhu, K.: Attractors for nonclassical diffusion equations with arbitrary polynomial growth nonlinearity. Nonlinear Anal., Real World Appl. 31, 23–37 (2016)

    MathSciNet  MATH  Google Scholar 

  69. Xie, Y., Li, Y., Zeng, Y.: Uniform attractors for nonclassical diffusion equations with memory. J. Funct. Spaces 2016, 5340489 (2016). https://doi.org/10.1155/2016/5340489

    Article  MathSciNet  MATH  Google Scholar 

  70. Wang, F., Wang, P., Yao, Z.: Approximate controllability of fractional partial differential equation. Adv. Differ. Equ. 2015, 367 (2015). https://doi.org/10.1186/s13662-015-0692-3

    Article  MathSciNet  MATH  Google Scholar 

  71. Liu, Y., Wu, J.: Multiple solutions of ordinary differential systems with min-max terms and applications to the fuzzy differential equations. Adv. Differ. Equ. 2015, 379 (2015). https://doi.org/10.1186/s13662-015-0708-z

    Article  MathSciNet  MATH  Google Scholar 

  72. Yan, L., Liu, J., Luo, Z.: Existence and multiplicity of solutions for second-order impulsive differential equations on the half-line. Adv. Differ. Equ. 2013, 293 (2013). https://doi.org/10.1186/1687-1847-2013-293

    Article  MathSciNet  MATH  Google Scholar 

  73. Huang, L., Lv, B.: Cores and independence numbers of Grassmann graphs. Graphs Comb. 33(6), 1607–1620 (2017)

    MathSciNet  MATH  Google Scholar 

  74. Liu, W.: An incremental approach to obtaining attribute reduction for dynamic decision systems. Open Math. 14, 875–888 (2016)

    MathSciNet  MATH  Google Scholar 

  75. Huang, C., Zhang, H., Cao, J., Hu, H.: Stability and Hopf bifurcation of a delayed prey–predator model with disease in the predator. Int. J. Bifurc. Chaos 29(7), 1–23 (2019). https://doi.org/10.1142/S0218127419500913

    Article  MathSciNet  MATH  Google Scholar 

  76. Wang, F., Yao, Z.: Approximate controllability of fractional neutral differential systems with bounded delay. Fixed Point Theory 17, 495–508 (2016)

    MathSciNet  MATH  Google Scholar 

  77. Zhou, S., Jiang, Y.: Finite volume methods for N-dimensional time fractional Fokker–Planck equations. Bull. Malays. Math. Sci. Soc. 42(6), 3167–3186 (2019)

    MathSciNet  MATH  Google Scholar 

  78. Huang, C., Wen, S. Li, M., Wen, F., Yang, X.: An empirical evaluation of the influential nodes for stock market network: Chinese A shares case. Finance Res. Lett. (2020). https://doi.org/10.1016/j.frl.2020.101517

    Article  Google Scholar 

  79. Liu, F., Feng, L., Vo, A., Li, J.: Unstructured-mesh Galerkin finite element method for the two-dimensional multi-term time-space fractional Bloch–Torrey equations on irregular convex domains. Comput. Math. Appl. 78(5), 1637–1650 (2019)

    MathSciNet  Google Scholar 

  80. Jin, Q., Li, L., Lang, G.: p-Regularity and p-regular modification in T-convergence spaces. Mathematics, 7(4), 370 (2019). https://doi.org/10.3390/math7040370

    Article  Google Scholar 

  81. Huang, L.: Endomorphisms and cores of quadratic forms graphs in odd characteristic. Finite Fields Appl. 55, 284–304 (2019)

    MathSciNet  MATH  Google Scholar 

  82. Huang, L., Lv, B., Wang, K.: Erdos–Ko–Rado theorem, Grassmann graphs and \(p^{s}\)-Kneser graphs for vector spaces over a residue class ring. J. Comb. Theory, Ser. A 164, 125–158 (2019)

    MathSciNet  MATH  Google Scholar 

  83. Li, Y., Vuorinen, M., Zhou, Q.: Characterizations of John spaces. Monatshefte Math. 188(3), 547–559 (2019)

    MathSciNet  MATH  Google Scholar 

  84. Huang, L., Lv, B., Wang, K.: The endomorphisms of Grassmann graphs. Ars Math. Contemp. 10(2), 383–392 (2016)

    MathSciNet  MATH  Google Scholar 

  85. Gong, X., Wen, F., He, Z., Yang, J., Yang, X.: Extreme return, extreme volatility and investor sentiment. Filomat 30(15), 3949–3961 (2016)

    MATH  Google Scholar 

  86. Jiang, Y., Huang, B.: A note on the value distribution of \(f^{1} (f^{(k)})^{n}\). Hiroshima Math. J. 46(2), 135–147 (2016)

    MathSciNet  MATH  Google Scholar 

  87. Huang, L., Huang, J., Zhao, K.: On endomorphisms of alternating forms graph. Discrete Math. 338(3), 110–121 (2015)

    MathSciNet  MATH  Google Scholar 

  88. Huang, C., Qiao, Y., Huang, L., Agarwal, R.P.: Dynamical behaviors of a food-chain model with stage structure and time delays. Adv. Differ. Equ. 2018, 186 (2018). https://doi.org/10.1186/s13662-018-1589-8

    Article  MathSciNet  MATH  Google Scholar 

  89. Huang, C., Cao, J., Wen, F., Yang, X.: Stability analysis of SIR model with distributed delay on complex networks. PLoS ONE 11(8), e0158813 (2016). https://doi.org/10.1371/journal.pone.0158813

    Article  Google Scholar 

  90. Zhou, Y., Wan, X., Huang, C., Yang, X.: Finite-time stochastic synchronization of dynamic networks with nonlinear coupling strength via quantized intermittent control. Appl. Math. Comput. 376, 125157 (2020). https://doi.org/10.1016/j.amc.2020.125157

    Article  MathSciNet  MATH  Google Scholar 

  91. Wang, W., Huang, C., Huang, C., Cao, J., Lu, J., Wang, L.: Bipartite formation problem of second-order nonlinear multi-agent systems with hybrid impulses. Appl. Math. Comput. 370, 124926 (2020). https://doi.org/10.1016/j.amc.2019.124926

    Article  MathSciNet  MATH  Google Scholar 

  92. Wang, W., Liu, F., Chen, W.: Exponential stability of pseudo almost periodic delayed Nicholson-type system with patch structure. Math. Methods Appl. Sci. 42(2), 592–604 (2019)

    MathSciNet  MATH  Google Scholar 

  93. Wang, W.: Finite-time synchronization for a class of fuzzy cellular neural networks with time-varying coefficients and proportional delays. Fuzzy Sets Syst. 338, 40–49 (2018)

    MathSciNet  MATH  Google Scholar 

  94. Wang, W., Shi, C., Chen, W.: Stochastic Nicholson-type delay differential system. Int. J. Control, 1–14 (2019). https://doi.org/10.1080/00207179.2019.1651941

    Article  Google Scholar 

  95. Qian, C., Hu, Y.: Novel stability criteria on nonlinear density-dependent mortality Nicholson’s blowflies systems in asymptotically almost periodic environments. J. Inequal. Appl. (2020). https://doi.org/10.1186/s13660-019-2275-4

    Article  MathSciNet  Google Scholar 

Download references

Acknowledgements

The authors would like to thank the anonymous referees and the editor for very helpful suggestions and comments which led to improvements of the original paper.

Availability of data and materials

Data sharing not applicable to this article as no datasets were generated or analyzed during the current study.

Funding

This work was supported by Natural Science Foundation of Hunan Province of China (Nos. 2019JJ40141, 2019JJ40142), and Scientific Research Fund of Hunan Provincial Education Department (19A267).

Author information

Authors and Affiliations

Authors

Contributions

The two authors contributed equally to this work. All authors read and approved the final manuscript.

Corresponding author

Correspondence to Luogen Yao.

Ethics declarations

Competing interests

The authors declare that they have no competing interests.

Rights and permissions

Open Access This article is licensed under a Creative Commons Attribution 4.0 International License, which permits use, sharing, adaptation, distribution and reproduction in any medium or format, as long as you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons licence, and indicate if changes were made. The images or other third party material in this article are included in the article’s Creative Commons licence, unless indicated otherwise in a credit line to the material. If material is not included in the article’s Creative Commons licence and your intended use is not permitted by statutory regulation or exceeds the permitted use, you will need to obtain permission directly from the copyright holder. To view a copy of this licence, visit http://creativecommons.org/licenses/by/4.0/.

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Yao, L., Cao, Q. Anti-periodicity on high-order inertial Hopfield neural networks involving mixed delays. J Inequal Appl 2020, 182 (2020). https://doi.org/10.1186/s13660-020-02444-3

Download citation

  • Received:

  • Accepted:

  • Published:

  • DOI: https://doi.org/10.1186/s13660-020-02444-3

Keywords