Neural Computing and Applications

, Volume 24, Issue 5, pp 1047–1058

Anti-periodic solutions for HCNNs with time-varying delays in the leakage terms

Authors

    • Department of MathematicsXiangnan College
Original Article

DOI: 10.1007/s00521-012-1330-6

Cite this article as:
Xu, Y. Neural Comput & Applic (2014) 24: 1047. doi:10.1007/s00521-012-1330-6

Abstract

In this paper, a class of high-order cellular neural networks model is considered with the introduction of time-varying delays in the leakage terms. By using differential inequality techniques, some very verifiable and practical delay-dependent criteria on the existence and global exponential stability of anti-periodic solution for the model are derived. Even for the model without leakage delays, the criteria are shown to be less conservative than many recent publications. Moreover, some examples and remarks are given to demonstrate the feasibility of our method.

Keywords

High-order cellular neural networksAnti-periodic solutionExponential stabilityTime-varying delayLeakage term

1 Introduction

As pointed out in [1], high-order cellular neural networks (HCNNs), which include both the Cohen–Grossberg neural network and the Hopfield neural network as special cases, allow high-order interactions between neurons and, therefore, have stronger approximation property, faster convergence rate, greater storage capacity, and higher fault tolerance than the traditional first-order neural networks. Furthermore, in the past decade, HCNNs have attracted many attentions due to its wide range of applications in many fields such as signal and image processing, pattern recognition, optimization, and many other subjects. There have been extensive results on the problem of global stability of periodic solutions and anti-periodic solutions of HCNNs in the literature (see [26]). Recently, some attention has been paid to neural networks with time delay in the leakage (or “forgetting") term (see [713]). However, to the best of our knowledge, few authors have investigated the problem on anti-periodic solutions for HCNNs with delays in the leakage terms. Motivated by the above argument, in this present paper, we shall consider the existence and exponential stability of the anti-periodic solutions for the following HCNNs with time-varying delays in the leakage terms:
$$ \begin{aligned} x'_{i}(t) &= -c_{i}(t) x_{i}(t-\delta_i(t)) +\sum^n_{j=1}a_{ij}(t)f_{j}(x_{j}(t-\tau_{ij}(t))) \\ &\quad+\sum\limits_{j=1}^{n}\sum\limits_{l=1}^{n}b_{ijl}(t)g_{j}(x_{j}(t-\alpha_{ijl}(t)))g_{l}(x_{l}(t-\beta_{ijl}(t)))\\ &\quad+\sum\limits^n_{j=1}\sum\limits^n_{l=1}d_{ijl}(t) \int\limits_{0}^{\infty}\sigma_{ijl}(u)h_{j}(x_{j}(t-u))\hbox{d}u\quad \times \int\limits_{0}^{\infty}\nu_{ijl} (u) h_{l}(x_{l}(t-u))\hbox{d}u +I_{i}(t),\quad i=1, 2, \ldots, n, \end{aligned} $$
(1.1)
in which n corresponds to the number of units in a neural network, xi(t) corresponds to the state vector of the ith unit at the time tci(t) represents the rate with which the ith unit will reset its potential to the resting state in isolation when disconnected from the network and external inputs, aij(t), bijl(t), and dijl(t) are the first- and second-order connection weights of the neural network, δi(t) ≥ 0 corresponds to the time-varying leakage delays, αijl(t) ≥ 0, βijl(t) ≥ 0, and τij(t) ≥ 0 correspond to the transmission delays, σijl(u) and νijl (u) correspond to the transmission delay kernels, Ii(t) denotes the external inputs at time tfjgj, and hj are the activation functions of signal transmission.

The main purpose of this paper is to give the conditions for the existence and exponential stability of the anti-periodic solutions for system (1.1). By applying differential inequality techniques, we derive some new sufficient conditions ensuring the existence, uniqueness, and exponential stability of the anti-periodic solution for system (1.1), which are new and complement previously known results. Moreover, an example is also provided to illustrate the effectiveness of our results.

Let \(u(t):R\longrightarrow R \) be continuous in t. u(t) is said to be T-anti-periodic on R if
$$ u(t+T)=-u(t)\quad \hbox{for\,all}\quad t\in R. $$
Throughout this paper, for \(i, j, l=1, 2, \ldots, n\), it will be assumed that \(c_{i}, I_{i}, a_{ij}, b_{ijl}, d_{ijl}:R\rightarrow R\) and \(\delta_{i}, \tau_{ij}, \alpha_{ijl}, \beta_{ijl}:R\rightarrow [0, +\infty)\) are bounded continuous functions, \(\sigma_{ijl}, \nu_{ijl}:[0, +\infty)\rightarrow R\) are continuous functions, |σijl (t)|eκt and |νijl (t)|eκt are integrable on \([0, +\infty)\) for a certain positive constant κ, and
$$ c_{i}(t+T) = c_{i}(t ) , a_{ij}(t+T)f_{j}(v)=-a_{ij}(t)f_{j}(-v), $$
(1.2)
$$ b_{ijl}(t+T)g_{j}(v_{j})g_{l}(v_{l}) =-b_{ijl}(t )g_{j}(-v_{j})g_{l}(-v_{l_{}}) , $$
(1.3)
$$ \begin{aligned} &d_{ijl}(t+T) \int\limits_{0}^{\infty}\sigma_{ijl} (u)h_{j}(v_{j}(t-u))\hbox{d}u\int\limits_{0}^{\infty}\nu_{ijl} (u) h_{l}( v_{l}(t-u))\hbox{d}u \\ &=-d_{ijl}(t)\int\limits_{0}^{\infty}\sigma_{ijl} (u)h_{j}(-v_{j}(t-u))\hbox{d}u\int\limits_{0}^{\infty}\nu_{ijl} (u) h_{l}(-v_{l}(t-u))\hbox{d}u , \end{aligned} $$
(1.4)
$$ \delta_{i}(t+T)=\delta_{i}(t),\quad \tau_{ij}(t+T)=\tau_{ij}(t),\quad I_{i}(t+T)=-I_{i}(t), $$
(1.5)
$$ \alpha_{ijl}(t+T)=\alpha_{ijl}(t),\quad \beta_{ijl}(t+T)=\beta_{ijl}(t), $$
(1.6)
where \( t,v \in R , v_{j}\) and vl are real-valued bounded continuous function defined on R.
For bounded continuous functions f, we set
$$ f^{-}= \inf\limits_{t\in R}|f(t)|, \quad f^{+}= \sup\limits_{t\in R}|f(t)| . $$

In order to investigate the anti-periodic solution of HCNNs (1.1), we also give some usual assumptions.

(H1)

there exist nonnegative constants LjfLjgLjhMjg and Mjh such that
$$ |f_{j}(u )-f_{j}(v )| \leq L^{f}_{j}|u -v |,\quad |g_{j}(u )-g_{j}(v )| \leq L^{g}_{j}|u -v |,\quad |h_{j}(u )-h_{j}(v )| \leq L^{h}_{j}|u -v |, $$
and
$$ |g_{j}(u )|\leq M^{g}_{j},\quad |h_{j}(u )|\leq M^{h}_{j}, $$
where \(u, v \in R,\, j=1, 2, \ldots, n .\)

(H2)

there exist positive constants \(\xi_{1}, \xi_{2}, \ldots, \xi_{n}\) and η such that for all t > 0, the following inequality holds
$$ \begin{aligned} &-\left[ c_i (t) -c_i (t) \delta_{i}(t) c_i ^{+}\right] \xi_{i} + \sum\limits_{j = 1}^n \left(| a_{ij}(t)| +a_{ij}^{+}c_i (t) \delta_{i}(t)\right) L^{f}_{j} \xi_{j} \\ & \quad+ \sum\limits_{j=1}^{n}\sum\limits_{l=1}^{n} \left(|b_{ijl}(t)| +b_{ijl}^{+}c_i (t) \delta_{i}(t)\right)\left(L^{g}_{j}M^{g}_{l} \xi_{j}+M^{g}_{j}L^{g}_{l} \xi_{l}\right) \\ &\quad+\sum\limits^n_{j=1}\sum\limits^n_{l=1}\left[\left(|d_{ijl}(t)|+d_{ijl}^{+}c_i (t) \delta_{i}(t)\right) \int\limits_{0}^{\infty}|\sigma_{ijl} (u)| \hbox{d}u \int\limits_{0}^{\infty}|\nu_{ijl} (u)|\hbox{d}u \left(L^{h}_{j}M^{h}_{l} \xi_{j}+ M^{h}_{j}L^{h}_{l} \xi_{l}\right)\right] \\ &\quad< -\eta,\quad t \geq0, \quad i= 1, 2,\ldots,n, \end{aligned} $$
The initial conditions associated with system (1.1) are of the form
$$ x_{i}(s)=\varphi_{i}(s),\quad s\in (-\infty, 0],\quad i=1,2,\ldots,n, $$
(1.7)
where \(\varphi_{i}(\cdot)\) denotes real-valued bounded continuous function defined on \((-\infty, 0]. \)

The rest of this paper is organized as follows. In Sect. 2, we give some preliminary lemmas. In Sect. 3, we present our main results. In Sect. 4, we present some examples and remarks to illustrate the effectiveness of our results.

2 Preliminary lemmas

In this section, we present two important lemmas, which are useful to prove our main results in Sect. 3.

Lemma 2.1

Let (H1) and (H2) hold. Suppose that\(x(t)= (x_{1}(t), x_{2}(t),\ldots, x_{n}(t))^{T} \)is a solution of system (1.1) with initial conditions
$$ x_{i}(s)= \varphi _{i}(s),\quad | \varphi _{i}(s)|<\xi_{i}\frac{\gamma}{\eta},\quad s\in (-\infty, 0],\quad i=1,2,\ldots,n, $$
(2.1)
where
$$ \begin{aligned} \gamma&=2+\max\limits_{1\leq i \leq n}\left\{\sum_{j=1}^{n} a_{ij}^{+} |f_{j}(0)|\left(1+c_{i} ^{+} \delta_{i} ^{+}\right)+\sum\limits^n_{j=1}\sum\limits^n_{l=1} b_{ijl} ^{+}|g_{j}(0)|M^{g}_{l}\left(1+c_{i} ^{+} \delta_{i} ^{+}\right) \right. \\ &\left.\quad+\sum\limits^n_{j=1}\sum\limits^n_{l=1} d_{ijl} ^{+} \int\limits_{0}^{\infty} |\sigma_{ijl} (u)| \hbox{d}u\int\limits_{0}^{\infty}|\nu_{ijl} (u)| \hbox{d}u |h_{j}(0)|M^{h}_{l}\left(1+c_{i} ^{+} \delta_{i} ^{+}\right) + I_{i}^{+}\left(1+c_{i} ^{+} \delta_{i} ^{+}\right) \right\}. \end{aligned} $$
(2.2)
Then,
$$ | x _{i}(t)| <\xi_{i}\frac{\gamma}{\eta},\quad \hbox{for\,all}\quad t\geq 0,\quad i=1,2,\ldots,n. $$
(2.3)

Proof

Assume, by way of contradiction, that (2.3) does not hold. Then, we may assume that there exist \(i\in \{1,2,\ldots,n \}\) and t* > 0 such that
$$ \left| {{x_i}(t_{*} )} \right| = \xi_{i}\frac{\gamma}{\eta}\quad \hbox{and}\quad\left| {{x_j}(t)} \right| < \xi_{j}\frac{\gamma}{\eta}\quad\hbox{for}\quad\hbox{all}\quad t \in (-\infty, t_{*} ),\quad j = 1,2, \ldots ,n. $$
(2.4)
From system (1.1), we derive
$$ \begin{aligned} x'_i(t) &= -c_{i}(t) x_{i}(t-\delta_i(t)) +\sum^n_{j=1}a_{ij}(t)f_{j}(x_{j}(t-\tau_{ij}(t))) \\ &\quad+\sum\limits_{j=1}^{n}\sum\limits_{l=1}^{n}b_{ijl}(t)g_{j}(x_{j}(t-\alpha_{ijl}(t)))g_{l}(x_{l}(t-\beta_{ijl}(t)))\\ &\quad+\sum\limits^n_{j=1}\sum\limits^n_{l=1}d_{ijl}(t) \int\limits_{0}^{\infty}\sigma_{ijl} (u)h_{j}(x_{j}(t-u))\hbox{d}u\quad\times \int\limits_{0}^{\infty}\nu_{ijl} (u) h_{l}(x_{l}(t-u))\hbox{d}u +I_{i}(t) \\ &= -c_{i}(t) x_{i}(t) +c_{i}(t)\left[x_{i}(t) - x_{i}(t-\delta_i(t))\right]+\sum^n_{j=1}a_{ij}(t)f_{j}(x_{j}(t-\tau_{ij}(t))) \\ &\quad+\sum\limits_{j=1}^{n}\sum\limits_{l=1}^{n}b_{ijl}(t)g_{j}(x_{j}(t-\alpha_{ijl}(t)))g_{l}(x_{l}(t-\beta_{ijl}(t)))\\ &\quad+\sum\limits^n_{j=1}\sum\limits^n_{l=1}d_{ijl}(t) \int\limits_{0}^{\infty}\sigma_{ijl} (u)h_{j}(x_{j}(t-u))\hbox{d}u\int\limits_{0}^{\infty}\nu_{ijl} (u) h_{l}(x_{l}(t-u))\hbox{d}u +I_{i}(t) \\ &= -c_{i}(t) x_{i}(t ) +c_{i}(t)\int\limits_{t -\delta_{i}(t )}^{t }x_{i}'(s ) \hbox{d}s +\sum^n_{j=1}a_{ij}(t)f_{j}(x_{j}(t-\tau_{ij}(t))) \\ &\quad+\sum\limits_{j=1}^{n}\sum\limits_{l=1}^{n}b_{ijl}(t)g_{j}(x_{j}(t-\alpha_{ijl}(t)))g_{l}(x_{l}(t-\beta_{ijl}(t)))\\ &\quad+\sum\limits^n_{j=1}\sum\limits^n_{l=1}d_{ijl}(t) \int\limits_{0}^{\infty}\sigma_{ijl} (u)h_{j}(x_{j}(t-u))\hbox{d}u\quad\times \int\limits_{0}^{\infty}\nu_{ijl} (u) h_{l}(x_{l}(t-u))\hbox{d}u \\ &\quad+I_{i}(t),\quad i = 1,2, \ldots ,n. \end{aligned} $$
(2.5)
Calculating the upper left derivative of \(\left| x_{i}(t) \right|\), together with (2.2), (2.4), (2.5), and (H1) − (H2), we obtain
$$ \begin{aligned} 0 &\leq D^{-} \left| {{x_i}(t_{*} )}\right|\\ &\leq -c _{i}(t_{*})| x_{i}(t _{*})|+\left|c_{i}(t_{*})\int\limits_{t_{*} -\delta_{i}(t_{*} )}^{t_{*} }x_{i}'(s) \hbox{d}s+\sum^n_{j=1}a_{ij}(t_{*})f_{j}(x_{j}(t_{*}-\tau_{ij}(t_{*})))\right. \\&\quad+\sum\limits_{j=1}^{n}\sum\limits_{l=1}^{n}b_{ijl}(t_{*})g_{j}(x_{j}(t_{*}-\alpha_{ijl}(t_{*})))g_{l}(x_{l}(t_{*}-\beta_{ijl}(t_{*})))\\&\quad+\sum\limits^n_{j=1}\sum\limits^n_{l=1}d_{ijl}(t_{*})\int\limits_{0}^{\infty}\sigma_{ijl}(u)h_{j}(x_{j}(t_{*}-u))\hbox{d}u\int\limits_{0}^{\infty}\nu_{ijl}(u) h_{l}(x_{l}(t_{*}-u))\hbox{d}u+I_{i}(t_{*})| \\ &\leq-c_{i}(t_{*}) |x_{i}(t _{*}) |+ c_{i}(t_{*})\int\limits_{t_{*}-\delta_{i}(t_{*} )}^{t_{*} } |x_{i}'(s) |\hbox{d}s \\&\quad+\sum^n_{j=1}|a_{ij}(t_{*})|(|f_{j}(x_{j}(t_{*}-\tau_{ij}(t_{*})))-f_{j}(0)|+|f_{j}(0)|)\\ &\quad+\sum\limits_{j=1}^{n}\sum\limits_{l=1}^{n}|b_{ijl}(t_{*})|(|g_{j}(x_{j}(t_{*}-\alpha_{ijl}(t_{*})))-g_{j}(0)|+|g_{j}(0)|)M^{g}_{l}\\&\quad+\left.\sum\limits^n_{j=1}\sum\limits^n_{l=1}|d_{ijl}(t_{*})|\int\limits_{0}^{\infty} |\sigma_{ijl} (u)|(|h_{j}(x_{j}(t_{*}-u))-h_{j}(0)|+|h_{j}(0)|)\hbox{d}u\int\limits_{0}^{\infty}|\nu_{ijl} (u)| M^{h}_{l}\hbox{d}u+|I_{i}(t_{*})\right| \\&\leq -c_{i}(t_{*}) |x_{i}(t _{*}) |+c_{i}(t_{*})\int\limits_{t_{*} -\delta_{i}(t_{*} )}^{t_{*} }|-c_{i}(s) x_{i}(s-\delta_i(s)) \\ &\quad+\sum^n_{j=1}a_{ij}(s)f_{j}(x_{j}(s-\tau_{ij}(s))) \\&\quad+\sum\limits_{j=1}^{n}\sum\limits_{l=1}^{n}b_{ijl}(s)g_{j}(x_{j}(s-\alpha_{ijl}(s)))g_{l}(x_{l}(s-\beta_{ijl}(s)))\\ &\quad+\sum\limits^n_{j=1}\sum\limits^n_{l=1}d_{ijl}(s)\int\limits_{0}^{\infty}\sigma_{ijl}(u)h_{j}(x_{j}(s-u))\hbox{d}u\quad\times \int\limits_{0}^{\infty}\nu_{ijl} (u)h_{l}(x_{l}(s-u))\hbox{d}u +I_{i}(s) |\hbox{d}s \\&\quad+\sum^n_{j=1}|a_{ij}(t_{*})|(L^{f}_{j}|x_{j}(t_{*}-\tau_{ij}(t_{*}))|+|f_{j}(0)|) \\ &\quad+\sum\limits_{j=1}^{n}\sum\limits_{l=1}^{n}|b_{ijl}(t_{*})|(L^{g}_{j}|x_{j}(t_{*}-\alpha_{ijl}(t_{*}))|+|g_{j}(0)|)M^{g}_{l} \\&\quad+\sum\limits^n_{j=1}\sum\limits^n_{l=1}|d_{ijl}(t_{*})|\int\limits_{0}^{\infty} |\sigma_{ijl}(u)|(L^{h}_{j}|x_{j}(t_{*}-u))|+|h_{j}(0)|)\hbox{d}u\int\limits_{0}^{\infty}|\nu_{ijl} (u)|M^{h}_{l}\hbox{d}u+|I_{i}(t_{*})| \\ &\leq -c_{i}(t_{*})|x_{i}(t _{*}) |+ c_{i}(t_{*}) \int\limits_{t_{*} -\delta_{i}(t_{*})}^{t_{*} } [ c_{i} ^{+} |x_{i}(s-\delta_i(s))|\\&\quad+\sum^n_{j=1} a_{ij} ^{+}(L^{f}_{j}|x_{j}(s-\tau_{ij}(s))|+|f_{j}(0)|) \\ &\quad+\sum\limits_{j=1}^{n}\sum\limits_{l=1}^{n} b_{ijl}^{+}(L^{g}_{j}|x_{j}(s-\alpha_{ijl}(s)) |+|g_{j}(0)|)M^{g}_{l} \\&\quad+\sum\limits^n_{j=1}\sum\limits^n_{l=1} d_{ijl} ^{+}\int\limits_{0}^{\infty} |\sigma_{ijl}(u)|(L^{h}_{j}|x_{j}(s-u))|+|h_{j}(0)|)\hbox{d}u\int\limits_{0}^{\infty}|\nu_{ijl} (u)|M^{h}_{l}\hbox{d}u+I^{+}_{i} ]\hbox{d}s \\&\quad+\sum^n_{j=1}|a_{ij}(t_{*})|(L^{f}_{j}|x_{j}(t_{*}-\tau_{ij}(t_{*}))|+|f_{j}(0)|) \\ &\quad+\sum\limits_{j=1}^{n}\sum\limits_{l=1}^{n}|b_{ijl}(t_{*})|(L^{g}_{j}|x_{j}(t_{*}-\alpha_{ijl}(t_{*}))|+|g_{j}(0)|)M^{g}_{l} \\&\quad+\sum\limits^n_{j=1}\sum\limits^n_{l=1}|d_{ijl}(t_{*})|\int\limits_{0}^{\infty}|\sigma_{ijl}(u)|(L^{h}_{j}|x_{j}(t_{*}-u))|+|h_{j}(0)|)\quad\times\hbox{d}u\int\limits_{0}^{\infty}|\nu_{ijl} (u)| M^{h}_{l}\hbox{d}u+|I_{i}(t_{*})|\\ &\leq -[c_{i}(t_{*}) -c_{i}(t_{*})\delta_{i}(t_{*} )c_{i} ^{+} ]|x_{i}(t _{*}) |+\sum_{j=1}^{n}(|a_{ij}(t_{*})|+a_{ij}^{+}c_{i}(t_{*})\delta_{i}(t_{*} )) L ^{f}_{j}\xi_{j}\frac{\gamma}{\eta} \\&\quad+\sum\limits_{j=1}^{n}\sum\limits_{l=1}^{n}(|b_{ijl}(t_{*})|+b_{ijl}^{+}c_{i}(t_{*})\delta_{i}(t_{*} )) L^{g}_{j}M^{g}_{l}\xi_{j}\frac{\gamma}{\eta} \\&\quad+\sum\limits^n_{j=1}\sum\limits^n_{l=1}(|d_{ijl}(t_{*})|+d_{ijl}^{+}c_{i}(t_{*})\delta_{i}(t_{*} )) \int\limits_{0}^{\infty} |\sigma_{ijl} (u)|\hbox{d}u\int\limits_{0}^{\infty}|\nu_{ijl} (u)|duL^{h}_{j}M^{h}_{l}\xi_{j}\frac{\gamma}{\eta} \\ &\quad+\sum_{j=1}^{n} a_{ij}^{+} |f_{j}(0)|(1+c_{i} ^{+} \delta_{i}^{+})+\sum\limits^n_{j=1}\sum\limits^n_{l=1} b_{ijl}^{+}|g_{j}(0)|M^{g}_{l}(1+c_{i} ^{+} \delta_{i} ^{+}) \\&\quad+\sum\limits^n_{j=1}\sum\limits^n_{l=1} d_{ijl} ^{+}\int\limits_{0}^{\infty} |\sigma_{ijl} (u)|\hbox{d}u\quad\times\int\limits_{0}^{\infty}|\nu_{ijl} (u)| \hbox{d}u|h_{j}(0)|M^{h}_{l}(1+c_{i} ^{+} \delta_{i} ^{+}) \\&\quad+I_{i}^{+}(1+c_{i} ^{+} \delta_{i} ^{+}) \\ &\leq\left\{-[c_{i}(t_{*}) -c_{i}(t_{*}) \delta_{i}(t_{*} )c_{i} ^{+}]\xi_{i} + \sum_{j=1}^{n}(|a_{ij}(t_{*})|+a_{ij}^{+}c_{i}(t_{*})\delta_{i}(t_{*} )) L ^{f}_{j}\xi_{j} \right. \\ &\quad+\sum\limits_{j=1}^{n}\sum\limits_{l=1}^{n}(|b_{ijl}(t_{*})|+b_{ijl}^{+}c_{i}(t_{*})\delta_{i}(t_{*} )) L^{g}_{j}M^{g}_{l}\xi_{j} \\&\quad\left.+\sum\limits^n_{j=1}\sum\limits^n_{l=1}(|d_{ijl}(t_{*})|+d_{ijl}^{+}c_{i}(t_{*})\delta_{i}(t_{*} )) \int\limits_{0}^{\infty} |\sigma_{ijl} (u)|\hbox{d}u\int\limits_{0}^{\infty}|\nu_{ijl} (u)|duL^{h}_{j}M^{h}_{l}\xi_{j} \right\}\frac{\gamma}{\eta} + (\gamma-2)\\ &< -\eta\frac{\gamma}{\eta}+ \gamma= 0. \end{aligned} $$
It is a contradiction and shows that (2.3) holds. The proof is now completed. \(\square\)

Remark 2.1

In view of the boundedness of this solution, from the theory of functional differential equations with infinite delay in [14], it follows that the solution of system (1.1) with initial conditions (2.1) can be defined on \([ 0, +\infty). \)

Lemma 2.2

Suppose that (H1) − (H2) are true. Let\(x^{*}(t)=(x^{*}_{1}(t), x^{*}_{2}(t),\ldots,x^{*}_{n}(t))^{T} \)be the solution of system (1.1) with initial value\( \varphi^{*}=(\varphi^{*}_{1}(t), \varphi^{*}_{2}(t), \ldots, \varphi^{*}_{n}(t))^{T} , \)and\( x(t)=(x_{1}(t), x_{2}(t),\ldots,x_{n}(t))^{T} \)be the solution of system (1.1) with initial value\( \varphi=(\varphi _{1}(t), \varphi _{2}(t), \ldots, \varphi _{n}(t))^{T}. \)Then, there exists a positive constant λ such that
$$ x_{ i}(t)-x^{*}_{ i}(t) =O\left(e^{-\lambda t}\right),\quad i= 1, 2, \ldots, n. $$

Proof

Let y(t) = x(t) − x*(t). Then,
$$ \begin{aligned} y_i^\prime (t) &= - c_i (t)y_i (t-\delta_i(t)) +\sum^n_{j=1}a_{ij}(t)\left(f_{j}(y_{j}(t-\tau_{ij}(t))+x^{*}_{j}(t-\tau_{ij}(t))) -f_{j}\left(x^{*}_{j}(t-\tau_{ij}(t))\right)\right) \\ &\quad + \sum\limits_{j=1}^{n}\sum\limits_{l=1}^{n}b_{ijl}(t)(g_{j}(y_{j}(t-\alpha_{ijl}(t))+x^{*}_{j}(t-\alpha_{ijl}(t))) g_{l}(y_{l}(t-\beta_{ijl}(t))+x^{*}_{l}(t-\beta_{ijl}(t))) \\ &\quad-g_{j}\left(x^{*}_{j}(t-\alpha_{ijl}(t))\right)g_{l}\left(x^{*}_{l}(t-\beta_{ijl}(t)))\right)+\sum\limits^n_{j=1}\sum\limits^n_{l=1}d_{ijl}(t)\\ &\quad\cdot\left(\int\limits_{0}^{\infty}\sigma_{ijl} (u)h_{j}(y_{j}(t-u)+x^{*}_{j}(t-u))\hbox{d}u \int\limits_{0}^{\infty}\nu_{ijl} (u) h_{l}(y_{l}(t-u)+x^{*}_{l}(t-u))\hbox{d}u \right. \\ &\quad\left. -\int\limits_{0}^{\infty}\sigma_{ijl} (u)h_{j}(x^{*}_{j}(t-u))\hbox{d}u\int\limits_{0}^{\infty}\nu_{ijl} (u) h_{l}(x^{*}_{l}(t-u))\hbox{d}u\right),\quad i= 1,2,\ldots,n. \end{aligned} $$
(2.6)
Define continuous functions \(\Upgamma_{i} (\omega)\) by setting
$$ \begin{aligned} \Upgamma_{i}(\omega)&=-\left[c_i (t){e^{\omega \delta_i(t)}}-\omega -c_i (t){e^{\omega \delta_i(t)}}\delta_{i}(t )\left(\omega+c_i ^{+}{e^{\omega \delta_i ^{+}}}\right)\right] \xi_{i} \\ &\quad + \sum\limits_{j = 1}^n \left(| a_{ij}(t)|e^{\omega \tau _{ij} (t ) } +a_{ij}^{+}c_i (t){e^{\omega \delta_i(t)}}\delta_{i}(t)e^{\omega \tau _{ij} ^{+}}\right) L^{f}_{j} \xi_{j} \\ &\quad + \sum\limits_{j=1}^{n}\sum\limits_{l=1}^{n}\left[\left(|b_{ijl}(t)|e^{\omega \alpha_{ijl}(t)} +b_{ijl}^{+}c_i (t){e^{\omega \delta_i(t)}}\delta_{i}(t)e^{\omega \alpha_{ijl} ^{+} }\right)L^{g}_{j}M^{g}_{l} \xi_{j}\right.\\ &\quad\left.+\left(|b_{ijl}(t)|e^{\omega \beta_{ijl}(t)} +b_{ijl}^{+}c_i (t){e^{\omega \delta_i(t)}}\delta_{i}(t)e^{\omega\beta_{ijl} ^{+}}\right) M^{g}_{j}L^{g}_{l}\xi_{l}\right] \\ &\quad+\sum\limits^n_{j=1}\sum\limits^n_{l=1}\left[(|d_{ijl}(t)|+d_{ijl}^{+}c_i (t){e^{\omega \delta_i(t)}}\delta_{i}(t ))\left( \int\limits_{0}^{\infty}|\sigma_{ijl} (u)|e^{\omega u} \hbox{d}u \int\limits_{0}^{\infty}|\nu_{ijl} (u)|\hbox{d}u L^{h}_{j}M^{h}_{l} \xi_{j} \right.\right. \\ &\quad\left.\left. +\int\limits_{0}^{\infty}|\sigma_{ijl} (u)| \hbox{d}u\int\limits_{0}^{\infty}|\nu_{ijl} (u)|e^{\omega u}\hbox{d}uM^{h}_{j}L^{h}_{l} \xi_{l}\right)\right] , \, \kappa\geq\omega \geq0, t \geq0,\quad i= 1, 2,\ldots, n. \end{aligned} $$
Then,
$$ \begin{aligned} \Upgamma_{i}(0)&=-\left[ c_i (t) -c_i (t) \delta_{i}(t ) c_i ^{+}\right] \xi_{i} + \sum\limits_{j = 1}^n \left(| a_{ij}(t )| +a_{ij}^{+}c_i (t) \delta_{i}(t)\right) L^{f}_{j} \xi_{j} \\ &\quad + \sum\limits_{j=1}^{n}\sum\limits_{l=1}^{n} \left(|b_{ijl}(t)| +b_{ijl}^{+}c_i (t) \delta_{i}(t ) )(L^{g}_{j}M^{g}_{l} \xi_{j}+M^{g}_{j}L^{g}_{l} \xi_{l}\right) \\ &\quad+\sum\limits^n_{j=1}\sum\limits^n_{l=1}\left[\left(|d_{ijl}(t)|+d_{ijl}^{+}c_i (t) \delta_{i}(t)\right) \int\limits_{0}^{\infty}|\sigma_{ijl} (u)| \hbox{d}u \int\limits_{0}^{\infty}|\nu_{ijl} (u)|\hbox{d}u (L^{h}_{j}M^{h}_{l} \xi_{j}+ M^{h}_{j}L^{h}_{l} \xi_{l})\right] \\ &< -\eta,\quad t \geq 0, \quad i= 1, 2, \ldots , n , \end{aligned} $$
which, together with the continuity of \(\Upgamma_{i} (\omega),\) implies that we can choose two positive constants λ < κ and \(\overline{\eta} \) such that
$$ \begin{aligned} -\overline{\eta}>\Upgamma_{i} (\lambda) &=-\left[ c_i (t){e^{\lambda \delta_i(t)}}-\lambda -c_i (t){e^{\lambda \delta_i(t)}}\delta_{i}(t)\left(\lambda+c_i ^{+}{e^{\lambda \delta_i ^{+}}}\right)\right] \xi_{i} \\ &\quad+\sum\limits_{j = 1}^n\left(| a_{ij}(t )|e^{\lambda \tau _{ij} (t ) } +a_{ij}^{+}c_i (t){e^{\lambda \delta_i(t)}}\delta_{i}(t)e^{\lambda \tau _{ij} ^{+}}\right) L^{f}_{j} \xi_{j} \\ &\quad +\sum\limits_{j=1}^{n}\sum\limits_{l=1}^{n}\left[\left(|b_{ijl}(t)|e^{\lambda \alpha_{ijl}(t)} +b_{ijl}^{+}c_i (t){e^{\lambda \delta_i(t)}}\delta_{i}(t )e^{\lambda \alpha_{ijl} ^{+} }\right)L^{g}_{j}M^{g}_{l} \xi_{j}\right. \\ &\left.\quad+\left(|b_{ijl}(t)|e^{\lambda \beta_{ijl}(t)} +b_{ijl}^{+}c_i (t){e^{\lambda \delta_i(t)}}\delta_{i}(t )e^{\lambda \beta_{ijl}^{+} }\right) M^{g}_{j}L^{g}_{l}\xi_{l}\right] \\ &\quad+\sum\limits^n_{j=1}\sum\limits^n_{l=1}\left[\left(|d_{ijl}(t)|+d_{ijl}^{+}c_i (t){e^{\lambda \delta_i(t)}}\delta_{i}(t)\right)\left( \int\limits_{0}^{\infty}|\sigma_{ijl} (u)|e^{\lambda u} \hbox{d}u \int\limits_{0}^{\infty}|\nu_{ijl} (u)|\hbox{d}u \right.\right. \\ &\quad \left.\left. \cdot L^{h}_{j}M^{h}_{l} \xi_{j}+\int\limits_{0}^{\infty}|\sigma_{ijl} (u)| \hbox{d}u\int\limits_{0}^{\infty}|\nu_{ijl} (u)|e^{\lambda u}\hbox{d}uM^{h}_{j}L^{h}_{l} \xi_{l}\right)\right] ,\,t \geq 0,\quad i= 1, 2,\ldots,n . \end{aligned} $$
(2.7)
Let
$$ Y_i (t) = y_i (t) {e^{\lambda t}},\quad i = 1,2, \ldots ,n. $$
Then,
$$ \begin{aligned} Y_i^\prime (t) &= \lambda {Y_i}(t) - c_i (t){e^{\lambda t}}y_i (t-\delta_i(t)) \\ &\quad +e^{\lambda t}\left\{ \sum^n_{j=1}a_{ij}(t) (f_{j}(y_{j}(t-\tau_{ij}(t))+x^{*}_{j}(t-\tau_{ij}(t))) -f_{j}(x^{*}_{j}(t-\tau_{ij}(t)))) \right. \\ &\quad +\sum\limits_{j=1}^{n}\sum\limits_{l=1}^{n}b_{ijl}(t)(g_{j}(y_{j} (t-\alpha_{ijl}(t))+x^{*}_{j}(t-\alpha_{ijl}(t))) g_{l}(y_{l}(t-\beta_{ijl}(t)) \\ &\quad+x^{*}_{l}(t-\beta_{ijl}(t)))-g_{j}(x^{*}_{j} (t-\alpha_{ijl}(t)))g_{l}(x^{*}_{l}(t-\beta_{ijl}(t))))\\ &\quad+\sum\limits^n_{j=1}\sum\limits^n_{l=1}d_{ijl}(t) \left(\int\limits_{0}^{\infty}\sigma_{ijl} (u)h_{j}(y_{j}(t-u)+x^{*}_{j}(t-u))\hbox{d}u \right. \\ &\quad \cdot \int\limits_{0}^{\infty}\nu_{ijl} (u) h_{l}(y_{l}(t-u)+x^{*}_{l}(t-u))\hbox{d}u \\ &\quad\left.\left. -\int\limits_{0}^{\infty}\sigma_{ijl} (u)h_{j}(x^{*}_{j}(t-u))\hbox{d}u\int\limits_{0}^{\infty}\nu_{ijl} (u) h_{l}(x^{*}_{l}(t-u))\hbox{d}u\right)\right\} \\ &= \lambda {Y_i}(t) - c_i (t){e^{\lambda \delta_i(t)}} Y_i (t ) +c_i (t){e^{\lambda \delta_i(t)}} [Y_i (t)- Y _i (t-\delta_i(t)) ] \\ &\quad +e^{\lambda t}\left\{\sum^n_{j=1}a_{ij}(t) (f_{j}(y_{j}(t-\tau_{ij}(t))+x^{*}_{j}(t-\tau_{ij}(t))) -f_{j}(x^{*}_{j}(t-\tau_{ij}(t)))) \right. \\ &\quad + \sum\limits_{j=1}^{n}\sum\limits_{l=1}^{n}b_{ijl}(t)(g_{j}(y_{j}(t-\alpha_{ijl}(t))+x^{*}_{j}(t-\alpha_{ijl}(t))) g_{l}(y_{l}(t-\beta_{ijl}(t)) \\ &\quad+x^{*}_{l}(t-\beta_{ijl}(t)))-g_{j}(x^{*}_{j}(t-\alpha_{ijl}(t)))g_{l}(x^{*}_{l}(t-\beta_{ijl}(t))))\\ &\quad+\sum\limits^n_{j=1}\sum\limits^n_{l=1}d_{ijl}(t)\left(\int\limits_{0}^{\infty}\sigma_{ijl} (u)h_{j}(y_{j}(t-u)+x^{*}_{j}(t-u))\hbox{d}u\right.\\ &\quad\cdot \int\limits_{0}^{\infty}\nu_{ijl} (u) h_{l}(y_{l}(t-u)+x^{*}_{l}(t-u))\hbox{d}u \\ &\quad\left.\left.-\int\limits_{0}^{\infty}\sigma_{ijl}(u)h_{j}(x^{*}_{j}(t-u))\hbox{d}u\int\limits_{0}^{\infty}\nu_{ijl} (u) h_{l}(x^{*}_{l}(t-u))\hbox{d}u\right)\right\} \\ &= \lambda {Y_i}(t) - c_i (t){e^{\lambda \delta_i(t)}} Y_i (t ) +c_i (t){e^{\lambda \delta_i(t)}} \int\limits_{t -\delta_{i}(t )}^{t }Y_{i}'(s ) \hbox{d}s \\ &\quad +e^{\lambda t}\left\{ \sum^n_{j=1}a_{ij}(t) (f_{j}(y_{j}(t-\tau_{ij}(t))+x^{*}_{j}(t-\tau_{ij}(t))) -f_{j}(x^{*}_{j}(t-\tau_{ij}(t))))\right. \\ &\quad + \sum\limits_{j=1}^{n}\sum\limits_{l=1}^{n}b_{ijl}(t)(g_{j}(y_{j}(t-\alpha_{ijl}(t))+x^{*}_{j}(t-\alpha_{ijl}(t))) g_{l}(y_{l}(t-\beta_{ijl}(t)) \\ &\quad+x^{*}_{l}(t-\beta_{ijl}(t)))-g_{j}(x^{*}_{j}(t-\alpha_{ijl}(t)))g_{l}(x^{*}_{l}(t-\beta_{ijl}(t)))) \\ &\quad+\sum\limits^n_{j=1}\sum\limits^n_{l=1}d_{ijl}(t) \left(\int\limits_{0}^{\infty}\sigma_{ijl} (u)h_{j}(y_{j}(t-u)+x^{*}_{j}(t-u))\hbox{d}u\right.\\ &\quad\cdot \int\limits_{0}^{\infty}\nu_{ijl} (u) h_{l}(y_{l}(t-u)+x^{*}_{l}(t-u))\hbox{d}u \\ &\quad\left.\left.-\int\limits_{0}^{\infty}\sigma_{ijl} (u)h_{j}(x^{*}_{j}(t-u))\hbox{d}u\int\limits_{0}^{\infty}\nu_{ijl} (u) h_{l}(x^{*}_{l}(t-u))\hbox{d}u\right)\right\} \\ &= \lambda {Y_i}(t) - c_i (t){e^{\lambda \delta_i(t)}} Y_i (t ) +c_i (t){e^{\lambda \delta_i(t)}} \int\limits_{t -\delta_{i}(t )}^{t }\left\{\lambda {Y_i}(s) - c_i (s){e^{\lambda s}}y_i (s-\delta_i(s)) \right. \\ &\quad + e^{\lambda s} \sum^n_{j=1}a_{ij}(s) (f_{j}(y_{j}(s-\tau_{ij}(s))+x^{*}_{j}(s-\tau_{ij}(s))) -f_{j}(x^{*}_{j}(s-\tau_{ij}(s)))) \\ &\quad + e^{\lambda s} \sum\limits_{j=1}^{n}\sum\limits_{l=1}^{n}b_{ijl}(s)(g_{j}(y_{j}(s-\alpha_{ijl}(s))+x^{*}_{j}(s-\alpha_{ijl}(s))) g_{l}(y_{l}(s-\beta_{ijl}(s)) \\ &\quad+x^{*}_{l}(s-\beta_{ijl}(s)))-g_{j}(x^{*}_{j}(s-\alpha_{ijl}(s)))g_{l}(x^{*}_{l}(s-\beta_{ijl}(s))))\\ &\quad+e^{\lambda s} \sum\limits^n_{j=1}\sum\limits^n_{l=1}d_{ijl}(s) \left(\int\limits_{0}^{\infty}\sigma_{ijl} (u)h_{j}(y_{j}(s-u)+x^{*}_{j}(s-u))\hbox{d}u\right.\\ &\cdot \int\limits_{0}^{\infty}\nu_{ijl} (u) h_{l}(y_{l}(s-u)+x^{*}_{l}(s-u))\hbox{d}u \\ &\quad\left.\left.-\int\limits_{0}^{\infty}\sigma_{ijl} (u)h_{j}(x^{*}_{j}(s-u))\hbox{d}u\int\limits_{0}^{\infty}\nu_{ijl} (u) h_{l}(x^{*}_{l}(s-u))\hbox{d}u\right)\right\} \hbox{d}s \\ &\quad +e^{\lambda t}\left\{ \sum^n_{j=1}a_{ij}(t) (f_{j}(y_{j}(t-\tau_{ij}(t))+x^{*}_{j}(t-\tau_{ij}(t))) -f_{j}(x^{*}_{j}(t-\tau_{ij}(t)))) \right. \\ &\quad + \sum\limits_{j=1}^{n}\sum\limits_{l=1}^{n}b_{ijl}(t)(g_{j}(y_{j}(t-\alpha_{ijl}(t))+x^{*}_{j}(t-\alpha_{ijl}(t))) g_{l}(y_{l}(t-\beta_{ijl}(t)) \\ &\quad+x^{*}_{l}(t-\beta_{ijl}(t)))-g_{j}(x^{*}_{j}(t-\alpha_{ijl}(t)))g_{l}(x^{*}_{l}(t-\beta_{ijl}(t))))\\ &\quad+\sum\limits^n_{j=1}\sum\limits^n_{l=1}d_{ijl}(t) \left(\int\limits_{0}^{\infty}\sigma_{ijl} (u)h_{j}(y_{j}(t-u)+x^{*}_{j}(t-u))\hbox{d}u\right.\\ &\quad\cdot \int\limits_{0}^{\infty}\nu_{ijl} (u) h_{l}(y_{l}(t-u)+x^{*}_{l}(t-u))\hbox{d}u \\ &\quad\left.\left.-\int\limits_{0}^{\infty}\sigma_{ijl} (u)h_{j}(x^{*}_{j}(t-u))\hbox{d}u\int\limits_{0}^{\infty}\nu_{ijl} (u) h_{l}(x^{*}_{l}(t-u))\hbox{d}u\right)\right\}, \end{aligned} $$
(2.8)
where \(i=1,2,\ldots,n.\)
We define a positive constant M as follows:
$$ M = \max\limits_{1\le i\le n}\left\{\sup\limits_{s\in(- \infty ,0]}|Y_i(s)| \right\} . $$
Let K be a positive number such that
$$ |Y_i(t)|\leq M< K\xi_{i}\quad \hbox{for\,all} \quad t \in(- \infty , 0],\quad i = 1, 2, \ldots , n. $$
(2.9)
We claim that
$$ |Y_i(t)|< K\xi_{i},\quad \hbox{for\,all} \quad t >0, \quad i = 1, 2, \ldots ,n. $$
(2.10)
Otherwise, there must exist \(i \in \{ 1, 2, \ldots, n \}\) and θ > 0 such that one of the following two cases occurs.
$$ (1)\quad {Y_i}(\theta ) = K\xi_{i},\quad |Y_j (t)| < K\xi_{j}\quad \hbox{for\,all}\,t \in (- \infty, \theta ),\quad j = 1,2, \ldots,n; $$
(2.11)
$$ (2)\quad {Y_i}(\theta ) = -K\xi_{i}, \quad |Y_j (t)| < K\xi_{j}\quad \hbox{for\,all}\,t \in (- \infty, \theta ),\quad j = 1,2, \ldots,n. $$
(2.12)

Now, we consider two cases.

Case (i)

If (2.11) holds, then from (2.7), (2.8), and (H1) − (H2), we have
$$ \begin{aligned} 0 &\leq Y_i '(\theta ) \\ &= \lambda {Y_i}(\theta) - c_i (\theta){e^{\lambda \delta_i(\theta)}} Y_i (\theta ) +c_i (\theta){e^{\lambda \delta_i(\theta)}} \int\limits_{\theta -\delta_{i}(\theta )}^{\theta }\left\{\lambda {Y_i}(s) - c_i (s){e^{\lambda s}}y_i (s-\delta_i(s)) \right. \\ &\quad + e^{\lambda s} \sum^n_{j=1}a_{ij}(s) (f_{j}(y_{j}(s-\tau_{ij}(s))+x^{*}_{j}(s-\tau_{ij}(s))) -f_{j}(x^{*}_{j}(s-\tau_{ij}(s)))) \\ &\quad + e^{\lambda s} \sum\limits_{j=1}^{n}\sum\limits_{l=1}^{n}b_{ijl}(s)(g_{j}(y_{j}(s-\alpha_{ijl}(s))+x^{*}_{j}(s-\alpha_{ijl}(s))) g_{l}(y_{l}(s-\beta_{ijl}(s)) \\ &\quad+x^{*}_{l}(s-\beta_{ijl}(s)))-g_{j}(x^{*}_{j}(s-\alpha_{ijl}(s)))g_{l}(x^{*}_{l}(s-\beta_{ijl}(s)))) \\ &\quad+e^{\lambda s} \sum\limits^n_{j=1}\sum\limits^n_{l=1}d_{ijl}(s) \left(\int\limits_{0}^{\infty}\sigma_{ijl} (u)h_{j}(y_{j}(s-u)+x^{*}_{j}(s-u))\hbox{d}u\right. \\ &\quad \cdot \int\limits_{0}^{\infty}\nu_{ijl} (u) h_{l}(y_{l}(s-u)+x^{*}_{l}(s-u))\hbox{d}u \\ &\quad\left.\left.-\int\limits_{0}^{\infty}\sigma_{ijl} (u)h_{j}(x^{*}_{j}(s-u))\hbox{d}u\int\limits_{0}^{\infty}\nu_{ijl} (u) h_{l}(x^{*}_{l}(s-u))\hbox{d}u\right) \right\} \hbox{d}s \\ &\quad +{e^{\lambda \theta}}\left\{ \sum^n_{j=1}a_{ij}(\theta) (f_{j}(y_{j}(\theta-\tau_{ij}(\theta))+x^{*}_{j}(\theta-\tau_{ij}(\theta))) -f_{j}(x^{*}_{j}(\theta-\tau_{ij}(\theta)))) \right. \\ &\quad + \sum\limits_{j=1}^{n}\sum\limits_{l=1}^{n}b_{ijl}(\theta)(g_{j}(y_{j}(\theta-\alpha_{ijl}(\theta))+x^{*}_{j}(\theta-\alpha_{ijl}(\theta))) g_{l}(y_{l}(\theta-\beta_{ijl}(\theta)) \\ &\quad+x^{*}_{l}(\theta-\beta_{ijl}(\theta)))-g_{j}(x^{*}_{j}(\theta-\alpha_{ijl}(\theta)))g_{l}(x^{*}_{l}(\theta-\beta_{ijl}(\theta)))) \\ &\quad+\sum\limits^n_{j=1}\sum\limits^n_{l=1}d_{ijl}(\theta) \left(\int\limits_{0}^{\infty}\sigma_{ijl} (u)h_{j}(y_{j}(\theta-u)+x^{*}_{j}(\theta-u))\hbox{d}u \right. \\ &\quad\cdot \int\limits_{0}^{\infty}\nu_{ijl} (u) h_{l}(y_{l}(\theta-u)+x^{*}_{l}(\theta-u))\hbox{d}u \\ &\quad\left.\left.-\int\limits_{0}^{\infty}\sigma_{ijl} (u)h_{j}(x^{*}_{j}(\theta-u))\hbox{d}u\int\limits_{0}^{\infty}\nu_{ijl} (u) h_{l}(x^{*}_{l}(\theta-u))\hbox{d}u\right)\right\} \\ &\leq \lambda {Y_i}(\theta) - c_i (\theta){e^{\lambda \delta_i(\theta)}} Y_i (\theta ) +c_i (\theta){e^{\lambda \delta_i(\theta)}} \int\limits_{\theta -\delta_{i}(\theta )}^{\theta }\left\{\lambda {Y_i}(\theta) + c_i^{+}{e^{\lambda \delta_i(s)}}|Y_i (s-\delta_i(s))| \right. \\ &\quad + \sum^n_{j=1}a_{ij}^{+}L_{j}^{f}e^{\lambda \tau_{ij}(s)} |Y_{j}(s-\tau_{ij}(s))| + e^{\lambda s} \sum\limits_{j=1}^{n}\sum\limits_{l=1}^{n}b_{ijl}^{+}(|g_{j}(y_{j}(s-\alpha_{ijl}(s))+x^{*}_{j}(s-\alpha_{ijl}(s)))\\ &\quad \cdot g_{l}(y_{l}(s-\beta_{ijl}(s))+x^{*}_{l}(s-\beta_{ijl}(s)))-g_{j}(x^{*}_{j}(s-\alpha_{ijl}(s))) g_{l}(y_{l}(s-\beta_{ijl}(s)) \\ &\quad+x^{*}_{l}(s-\beta_{ijl}(s)))|+|g_{j}(x^{*}_{j}(s-\alpha_{ijl}(s))) g_{l}(y_{l}(s-\beta_{ijl}(s)) \\ &\quad+x^{*}_{l}(s-\beta_{ijl}(s)))-g_{j}(x^{*}_{j}(s-\alpha_{ijl}(s)))g_{l}(x^{*}_{l}(s-\beta_{ijl}(s)))|)\\ &\quad+e^{\lambda s} \sum\limits^n_{j=1}\sum\limits^n_{l=1}d_{ijl}^{+} \left(\left|\int\limits_{0}^{\infty}\sigma_{ijl} (u)h_{j}(y_{j}(s-u)+x^{*}_{j}(s-u))\hbox{d}u \right. \right. \\ &\quad\cdot \int\limits_{0}^{\infty}\nu_{ijl} (u) h_{l}(y_{l}(s-u)+x^{*}_{l}(s-u))\hbox{d}u \\ &\quad\left. -\int\limits_{0}^{\infty}\sigma_{ijl} (u)h_{j}( x^{*}_{j}(s-u))\hbox{d}u \int\limits_{0}^{\infty}\nu_{ijl} (u) h_{l}(y_{l}(s-u)+x^{*}_{l}(s-u))\hbox{d}u\right|\\ &\quad +\left|\int\limits_{0}^{\infty}\sigma_{ijl} (u)h_{j}( x^{*}_{j}(s-u))\hbox{d}u \int\limits_{0}^{\infty}\nu_{ijl} (u) h_{l}(y_{l}(s-u)+x^{*}_{l}(s-u))\hbox{d}u \right. \\ &\quad\left.\left.\left.-\int\limits_{0}^{\infty}\sigma_{ijl} (u)h_{j}(x^{*}_{j}(s-u))\hbox{d}u\int\limits_{0}^{\infty}\nu_{ijl} (u) h_{l}(x^{*}_{l}(s-u))\hbox{d}u\right|\right) \right\} \hbox{d}s \\ &\quad \left. + \sum^n_{j=1}\left|a_{ij}(\theta)\right| L^{f}_{j} e^{\lambda \tau_{ij}(\theta)} |Y_{j}(\theta-\tau_{ij}(\theta))|\right. \\ &\quad \left.+ \sum\limits_{j=1}^{n}\sum\limits_{l=1}^{n}|b_{ijl}(\theta)| e^{\lambda \theta}(|g_{j}(y_{j}(\theta-\alpha_{ijl}(\theta))+x^{*}_{j}(\theta-\alpha_{ijl}(\theta))) g_{l}(y_{l}(\theta-\beta_{ijl}(\theta)) \right.\\ &\quad+x^{*}_{l}(\theta-\beta_{ijl}(\theta)))-g_{j}( x^{*}_{j}(\theta-\alpha_{ijl}(\theta))) g_{l}(y_{l}(\theta-\beta_{ijl}(\theta))+x^{*}_{l}(\theta-\beta_{ijl}(\theta)))|\\ &\quad+|g_{j}( x^{*}_{j}(\theta-\alpha_{ijl}(\theta))) g_{l}(y_{l}(\theta-\beta_{ijl}(\theta))+x^{*}_{l}(\theta-\beta_{ijl}(\theta))) \\ &\quad-g_{j}(x^{*}_{j}(\theta-\alpha_{ijl}(\theta)))g_{l}(x^{*}_{l}(\theta-\beta_{ijl}(\theta)))|)\\ &\quad+\sum\limits^n_{j=1}\sum\limits^n_{l=1}|d_{ijl}(\theta)|e^{\lambda \theta } \left(\left|\int\limits_{0}^{\infty}\sigma_{ijl}(u)h_{j}(y_{j}(\theta-u)+x^{*}_{j}(\theta-u))\hbox{d}u \right.\right.\\ &\quad\cdot \int\limits_{0}^{\infty}\nu_{ijl} (u) h_{l}(y_{l}(\theta-u)+x^{*}_{l}(\theta-u))\hbox{d}u \\ &\quad-\int\limits_{0}^{\infty}\sigma_{ijl} (u)h_{j}( x^{*}_{j}(\theta-u))\hbox{d}u \int\limits_{0}^{\infty}\nu_{ijl} (u) h_{l}(y_{l}(\theta-u)+x^{*}_{l}(\theta-u))\hbox{d}u| \\ &\quad+|\int\limits_{0}^{\infty}\sigma_{ijl} (u)h_{j}( x^{*}_{j}(\theta-u))\hbox{d}u \int\limits_{0}^{\infty}\nu_{ijl} (u) h_{l}(y_{l}(\theta-u)+x^{*}_{l}(\theta-u))\hbox{d}u \\ &\quad\left.\left.-\int\limits_{0}^{\infty}\sigma_{ijl} (u)h_{j}(x^{*}_{j}(\theta-u))\hbox{d}u\int\limits_{0}^{\infty}\nu_{ijl} (u) h_{l}(x^{*}_{l}(\theta-u))\hbox{d}u\right|\right) \\ &\leq \lambda {Y_i}(\theta) - c_i (\theta){e^{\lambda \delta_i(\theta)}} Y_i (\theta ) +c_i (\theta){e^{\lambda \delta_i(\theta)}} \int\limits_{\theta -\delta_{i}(\theta )}^{\theta }\left\{\lambda {Y_i}(\theta) + c_i^{+}{e^{\lambda \delta_i(s)}}|Y_i (s-\delta_i(s))| \right. \\ &\quad + \sum^n_{j=1}a_{ij}^{+}L_{j}^{f}e^{\lambda \tau_{ij}(s)} |Y_{j}(s-\tau_{ij}(s))| + \sum\limits_{j=1}^{n}\sum\limits_{l=1}^{n}b_{ijl}^{+}( L^{g}_{j}e^{\lambda \alpha_{ijl}(s)}|Y_{j}(s-\alpha_{ijl}(s))| M^{g}_{l} \\ &\quad+M^{g}_{j}e^{\lambda \beta_{ijl}(s)}|Y_{l}(s-\beta_{ijl}(s))| L^{g}_{l}) \\ &\quad+ \sum\limits^n_{j=1}\sum\limits^n_{l=1}d_{ijl}^{+} \left( \int\limits_{0}^{\infty}|\sigma_{ijl} (u)|e^{\lambda u}L^{h}_{j}|Y_{j}(s-u)| \hbox{d}u \int\limits_{0}^{\infty}|\nu_{ijl} (u)|\hbox{d}u M^{h}_{l} \right.\\ &\quad \left.\left.+M^{h}_{j}\int\limits_{0}^{\infty}|\sigma_{ijl} (u)| \hbox{d}u\int\limits_{0}^{\infty}|\nu_{ijl} (u)|e^{\lambda u}L^{h}_{l}|Y_{l}(s-u)| \hbox{d}u \right) \right\} \hbox{d}s \\ &\quad + \sum^n_{j=1}|a_{ij}(\theta)| L^{f}_{j} e^{\lambda \tau_{ij}(\theta)} |Y_{j}(\theta-\tau_{ij}(\theta))| + \sum\limits_{j=1}^{n}\sum\limits_{l=1}^{n}|b_{ijl}(\theta)| ( L^{g}_{j}e^{\lambda \alpha_{ijl}(\theta)}|Y_{j}(\theta-\alpha_{ijl}(\theta))| M^{g}_{l}\\ &\quad+M^{g}_{j}e^{\lambda \beta_{ijl}(\theta)}|Y_{l}(\theta-\beta_{ijl}(\theta))|L^{g}_{l} )\\ &\quad+\sum\limits^n_{j=1}\sum\limits^n_{l=1}|d_{ijl}(\theta)|\left( \int\limits_{0}^{\infty}|\sigma_{ijl} (u)|e^{\lambda u}L^{h}_{j}|Y_{j}(\theta-u)| \hbox{d}u \int\limits_{0}^{\infty}|\nu_{ijl} (u)|\hbox{d}u M^{h}_{l} \right. \\ &\quad \left.+M^{h}_{j}\int\limits_{0}^{\infty}|\sigma_{ijl} (u)| \hbox{d}u\int\limits_{0}^{\infty}|\nu_{ijl} (u)|e^{\lambda u}L^{h}_{l}|Y_{l}(\theta-u)| \hbox{d}u \right) \\ &\leq -\left[ c_i (\theta){e^{\lambda \delta_i(\theta)}}-\lambda -c_i (\theta){e^{\lambda \delta_i(\theta)}}\delta_{i}(\theta )(\lambda+c_i ^{+}{e^{\lambda \delta_i ^{+}}})\right]K\xi_{i} \\ &\quad + \sum\limits_{j = 1}^n (| a_{ij}(\theta )|e^{\lambda \tau _{ij} (\theta ) } +a_{ij}^{+}c_i (\theta){e^{\lambda \delta_i(\theta)}}\delta_{i}(\theta )e^{\lambda \tau _{ij} ^{+} }) L^{f}_{j} K\xi_{j} \\ &\quad + \sum\limits_{j=1}^{n}\sum\limits_{l=1}^{n}[(|b_{ijl}(\theta)|e^{\lambda \alpha_{ijl}(\theta)} +b_{ijl}^{+}c_i (\theta){e^{\lambda \delta_i(\theta)}}\delta_{i}(\theta )e^{\lambda \alpha_{ijl} ^{+} })L^{g}_{j}M^{g}_{l}K\xi_{j} \\ &\quad+(|b_{ijl}(\theta)|e^{\lambda \beta_{ijl}(\theta)} +b_{ijl}^{+}c_i (\theta){e^{\lambda \delta_i(\theta)}}\delta_{i}(\theta )e^{\lambda \beta_{ijl} ^{+} })M^{g}_{j}L^{g}_{l}K\xi_{l}] \\ &\quad+\sum\limits^n_{j=1}\sum\limits^n_{l=1}\left[(|d_{ijl}(\theta)|+d_{ijl}^{+}c_i (\theta){e^{\lambda \delta_i(\theta)}}\delta_{i}(\theta ))\left( \int\limits_{0}^{\infty}|\sigma_{ijl} (u)|e^{\lambda u} \hbox{d}u \int\limits_{0}^{\infty}|\nu_{ijl} (u)|\hbox{d}u L^{h}_{j}M^{h}_{l}K\xi_{j} \right. \right. \\ &\quad\left.\left. +\int\limits_{0}^{\infty}|\sigma_{ijl} (u)| \hbox{d}u\int\limits_{0}^{\infty}|\nu_{ijl} (u)|e^{\lambda u} \hbox{d}uM^{h}_{j}L^{h}_{l} K\xi_{l} \right)\right] \\ &= \left\{-[ c_i (\theta){e^{\lambda \delta_i(\theta)}}-\lambda -c_i (\theta){e^{\lambda \delta_i(\theta)}}\delta_{i}(\theta )(\lambda+c_i ^{+}{e^{\lambda \delta_i ^{+}}})] \xi_{i} \right. \\ &\quad + \sum\limits_{j = 1}^n (| a_{ij}(\theta )|e^{\lambda \tau _{ij} (\theta ) } +a_{ij}^{+}c_i (\theta){e^{\lambda \delta_i(\theta)}}\delta_{i}(\theta )e^{\lambda \tau _{ij} ^{+} }) L^{f}_{j} \xi_{j} \\ &\quad + \sum\limits_{j=1}^{n}\sum\limits_{l=1}^{n}\left[(|b_{ijl}(\theta)|e^{\lambda \alpha_{ijl}(\theta)} +b_{ijl}^{+}c_i (\theta){e^{\lambda \delta_i(\theta)}}\delta_{i}(\theta )e^{\lambda \alpha_{ijl} ^{+} })L^{g}_{j}M^{g}_{l} \xi_{j} \right.\\ &\quad\left.+(|b_{ijl}(\theta)|e^{\lambda \beta_{ijl}(\theta)} +b_{ijl}^{+}c_i (\theta){e^{\lambda \delta_i(\theta)}}\delta_{i}(\theta )e^{\lambda \beta_{ijl} ^{+} }) M^{g}_{j}L^{g}_{l}\xi_{l}\right] \\ &\quad+\sum\limits^n_{j=1}\sum\limits^n_{l=1}\left[(|d_{ijl}(\theta)|+d_{ijl}^{+}c_i (\theta){e^{\lambda \delta_i(\theta)}}\delta_{i}(\theta ))\left( \int\limits_{0}^{\infty}|\sigma_{ijl} (u)|e^{\lambda u} \hbox{d}u \int\limits_{0}^{\infty}|\nu_{ijl} (u)|\hbox{d}u L^{h}_{j}M^{h}_{l} \xi_{j} \right.\right.\\ &\quad \left.\left.\left.+\int\limits_{0}^{\infty}|\sigma_{ijl} (u)| \hbox{d}u\int\limits_{0}^{\infty}|\nu_{ijl} (u)|e^{\lambda u} \hbox{d}uM^{h}_{j}L^{h}_{l} \xi_{l} \right)\right]\right\}K \\ &< -\overline{\eta} K < 0, \end{aligned} $$
which is a contradiction and implies that (2.11) does not hold.

Case (ii)

If (2.12) holds, then, together with (2.7), (2.8) and (H1) − (H2), using a similar argument as in case (i), we can show that (2.12) is not true. Therefore, (2.10) holds. Consequently, we know that
$$ |Y_i (t)| = \left| {{y_i}(t)} \right|{e^{\lambda t}} < K\xi_{i},\quad \hbox{for\,all} \quad t > 0,\quad i = 1,2, \ldots ,n. $$
That is
$$ \left| {{x_i}(t) - x_i^*(t)} \right| \le K\xi_{i} {e^{ - \lambda t}},\quad \hbox{for\,all}\quad t > 0,\quad \hbox{and} \quad i = 1,2, \ldots ,n. $$
The proof of Lemma 2.2 is completed. \(\square\)

Remark 2.2

If \(x^{*}(t)=(x^{*}_{1}(t), x^{*}_{2}(t),\ldots,x^{*}_{n}(t))^{T} \) be the T-anti-periodic solution of system (1.1), it follows from Lemma 2.2 that x*(t) is globally exponentially stable.

3 Main results

In this section,we present our main result that there exists the exponentially stable anti-periodic solution of HCNNs (1.1).

Theorem 3.1

Suppose that (H1) and (H2) are satisfied. Then, system (1.1) has exactly oneT-anti-periodic solutionx*(t). Moreover, x*(t) is globally exponentially stable.

Proof

Let \(v(t)= (v_{1}(t), v_{2}(t),\ldots, v_{n}(t))^{T} \) is a solution of system (1.1) with initial conditions (2.1). By Lemma 2.1, the solution v(t) is bounded and exists on \([0, +\infty). \) From (1.1)–(1.5), we have
$$ \begin{aligned} &((-1)^{k+1} v_{i} (t + (k+1)T))^{\prime} \\ &=(-1)^{k+1}v^{\prime}_{i} (t + (k+1)T) \\ &=(-1)^{k+1}\{ -c_{i}(t+ (k+1)T) v_{i}(t+ (k+1)T-\delta_i(t+ (k+1)T)) \\ &\quad+\sum^n_{j=1}a_{ij}(t+ (k+1)T)f_{j}(v_{j}(t+ (k+1)T-\tau_{ij}(t+ (k+1)T))) \\ &\quad+ \sum\limits_{j=1}^{n}\sum\limits_{l=1}^{n}b_{ijl}(t+ (k+1)T)g_{j}(v_{j}(t+ (k+1)T-\alpha_{ijl}(t+ (k+1)T))) \\ & \cdot g_{l}(v_{l}(t+ (k+1)T-\beta_{ijl}(t+ (k+1)T))) \\ &\quad+\sum\limits^n_{j=1}\sum\limits^n_{l=1}d_{ijl}(t+ (k+1)T) \int\limits_{0}^{\infty}\sigma_{ijl} (u)h_{j}(v_{j}(t+ (k+1)T-u))\hbox{d}u \\ & \cdot \int\limits_{0}^{\infty}\nu_{ijl} (u) h_{l}(v_{l}(t+ (k+1)T-u))\hbox{d}u +I_{i}(t+ (k+1)T)\} \\ &=(-1)^{k+1}\{-c_{i}(t ) v_{i}(t+ (k+1)T-\delta_i(t )) \\ &\quad+\sum^n_{j=1}a_{ij}(t+ (k+1)T)f_{j}(v_{j}(t+ (k+1)T-\tau_{ij}(t ))) \\ &\quad+ \sum\limits_{j=1}^{n}\sum\limits_{l=1}^{n}b_{ijl}(t+ (k+1)T)g_{j}(v_{j}(t+ (k+1)T-\alpha_{ijl}(t ))) \\ & \cdot g_{l}(v_{l}(t+ (k+1)T-\beta_{ijl}(t ))) \\ &\quad+\sum\limits^n_{j=1}\sum\limits^n_{l=1}d_{ijl}(t+ (k+1)T) \int\limits_{0}^{\infty}\sigma_{ijl} (u)h_{j}(v_{j}(t+ (k+1)T-u))\hbox{d}u \\ & \cdot \int\limits_{0}^{\infty}\nu_{ijl} (u) h_{l}(v_{l}(t+ (k+1)T-u))\hbox{d}u +I_{i}(t+ (k+1)T)\} \\ &= -c_{i}(t ) (-1)^{k+1}v_{i}(t+ (k+1)T-\delta_i(t )) \\ &\quad+\sum^n_{j=1}a_{ij}(t )f_{j}((-1)^{k+1}v_{j}(t+ (k+1)T-\tau_{ij}(t ))) \\ &\quad+ \sum\limits_{j=1}^{n}\sum\limits_{l=1}^{n}b_{ijl}(t )g_{j}((-1)^{k+1}v_{j}(t+ (k+1)T-\alpha_{ijl}(t ))) \\ & \cdot g_{l}((-1)^{k+1}v_{l}(t+ (k+1)T-\beta_{ijl}(t ))) \\ &\quad+\sum\limits^n_{j=1}\sum\limits^n_{l=1}d_{ijl}(t ) \int\limits_{0}^{\infty}\sigma_{ijl} (u)h_{j}((-1)^{k+1}v_{j}(t+ (k+1)T-u))\hbox{d}u \\ & \cdot \int\limits_{0}^{\infty}\nu_{ijl} (u) h_{l}((-1)^{k+1}v_{l}(t+ (k+1)T-u))\hbox{d}u +I_{i}(t ), \end{aligned} $$
(3.1)
for all \(i=1, 2, \ldots,n,\quad k=0, 1, 2, \ldots . \) Thus, for any natural number k, (−1)k+1v (t + (k + 1)T) are the solutions of system (1.1) for t + (k + 1)T ≥ 0. Hence, −v(t + T) is also a solution of (1.1) with initial values
$$ -v_{i }(s + T) , s\in (-\infty, 0],\,\,i = 1,2, \ldots ,n. $$
Then, by the proof of Lemma 2.2, there exists a constant K > 0 such that for any natural number k
$$ \begin{aligned} &\left| (-1)^{k+1} v_{i }(t + (k + 1)T) - (-1)^{k }v_{i }(t + kT) \right| \\ &=\left| v_{i }(t + (k + 1)T) +v_{i }(t + kT) \right| \\ &= \left|v_{i }(t + kT) - (-v_{i }(t + k T+T) ) \right|\\ &\leq K \xi_{i}{e^{ - \lambda (t + kT)}} \\ &= K \xi_{i} e^{ - \lambda t } \left(\frac{1}{e^{ \lambda T }}\right) ^{k},\quad t + kT \geq 0, \quad i = 1,2, \ldots ,n. \end{aligned} $$
(3.2)
Moreover, for any natural number m, we obtain
$$ (-1)^{m+1} v_{i} \left(t + (m+1)T\right) = v_{i} (t) +\sum\limits_{k=0}^{m}\left[(-1)^{k+1} v _{i}(t + (k+1)T)-(-1)^{k} v_{i} (t + kT)\right], $$
(3.3)
where \(i =1,2,\ldots,n. \)

In view of (3.2) and (3.3) that {(−1)mv(t + mT)} uniformly converges to a continuous function, \(x^{*}(t)=(x^{*}_{1}(t), x^{*}_{2}(t),\ldots,x^{*}_{n}(t))^{T}\) on any compact set of R.

Now, we will show that x*(t) is T-anti-periodic solution of system (1.1). First, x*(t) is T-anti-periodic, since
$$ x^{*}(t+T)=\lim\limits_{m\rightarrow \infty}(-1)^{m } v (t +T+ m T)=-\lim\limits_{(m+1)\rightarrow \infty}(-1)^{m+1 } v (t + (m +1)T)=-x^{*}(t ). $$
Next, we prove that x*(t) is a solution of (1.1). In fact, together with the continuity of the right side of (1.1), (3.1) implies that \( \{((-1)^{m+1} v (t + (m+1)T))^{\prime}\}\) uniformly converges to a continuous function on any compact set of R. Thus, letting \(m \longrightarrow +\infty, \) we obtain
$$ \begin{aligned} \frac{d}{\hbox{d}t}\{x^{*}_{i}(t)\} &= -c_{i}(t) x^{*}_{i}(t-\delta_i(t)) +\sum^n_{j=1}a_{ij}(t)f_{j}(x^{*}_{j}(t-\tau_{ij}(t))) \\ &\quad+\sum\limits_{j=1}^{n}\sum\limits_{l=1}^{n}b_{ijl}(t)g_{j}(x^{*}_{j}(t-\alpha_{ijl}(t)))g_{l}(x^{*}_{l}(t-\beta_{ijl}(t)))\\ &\quad+\sum\limits^n_{j=1}\sum\limits^n_{l=1}d_{ijl}(t)\int\limits_{0}^{\infty}\sigma_{ijl} (u)h_{j}(x^{*}_{j}(t-u))\hbox{d}u\int\limits_{0}^{\infty}\nu_{ijl} (u)h_{l}(x^{*}_{l}(t-u))\hbox{d}u \\ &\quad+I_{i}(t),\quad i=1, 2, \ldots, n. \end{aligned} $$
(3.4)

Therefore, x*(t) is a solution of (1.1).

Finally, by Lemma 2.2, we can prove that x*(t) is globally exponentially stable. This completes the proof. \(\square\)

4 Examples and remarks

In this section, some examples and remarks are provided to demonstrate the effectiveness of our results.

Example 4.1

Consider the following HCNNs with time-varying delays in the leakage terms:
$$ \left\{ \begin{array}{l} x_{1}^{\prime}(t)= -1.5x_{1}\left(t-\frac{1}{1,000}|\sin t|\right)+\frac{1}{4} \sin t f_{1}(x_{1}(t-1))+\frac{1}{36}\sin ^{3}tf_{2}(x_{2}(t-1))\\\quad\quad\quad\quad + \frac{1}{72}\sin t g^{2}_{1}(x_{1}(t-1))+ \frac{1}{72}\sin t g_{1}(x_{1}(t-1))g_{2}(x_{2}(t-1)) \\ \quad\quad\quad\quad+ \frac{1}{72}\sin t g^{2}_{2}(x_{2}(t-1)) \\ \quad\quad\quad\quad+\frac{1}{80}\sin t \int\limits_{0}^{\infty}e^{-u} h_{1}(x_{1}(t-u))\hbox{d}u\int\limits_{0}^{\infty} e^{-u} h_{2}(x_{2}(t-u))\hbox{d}u + \sin t , \\ x_{2}^{\prime}(t) = -1.5x_{2}\left(t-\frac{1}{1,000}|\cos t|\right)+\frac{1}{36}\sin ^{5}t f_{1}(x_{1}(t-1))+\frac{1}{4} \sin ^{3} t f_{2}(x_{2}(t-1))\\ \quad\quad\quad\quad+ \frac{1}{72}\sin ^{3}t g^{2}_{1}(x_{1}(t-1))+ \frac{1}{72}\sin ^{3}t g_{1}(x_{1}(t-1))g_{2}(x_{2}(t-1)) \\\quad\quad\quad\quad + \frac{1}{72}\sin ^{3}t g^{2}_{2}(x_{2}(t-1)) \\\quad\quad\quad\quad +\frac{1}{80}\sin ^{7} t \int\limits_{0}^{\infty}e^{-u} h_{1}(x_{1}(t-u))\hbox{d}u\int\limits_{0}^{\infty} e^{-u} h_{2}(x_{2}(t-u))\hbox{d}u+ 2\sin t, \\ \end{array} \right. $$
(4.1)
where
$$ \begin{aligned} f_{i}(x) &= \frac{1}{2}(|x|+\cos x), g_{i}(x)= h_{i}(x)= |\arctan x|+\cos x, c_{i}(t)= 1.5, I_{i}(t)=i\sin t,\quad i=1,2,\\ \delta_{1}(t)&=\frac{1}{1,000}|\sin t|, \delta_{2}(t)=\frac{1}{1,000}|\cos t|, a_{11}(t)=\frac{1}{4}\sin t, a_{12}(t)=\frac{1}{36}\sin ^{3} t, \\ a_{21}(t)&=\frac{1}{36}\sin^{5} t, a_{22}(t)=\frac{1}{4}\sin ^{3}t, b_{111}(t)=b_{112}(t)=b_{122}(t)=\frac{1}{72}\sin t,\\ b_{211}(t)&=b_{212}(t)=b_{222}(t)=\frac{1}{72}\sin ^{3}t,d_{112}(t)=\frac{1}{80}\sin t, d_{212}(t)=\frac{1}{80}\sin ^{7}t. \end{aligned} $$
Noting that
$$ L^{f}_{i}=1,\quad L^{g}_{i}=L^{h}_{i}=2, \quad M^{g}_{i}=M^{h}_{i}= \frac{\pi}{2}+1,\quad i=1,2. $$
Therefore,
$$ \begin{aligned} &-\left[ c_i (t) -c_i (t) \delta_{i}(t ) c_i ^{+}\right] \xi_{i} + \sum\limits_{j = 1}^2 \left(| a_{ij}(t )| +a_{ij}^{+}c_i (t) \delta_{i}(t)\right ) L^{f}_{j} \xi_{j} \\ &\quad +\sum\limits_{j=1}^{2}\sum\limits_{l=1}^{2} \left(|b_{ijl}(t)| +b_{ijl}^{+}c_i (t) \delta_{i}(t)\right)\left(L^{g}_{j}M^{g}_{l} \xi_{j}+M^{g}_{j}L^{g}_{l} \xi_{l}\right) \\ &\quad+\sum\limits^2_{j=1}\sum\limits^2_{l=1}\left[\left(|d_{ijl}(t)|+d_{ijl}^{+}c_i (t) \delta_{i}(t)\right) \int\limits_{0}^{\infty}|\sigma_{ijl} (u)| \hbox{d}u \int\limits_{0}^{\infty}|\nu_{ijl} (u)|\hbox{d}u \left(L^{h}_{j}M^{h}_{l} \xi_{j}+ M^{h}_{j}L^{h}_{l} \xi_{l}\right)\right] \\ &<-\left[1.5-1.5\times \frac{1}{1,000} \times 1.5\right]+\left[\left(\frac{1}{4} +\frac{1}{4}\times \frac{1}{1,000} \times 1.5\right)+\left(\frac{1}{36}+\frac{1}{36}\times \frac{1}{1,000} \times 1.5\right)\right] \\ &\quad +\left(\frac{1}{72}+\frac{1}{72}\times \frac{1}{1,000} \times 1.5\right)\times\left(\frac{\pi}{2}+1\right)\times 4\times 3 +\left(\frac{1}{80}+\frac{1}{80}\times \frac{1}{1,000} \times 1.5\right)\times\left(\frac{\pi}{2}+1\right)\times 4\\ &< -0.5, \quad t \geq 0, \quad \xi_{i}=1,\quad i= 1, 2, \end{aligned} $$
which implies that system (4.1) satisfies all the conditions in Theorem 3.1. Hence, system (4.1) has exactly one π − anti-periodic solution. Moreover, the π − anti-periodic solution is globally exponentially stable.

Remark 4.1

Since the results in [713] only consider HCNNs with constant delays in the leakage terms, and the results on neural networks with leakage delays in [1517] give no opinions about the problem of anti-periodic solutions, one can find that the results in these references cannot be applicable to system (4.1) to obtain the existence and exponential stability of the anti-periodic solutions. This implies that the results of this paper are essentially new. The fact is verified by the numerical simulation in Fig. 1.
https://static-content.springer.com/image/art%3A10.1007%2Fs00521-012-1330-6/MediaObjects/521_2012_1330_Fig1_HTML.gif
Fig. 1

Numerical solution x(t) = (x1(t), x2(t))T of system (4.1) for initial value \(\varphi(t)\equiv (0.1,0.3)^T\)

Remark 4.2

Let
$$ f_{i}(x) = \frac{1}{2}(|x|+\cos x),\quad g_{i}(x)= h_{i}(x)= |\arctan x|+\cos x,\quad i=1,2. $$
Consider the following delayed HCNNs without delays in the leakage terms:
$$ \left\{ \begin{array}{ll} x_{1}^{\prime}(t)= -1.5x_{1}(t )+\frac{1}{4} \sin t f_{1}(x_{1}(t-1))+\frac{1}{36}\sin ^{3}tf_{2}(x_{2}(t-1)) \\\quad\quad\quad\quad + \frac{1}{72}\sin t g^{2}_{1}(x_{1}(t-1))\\ \quad\quad\quad\quad+\frac{1}{72}\sin t g_{1}(x_{1}(t-1))g_{2}(x_{2}(t-1)) + \frac{1}{72}\sin t g^{2}_{2}(x_{2}(t-1)) \\\quad\quad\quad\quad +\frac{1}{80}\sin t \int\limits_{0}^{\infty}e^{-u} h_{1}(x_{1}(t-u))\hbox{d}u\int\limits_{0}^{\infty} e^{-u} h_{2}(x_{2}(t-u))\hbox{d}u + \sin t , \\ x_{2}^{\prime}(t) = -1.5x_{2}(t )+\frac{1}{36}\sin ^{5}t f_{1}(x_{1}(t-1))+\frac{1}{4} \sin ^{3} t f_{2}(x_{2}(t-1)) \\\quad\quad\quad\quad + \frac{1}{72}\sin ^{3}t g^{2}_{1}(x_{1}(t-1))\\\quad\quad\quad\quad + \frac{1}{72}\sin ^{3}t g_{1}(x_{1}(t-1))g_{2}(x_{2}(t-1)) + \frac{1}{72}\sin ^{3}t g^{2}_{2}(x_{2}(t-1)) \\\quad\quad\quad\quad +\frac{1}{80}\sin ^{7} t \int\limits_{0}^{\infty}e^{-u} h_{1}(x_{1}(t-u))\hbox{d}u\int\limits_{0}^{\infty} e^{-u} h_{2}(x_{2}(t-u))\hbox{d}u+ 2\sin t. \\ \end{array} \right. $$
(4.2)
In view of \( \delta _{1}^{ + } = \delta _{2}^{ + } = 0, \) by using a similar argument as in example 4.1, we can show that system (4.2) satisfy all the conditions in Theorem 3.1. Hence, system (4.2) also has exactly one π − anti-periodic solution, which is globally exponentially stable. The fact is verified by the numerical simulation in Fig. 2. Recently, the references [2, 3, 5, 6] also considered the problem on anti-periodic solutions for HCNNs (1.1) with \(\delta _{1}(t)\equiv\delta _{2}(t)\equiv\cdots\equiv\delta _{n}(t)\equiv 0, \) and the following fundamental condition:
$$ f_{j}(0)=g_{j}(0)= h_{i}(0 )=0,\quad j=1, 2, \ldots, n. $$
(4.3)
However, noting \(f_{i}(x) = \frac{1}{2}(|x|+\cos x), g_{i}(x)= h_{i}(x)= |\arctan x|+\cos x,\quad i=1,2 , \) one can find that system (4.2) do not satisfy the above fundamental condition (4.3). This implies that our result on anti-periodic solutions for HCNNs (1.1) without delays in the leakage terms also extends the known one in [2, 3, 5, 6]. Moreover, in this present paper, we proposed a new approach to prove the existence and exponential stability of the anti-periodic solutions for HCNNs with time-varying delays in the leakage terms. Thus, the results of this paper substantially extend and improve some important results in the literature.
https://static-content.springer.com/image/art%3A10.1007%2Fs00521-012-1330-6/MediaObjects/521_2012_1330_Fig2_HTML.gif
Fig. 2

Numerical solution x(t) = (x1(t), x2(t))T of system (4.2) for initial value \(\varphi(t)\equiv (0.2,0.6)^T\)

Acknowledgments

The authors would like to express the sincere appreciation to the reviewers for their helpful comments in improving the presentation and quality of the paper. In particular, the authors expresses the sincere gratitude to Prof. Bingwen Liu for the helpful discussion when this work is carried out. This work was supported by the Construct Program of the Key Discipline in Hunan Province, the Science and Technology Planning project of Technology Department of Hunan Province (Grant no. 2012FJ4300), the Science and Technology Planning project of Chenzhou City in Hunan Province (Grant no. [2011]29), and the Natural Scientific Research Fund of Hunan Provincial Education Department (Grant no. 11C1186).

Copyright information

© Springer-Verlag London 2013