In this section, we first introduce some notations and lemmas. Denote
$$\bar{s}_{ji}=\sup_{t\in R}\bigl\vert s_{ji}(t)\bigr\vert ,\qquad \bar{t}_{ij}=\sup _{t\in R}\bigl\vert t_{ij}(t)\bigr\vert ,\qquad \bar{c}_{i}=\sup_{t\in R}\bigl\vert c_{i}(t)\bigr\vert ,\qquad \bar{d}_{j}=\sup _{t\in R}\bigl\vert d_{j}(t)\bigr\vert . $$
For any vector \(U=(u_{1},u_{2},\ldots,u_{m})^{T}\) and matrix \(M=(m_{ij})_{m\times{m}}\), define the following norm:
$$\|U\|= \Biggl(\sum_{i=1}^{m}u_{i}^{2} \Biggr)^{\frac{1}{2}},\qquad \|M\|= \Biggl(\sum_{i,j=1}^{m}m_{ij}^{2} \Biggr)^{\frac{1}{2}}. $$
Let
$$\begin{aligned}& \varphi(s)=\bigl(\varphi_{1}(s),\varphi_{2}(s), \ldots, \varphi_{m}(s)\bigr)^{T},\quad \varphi_{i}(s)\in{C} \bigl([-\delta,0],R\bigr),i=1,2, \ldots,m, \\& \psi(s)=\bigl(\psi_{1}(s),\psi_{2}(s), \ldots, \psi_{m}(s)\bigr)^{T},\quad \psi_{i}(s)\in{C} \bigl([-\tau,0],R\bigr),i=1,2, \ldots,m, \end{aligned}$$
where \(\tau=\max_{1\leq{i,j}\leq{m}}\{\tau_{ij}\}\), \(\delta=\max_{1\leq{i,j}\leq {m}}\{\delta_{ji}\}\). Define
$$\|\varphi\|=\sup_{-\delta\leq{s}\leq0} \Biggl(\sum _{i=1}^{m}\bigl\vert \varphi _{i}(s)\bigr\vert ^{2} \Biggr)^{\frac{1}{2}},\qquad \|\psi\|=\sup _{-\tau\leq{s}\leq0} \Biggl(\sum_{i=1}^{m} \bigl\vert \psi_{i}(s)\bigr\vert ^{2} \Biggr)^{\frac{1}{2}}. $$
The initial conditions of (1.3) are given by
$$ \left \{ \textstyle\begin{array}{l} x_{i0}(s)=\varphi_{i}(s),\quad -\sigma\leq s\leq0, \\ y_{i0}(s)=\psi_{i}(s),\quad -\tau\leq s\leq0. \end{array}\displaystyle \right . $$
(2.1)
Let \(x(t)=(x_{1}(t),x_{2}(t), \ldots,x_{m}(t))^{T}\), \(y(t)=(y_{1}(t),y_{2}(t), \ldots,y_{m}(t))^{T} \) be the solution of system (1.3) with initial conditions (2.1). We say the solution \(x(t)=(x_{1}(t),x_{2}(t), \ldots,x_{m}(t))^{T}\) is T-anti-periodic on \(R^{m}\) if \(x_{i}(t+T)=-x_{i}(t)\) (\(i=1,2, \ldots,m\)) for all \(t\in{R}\), where T is a positive constant.
Throughout this paper, we assume that the following conditions hold.
-
(H1)
\(f_{i}, g_{i}\in C(R^{2},R)\), \(i,j=1,2,\ldots,m\), there exist constants \(\alpha_{if}>0\), \(\alpha_{ig}>0\), \(\beta_{if}>0\), and \(\beta_{ig}>0\) such that
$$\left \{ \textstyle\begin{array}{lc} |f_{i}(u_{i}, u_{j} )-f_{i}(\bar{u}_{i}, \bar{u}_{j})| \leq\alpha_{if}|u_{i} -\bar {u}_{i}|+\beta_{if}|u_{j} -\bar{u}_{j}|, \qquad |f_{i}(u,v)|\leq F_{i}, \quad i\neq j, \\ |g_{i}(u_{i}, u_{j} )-g_{i}(\bar{u}_{i}, \bar{u}_{j})| \leq\alpha_{ig}|u_{i} -\bar {u}_{i}|+\beta_{ig}|u_{j} -\bar{u}_{j}|,\qquad |g_{i}(u,v)|\leq G_{i},\quad i\neq j \end{array}\displaystyle \right . $$
for all \(u_{i}, u_{j}, \bar{u}_{i}, \bar{u}_{j}, u, v \in{R} \).
-
(H2)
For all \(t,u,v\in{R}\),
$$\left \{ \textstyle\begin{array}{l} s_{ji}(t+T)f_{j}(u,v)=-s_{ji}(t)f_{j}(-u,-v), \\ t_{ij}(t+T)g_{i}(u,v)=-t_{ij}(t)g_{i}(-u,-v), \\ c_{i}(t+T)=-c_{i}(t),\qquad d_{j}(t+T)=-d_{j}(t), \end{array}\displaystyle \right . $$
where \(i,j=1,2,\ldots,m \) and T is a positive constant.
Definition 2.1
The solution \((x^{*}(t),y^{*}(t))^{T}\) of model (1.3) is said to globally exponentially stable if there exist constants \(\beta>0\) and \(M>1\) such that
$$\sum_{i=1}^{m}\bigl\vert x_{i}(t)-x_{i}^{*}(t)\bigr\vert ^{2}+\sum _{j=1}^{m}\bigl\vert y_{j}(t)-y_{j}^{*}(t) \bigr\vert ^{2}\leq {M}e^{-\beta t}\bigl\Vert \varphi-\varphi^{*} \bigr\Vert ^{2} $$
for each solution \((x(t),y(t))^{T}\) of model (1.3).
Lemma 2.1
Let
$$A=\left ( \textstyle\begin{array}{@{}c@{\quad}c@{}} -a_{i} & 0 \\ 0 & -b_{j} \end{array}\displaystyle \right ),\qquad \alpha=\min _{1\leq i,j\leq m}\{a_{i},b_{i}\}, $$
then
$$\|\exp A t\|\leq\sqrt{2}e^{-\alpha t}, \quad \forall t\geq0. $$
Proof
Note that
$$A=\left ( \textstyle\begin{array}{@{}c@{\quad}c@{}} -a_{i} & 0 \\ 0 & -b_{j} \end{array}\displaystyle \right ), $$
then
$$\exp A t=\left ( \textstyle\begin{array}{@{}c@{\quad}c@{}} e^{-a_{i}t} & 0 \\ 0 & e^{-b_{j}t} \end{array}\displaystyle \right ), $$
in view of the definition of matrix norm, we have
$$\|\exp A t\|= \bigl(e^{ -2a_{1}t}+e^{ -2a_{2}t} \bigr)^{\frac{1}{2}}\leq \sqrt {2}e^{-\alpha t}. $$
□
Lemma 2.2
Suppose that
$$({\mathrm{H}3})\quad \left \{ \textstyle\begin{array}{l} -2a_{i}+\sum_{j=1}^{m}\bar{s}_{ji} (\alpha_{jf}^{2\epsilon_{j}}+\alpha _{jf}^{2(1-\epsilon_{j})}+\beta_{jf}^{2\varepsilon_{j}} ) + \sum_{j=1}^{m}\bar{t}_{ij}\alpha_{ig}^{2(1-\xi_{i})}< 0, \\ -2b_{j}+\sum_{i=1}^{m}\bar{t}_{ij} (\alpha_{ig}^{2\xi_{i}}+\beta _{ig}^{2\varsigma_{i}}+\beta_{ig}^{2(1-\varsigma_{i})} ) +\sum_{i=1}^{m}\bar{s}_{ji}\beta_{jf}^{2(1-\varepsilon_{j})}< 0, \end{array}\displaystyle \right . $$
where
\(0\leq\epsilon_{j},\varepsilon_{j}, \xi_{i}, \varsigma_{i}<1\) (\(i,j=1,2,\ldots,m\)) are any constants. Then there exists
\(\beta>0\)
such that
$$\begin{aligned}& \beta-2a_{i}+\sum_{j=1}^{m} \bar{s}_{ji} \bigl(\alpha_{jf}^{2\epsilon _{j}}+ \alpha_{jf}^{2(1-\epsilon_{j})}+\beta_{jf}^{2\varepsilon_{j}} \bigr) + \sum_{j=1}^{m}\bar{t}_{ij} \alpha_{ig}^{2(1-\xi_{i})}e^{\beta\delta _{ij}}\leq0, \\& \beta-2b_{j}+\sum_{i=1}^{m} \bar{t}_{ij} \bigl(\alpha_{ig}^{2\xi_{i}}+\beta _{ig}^{2\varsigma_{i}}+\beta_{ig}^{2(1-\varsigma_{i})} \bigr) +\sum _{i=1}^{m}\bar{s}_{ji} \beta_{jf}^{2(1-\varepsilon_{j})}e^{\beta\tau _{ji}}\leq0. \end{aligned}$$
Proof
Let
$$\begin{aligned}& \varrho_{1i}(\beta) = \beta-2a_{i}+\sum _{j=1}^{m}\bar {s}_{ji} \bigl( \alpha_{jf}^{2\epsilon_{j}}+\alpha_{jf}^{2(1-\epsilon _{j})}+ \beta_{jf}^{2\varepsilon_{j}} \bigr) + \sum_{j=1}^{m} \bar{t}_{ij}\alpha_{ig}^{2(1-\xi_{i})}e^{\beta\delta _{ij}}, \\& \varrho_{2j}(\beta) = \beta-2b_{j}+\sum _{i=1}^{m}\bar{t}_{ij} \bigl(\alpha _{ig}^{2\xi_{i}}+\beta_{ig}^{2\varsigma_{i}}+ \beta_{ig}^{2(1-\varsigma _{i})} \bigr) +\sum_{i=1}^{m} \bar{s}_{ji}\beta_{jf}^{2(1-\varepsilon_{j})}e^{\beta\tau_{ji}}. \end{aligned}$$
Clearly, \(\varrho_{1i}(\beta)\), \(\varrho_{2j}(\beta)\) (\(i,j=1,2,\ldots,m\)) are continuously differential functions. One has
$$\left \{ \textstyle\begin{array}{l} \frac{d\varrho_{1i}(\beta)}{d\beta}=1+\sum_{j=1}^{m}\bar{t}_{ij}\alpha _{ig}^{2(1-\xi_{i})}\delta_{ij} e^{\beta\delta_{ij}}>0,\qquad \lim_{\beta\rightarrow{+\infty}}\varrho_{1i}(\beta)=+\infty,\qquad \varrho _{1i}(0)< 0, \\ \frac{d\varrho_{2j}(\beta)}{d\beta}=1+\sum_{i=1}^{m}\bar{s}_{ji}\beta _{jf}^{2(1-\varepsilon_{j})}\tau_{ji}e^{\beta\tau_{ji}}>0,\qquad \lim_{\beta\rightarrow{+\infty}}\varrho_{2j}(\beta)=+\infty,\qquad \varrho_{2j}(0)< 0. \end{array}\displaystyle \right . $$
According to the intermediate value theorem, we can conclude that there exist constants \(\beta_{i}^{*}>0\), \(\beta_{j}^{*}>0\) such that
$$\varrho_{1i}\bigl(\beta_{i}^{*}\bigr)=0,\qquad \varrho_{2j}\bigl(\beta_{j}^{*}\bigr)=0,\quad i,j=1,2,\ldots,m. $$
Let \(\beta_{0}=\min\{\beta_{1},\beta_{2},\ldots, \beta_{m}, \beta_{1}^{*},\beta_{2}^{*}, \ldots,\beta_{m}^{*}\}\), then it follows that \(\beta_{0}>0\) and
$$\varrho_{1i}(\beta_{0})\leq0,\qquad \varrho_{2j}( \beta_{0})\leq0, \quad i, j=1,2,\ldots,m. $$
The proof of Lemma 2.2 is complete. □
Lemma 2.3
Suppose that (H1) holds true. Then for any solution
\((x(t),y(t))^{T}\)
of model (1.3), there exists a constant
$$\gamma=\sqrt{2}\bigl(\|\varphi\|^{2}+\|\psi\|^{2}\bigr)+ \frac{\sqrt{2}}{\alpha} \Biggl[\sum_{j=1}^{m}( \bar{s}_{ji}F_{j}+\bar{c}_{i})+\sum _{i=1}^{m}(\bar {t}_{ij}G_{i}+ \bar{d}_{j}) \Biggr] $$
such that
$$\bigl\vert x_{i}(t)\bigr\vert \leq{\gamma},\qquad \bigl\vert y_{j}(t)\bigr\vert \leq{\gamma}, \quad i,j=1,2,\ldots,n, \forall t>0. $$
Proof
Let
$$\begin{aligned}& z_{ij}(t)=\left ( \textstyle\begin{array}{@{}c@{}} x_{i}(t)\\ y_{j}(t) \end{array}\displaystyle \right ), \\& A=\left ( \textstyle\begin{array}{@{}c@{\quad}c@{}} -a_{i} & 0 \\ 0 & -b_{j} \end{array}\displaystyle \right ),\qquad B_{ij}(t)= \left ( \textstyle\begin{array}{@{}c@{}} c_{i}(t) \\ d_{j}(t) \end{array}\displaystyle \right ), \\& F_{ij}\bigl(x_{i}(t),y_{j}(t)\bigr)=\left ( \textstyle\begin{array}{@{}c@{}} \sum_{j=1}^{m}s_{ji}(t)f_{j}[x_{j}(t),y_{j}(t-\tau_{ji})] \\ \sum_{i=1}^{m}t_{ij}(t)g_{i}[x_{i}(t-\delta_{ij}),y_{i}(t)] \end{array}\displaystyle \right ), \end{aligned}$$
then the model (1.3) takes the following form:
$$ z_{ij}'(t)\leq{A}z_{ij}(t)+F_{ij} \bigl(x_{i}(t),y_{j}(t)\bigr)+B_{ij}(t). $$
(2.2)
By (2.2), we get
$$z_{ij}(t)\leq{e^{At}}z_{ij}(0)+ \int _{0}^{t}e^{A(t-s)}\bigl[F_{ij} \bigl(x_{i}(s),y_{j}(s)\bigr)+B_{ij}(s)\bigr]\, ds. $$
In view of Lemma 2.1, we have
$$\begin{aligned} \begin{aligned}[b] \bigl\Vert z_{ij}(t)\bigr\Vert &\leq \sqrt{2}e^{-\alpha t}\bigl\Vert z_{ij}(0)\bigr\Vert +\sqrt{2} \int_{0}^{t}e^{\alpha (t-s)}\bigl[\bigl\Vert F_{ij}\bigl(x_{i}(s),y_{j}(s)\bigr)\bigr\Vert + \bigl\vert B_{ij}(s)\bigr\vert \bigr]\, ds \\ &\leq\sqrt{2}\bigl(\Vert \varphi \Vert ^{2}+\Vert \psi \Vert ^{2}\bigr)+\frac{\sqrt{2}}{\alpha} \bigl(1-e^{-\alpha t} \bigr) \Biggl[\sum _{j=1}^{m}(\bar{s}_{ji}F_{j}+ \bar{c}_{i})+\sum_{i=1}^{m}(\bar {t}_{ij}G_{i}+\bar{d}_{j}) \Biggr] \\ &\leq\sqrt{2}\bigl(\Vert \varphi \Vert ^{2}+\Vert \psi \Vert ^{2}\bigr)+\frac{\sqrt{2}}{\alpha} \Biggl[\sum_{j=1}^{m}( \bar{s}_{ji}F_{j}+\bar{c}_{i})+\sum _{i=1}^{m}(\bar {t}_{ij}G_{i}+ \bar{d}_{j}) \Biggr]. \end{aligned} \end{aligned}$$
Let
$$ \gamma=\sqrt{2}\bigl(\|\varphi\|^{2}+\|\psi \|^{2}\bigr)+\frac{\sqrt{2}}{\alpha} \Biggl[\sum_{j=1}^{m}( \bar{s}_{ji}F_{j}+\bar{c}_{i})+\sum _{i=1}^{m}(\bar {t}_{ij}G_{i}+ \bar{d}_{j}) \Biggr]. $$
(2.3)
Then it follows that \(|x_{i}(t)|\leq\gamma\), \(|y_{j}(t)|\leq\gamma\) for all \(t>0\). This completes the proof of Lemma 2.3. □