1 Introduction
In the past several decades, the dynamics of BAM neural networks has been widely investigated for their essential applications in classification, pattern recognition, optimization, signal and image processing, and so on [1‐41]. In 1987, Kosko [42] proposed the following BAM neural network:
where \(i=1,2,\ldots,n,t>0\). Here, \(a_{i}>0,b_{i}>0\) denote the time scales of the respective layers of the network; \(-a_{i}u_{i}(t)\) and \(-b_{i}v_{i}(t)\) stand for the stabilizing negative feedback of the model. Noticing that the leakage delay often appears in the negative feedback term of neural networks (see [43‐47]), Gopalsmay [48] studied the stability of the equilibrium and periodic solutions for the following BAM neural network:
where \(i=1,2,\ldots,n,t>0\). Since the delays in neural networks are usually time-varying in the real world, Liu [49] discussed the global exponential stability for the following general BAM neural network with time-varying leakage delays:
However, so far, there have been rare reports on the existence and exponential stability of anti-periodic solutions of neural networks, especially for neural networks with leakage delays. Furthermore, the existence of anti-periodic solutions can be applied to help better describe the dynamical properties of nonlinear systems [49‐65]. So we think that the investigation on the existence and stability of anti-periodic solutions for neural networks with leakage delays has significant value. Inspired by the ideas and considering the change of system parameters in time, we can modify neural network model (1.3) as follows:
The main objective of this article is to analyze the exponential stability behavior of anti-periodic oscillations of (1.4). Based on the fundamental solution matrix, Lyapunov function, and fundamental function sequences, we establish a sufficient condition ensuring the existence and global exponential stability of anti-periodic solutions of (1.4). The derived findings can be used directly to numerous specific networks. Besides, computer simulations are performed to support the obtained predictions. Our findings are a good complement to the work of Gopalsmay [48] and Liu [49].
$$\begin{aligned} \textstyle\begin{cases} \frac{du_{i}(t)}{dt}=-a_{i}u_{i}(t)+\sum_{j=1}^{n}a_{ij}f_{j}(v_{j}(t-\sigma _{j}(t)))+I_{i},\\ \frac{dv_{i}(t)}{dt}=-b_{i}v_{i}(t)+\sum_{j=1}^{n}b_{ij}g_{j}(u_{j}(t-\tau_{j}(t)))+J_{i}, \end{cases}\displaystyle \end{aligned}$$
(1.1)
$$\begin{aligned} \textstyle\begin{cases} \frac{dx_{i}(t)}{dt}=-a_{i}x_{i}(t-\tau_{i}^{(i)})+\sum_{j=1}^{n}a_{ij}f_{j}(y_{j}(t-\sigma_{j}^{(2)}))+I_{i},\\ \frac{dy_{i}(t)}{dt}=-b_{i}y_{i}(t-\tau_{i}^{(2)})+\sum_{j=1}^{n}b_{ij}g_{j}(x_{j}(t-\sigma_{j}^{(1)}))+J_{i}, \end{cases}\displaystyle \end{aligned}$$
(1.2)
$$\begin{aligned} \textstyle\begin{cases} \frac{dx_{i}(t)}{dt}=-a_{i}x_{i}(t-\delta_{i}(t))+\sum_{j=1}^{n}a_{ij}f_{j}(y_{j}(t-\sigma_{ij}(t))+I_{i},\\ \frac{dy_{i}(t)}{dt}=-b_{i}y_{i}(t-\eta_{i}(t)+\sum_{j=1}^{n}b_{ij}g_{j}(x_{j}(t-\tau _{ij}(t))+J_{i}. \end{cases}\displaystyle \end{aligned}$$
(1.3)
$$\begin{aligned} \textstyle\begin{cases} \frac{dx_{i}(t)}{dt}=-a_{i}x_{i}(t-\delta_{i}(t))+\sum_{j=1}^{n}a_{ij}(t)f_{j}(y_{j}(t-\sigma_{ij}(t)))+I_{i}(t),\\ \frac{dy_{i}(t)}{dt}=-b_{i}y_{i}(t-\eta_{i}(t))+\sum_{j=1}^{n}b_{ij}(t)g_{j}(x_{j}(t-\tau_{ij}(t)))+J_{i}(t). \end{cases}\displaystyle \end{aligned}$$
(1.4)
The paper is planned as follows. In Sect. 2, several notations and preliminary results are prepared. In Sect. 3, we give a sufficient condition for the existence and global exponential stability of anti-periodic solution of (1.4). In Sect. 4, we present an example to show the correctness of the obtained analytic findings.
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Remark 1.1
A time delay that exists in the negative feedback term (or called leakage term or forgetting term) of neural networks is called leakage delay. If there exists an anti-periodic solution in a dynamical system, then we can say that the system has anti-periodic oscillations.
2 Preliminary results
In this segment, several notations and lemmas will be given.
For any vector \(V=(v_{1},v_{2},\dots,v_{n})^{T}\) and matrix \(D=(d_{ij})_{n\times{n}}\), we define the norm as
respectively. Let
where \(\varphi_{i}(s)\in{C}([-\tau,0],R), \psi_{i}(s)\in{C}([-\tau,0],R),i=1,2,\ldots,n\), we define
We assume that system (1.4) always satisfies the following initial conditions:
Let \(x(t)=(x_{1}(t),x_{2}(t),\ldots,x_{n}(t))^{T}, y(t)=(y_{1}(t),y_{2}(t),\ldots,y_{n}(t))^{T}\) be the solution of system (1.4) with initial conditions (2.1). We say the solution \(x(t)=(x_{1}(t),x_{2}(t),\ldots,x_{n}(t))^{T}, y(t)=(y_{1}(t),y_{2}(t),\ldots,y_{n}(t))^{T}\) is T-anti-periodic on \(R^{2n}\) if \(x_{i}(t+T)=-x_{i}(t), y_{i}(t+T)=-y_{i}(t)\) for all \(t\in{R}\) and \(i=1,2,\dots,n\), where T is a positive constant.
$$\Vert V \Vert = \Biggl(\sum_{i=1}^{n}v_{i}^{2} \Biggr)^{\frac{1}{2}},\qquad \Vert D \Vert = \Biggl(\sum _{i=1}^{n}d_{ij}^{2} \Biggr)^{\frac{1}{2}}, $$
$$\begin{aligned} &\tau=\Bigl\{ \sup_{t\in{R}}\max_{1\leq{i}\leq{n}} \delta_{i}(t),\sup_{t\in {R}}\max_{1\leq{i}\leq{n}} \eta_{i}(t), \sup_{t\in{R}}\max_{1\leq{i,j}\leq {n}} \sigma_{ij}(t),\sup_{t\in{R}}\max_{1\leq{i,j}\leq{n}} \tau_{ij}(t)\Bigr\} , \\ &\varphi(s)=\bigl(\varphi_{1}(s),\varphi_{2}(s),\ldots, \varphi_{n}(s)\bigr)^{T},\qquad \psi(s)=\bigl(\psi_{1}(s), \psi_{2}(s),\ldots,\psi_{n}(s)\bigr)^{T}, \end{aligned}$$
$$\Vert \varphi \Vert =\sup_{-\tau\leq{s}\leq0} \Biggl(\sum _{i=1}^{n} \bigl\vert \varphi _{i}(s) \bigr\vert ^{2} \Biggr)^{\frac{1}{2}}, \qquad \Vert \psi \Vert =\sup _{-\tau\leq{s}\leq0} \Biggl(\sum_{i=1}^{n} \bigl\vert \psi_{i}(s) \bigr\vert ^{2} \Biggr)^{\frac{1}{2}}. $$
$$\begin{aligned} \textstyle\begin{cases} x_{i0}=\varphi_{i}(s),& s\in[-\tau,0],\\ y_{i0}=\psi_{i}(s),& s\in[-\tau,0]. \end{cases}\displaystyle \end{aligned}$$
(2.1)
Throughout this paper, for \(i,j=1,2,\ldots,n\), it will be assumed that there exist constants such that
and \(t-\delta_{i}(t)>0, t-\eta_{i}(t)>0\).
$$\begin{aligned} &\delta_{i}^{+}=\sup_{t\in{R}} \delta_{i}(t), \qquad \eta_{i}^{+}=\sup _{t\in{R}}\eta_{i}(t),\qquad a_{ij}^{+}= \sup_{t\in{R}}a_{ij}(t), \\ & b_{ij}^{+}= \sup_{t\in{R}}b_{ij}(t),\qquad \tau_{ij}^{+}= \sup_{t\in{R}}\tau_{ij}(t), \qquad \sigma_{ij}^{+}= \sup_{t\in{R}}\sigma_{ij}(t) \end{aligned}$$
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We also assume that the following conditions hold.
(H1) For \(j=1,2,\ldots,n\), there exist constants \(L_{jf}>0, L_{jg}>0, M_{j}^{f}>0\), and \(M_{j}^{g}>0\) such that
for all \(u, v \in{R}\).
$$\textstyle\begin{cases} \vert f_{j}(u )-f_{j}(v ) \vert \leq L_{jf} \vert u -v \vert , \quad \vert f_{j}(u) \vert < M_{j}^{f},\\ \vert g_{j}(u )-g_{j}(v ) \vert \leq L_{jg} \vert u -v \vert ,\quad \vert g_{j}(u) \vert < M_{j}^{g}, \end{cases} $$
(H2) For all \(t,u\in{R}\) and \(i,j=1,2,\ldots,n\),
where T is a positive constant.
$$\textstyle\begin{cases} a_{ij}(t+T)f_{j}(u)=-a_{ij}(t)f_{j}(-u),\qquad b_{ij}(t+T)g_{j}(u)=-b_{ij}(t)g_{j}(-u),\\ \delta_{i}(t+T)=\delta_{i}(t),\qquad \eta_{i}(t+T)=\eta_{i}(t),\qquad \sigma_{ij}(t+T)=\sigma_{ij}(t),\\ \tau_{ij}(t+T)=\tau_{ij}(t),\qquad I_{i}(t+T)=-I_{i}(t),\qquad J_{i}(t+T)=-J_{i}(t), \end{cases} $$
It is clear that the conditions can be fulfilled; for example, let \(a_{ij}(t)=0.2 \vert \cos t \vert , f_{j}(u)=u^{2j-1},i,j=1,2,\ldots,n\), then we have \(a_{ij}(t+T)f_{j}(u)=-a_{ij}(t)f_{j}(-u)\).
(H3) The following inequality holds:
where \(\alpha=\min_{1\leq{i}\leq{n}}\{a_{i},b_{i}\}, i=1,2,\dots,n\).
$$\frac{\sqrt{2}}{\alpha}\bigl(a_{i}\delta_{i}^{+}+b_{i} \eta_{i}^{+}\bigr)< 1, $$
Definition 2.1
The solution \((x^{*}(t),y^{*}(t))^{T}\) of system (1.4) is said to globally exponentially stable if there exist constants \(\beta>0\) and \(M>1\) such that
for each solution \((x(t),y(t))^{T}\) of system (1.4).
$$\sum_{i=1}^{n} \bigl\vert x_{i}(t)-x_{i}^{*}(t) \bigr\vert ^{2}+\sum _{i=1}^{n} \bigl\vert y_{i}(t)-y_{i}^{*}(t) \bigr\vert ^{2}\leq {M}e^{-\beta t}\bigl( \bigl\Vert \varphi- \varphi^{*} \bigr\Vert ^{2}+ \bigl\Vert \psi-\psi^{*} \bigr\Vert ^{2}\bigr) $$
Next, we present three important lemmas which are necessary for proving our main results in Sect. 3.
Lemma 2.1
Let
then we have
for all
\(t\geq0\).
$$A=\left ( \begin{matrix} -a_{i} & 0 \\ 0 & -b_{i} \end{matrix} \right ),\qquad \alpha=\min_{1\leq{i}\leq{n}}\{a_{i},b_{i} \}, $$
$$\Vert \exp A t \Vert \leq\sqrt{2}e^{-\alpha t} $$
Proof
Since
it follows that
By the definition of matrix norm, we get
□
$$A=\left ( \begin{matrix} -a_{i} & 0 \\ 0 & -b_{i} \end{matrix} \right ), $$
$$\exp A t=\left ( \begin{matrix} e^{-a_{i}t} & 0 \\ 0 & e^{-b_{i}t} \end{matrix} \right ). $$
$$\Vert \exp A t \Vert = \bigl(e^{ -2a_{i}t}+e^{ -2b_{i}t} \bigr)^{\frac{1}{2}}\leq\sqrt {2}e^{-\alpha t}. $$
Lemma 2.2
Assume that
where
\(0\leq\varepsilon_{i}\leq1\ (i=1,2,3,4,5,6)\)
are any constants. Then there exists
\(\beta>0\)
such that
$$\mathrm{(H4)} \textstyle\begin{cases} -2a_{i}+a_{i}^{2}\delta_{i}^{+}+\sum_{j=1}^{n}a_{i}\delta _{i}^{+}a_{ij}^{+}L_{jf}^{2\xi_{j}}+\sum_{j=1}^{n}a_{ij}^{+}L_{jf}^{2\xi_{j}}\\ \quad{}+\sum_{i=1}^{n}a_{i}^{2}\delta_{i}^{+}+\sum_{j=1}^{n}b_{i}\eta _{i}^{+}b_{ij}^{+}L_{jg}^{2(1-\varepsilon _{j})}+b_{ij}^{+}L_{jg}^{2(1-\varepsilon_{j})}< 0, \\ {-}2b_{i}+b_{i}^{2}\eta_{i}^{+}+\sum_{j=1}^{n}b_{i}\eta _{i}^{+}b_{ij}^{+}L_{jg}^{2\varepsilon_{j}}+\sum_{j=1}^{n}b_{ij}^{+}L_{jg}^{2\varepsilon_{j}}\\ \quad{} +\sum_{i=1}^{n}a_{i}^{2}\delta_{i}^{+}a_{ij}^{+}L_{jf}^{2(1-\xi _{j})}+a_{ij}^{+}L_{jf}^{2(1-\xi_{j})}+b_{i}\eta_{i}^{+}< 0, \end{cases} $$
$$\begin{aligned} & \beta-2a_{i}+a_{i}^{2}\delta_{i}^{+}+ \sum_{j=1}^{n}a_{i}\delta _{i}^{+}a_{ij}^{+}L_{jf}^{2\xi_{j}}+ \sum_{j=1}^{n}a_{ij}^{+}L_{jf}^{2\xi _{j}} \\ & \quad{}+\sum_{i=1}^{n}a_{i}^{2} \delta_{i}^{+}e^{\beta \delta_{i}^{+}}+\sum _{j=1}^{n}b_{i}\eta_{i}^{+}b_{ij}^{+}L_{jg}^{2(1-\varepsilon _{j})}e^{\beta\tau_{ij}^{+}}+b_{ij}^{+}L_{jg}^{2(1-\varepsilon_{j})}e^{\beta \tau_{ij}^{+}} \leq0, \\ & \beta-2b_{i}+b_{i}^{2}\eta_{i}^{+}+ \sum_{j=1}^{n}b_{i}\eta _{i}^{+}b_{ij}^{+}L_{jg}^{2\varepsilon_{j}}+ \sum_{j=1}^{n}b_{ij}^{+}L_{jg}^{2\varepsilon_{j}} \\ & \quad{}+\sum_{i=1}^{n}a_{i}^{2} \delta_{i}^{+}a_{ij}^{+}L_{jf}^{2(1-\xi_{j})}e^{\beta \delta_{ij}^{+}}+a_{ij}^{+}L_{jf}^{2(1-\xi_{j})}e^{\beta\delta _{ij}^{+}}+b_{i} \eta_{i}^{+}e^{\beta\eta_{i}^{+}}\leq0. \end{aligned}$$
Proof
Let
Obviously, \(\varrho_{ji}(\beta)\ (j=1,2;i=1,2,\ldots,n)\) is a continuously differential function. We can easily check that
By using the intermediate value theorem, we have that there exist constants \(\beta_{i}^{*}>0\ (i=1,2)\) such that
Let \(\beta_{0}=\min\{\beta_{1}^{*},\beta_{2}^{*}\}\), then it follows that \(\beta_{0}>0\) and
This completes the proof of Lemma 2.2. □
$$\begin{aligned} \varrho_{1i}(\beta)={}&\beta-2a_{i}+a_{i}^{2} \delta_{i}^{+}+\sum_{j=1}^{n}a_{i} \delta_{i}^{+}a_{ij}^{+}L_{jf}^{2\xi_{j}}+ \sum_{j=1}^{n}a_{ij}^{+}L_{jf}^{2\xi_{j}} \\ & {}+\sum_{i=1}^{n}a_{i}^{2} \delta_{i}^{+}e^{\beta \delta_{i}^{+}}+\sum _{j=1}^{n}b_{i}\eta_{i}^{+}b_{ij}^{+}L_{jg}^{2(1-\varepsilon _{j})}e^{\beta\tau_{ij}^{+}}+b_{ij}^{+}L_{jg}^{2(1-\varepsilon_{j})}e^{\beta \tau_{ij}^{+}}, \\ \varrho_{2i}(\beta)={}&\beta-2b_{i}+b_{i}^{2} \eta_{i}^{+}+\sum_{j=1}^{n}b_{i} \eta _{i}^{+}b_{ij}^{+}L_{jg}^{2\varepsilon_{j}}+ \sum_{j=1}^{n}b_{ij}^{+}L_{jg}^{2\varepsilon_{j}} \\ & {}+\sum_{i=1}^{n}a_{i}^{2} \delta_{i}^{+}a_{ij}^{+}L_{jf}^{2(1-\xi_{j})}e^{\beta \delta_{ij}^{+}}+a_{ij}^{+}L_{jf}^{2(1-\xi_{j})}e^{\beta\delta _{ij}^{+}}+b_{i} \eta_{i}^{+}e^{\beta\eta_{i}^{+}}. \end{aligned}$$
$$\textstyle\begin{cases} \frac{d\varrho_{1i}(\beta)}{d\beta}=1+\delta_{i}^{+}\sum_{i=1}^{n}a_{i}^{2}\delta_{i}^{+}e^{\beta \delta_{i}^{+}}+\tau_{ij}^{+}\sum_{j=1}^{n}b_{i}\eta _{i}^{+}b_{ij}^{+}L_{jg}^{2(1-\varepsilon_{j})}e^{\beta\tau_{ij}^{+}}+\tau _{ij}^{+}b_{ij}^{+}L_{jg}^{2(1-\varepsilon_{j})}e^{\beta\tau _{ij}^{+}}>0,\\ \lim_{\beta\rightarrow{+\infty}}\varrho_{1i}(\beta)=+\infty,\qquad \varrho _{1i}(0)< 0,\\ \frac{d\varrho_{2i}(\beta)}{d\beta}=1+\delta_{ij}^{+}\sum_{i=1}^{n}a_{i}^{2}\delta_{i}^{+}a_{ij}^{+}L_{jf}^{2(1-\xi_{j})}e^{\beta \delta_{ij}^{+}}+\delta_{ij}^{+}a_{ij}^{+}L_{jf}^{2(1-\xi_{j})}e^{\beta \delta_{ij}^{+}}+\eta_{i}^{+}b_{i}\eta_{i}^{+}e^{\beta\eta_{i}^{+}}>0,\\ \lim_{\beta\rightarrow{+\infty}}\varrho_{2i}(\beta)=+\infty,\qquad \varrho_{2i}(0)< 0. \end{cases} $$
$$\varrho_{j}\bigl(\beta_{i}^{*}\bigr)=0,\quad j=1,2. $$
$$\varrho_{j}(\beta_{0})\leq0,\quad j=1,2. $$
Lemma 2.3
Assume that (H1), (H3), and (H4) are satisfied. Then, for any solution
\((x_{1}(t),x_{2}(t),\ldots,x_{n}(t), y_{1},y_{2},\ldots, y_{n})^{T}\)
of system (1.4), there exists a constant
such that
for all
\(t>0\).
$$\begin{aligned} \sigma^{*}={}& \biggl[1-\frac{\sqrt{2}}{\alpha}\bigl(a_{i} \delta_{i}^{+}+b_{i}\eta _{i}^{+} \bigr) \biggr]^{-1} \bigl[a_{i}\delta_{i}^{+} \bigl(a_{ij}^{+}M_{j}^{f}+I_{i}^{+} \bigr) \\ & {}+a_{ij}^{+}M_{j}^{f}+I_{i}^{+}+b_{i} \eta_{i}^{+} \bigl(b_{ij}^{+}M_{j}^{g}+J_{i}^{+} \bigr)+b_{ij}^{+}M_{j}^{g}+J_{i}^{+} \bigr] \end{aligned}$$
$$\bigl\vert x_{i}(t) \bigr\vert \leq{\sigma^{*}},\qquad \bigl\vert y_{i}(t) \bigr\vert \leq{\sigma^{*}},\quad i=1,2,\ldots,n, $$
Proof
From (1.4), we have
Then we have
Thus
Let
where
then system (1.4) can be written in the following equivalent form:
Solving inequality (2.5), we have
It follows from Lemma 2.1 that
Then
Let
Then it follows that \(\vert x_{i}(t) \vert \leq\sigma^{*}, \vert y_{i}(t) \vert \leq\sigma^{*}, i=1,2,\ldots,n\), for all \(t>0\). This completes the proof of Lemma 2.3. □
$$\begin{aligned} \textstyle\begin{cases} \frac{dx_{i}(t)}{dt}=-a_{i}x_{i}(t)+a_{i}[x_{i}(t)-x_{i}(t-\delta_{i}(t))]+\sum_{j=1}^{n}a_{ij}(t)f_{j}(y_{j}(t-\sigma_{ij}(t)))+I_{i}(t),\\ \frac{dy_{i}(t)}{dt}=-b_{i}y_{i}(t)+b_{i}[y_{i}(t)-y_{i}(t-\eta_{i}(t))]+\sum_{j=1}^{n}b_{ij}(t)g_{j}(x_{j}(t-\tau_{ij}(t)))+J_{i}(t). \end{cases}\displaystyle \end{aligned}$$
(2.2)
$$\begin{aligned} \textstyle\begin{cases} \frac{dx_{i}(t)}{dt}=-a_{i}x_{i}(t)+a_{i}\int_{t-\delta _{i}(t)}^{t}x_{i}^{\prime}(s)\,ds+\sum_{j=1}^{n}a_{ij}(t)f_{j}(y_{j}(t-\sigma _{ij}(t)))+I_{i}(t),\\ \frac{dy_{i}(t)}{dt}=-b_{i}y_{i}(t)+b_{i}\int_{t-\eta_{i}(t)}^{t}y_{i}^{\prime}(s)\,ds+\sum_{j=1}^{n}b_{ij}(t)g_{j}(x_{j}(t-\tau_{ij}(t)))+J_{i}(t). \end{cases}\displaystyle \end{aligned}$$
(2.3)
$$\begin{aligned} \textstyle\begin{cases} \frac{dx_{i}(t)}{dt}=-a_{i}x_{i}(t)+a_{i}\int_{t-\delta_{i}(t)}^{t} [-a_{i}x_{i}(s-\delta_{i}(s))\\ \phantom{\frac{dx_{i}(t)}{dt}=}{}+\sum_{j=1}^{n}a_{ij}(s)f_{j}(y_{j}(s-\sigma _{ij}(s)))+I_{i}(s) ]\,ds\\ \phantom{\frac{dx_{i}(t)}{dt}=}{}+\sum_{j=1}^{n}a_{ij}(t)f_{j}(y_{j}(t-\sigma_{ij}(t)))+I_{i}(t),\\ \frac{dy_{i}(t)}{dt}=-b_{i}y_{i}(t)+b_{i}\int_{t-\eta_{i}(t)}^{t} [-b_{i}y_{i}(s-\eta_{i}(s))\\ \phantom{\frac{dy_{i}(t)}{dt}=}{}+\sum_{j=1}^{n}b_{ij}(s)g_{j}(x_{j}(s-\tau _{ij}(s)))+J_{i}(s) ]\,ds\\ \phantom{\frac{dx_{i}(t)}{dt}=}{}+\sum_{j=1}^{n}b_{ij}(t)g_{j}(x_{j}(t-\tau_{ij}(t)))+J_{i}(t). \end{cases}\displaystyle \end{aligned}$$
(2.4)
$$\begin{aligned} &z_{ii}(t)=\left ( \begin{matrix} x_{i}(t) \\ y_{i}(t) \end{matrix} \right ),\qquad A=\left ( \begin{matrix} -a_{i} & 0 \\ 0 & -b_{i} \end{matrix} \right ),\\ & f \bigl(x_{i}(t),y_{i}(t)\bigr)=\left ( \begin{matrix} f_{1}(x_{i}(t),y_{i}(t) \\ f_{2}(x_{i}(t),y_{i}(t) \end{matrix} \right),\qquad I_{ii}(t)= \left( \begin{matrix} I_{i}(t) \\ J_{i}(t) \end{matrix} \right), \end{aligned}$$
$$\begin{aligned} f_{1}(x_{i}(t),y_{i}(t)={}&a_{i} \int_{t-\delta_{i}(t)}^{t} \Biggl[-a_{i}x_{i} \bigl(s-\delta _{i}(s)\bigr)+\sum_{j=1}^{n}a_{ij}(s)f_{j} \bigl(y_{j}\bigl(s-\sigma_{ij}(s)\bigr)\bigr)+I_{i}(s) \Biggr]\,ds \\ & {}+\sum_{j=1}^{n}a_{ij}(t)f_{j} \bigl(y_{j}\bigl(t-\sigma_{ij}(t)\bigr)\bigr), \\ f_{2}(x_{i}(t),y_{i}(t)={}& b_{i} \int_{t-\eta_{i}(t)}^{t} \Biggl[-b_{i}y_{i} \bigl(s-\eta _{i}(s)\bigr)+\sum_{j=1}^{n}b_{ij}(s)g_{j} \bigl(x_{j}\bigl(s-\tau_{ij}(s)\bigr)\bigr)+J_{i}(s) \Biggr]\,ds \\ & {}+\sum_{j=1}^{n}b_{ij}(t)g_{j} \bigl(x_{j}\bigl(t-\tau_{ij}(t)\bigr)\bigr), \end{aligned}$$
$$\begin{aligned} z_{ii}^{\prime}(t)\leq{A}z_{ii}(t)+f \bigl(x_{i}(t),y_{i}(t)\bigr)+I_{ii}(t). \end{aligned}$$
(2.5)
$$z_{ii}(t)\leq{e^{At}}z_{ii}(0)+ \int_{0}^{t}e^{A(t-s)}\bigl[f \bigl(x_{i}(s),y_{i}(s)\bigr)+I_{ii}(t)\bigr]\,ds. $$
$$\begin{aligned} \bigl\Vert z_{ii}(t) \bigr\Vert \leq {}&\sqrt{2}e^{-\alpha t} \bigl\Vert z_{ii}(0) \bigr\Vert +\sqrt{2} \int_{0}^{t}e^{\alpha (t-s)}\bigl[ \bigl\Vert f \bigl(x_{i}(s),y_{i}(s)\bigr) \bigr\Vert + \bigl\vert I_{ii}(s) \bigr\vert \bigr]\,ds \\ \leq{}& \sqrt{2} \Vert \varphi \Vert ^{2}+\frac{\sqrt{2}}{\alpha} \bigl(1-e^{-\alpha t} \bigr) \\ &{} \times \bigl[a_{i}\delta_{i}^{+} \bigl( \bigl\vert x_{i}\bigl(t-\delta _{i}(t)\bigr) \bigr\vert +a_{ij}^{+}M_{j}^{f}+I_{i}^{+} \bigr)+a_{ij}^{+}M_{j}^{f}+I_{i}^{+} \\ &{} +b_{i}\eta_{i}^{+} \bigl( \bigl\vert y_{i}\bigl(t-\eta _{i}(t)\bigr) \bigr\vert +b_{ij}^{+}M_{j}^{g}+J_{i}^{+} \bigr)+b_{ij}^{+}M_{j}^{g}+J_{i}^{+} \bigr] \\ \leq {}&\sqrt{2} \Vert \varphi \Vert ^{2}+\frac{\sqrt{2}}{\alpha} \bigl(1-e^{-\alpha t} \bigr) \\ &{} \times \bigl[a_{i}\delta_{i}^{+} \bigl( \bigl\Vert z_{ii}(t) \bigr\Vert +a_{ij}^{+}M_{j}^{f}+I_{i}^{+} \bigr)+a_{ij}^{+}M_{j}^{f}+I_{i}^{+} \\ &{} + b_{i}\eta_{i}^{+} \bigl(\Vert z_{ii}(t) \Vert +b_{ij}^{+}M_{j}^{g}+J_{i}^{+} \bigr)+b_{ij}^{+}M_{j}^{g}+J_{i}^{+} \bigr] \\ \leq{}& \sqrt{2} \Vert \varphi \Vert ^{2}+\frac{\sqrt{2}}{\alpha} \\ &{} \times \bigl[a_{i}\delta_{i}^{+} \bigl( \bigl\Vert z_{ii}(t) \bigr\Vert +a_{ij}^{+}M_{j}^{f}+I_{i}^{+} \bigr)+a_{ij}^{+}M_{j}^{f}+I_{i}^{+} \\ &{} + b_{i}\eta_{i}^{+} \bigl( \bigl\Vert z_{ii}(t) \bigr\Vert +b_{ij}^{+}M_{j}^{g}+J_{i}^{+} \bigr)+b_{ij}^{+}M_{j}^{g}+J_{i}^{+} \bigr]. \end{aligned}$$
(2.6)
$$\begin{aligned} \bigl\Vert z_{ii}(t) \bigr\Vert \leq{}& \biggl[1- \frac{\sqrt{2}}{\alpha}\bigl(a_{i}\delta _{i}^{+}+b_{i} \eta_{i}^{+}\bigr) \biggr]^{-1} \bigl[a_{i}\delta_{i}^{+} \bigl(a_{ij}^{+}M_{j}^{f}+I_{i}^{+} \bigr) \\ & {}+a_{ij}^{+}M_{j}^{f}+I_{i}^{+}+b_{i} \eta_{i}^{+} \bigl(b_{ij}^{+}M_{j}^{g}+J_{i}^{+} \bigr)+b_{ij}^{+}M_{j}^{g}+J_{i}^{+} \bigr]. \end{aligned}$$
$$\begin{aligned} \sigma^{*}={}& \biggl[1-\frac{\sqrt{2}}{\alpha}\bigl(a_{i} \delta_{i}^{+}+b_{i}\eta _{i}^{+} \bigr) \biggr]^{-1} \bigl[a_{i}\delta_{i}^{+} \bigl(a_{ij}^{+}M_{j}^{f}+I_{i}^{+} \bigr) \\ & {}+a_{ij}^{+}M_{j}^{f}+I_{i}^{+}+b_{i} \eta_{i}^{+} \bigl(b_{ij}^{+}M_{j}^{g}+J_{i}^{+} \bigr)+b_{ij}^{+}M_{j}^{g}+J_{i}^{+} \bigr]. \end{aligned}$$
3 Main results
In this section, we present our main result that there exists an exponentially stable anti-periodic solution of (1.4).
Theorem 3.1
Assume that (H1)–(H4) hold true. Then any solution
\((x^{*}(t),y^{*}(t))^{T}\)
of system (1.4) is globally exponentially stable.
Proof
Let \(u_{i}(t)=x_{i}(t)-x_{i}^{*}(t), v_{i}(t)=y_{i}(t)-y_{i}^{*}(t), i=1,2,\ldots,n\). It follows from system (2.4) that
which leads to
In view of condition (H1), we get
Then
where \(0\leq\xi_{j},\varepsilon_{j}\leq1,j=1,2,\ldots,n\).
$$\begin{aligned} \textstyle\begin{cases} \frac{du_{i}(t)}{dt}=-a_{i}u_{i}(t)+a_{i}\int_{t-\delta_{i}(t)}^{t} [-a_{i}u_{i}(s-\delta_{i}(s))\\ \phantom{\frac{du_{i}(t)}{dt}=}{}+\sum_{j=1}^{n}a_{ij}(s)(f_{j}(y_{j}(s-\sigma_{ij}(s)))-f_{j}(y_{j}^{*}(s-\sigma _{ij}(s)))) ]\,ds\\ \phantom{\frac{du_{i}(t)}{dt}=}{}+\sum_{j=1}^{n}a_{ij}(t)(f_{j}(y_{j}(t-\sigma_{ij}(t)))-f_{j}(y_{j}^{*}(t-\sigma _{ij}(t)))),\\ \frac{dv_{i}(t)}{dt}=-b_{i}v_{i}(t)+b_{i}\int_{t-\eta_{i}(t)}^{t} [-b_{i}v_{i}(s-\eta_{i}(s))\\ \phantom{\frac{dv_{i}(t)}{dt}=}{}+\sum_{j=1}^{n}b_{ij}(s)(g_{j}(x_{j}(s-\tau_{ij}(s)))-g_{j}(x_{j}^{*}(s-\tau _{ij}(s)))) ]\,ds\\ \phantom{\frac{dv_{i}(t)}{dt}=}{}+\sum_{j=1}^{n}b_{ij}(t)(g_{j}(x_{j}(t-\tau_{ij}(t)))-g_{j}(x_{j}(t-\tau_{ij}(t)))), \end{cases}\displaystyle \end{aligned}$$
(3.1)
$$\begin{aligned} \textstyle\begin{cases} \frac{1}{2}\frac{du_{i}^{2}(t)}{dt}=-a_{i}u_{i}^{2}(t)+a_{i}u_{i}(t)\int_{t-\delta _{i}(t)}^{t} [-a_{i}u_{i}(s-\delta_{i}(s))\\ \phantom{\frac{1}{2}\frac{du_{i}^{2}(t)}{dt}=}{}+\sum_{j=1}^{n}a_{ij}(s)(f_{j}(y_{j}(s-\sigma_{ij}(s)))-f_{j}(y_{j}^{*}(s-\sigma _{ij}(s)))) ]\,ds\\ \phantom{\frac{1}{2}\frac{du_{i}^{2}(t)}{dt}=}{}+u_{i}(t)\sum_{j=1}^{n}a_{ij}(t)(f_{j}(y_{j}(t-\sigma _{ij}(t)))-f_{j}(y_{j}^{*}(t-\sigma_{ij}(t)))),\\ \frac{1}{2}\frac{dv_{i}^{2}(t)}{dt}=-b_{i}v_{i}^{2}(t)+b_{i}v_{i}(t)\int_{t-\eta _{i}(t)}^{t} [-b_{i}v_{i}(s-\eta_{i}(s))\\ \phantom{\frac{1}{2}\frac{dv_{i}^{2}(t)}{dt}=}{}+\sum_{j=1}^{n}b_{ij}(s)(g_{j}(x_{j}(s-\tau_{ij}(s)))-g_{j}(x_{j}^{*}(s-\tau _{ij}(s)))) ]\,ds\\ \phantom{\frac{1}{2}\frac{dv_{i}^{2}(t)}{dt}=}{}+v_{i}(t)\sum_{j=1}^{n}b_{ij}(t)(g_{j}(x_{j}(t-\tau_{ij}(t)))-g_{j}(x_{j}(t-\tau_{ij}(t)))). \end{cases}\displaystyle \end{aligned}$$
(3.2)
$$\begin{aligned} \textstyle\begin{cases} \frac{du_{i}^{2}(t)}{dt}\leq-2a_{i}u_{i}^{2}(t)+a_{i}u_{i}(t)\int_{t-\delta _{i}(t)}^{t} [-a_{i}u_{i}(s-\delta_{i}(s))\\ \phantom{\frac{du_{i}^{2}(t)}{dt}\leq}{}+\sum_{j=1}^{n}a_{ij}(s)(f_{j}(y_{j}(s-\sigma_{ij}(s)))-f_{j}(y_{j}^{*}(s-\sigma _{ij}(s)))) ]\,ds\\ \phantom{\frac{du_{i}^{2}(t)}{dt}\leq}{}+u_{i}(t)\sum_{j=1}^{n}a_{ij}(t)(f_{j}(y_{j}(t-\sigma _{ij}(t)))-f_{j}(y_{j}^{*}(t-\sigma_{ij}(t)))),\\ \frac{dv_{i}^{2}(t)}{dt}\leq-2b_{i}v_{i}^{2}(t)+b_{i}v_{i}(t)\int_{t-\eta_{i}(t)}^{t} [-b_{i}v_{i}(s-\eta_{i}(s))\\ \phantom{\frac{dv_{i}^{2}(t)}{dt}\leq}{}+\sum_{j=1}^{n}b_{ij}(s)(g_{j}(x_{j}(s-\tau_{ij}(s)))-g_{j}(x_{j}^{*}(s-\tau _{ij}(s)))) ]\,ds\\ \phantom{\frac{dv_{i}^{2}(t)}{dt}\leq}{}+v_{i}(t)\sum_{j=1}^{n}b_{ij}(t)(g_{j}(x_{j}(t-\tau_{ij}(t)))-g_{j}(x_{j}(t-\tau_{ij}(t)))). \end{cases}\displaystyle \end{aligned}$$
(3.3)
$$\begin{aligned} \textstyle\begin{cases} \frac{du_{i}^{2}(t)}{dt}\leq-2a_{i}u_{i}^{2}(t)+a_{i}\delta_{i}^{+} [a_{i}(u_{i}^{2}+u_{i}^{2}(t-\delta_{i}(t)))\\ \phantom{\frac{du_{i}^{2}(t)}{dt}\leq}{}+\sum_{j=1}^{n}a_{ij}^{+}(L_{jf}^{2\xi_{j}}u_{i}^{2}(t)+L_{jf}^{2(1-\xi _{j})}v_{j}^{2}(t-\delta_{ij}(t))) ]\\ \phantom{\frac{du_{i}^{2}(t)}{dt}\leq}{}+\sum_{j=1}^{n}a_{ij}^{+}(L_{jf}^{2\xi_{j}}u_{i}^{2}(t)+L_{jf}^{2(1-\xi _{j})}v_{j}^{2}(t-\delta_{ij}(t))),\\ \frac{dv_{i}^{2}(t)}{dt}\leq-2b_{i}v_{i}^{2}(t)+b_{i}\eta_{i}^{+} [b_{i}(v_{i}^{2}+v_{i}^{2}(t-\eta_{i}(t)))\\ \phantom{\frac{dv_{i}^{2}(t)}{dt}\leq}{}+\sum_{j=1}^{n}b_{ij}^{+}(L_{jg}^{2\varepsilon _{j}}v_{i}^{2}(t)+L_{jg}^{2(1-\varepsilon_{j})}u_{j}^{2}(t-\tau_{ij}(t))) ]\\ \phantom{\frac{dv_{i}^{2}(t)}{dt}\leq}{}+\sum_{j=1}^{n}b_{ij}^{+}(L_{jg}^{\varepsilon _{j}}v_{i}^{2}(t)+L_{jg}^{2(1-\varepsilon_{j})}u_{j}^{2}(t-\tau_{ij}(t))), \end{cases}\displaystyle \end{aligned}$$
(3.4)
Now we consider the following Lyapunov function:
where β is given by Lemma 2.2. Differentiating \(V(t)\) along solutions to system (1.4), together with (3.3), we have
It follows from Lemma 2.2 that \(\frac{dV(t)}{dt}\leq0\), which implies that \(V(t)\leq{V(0)}\) for all \(t>0\). Thus
Let
and choose
Then Eq. (3.7) can be rewritten as
for all \(t>0\). Then
for all \(t>0\). Thus the solution \((x(t),y(t))^{T}\) of system (1.4) is globally exponentially stable. □
$$\begin{aligned} V(t)={}&e^{\beta t}\sum_{i=1}^{n}u_{i}^{2}(t)+e^{\beta t} \sum_{i=1}^{n}v_{i}^{2}(t) \\ &{}+\sum_{i=1}^{n}a_{i}^{2} \delta_{i}^{+} \int_{t-\delta_{i}(t)}^{t}e^{\beta(s+\delta _{i}(t))}u_{i}^{2}(s)\,ds \\ & {}+\sum_{i=1}^{n}\sum _{j=1}^{n}a_{i}\delta_{i}^{+}a_{ij}^{+}L_{jf}^{2(1-\xi _{j})} \int_{t-\delta_{ij}(t)}^{t}e^{\beta(s+\delta_{ij}(t))}v_{j}^{2}(s)\,ds \\ & {}+\sum_{i=1}^{n}a_{ij}^{+}L_{jf}^{2(1-\xi_{j})} \int_{t-\delta _{ij}(t)}^{t}e^{\beta(s+\delta_{ij}(t))}v_{j}^{2}(s)\,ds \\ & {}+\sum_{i=1}^{n}b_{i}^{2} \eta_{i}^{+} \int_{t-\eta_{i}(t)}^{t}e^{\beta(s+\eta _{i}(t))}v_{i}^{2}(s)\,ds \\ & {}+\sum_{i=1}^{n}\sum _{j=1}^{n}b_{i}\eta _{i}^{+}b_{ij}^{+}L_{jg}^{2(1-\varepsilon_{j})} \int_{t-\tau _{ij}(t)}^{t}e^{\beta(s+\tau_{ij}(t))}u_{j}^{2}(s)\,ds \\ & {}+\sum_{i=1}^{n}b_{ij}^{+}L_{jg}^{2(1-\varepsilon_{j})} \int_{t-\tau _{ij}(t)}^{t}e^{\beta(s+\tau_{ij}(t))}u_{j}^{2}(s)\,ds, \end{aligned}$$
(3.5)
$$\begin{aligned} \frac{dV(t)}{dt}\leq{}&\beta{e^{\beta t}} \Biggl[\sum _{i=1}^{n}u_{i}^{2}(t)+\sum _{i=1}^{n}v_{i}^{2}(t) \Biggr] \\ & {}+e^{\beta t}\sum_{j=1}^{n} \Biggl\{ -2a_{i}u_{i}^{2}(t)+a_{i} \delta_{i}^{+} \Biggl[a_{i}\bigl(u_{i}^{2}+u_{i}^{2} \bigl(t-\delta_{i}(t)\bigr)\bigr) \\ & {}+\sum_{j=1}^{n}a_{ij}^{+} \bigl(L_{jf}^{2\xi_{j}}u_{i}^{2}(t)+L_{jf}^{2(1-\xi _{j})}v_{j}^{2} \bigl(t-\delta_{ij}(t)\bigr)\bigr) \Biggr] \\ & {}+\sum_{j=1}^{n}a_{ij}^{+} \bigl(L_{jf}^{2\xi_{j}}u_{i}^{2}(t)+L_{jf}^{2(1-\xi _{j})}v_{j}^{2} \bigl(t-\delta_{ij}(t)\bigr)\bigr) \Biggr\} \\ & {}+e^{\beta t}\sum_{j=1}^{n} \Biggl\{ -2b_{i}v_{i}^{2}(t)+b_{i} \eta_{i}^{+} \Biggl[b_{i}\bigl(v_{i}^{2}+v_{i}^{2} \bigl(t-\eta_{i}(t)\bigr)\bigr) \\ & {}+\sum_{j=1}^{n}b_{ij}^{+} \bigl(L_{jg}^{2\varepsilon _{j}}v_{i}^{2}(t)+L_{jg}^{2(1-\varepsilon_{j})}u_{j}^{2} \bigl(t-\tau_{ij}(t)\bigr)\bigr) \Biggr] \\ &{} +\sum_{j=1}^{n}b_{ij}^{+} \bigl(L_{jg}^{\varepsilon _{j}}v_{i}^{2}(t)+L_{jg}^{2(1-\varepsilon_{j})}u_{j}^{2} \bigl(t-\tau_{ij}(t)\bigr)\bigr) \Biggr\} \\ & {}+\sum_{i=1}^{n}a_{i}^{2} \delta_{i}^{+}\bigl[e^{\beta(t+\delta_{i}(t))} u_{i}^{2}(t)-e^{\beta t}u_{i}^{2} \bigl(t-\delta_{i}(t)\bigr)\bigr] \\ & {}+\sum_{i=1}^{n}\sum _{j=1}^{n}a_{i}\delta_{i}^{+}a_{ij}^{+}L_{jf}^{2(1-\xi _{j})} \bigl[e^{\beta(t+\delta_{ij}(t))}v_{j}^{2}(t)-e^{\beta t}v_{j}^{2} \bigl(t-\delta _{ij}(t)\bigr)\bigr] \\ & {}+\sum_{i=1}^{n}a_{ij}^{+}L_{jf}^{2(1-\xi_{j})} \bigl[e^{\beta(t+\delta _{ij}(t))}v_{j}^{2}(t)-e^{\beta(t)}v_{j}^{2} \bigl(t-\delta_{ij}(t)\bigr)\bigr] \\ & {}+\sum_{i=1}^{n}b_{i}^{2} \eta_{i}^{+}\bigl[e^{\beta(t+\eta_{i}(t))}v_{i}^{2}(t)-e^{\beta t}v_{i}^{2} \bigl(t-\eta_{i}(t)\bigr)\bigr] \\ & {}+\sum_{i=1}^{n}\sum _{j=1}^{n}b_{i}\eta _{i}^{+}b_{ij}^{+}L_{jg}^{2(1-\varepsilon_{j})} \bigl[e^{\beta(t+\tau _{ij}(t))}u_{j}^{2}(t)-e^{\beta t}u_{j}^{2} \bigl(t-\tau_{ij}(t)\bigr)\bigr] \\ & {}+\sum_{i=1}^{n}b_{ij}^{+}L_{jg}^{2(1-\varepsilon_{j})} \bigl[e^{\beta(t+\tau _{ij}(t))}u_{j}^{2}(t)-e^{\beta t}u_{j}^{2} \bigl(t-\tau_{ij}(t)\bigr)\bigr] \\ ={}&e^{\beta t} \Biggl[\beta-2a_{i}+a_{i}^{2} \delta_{i}^{+}+\sum_{j=1}^{n}a_{i} \delta _{i}^{+}a_{ij}^{+}L_{jf}^{2\xi_{j}}+ \sum_{j=1}^{n}a_{ij}^{+}L_{jf}^{2\xi _{j}}+ \sum_{i=1}^{n}a_{i}^{2} \delta_{i}^{+}e^{\beta\delta_{i}(t)} \\ & {}+\sum_{j=1}^{n}b_{i} \eta_{i}^{+}b_{ij}^{+}L_{jg}^{2(1-\varepsilon _{j})}e^{\beta\tau_{ij}(t)}+b_{ij}^{+}L_{jg}^{2(1-\varepsilon_{j})}e^{\beta \tau_{ij}(t)} \Biggr]x_{i}^{2}(t) \\ & {}+e^{\beta t} \Biggl[\beta-2b_{i}+b_{i}^{2} \eta_{i}^{+}+\sum_{j=1}^{n}b_{i} \eta _{i}^{+}b_{ij}^{+}L_{jg}^{2\varepsilon_{j}}+ \sum_{j=1}^{n}b_{ij}^{+}L_{jg}^{2\varepsilon_{j}} \\ & {}+\sum_{i=1}^{n}a_{i}^{2} \delta_{i}^{+}a_{ij}^{+}L_{jf}^{2(1-\xi_{j})}e^{\beta \delta_{ij}(t)}+a_{ij}^{+}L_{jf}^{2(1-\xi_{j})}e^{\beta\delta _{ij}(t)}+b_{i} \eta_{i}^{+}e^{\beta\eta_{i}(t)} \Biggr]y_{i}^{2}(t) \\ \leq{}&e^{\beta t} \Biggl[\beta-2a_{i}+a_{i}^{2} \delta_{i}^{+}+\sum_{j=1}^{n}a_{i} \delta _{i}^{+}a_{ij}^{+}L_{jf}^{2\xi_{j}}+ \sum_{j=1}^{n}a_{ij}^{+}L_{jf}^{2\xi _{j}}+ \sum_{i=1}^{n}a_{i}^{2} \delta_{i}^{+}e^{\beta\delta_{i}^{+}} \\ & {}+\sum_{j=1}^{n}b_{i} \eta_{i}^{+}b_{ij}^{+}L_{jg}^{2(1-\varepsilon _{j})}e^{\beta\tau_{ij}^{+}}+b_{ij}^{+}L_{jg}^{2(1-\varepsilon_{j})}e^{\beta \tau_{ij}^{+}} \Biggr]x_{i}^{2}(t) \\ & {}+e^{\beta t} \Biggl[\beta-2b_{i}+b_{i}^{2} \eta_{i}^{+}+\sum_{j=1}^{n}b_{i} \eta _{i}^{+}b_{ij}^{+}L_{jg}^{2\varepsilon_{j}}+ \sum_{j=1}^{n}b_{ij}^{+}L_{jg}^{2\varepsilon_{j}} \\ & {}+\sum_{i=1}^{n}a_{i}^{2} \delta_{i}^{+}a_{ij}^{+}L_{jf}^{2(1-\xi_{j})}e^{\beta \delta_{ij}^{+}}+a_{ij}^{+}L_{jf}^{2(1-\xi_{j})}e^{\beta\delta _{ij}^{+}}+b_{i} \eta_{i}^{+}e^{\beta\eta_{i}^{+}} \Biggr]y_{i}^{2}(t). \end{aligned}$$
(3.6)
$$\begin{aligned} & e^{\beta t} \Biggl[\sum_{i=1}^{n}u_{i}^{2}(t)+ \sum_{i=1}^{n}v_{i}^{2}(t) \Biggr] \\ &\quad \leq\sum_{i=1}^{n}u_{i}^{2}(0)+ \sum_{i=1}^{n}v_{i}^{2}(0) \\ & \qquad{}+\sum_{i=1}^{n}a_{i}^{2} \delta_{i}^{+} \int_{-\delta_{i}(0)}^{0}e^{\beta(s+\delta _{i}(0))}u_{i}^{2}(s)\,ds \\ & \qquad{}+\sum_{i=1}^{n}\sum _{j=1}^{n}a_{i}\delta_{i}^{+}a_{ij}^{+}L_{jf}^{2(1-\xi _{j})} \int_{-\delta_{ij}(0)}^{0}e^{\beta(s+\delta_{ij}(0))}v_{j}^{2}(s)\,ds \\ & \qquad{}+\sum_{i=1}^{n}a_{ij}^{+}L_{jf}^{2(1-\xi_{j})} \int_{-\delta _{ij}(0)}^{0}e^{\beta(s+\delta_{ij}(0))}v_{j}^{2}(s)\,ds \\ & \qquad{}+\sum_{i=1}^{n}b_{i}^{2} \eta_{i}^{+} \int_{-\eta_{i}(0)}^{0}e^{\beta(s+\eta _{i}(0))}v_{i}^{2}(s)\,ds \\ & \qquad{}+\sum_{i=1}^{n}\sum _{j=1}^{n}b_{i}\eta _{i}^{+}b_{ij}^{+}L_{jg}^{2(1-\varepsilon_{j})} \int_{-\tau _{ij}(0)}^{0}e^{\beta(s+\tau_{ij}(0))}u_{j}^{2}(s)\,ds \\ & \qquad{}+\sum_{i=1}^{n}b_{ij}^{+}L_{jg}^{2(1-\varepsilon_{j})} \int_{-\tau _{ij}(0)}^{t}e^{\beta(s+\tau_{ij}(0))}u_{j}^{2}(s)\,ds \\ &\quad \leq \bigl\Vert \varphi-\varphi^{*} \bigr\Vert ^{2}+ \bigl\Vert \psi-\psi^{*} \bigr\Vert ^{2}+\sum_{i=1}^{n}a_{i}^{2} \delta _{i}^{+}\frac{1}{\beta}e^{\beta\delta_{i}^{+}} \bigl\Vert \varphi-\varphi ^{*} \bigr\Vert ^{2} \\ & \qquad{}+\sum_{i=1}^{n}\sum _{j=1}^{n}a_{i}\delta_{i}^{+}a_{ij}^{+}L_{jf}^{2(1-\xi _{j})} \frac{1}{\beta}e^{\beta\delta_{ij}^{+}} \bigl\Vert \psi-\psi^{*} \bigr\Vert ^{2} \\ & \qquad{}+\sum_{i=1}^{n}a_{ij}^{+}L_{jf}^{2(1-\xi_{j})} \frac{1}{\beta}e^{\beta \delta_{ij}^{+}} \bigl\Vert \psi-\psi^{*} \bigr\Vert ^{2} \\ & \qquad{}+\sum_{i=1}^{n}b_{i}^{2} \eta_{i}^{+}\frac{1}{\beta}e^{\beta\eta _{i}^{+}} \bigl\Vert \psi-\psi^{*} \bigr\Vert ^{2} \\ & \qquad{}+\sum_{i=1}^{n}\sum _{j=1}^{n}b_{i}\eta _{i}^{+}b_{ij}^{+}L_{jg}^{2(1-\varepsilon_{j})} \frac{1}{\beta}e^{\beta\tau _{ij}^{+}} \bigl\Vert \varphi-\varphi^{*} \bigr\Vert ^{2} \\ & \qquad{}+\sum_{i=1}^{n}b_{ij}^{+}L_{jg}^{2(1-\varepsilon_{j})} \frac{1}{\beta }e^{\beta\tau_{ij}^{+}} \bigl\Vert \varphi-\varphi^{*} \bigr\Vert ^{2} \\ &\quad= \Biggl[1+\sum_{i=1}^{n}a_{i}^{2} \delta_{i}^{+}\frac{1}{\beta}e^{\beta\delta _{i}^{+}}+\sum _{i=1}^{n}\sum_{j=1}^{n}b_{i} \eta _{i}^{+}b_{ij}^{+}L_{jg}^{2(1-\varepsilon_{j})} \frac{1}{\beta}e^{\beta\tau _{ij}^{+}} \\ & \qquad{} +\sum_{i=1}^{n}b_{ij}^{+}L_{jg}^{2(1-\varepsilon_{j})} \frac {1}{\beta}e^{\beta\tau_{ij}^{+}} \Biggr] \bigl\Vert \varphi-\varphi^{*} \bigr\Vert ^{2} \\ & \qquad{}+ \Biggl[1+\sum_{i=1}^{n}\sum _{j=1}^{n}a_{i}\delta _{i}^{+}a_{ij}^{+}L_{jf}^{2(1-\xi_{j})} \frac{1}{\beta}e^{\beta\delta _{ij}^{+}} \\ & \qquad{}+\sum_{i=1}^{n}a_{ij}^{+}L_{jf}^{2(1-\xi_{j})} \frac{1}{\beta}e^{\beta \delta_{ij}^{+}}+\sum_{i=1}^{n}b_{i}^{2} \eta_{i}^{+}\frac{1}{\beta }e^{\beta\eta_{i}^{+}} \Biggr] \bigl\Vert \psi-\psi^{*} \bigr\Vert ^{2}. \end{aligned}$$
(3.7)
$$\begin{aligned} &\theta_{1}=1+\sum_{i=1}^{n}a_{i}^{2} \delta_{i}^{+}\frac{1}{\beta}e^{\beta\delta _{i}^{+}}+\sum _{i=1}^{n}\sum_{j=1}^{n}b_{i} \eta _{i}^{+}b_{ij}^{+}L_{jg}^{2(1-\varepsilon_{j})} \frac{1}{\beta}e^{\beta\tau_{ij}^{+}} +\sum_{i=1}^{n}b_{ij}^{+}L_{jg}^{2(1-\varepsilon_{j})} \frac{1}{\beta }e^{\beta\tau_{ij}^{+}}, \\ &\theta_{2}=1+\sum_{i=1}^{n}\sum _{j=1}^{n}a_{i}\delta _{i}^{+}a_{ij}^{+}L_{jf}^{2(1-\xi_{j})} \frac{1}{\beta}e^{\beta\delta _{ij}^{+}}+\sum_{i=1}^{n}a_{ij}^{+}L_{jf}^{2(1-\xi_{j})} \frac{1}{\beta }e^{\beta\delta_{ij}^{+}} \sum_{i=1}^{n}b_{i}^{2} \eta_{i}^{+}\frac{1}{\beta}e^{\beta\eta_{i}^{+}} \end{aligned}$$
$$M=\max\{\theta_{1},\theta_{2}\}>1. $$
$$\sum_{i=1}^{n}u_{i}^{2}(t)+ \sum_{i=1}^{n}v_{i}^{2}(t) \leq{M}e^{-\beta t} \bigl[ \bigl\Vert \varphi-\varphi^{*} \bigr\Vert ^{2}+ \bigl\Vert \psi-\psi^{*} \bigr\Vert ^{2} \bigr] $$
$$\sum_{i=1}^{n} \bigl\vert x_{i}(t)-x_{i}^{*}(t) \bigr\vert ^{2}+\sum _{i=1}^{n} \bigl\vert y_{i}(t)-y_{i}^{*}(t) \bigr\vert ^{2}\leq {M}e^{-\beta t} \bigl[ \bigl\Vert \varphi- \varphi^{*} \bigr\Vert ^{2}+ \bigl\Vert \psi-\psi^{*} \bigr\Vert ^{2} \bigr] $$
Theorem 3.2
Assume that (H1)–(H4) are satisfied. Then system (1.4) has exactly one
T-anti-periodic solution which is globally stable.
Proof
It follows from system (1.4) and (H2) that for each \(k\in{N}\), we have
Let
Obviously, for any \(k\in{N},(\bar{x}(t), \bar{y}(t))\) is also a solution of system (1.4). If the initial function \(\varphi_{i}(s), \psi_{i}(s)\ (i=1,2,\ldots,n)\) is bounded, it follows from Theorem 3.1 that there exists a constant \(\gamma>1\) such that
where \(t+kT>0,i=1,2,\ldots,n\). Since for any \(k\in{N}\) we have
then
By Lemma 2.3, we know that the solutions of system (1.4) are bounded. In view of (3.10) and (3.12), we can easily know that \(\{(-1)^{k+1} x_{i}(t + (k+1)T)\}\) uniformly converges to a continuous function \(x^{*}(t)=(x^{*}_{1}(t),x^{*}_{2}(t), \ldots, x^{*}_{n}(t))^{T}\) on any compact set of \({R^{n}}\). In a similar way, we can easily prove that \(\{(-1)^{k+1} y_{i}(t + (k+1)T)\}\) uniformly converges to a continuous function \(y^{*}(t)=(y^{*}_{1}(t),y^{*}_{2}(t), \ldots, y^{*}_{n}(t))^{T}\) on any compact set of \({R^{n}}\).
$$\begin{aligned} & \frac{d}{dt} \bigl[(-1)^{k+1}x_{i}\bigl(t+(k+1)T \bigr) \bigr] \\ & \quad=(-1)^{k+1} [-a_{i}x_{i}\bigl(t+(k+1)T- \delta_{i}\bigl(t+(k+1)T\bigr)\bigr) \\ & \qquad{}+\sum_{j=1}^{n}a_{ij} \bigl(t+(k+1)T\bigr)f_{j}\bigl(y_{j}\bigl(t+(k+1)T-\sigma _{ij}\bigl(t+(k+1)T\bigr)\bigr)\bigr) \\ & \qquad{}+I_{i}\bigl(t+(k+1)T\bigr) \\ & \quad=-a_{i}(-1)^{k+1}x_{i}\bigl(t+(k+1)T- \delta_{i}(t)\bigr) \\ & \qquad{}+\sum_{j=1}^{n}a_{ij}(t)f_{j} \bigl((-1)^{k+1}y_{j}\bigl(t+(k+1)T-\sigma _{ij}(t) \bigr)\bigr)+I_{i}(t), \end{aligned}$$
(3.8)
$$\begin{aligned} & \frac{d}{dt} \bigl[(-1)^{k+1}y_{i} \bigl(t+(k+1)T\bigr) \bigr] \\ & \quad=(-1)^{k+1} \Biggl[-b_{i}y_{i}\bigl(t+(k+1)T- \eta_{i}\bigl(t+(k+1)T\bigr)\bigr) \\ & \qquad{}+\sum_{j=1}^{n}b_{ij} \bigl(t+(k+1)T\bigr)g_{j}\bigl(x_{j}\bigl(t+(k+1)T-\tau _{ij}\bigl(t+(k+1)T\bigr)\bigr)\bigr) \\ & \qquad{}+J_{i}\bigl(t+(k+1)T\bigr) \Biggr] \\ & \quad=-b_{i}(-1)^{k+1}y_{i}\bigl(t+(k+1)T- \eta_{i}(t)\bigr) \\ & \qquad{}+\sum_{j=1}^{n}b_{ij}(t)g_{j} \bigl((-1)^{k+1}x_{j}\bigl(t+(k+1)T-\tau _{ij}(t) \bigr)\bigr)+J_{i}(t). \end{aligned}$$
(3.9)
$$\begin{aligned} &\bar{x}(t)=\bigl((-1)^{k+1}x_{1}\bigl(t+(k+1)T \bigr),(-1)^{k+1}x_{2}\bigl(t+(k+1)T\bigr),\ldots ,(-1)^{k+1}x_{n}\bigl(t+(k+1)T\bigr)\bigr)^{T}, \\ &\bar{y}(t)=\bigl((-1)^{k+1}y_{1}\bigl(t+(k+1)T \bigr),(-1)^{k+1}y_{2}\bigl(t+(k+1)T\bigr),\ldots ,(-1)^{k+1}y_{n}\bigl(t+(k+1)T\bigr)\bigr)^{T}. \end{aligned}$$
$$\begin{aligned} & \bigl\vert (-1)^{k+1}x_{i}\bigl(t+(k+1)T \bigr)-(-1)^{k}x_{i}(t+kT) \bigr\vert \\ & \quad \leq{M}e^{-\beta(t+kT)}\sup_{-\tau\leq{s}\leq0}\sum _{i=1}^{4} \bigl\vert x_{i}(t+T)+x_{i}(s) \bigr\vert ^{2} \\ & \quad \leq\gamma e^{-\beta(t+kT)}, \end{aligned}$$
(3.10)
$$\begin{aligned} (-1)^{k+1} x_{i} \bigl(t + (k+1)T\bigr) = x_{i} (t ) +\sum_{j=0}^{k}\bigl[(-1)^{j+1} x _{i}\bigl(t + (j+1)T\bigr)-(-1)^{j} x_{i} (t + jT)\bigr], \end{aligned}$$
(3.11)
$$\begin{aligned} (-1)^{k+1} x_{i} \bigl(t + (k+1)T\bigr) \leq \bigl\vert x_{i} (t ) \bigr\vert +\sum_{j=0}^{k} \bigl\vert (-1)^{j+1} x _{i}\bigl(t + (j+1)T \bigr)-(-1)^{j} x_{i} (t + jT) \bigr\vert . \end{aligned}$$
(3.12)
Now we show that \(x^{*}(t)\) is a T-anti-periodic solution of (1.4). Firstly, \(x^{*}(t)\) is T-anti-periodic since
Then we can conclude that \(x_{i}^{*}(t)\) is T-anti-periodic on R. Similarly, \(y_{i}^{*}(t)\) is also T-anti-periodic on R. Thus we can conclude that \((x^{*}(t),y^{*}(t))^{T}\) is the solution of system (1.4).
$$\begin{aligned} x^{*}(t+T) &=\lim_{k\to\infty}(-1)^{k }x(t +T+ kT) \\ &=-\lim_{(k+1)\to\infty}(-1)^{k+1 } x\bigl(t +(k +1)T \bigr)=-x^{*}(t ). \end{aligned}$$
(3.13)
In fact, together with the continuity of the right-hand side of system (1.4), let \(k\to\infty\), we can easily get
Therefore, \((x^{*}(t),y^{*}(t))^{T}\) is a solution of (1.4). Finally, by applying Theorem 3.1, it is easy to check that \((x^{*}(t),y^{*}(t))^{T}\) is globally exponentially stable. This completes the proof of Theorem 3.2. □
$$\begin{aligned} \textstyle\begin{cases} \frac{dx_{i}^{*}(t)}{dt}=-a_{i}x_{i}^{*}(t-\delta_{i}(t))+\sum_{j=1}^{n}a_{ij}(t)f_{j}(y_{j}^{*}(t-\sigma_{ij}(t)))+I_{i}(t),\\ \frac{dy_{i}^{*}(t)}{dt}=-b_{i}y_{i}^{*}(t-\eta_{i}(t))+\sum_{j=1}^{n}b_{ij}(t)g_{j}(x_{j}^{*}(t-\tau_{ij}(t)))+J_{i}(t). \end{cases}\displaystyle \end{aligned}$$
(3.14)
4 An example
In this section, we give an example to illustrate our main results derived in the previous sections. Consider the following BAM neural network with time-varying delays in the leakage terms:
where
Set \(f_{j}(u)=g_{j}(u)=\frac{1}{2}( \vert u+1 \vert - \vert u-1 \vert ),j=1,2\). Then \(L_{jf}=g_{jg}=M_{j}^{f}=M_{j}^{g}=1, \delta_{1}^{+}=\delta_{2}^{+}=0.05, a_{1j}^{+}=0.3, a_{2j}^{+}=0.5, \eta_{1}^{+}=\eta_{2}^{+}=0.04, b_{1j}^{+}=0.03, b_{2j}^{+}=0.05, j=1,2\). It is easy to verify that
and
Then all the conditions (H1)–(H4) hold. Thus system (4.1) has exactly one π-anti-periodic solution which is globally exponentially stable. The results are illustrated in Fig. 1.
$$\begin{aligned} \textstyle\begin{cases} \frac{dx_{1}(t)}{dt}=-a_{1}x_{1}(t-\delta_{1}(t))+\sum_{j=1}^{2}a_{1j}(t)f_{j}(y_{j}(t-\sigma_{1j}(t)))+I_{1}(t),\\ \frac{dx_{2}(t)}{dt}=-a_{2}x_{2}(t-\delta_{2}(t))+\sum_{j=1}^{2}a_{2j}(t)f_{j}(y_{j}(t-\sigma_{2j}(t)))+I_{2}(t),\\ \frac{dy_{1}(t)}{dt}=-b_{1}y_{1}(t-\eta_{1}(t))+\sum_{j=1}^{2}b_{1j}(t)g_{j}(x_{j}(t-\tau_{1j}(t)))+J_{1}(t),\\ \frac{dy_{2}(t)}{dt}=-b_{2}y_{2}(t-\eta_{2}(t))+\sum_{j=1}^{2}b_{2j}(t)g_{j}(x_{j}(t-\tau_{2j}(t)))+J_{2}(t), \end{cases}\displaystyle \end{aligned}$$
(4.1)
$$\begin{aligned} & \left [ \begin{matrix} \delta_{1}(t) & \delta_{2}(t) \\ \eta_{1}(t)& \eta_{2}(t) \end{matrix} \right ]=\left [ \begin{matrix} 0.05 \vert \sin t \vert & 0.05 \vert \sin t \vert \\ 0.04 \vert \cos t \vert & 0.04 \vert \cos t \vert \end{matrix} \right ], \\ & \left [ \begin{matrix} a_{11}(t) & a_{12}(t) \\ a_{21}(t)& a_{22}(t) \end{matrix} \right ]=\left [ \begin{matrix} 0.3 \vert \cos t \vert & 0.3 \vert \cos t \vert \\ 0.5 \vert \sin t \vert & 0.5 \vert \sin t \vert \end{matrix} \right ], \\ &\left [ \begin{matrix} b_{11}(t) & b_{12}(t) \\ b_{21}(t)& b_{22}(t) \end{matrix} \right ]=\left [ \begin{matrix} 0.03 \vert \cos t \vert & 0.03 \vert \cos t \vert \\ 0.05 \vert \sin t \vert & 0.05 \vert \sin t \vert \end{matrix} \right ], \\ & \left [ \begin{matrix} I_{1}(t) & I_{2}(t) \\ J_{1}(t)& J_{2}(t) \end{matrix} \right ]=\left [ \begin{matrix} 0.5\cos t& 0.5\cos t \\ 0.5\sin t & 0.5\sin t \end{matrix} \right ],\qquad \left [ \begin{matrix} a_{1} & a_{2} \\ b_{1}& b_{2} \end{matrix} \right ]=\left [ \begin{matrix} 2& 2 \\ 2 & 2 \end{matrix} \right ]. \end{aligned}$$
$$\frac{\sqrt{2}}{\alpha}\bigl(a_{i}\delta_{i}^{+}+b_{i} \eta_{i}^{+}\bigr)\approx0.12726< 1 $$
$$\begin{aligned} & {-}2a_{1}+a_{1}^{2}\delta_{1}^{+}+ \sum_{j=1}^{2}a_{1}\delta _{1}^{+}a_{1j}^{+}L_{jf}^{2\xi_{j}}+ \sum_{j=1}^{2}a_{1j}^{+}L_{jf}^{2\xi_{j}} \\ & \quad{}+\sum_{i=1}^{2}a_{i}^{2} \delta_{i}^{+}+\sum_{j=1}^{2}b_{1} \eta _{1}^{+}b_{1j}^{+}L_{jg}^{2(1-\varepsilon _{j})}+b_{1j}^{+}L_{jg}^{2(1-\varepsilon_{j})}=-2.862< 0, \\ & {-}2a_{2}+a_{2}^{2}\delta_{2}^{+}+ \sum_{j=1}^{2}a_{2}\delta _{2}^{+}a_{2j}^{+}L_{jf}^{2\xi_{j}}+ \sum_{j=1}^{2}a_{2j}^{+}L_{jf}^{2\xi_{j}} \\ & \quad{}+\sum_{i=1}^{2}a_{i}^{2} \delta_{i}^{+}+\sum_{j=1}^{2}b_{2} \eta _{2}^{+}b_{2j}^{+}L_{jg}^{2(1-\varepsilon _{j})}+b_{2j}^{+}L_{jg}^{2(1-\varepsilon_{j})}=-2.04< 0, \\ & {-}2b_{1}+b_{1}^{2}\eta_{1}^{+}+ \sum_{j=1}^{2}b_{1}\eta _{1}^{+}b_{1j}^{+}L_{jg}^{2\varepsilon_{j}}+ \sum_{j=1}^{2}b_{1j}^{+}L_{jg}^{2\varepsilon_{j}} \\ & \quad{}+\sum_{i=1}^{2}a_{1}^{2} \delta_{1}^{+}a_{1j}^{+}L_{jf}^{2(1-\xi _{j})}+a_{1j}^{+}L_{jf}^{2(1-\xi_{j})}+b_{1} \eta_{1}^{+}=-3.256< 0, \\ & {-}2b_{2}+b_{2}^{2}\eta_{2}^{+}+ \sum_{j=1}^{2}b_{2}\eta _{2}^{+}b_{2j}^{+}L_{jg}^{2\varepsilon_{j}}+ \sum_{j=1}^{2}b_{2j}^{+}L_{jg}^{2\varepsilon_{j}} \\ & \quad{}+\sum_{i=1}^{2}a_{2}^{2} \delta_{2}^{+}a_{2j}^{+}L_{jf}^{2(1-\xi _{j})}+a_{2j}^{+}L_{jf}^{2(1-\xi_{j})}+b_{2} \eta_{2}^{+}=-3.024< 0. \end{aligned}$$
Figure 1
Transient response of state variables \(x_{1}(t),x_{2}(t),y_{1}(t)\), and \(y_{2}(t)\)
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5 Conclusions
In this paper, we have investigated the asymptotic behavior of BAM neural networks with time-varying delays in the leakage terms. Applying the fundamental solution matrix of coefficient matrix, we obtained a series of new sufficient conditions to guarantee the existence and global exponential stability of an anti-periodic solution for the BAM neural networks with time-varying delays in the leakage terms. The obtained conditions are easy to check in practice. Finally, an example is included to illustrate the feasibility and effectiveness.
Acknowledgements
The first author was supported by the National Natural Science Foundation of China (No. 61673008) and Project of High-level Innovative Talents of Guizhou Province ([2016]5651), Major Research Project of the Innovation Group of the Education Department of Guizhou Province ([2017]039), Project of Key Laboratory of Guizhou Province with Financial and Physical Features ([2017]004) and Foundation of Science and Technology of Guizhou Province (2018). The authors would like to thank the referees and the editor for helpful suggestions incorporated into this paper.
Competing interests
The authors declare that they have no competing interests.
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