\documentclass[reqno]{amsart}
\usepackage{hyperref}

\AtBeginDocument{{\noindent\small
\emph{Electronic Journal of Differential Equations},
Vol. 2012 (2012), No. 38, pp. 1--22.\newline
ISSN: 1072-6691. URL: http://ejde.math.txstate.edu or http://ejde.math.unt.edu
\newline ftp ejde.math.txstate.edu}
\thanks{\copyright 2012 Texas State University - San Marcos.}
\vspace{9mm}}

\begin{document}
\title[\hfilneg EJDE-2012/38\hfil Existence and stability of periodic solutions]
{Existence and stability of periodic solutions of BAM high-order
 Hopfield neural networks with impulses and delays on time scales}

\author[W. Yang \hfil EJDE-2012/38\hfilneg]
{Wengui Yang}

\address{Wengui Yang \newline
Ministry of Public Education, Sanmenxia Polytechnic,
Sanmenxia 472000, China}
\email{yangwg8088@163.com}

\thanks{Submitted November 9, 2011. Published March 14, 2012.}
\subjclass[2000]{34K13, 34K20, 92B20}
\keywords{Bidirectional associative memory; impulses and delays;
\hfill\break\indent high-order Hopfield neural networks;
existence and stability; time scales}

\begin{abstract}
 By using Mawhins's continuation theorem of coincidence degree theory and
 constructing some suitable Lyapunov functions, the periodicity and the
 exponential stability for a class of bidirectional associative memory
 (BAM) high-order Hopfield neural networks with  impulses and delays on
 time scales are investigated.  An example illustrates our  results.
\end{abstract}

\maketitle
\numberwithin{equation}{section}
\newtheorem{theorem}{Theorem}[section]
\newtheorem{lemma}[theorem]{Lemma}
\newtheorem{remark}[theorem]{Remark}
\newtheorem{example}[theorem]{Example}
\allowdisplaybreaks

\section{Introduction}

The bidirectional associative memory (BAM) neural network models were
introduced by Kosko \cite{k1,k2}.
It is a special class of recurrent neural networks that can store bipolar vector pairs.
The BAM neural network is composed of neurons arranged in two neural fields; i.e.,
 the $F_X$-field and the $F_Y$-field. The neurons in one field are fully interconnected
to the neurons in the other field, while there is no interconnection among neurons
in the same neural field. Through iterations of forward and backward information
flows between the two neural fields, it performs a two-way associative search for
stored bipolar vector pairs and generalizes the single-field auto-associative Hebbian
correlation to a two-field pattern-matched hetero-associative circuits.
Therefore, BAM neural networks possesses good application prospects in many fields
such as pattern recognition, parallel computation, image and signal processing,
optimization automatic control and artificial intelligence. Recently, BAM neural networks
have attracted the attention of many scientists (e.g., mathematicians, physicists,
 computer scientists and so on) and many results for  BAM neural networks with or
without axonal signal transmission delays including stability and periodic solutions
have been obtained \cite{b1,b9,b3,b4,b2,b5,b7,b8,b6} and the references quoted therein.
For example, in Refs. \cite{c2,c3,c4,c1}, the authors discussed the problem of the
stability of the BAM neural networks with or without delays, and obtained some sufficient
conditions to ensure the stability of equilibrium point.
Li \cite{c5}, Ho et al \cite{c6}, Li and Yang \cite{c7}, Chen and Cui \cite{c8},
and Xia et al \cite{c9} discussed the existence and exponential stability of the
equilibrium point of several classes of impulsive BAM neural networks using different
methods, such as linear matrix inequality (LMI), Fixed point theorem, Halanay inequality,
 Lyapunov functional method, M-matrix theory and Topological degree methods, respectively.

On the other hand, due to the fact that high-order Hopfield  neural networks (HHNNs)
have stronger approximation property, faster convergence rate, greater storage capacity,
and higher fault tolerance than lower-order neural networks, high-order neural networks
have been the object of intensive analysis by numerous authors in the recent years.
We refer the reader to \cite{1,2,5,6,7,8,3,4,9}. However, to the best of the author's
knowledge, few results have been obtained the periodicity and the exponential stability
for a class of BAM high-order Hopfield neural networks with impulses and delays on time
scales.

The objective of this paper is to investigate the
existence and stability of periodic solutions of  BAM  HHNNs with impulses and delays
on time scales
\begin{equation} \label{e1.1}
\begin{gathered}
\begin{aligned}
x_i^\Delta(t)&= -c_i(t)x_i(t)+\sum_{j=1}^ma_{ij}(t)f_j(y_j(t-\tau_{ij}(t)))\\
&\quad +\sum_{j=1}^m\sum_{l=1}^mb_{ijl}(t)g_j(y_j(t-\sigma_{ijl}(t)))
g_l(y_l(t-\upsilon_{ijl}(t)))+I_i(t),
\quad t\neq t_k,
\end{aligned}\\
\Delta x_i(t_k)= x_i(t_k^+)-x_i(t_k^-)=\alpha_{ik}(x_i(t_k)),\quad
 i=1,2,\dots,n,\; k=1,2,\dots,
\\
\begin{aligned}
y_j^\Delta(t)&= -d_j(t)y_j(t)+\sum_{i=1}^ne_{ji}(t)p_i(x_i(t-\hat{\tau}_{ji}(t)))\\
&\quad +\sum_{i=1}^n\sum_{r=1}^nh_{jir}(t)q_i(x_i(t-\hat{\sigma}_{jir}(t)))
q_r(x_r(t-\hat{\upsilon}_{jir}(t)))+J_j(t),\quad t\neq t_k,
\end{aligned}\\
\Delta y_j(t_k)= y_j(t_k^+)-y_j(t_k^-)=\beta_{jk}(y_j(t_k)),\quad  j=1,2,\dots,m,\;
 k=1,2,\dots,
\end{gathered}
\end{equation}
where $\mathbb{T}$ is an $\omega$-periodic time scale which has the subspace topology
inherited from the standard topology on $\mathbb{R}$. And $x_i$ and $y_j$ are the
activations of the $i$-th neuron in $F_X$-layer and the $j$th neuron in $F_Y$-layer,
respectively; $a_i(t)>0$ and $b_j(t)>0$ represent the rate with which the $i$th neuron
from $F_X$-layer and the $j$th neuron from $F_Y$-layer will reset their potential to
the resting state in isolation when disconnected from the networks and external inputs,
respectively; $\tau_{ij}$, $\sigma_{ijl}$, $\upsilon_{ijl}$, and $\hat{\tau}_{ij}$,
 $\hat{\sigma}_{ijl}$, $\hat{\upsilon}_{ijl}$ represent
the axonal signal transmission delays; $I_i(t)$ and $J_j(t)$ are the external inputs
on the neurons. Here, $\Delta x_i(t_k)=x_i(t_k^+)-x_i(t_k^-)$ and
$\Delta y_j(t_k)=y_j(t_k^+)-y_j(t_k^-)$ are the impulses at moments $t_k$ and
$t_1<t_2<\cdots$ is a strictly increasing sequence such that $\lim_{k\to\infty}t_k=\infty$.

System \eqref{e1.1} is supplemented with initial values
\begin{gather*}
x_i(t)=\phi_{x_i}(s),\quad s\in[-\theta,0]\cap\mathbb{T},
\quad\theta=\max\{\tau,\sigma,\upsilon\},\\
y_j(t)=\phi_{y_j}(s),\quad s\in[-\hat{\theta},0]\cap\mathbb{T},
\quad\hat{\theta}=\max\{\hat{\tau},\hat{\sigma},\hat{\upsilon}\},\\
\tau=\max_{1\leq i\leq n,1\leq j\leq m}
\big\{\max_{t\in[0,\omega]\cap\mathbb{T}}\tau_{ij}(t)\big\},
\quad\sigma=\max_{1\leq i\leq n,1\leq j,l\leq m}
\big\{\max_{t\in[0,\omega]\cap\mathbb{T}}\sigma_{ijl}(t)\big\},\\
\upsilon=\max_{1\leq i\leq n,1\leq j,l\leq m}
\big\{\max_{t\in[0,\omega]\cap\mathbb{T}}\upsilon_{ijl}(t)\big\},
\quad\hat{\tau}=\max_{1\leq i\leq n,1\leq j\leq m}
\big\{\max_{t\in[0,\omega]\cap\mathbb{T}}\hat{\tau}_{ij}(t)\big\},\\
\hat{\sigma}=\max_{1\leq i,r\leq n,1\leq j\leq m}
\big\{\max_{t\in[0,\omega]\cap\mathbb{T}}\hat{\sigma}_{jir}(t)\big\},
\quad\hat{\upsilon}=\max_{1\leq i,r\leq n,1\leq j\leq m}
\big\{\max_{t\in[0,\omega]\cap\mathbb{T}}\hat{\upsilon}_{jir}(t)\big\},
\end{gather*}
where $\phi_{x_i}(\cdot)$ and $\phi_{y_j}(\cdot)$ denote continuous
 $\omega$-periodic function defined on
$[-\theta,0]\cap\mathbb{T}$ and $[-\hat{\theta},0]\cap\mathbb{T}$, respectively.

As usual in the theory of impulsive differential equations, at the
points of discontinuity $t_k$ of the solution
$t\to(x_1(t),\dots,x_n(t),y_1(t),\dots,y_m(t))^T$,
we assume that $(x_1(t_k^+),\dots,x_n(t_k^+),y_1(t_k^+),\dots,y_m(t_k^+))^T$ exists,
and 
$\big(x_1(t_k^-),\dots,x_n(t_k^-),y_1(t_k^-),\\ \dots,y_m(t_k^-)\big)^T
=(x_1(t_k),\dots,x_n(t_k),y_1(t_k),\dots,y_m(t_k))^T$.
It is clear that, in general,the derivatives $x_i'(t_k)$ and $y_j'(t_k)$ do not exist.
On the other hand, according to the first two and the
third equalities of \eqref{e1.1}, there exist the limits $x_i'(t_k^{\pm})$ and
$y_j'(t_k^{\pm})$. According to the above convention, we assume that
$x_i'(t_k)=x_i'(t_k^-)$ and $y_j'(t_k)=y_j'(t_k^-)$.

Throughout this paper, we assume the following.
\begin{itemize}

\item[(H1)] For $i,r=1,2,\dots,n$, $j,l=1,2,\dots,m$, $c_i(t)$,$d_j(t)$,
$a_{ij}(t)$, $e_{ji}(t)$, $b_{ijl}(t)$, $h_{jir}(t)$, $I_i(t)$, $J_j(t)$,
$\tau_{ij}(t)$, $\sigma_{ijl}(t)$, $\upsilon_{ijl}(t)$
$\hat{\tau}_{ji}(t)$, $\hat{\sigma}_{jir}(t)$, $\hat{\upsilon}_{jir}(t)$ are positive
continuous periodic functions with period $\omega>0$, and $c_i(t)$ and $d_j(t)$ are
regressive. And  assume that $t-\tau_{ij}(t)$, $t-\sigma_{ijl}(t)$,
$t-\upsilon_{ijl}(t)$ $t-\hat{\tau}_{ji}(t)$, $t-\hat{\sigma}_{jir}(t)$ and
 $t-\hat{\upsilon}_{jir}(t)$ belong to $\mathbb{T}$ for $t\in\mathbb{T}$.

\item[(H2)] There exist positive constants $M_j,N_j,\hat{M}_i,\hat{N}_i$, such that
$|f_j(x)|\leq M_j$, $|g_j(x)|\leq N_j$, $|p_i(x)|\leq \hat{M}_i$, $|q_i(x)|\leq \hat{N}_i$,
for $i=1,2,\dots,n$, $j=1,2,\dots,m$, $x\in\mathbb{R}$.

\item[(H3)] Functions $f_j(u),g_j(u),p_i(u),q_i(u)$ satisfy the Lipschitz condition;
 that is, there exist constants $L_j,H_j,\\\hat{L}_i,\hat{H}_i>0$ such that
$|f_j(u_1)-f_j(u_2)|\leq L_j|u_1-u_2|$, $|g_j(u_1)-g_j(u_2)|\leq
H_j|u_1-u_2|$, $|p_i(u_1)-p_i(u_2)|\leq \hat{L}_i|u_1-u_2|$, $|q_i(u_1)-q_i(u_2)|\leq
\hat{H}_i|u_1-u_2|$ for any $u_1,u_2\in\mathbb{R}$, for $i=1,2,\dots,n$, $j=1,2,\dots,m$.

\item[(H4)] There exists a positive integer $q$ such that for $k=1,2,\dots$,
$i=1,2,\dots,n$, $j=1,2,\dots,m$, $\{t_k, k=1,2,\dots\}\cap[0,\omega]={t_1,t_2,\dots,t_q}$,
$t_{k+q}=t_k+\omega$,  $\alpha_{i,k+q}(\cdot)=\alpha_{ik}(\cdot)$,
 $\beta_{j,k+q}(\cdot)=\beta_{jk}(\cdot)$.
\end{itemize}

For convenience, we shall use the following notations.
$$
\bar{f}=\frac{1}{\omega}\int_{\bar{k}}^{\bar{k}+\omega} f(t)dt,\quad
f^+=\max_{t\in[\bar{k},\bar{k}+\omega]\cap\mathbb{T}}|f(t)|,
\quad f^-=\min_{t\in[\bar{k},\bar{k}+\omega]\cap\mathbb{T}}|f(t)|,
$$
where $\bar{k}=\min\left\{[0,+\infty)\cap\mathbb{T}\right\}$, $f(t)$ is an
 $\omega$-periodic function.

\section{Preliminaries}

Some preliminary definitions and theorems on time scales can be found in \cite{10,11},
which are excellent references for the calculus of time scales. We will recall some
basic definitions and lemmas which are used in what follows.

Let $\mathbb{T}$ be a nonempty closed subset (time scale) of $\mathbb{R}$.
The forward and backward jump operators $\sigma,\rho:\mathbb{T}\to\mathbb{T}$ and
the graininess $\mu:\mathbb{T}:\mathbb{T}\to \mathbb{R}^+$ are defined by
$$
\sigma(t)=\inf\{s\in\mathbb{T}: s>t\},\quad
 \rho(t)=\sup\{s\in\mathbb{T}: s<t\},\quad  \mu(t)=\sigma(t)-t.
$$

A point $t\in\mathbb{T}$ is called left-dense if $t>\inf\mathbb{T}$ and $\rho(t)=t$,
left-scattered if left-scattered if $\rho(t)<t$, right-dense if $t<\sup\mathbb{T}$ and
$\sigma(t)=t$, and right-scattered if $\sigma(t)>t$. If $\mathbb{T}$ has a left-scattered
maximum $m$, then $\mathbb{T}^k=\mathbb{T}\backslash\{m\}$; otherwise
 $\mathbb{T}^k=\mathbb{T}$. If $\mathbb{T}$ has a right-scattered minimum $m$,
then $\mathbb{T}_k=\mathbb{T}\backslash\{m\}$; otherwise $\mathbb{T}_k=\mathbb{T}$.

A function $f:\mathbb{T}\to\mathbb{R}$ is right-dense continuous provided that it is
continuous at right-dense point in $\mathbb{T}$ and its left-side limits exist at
left-dense points in $\mathbb{T}$. If $f$ is continuous at each right-dense point
and each left-dense point, then $f$ is said to be continuous function
on $\mathbb{T}$.

For $y:\mathbb{T}\to\mathbb{R}$ and $t\in\mathbb{T}^k$, we define the delta
derivative of y(t), $y^\Delta(t)$ to be the number (if it exists) with the property
that for a given $\varepsilon>0$, there exists a neighborhood $U$ of $t$ such that
$$
|[y(\sigma(t))-y(s)]-y^\Delta(t)[\sigma(t)-s]|<\varepsilon|\sigma(t)-s|,\quad
\forall s\in U.
$$

If $y$ is continuous, then $y$ is right-dense continuous, and if $y$ is delta
differentiable at $t$, then $y$ is continuous at $t$. A function
$r:\mathbb{T}\to\mathbb{R}$ is called regressive if $1+\mu(t)r(t)\neq0$ for all
$t\in\mathbb{T}^k$.
A function $r$ from $\mathbb{T}$ to $\mathbb{R}$ is positively regressive if
$1+\mu(t)r(t)>0$ for every  $t\in\mathbb{T}$. Denote $\mathfrak{R}^+$ is the set
of positively regressive functions from $\mathbb{T}$ to $\mathbb{R}$, and
$\mathbb{T}^+=[0,+\infty)\cap\mathbb{T}$.

If $r$ is regressive function, then the generalized exponential function $e_r$ is defined
 by
$$
e_r(t,s)=\exp\Big\{\int_s^t\xi_{\mu(\tau)}(r(\tau))\Delta\tau\Big\}
\quad {\rm for}\quad s,t\in\mathbb{T},
$$
with the cylinder transformation
\[
\xi_h(z)=\begin{cases}
\frac{\log(1+hz)}{h}, &\text{if }h\neq0,\\
z, &\text{if } h=0.
\end{cases}
\]
Let $p,q: \mathbb{T}\to\mathbb{R}$ be two regressive functions, we define
$$
p\oplus q:=p+q+\mu pq, \quad \ominus p:=-\frac{p}{1+\mu p},\quad
p\ominus q:=p\oplus(\ominus q).
$$
The periodic solution
$z^\ast(t)=(x^\ast_1(t),\dots,x^\ast_n(t),y^\ast_1(t),\dots,y^\ast_m(t))^T$
of system \eqref{e1.1} is said to be exponentially stable if there exists a positive
constant $\vartheta$ such that for every
$\varrho\in\mathbb{T}$, there exist $N=N(\varrho)>1$ such that the solution
$z(t)=(x_1(t),\dots,x_n(t),\\y_1(t),\dots,y_m(t))^T$ of \eqref{e1.1} satisfies
\[
\|z(t)-z^\ast(t)\|\leq Ne_{-\vartheta}(t,\varrho)
\Big(\sum_{i=1}^n|\phi_{x_i}(\varrho)-x_i^\ast(\varrho)|
+\sum_{j=1}^m|\phi_{y_j}(\varrho)-y_j^\ast(\varrho)|\Big),
\]
where $\varrho\in[-\max\{\theta,\hat{\theta}\},0]\cap\mathbb{T}$.


\begin{lemma}[\cite{5}] \label{l2.1}
 If $f,g\in C(\mathbb{T},\mathbb{R})$, and $f(t)\leq g(t)$ on $[\bar{k},\bar{k}+\omega)$,
then
\[
\int_{\bar{k}}^{\bar{k}+\omega}f(t)\Delta t\leq\int_{\bar{k}}^{\bar{k}+\omega}g(t)\Delta t.
\]
\end{lemma}

\begin{lemma}[\cite{12}] \label{l2.2}
Let $t_1,t_2\in[\bar{k},\bar{k}+\omega]\cap\mathbb{T}$,
$t\in\mathbb{T}$. If $f:\mathbb{T}\to\mathbb{R}$ is $\omega$-periodic, then
$$
f(t)\leq f(t_1)+\int_{\bar{k}}^{\bar{k}+\omega}|f^\Delta(t)|\Delta t
\quad \text{and}\quad  f(t)\geq f(t_2)-\int_{\bar{k}}^{\bar{k}+\omega}|f^\Delta(t)|\Delta t.
$$
\end{lemma}

\begin{lemma}[\cite{13}] \label{l2.3}
 Let $a,b\in\mathbb{T}$. For rd-continuous functions $f,g:[a,b]\to\mathbb{R}$, one has
$$
\Big(\int_a^b|f(t)g(t)|\Delta t\Big)^2\leq
\Big(\int_a^b|f(t)|^2\Delta t\Big)\Big(\int_a^b|g(t)|^2\Delta t\Big).
$$
\end{lemma}


\begin{lemma}[\cite{14}] \label{l2.4}
  Let $X$ and $Z$ be two Banach spaces and let $L$ be a Fredholm mapping of index zero.
Let $\Omega\subset X$ be an open bounded set and let $N:\overline{\Omega}\to Z$ be a
continuous operator which is $L$-compact on $\Omega$. Assume that
\begin{itemize}
\item[(a)] for each $\lambda\in(0,1)$, $x\in\partial\Omega\cap\operatorname{Dom}L$,
$Lx\neq\lambda Nx$;

\item[(b)] for each $x\in\partial\Omega\cap\ker  L$, $QNx\neq0$;

\item[(c)] $\deg  (JNQx,\Omega\cap\ker  L,0)\neq0$,
where $JQN: \ker  L\to\ker  L$.
\end{itemize}
Then $Lx=Nx$ has at least one solution in $\overline{\Omega}\cap\operatorname{Dom} L$.
\end{lemma}

\begin{lemma} \label{l2.5}
 Let $r : \mathbb{T}\to\mathbb{R}$ be right-dense continuous and regressive.
 Let $a\in\mathbb{T}$ and $y_a\in\mathbb{R}$. The
unique solution of the initial value problem
\begin{gather*}
y^\Delta(t)=r(t)y(t)+h(t), \quad y(a)=y_a,\\
\Delta y(t_k)=y(t_k^+)-y(t_k^-)=\varphi_{k}(y(t_k)),\quad k=1,2,\dots,q,
\end{gather*}
is
$$
y(t)=e_r(t,a)y_a+\int_a^te_r(t,\sigma(s))h(s)\Delta s+\sum_{k : t_k\in[a,t)_\mathbb{T}}
e_r(t,t_k)\varphi_{k}(y(t_k)).
$$
\end{lemma}

\begin{proof}
The proof of Lemma \ref{l2.5} is similar to that of \cite[Lemma 2.7]{15},
it is omitted.
\end{proof}

According to (H1)--(H4) and $\bar{k}=\min\{[0,+\infty)\cap\mathbb{T}\}$,
 for system \eqref{e1.1}, finding the periodic solutions is equivalent to finding
those of the  boundary-value problem
\begin{equation} \label{e2.1}
\begin{gathered}
\begin{aligned}
x_i^\Delta(t)&= -c_i(t)x_i(t)+\sum_{j=1}^ma_{ij}(t)f_j(y_j(t-\tau_{ij}(t)))\\
&\quad +\sum_{j=1}^m\sum_{l=1}^mb_{ijl}(t)g_j(y_j(t-\sigma_{ijl}(t)))
 g_l(y_l(t-\upsilon_{ijl}(t)))+I_i(t),\\
&\quad  t\in[\bar{k},\bar{k}+\omega],\; t\neq t_k,
\end{aligned}
\\
\Delta x_i(t_k)= x_i(t_k^+)-x_i(t_k^-)=\alpha_{ik}(x_i(t_k)),
 x_i(\bar{k})=x_i(\bar{k}+\omega),\\
 i=1,2,\dots,n, k=1,2,\dots,q,
\\
\begin{aligned}
y_j^\Delta(t)&= -d_j(t)y_j(t)+\sum_{i=1}^ne_{ji}(t)p_i(x_i(t-\hat{\tau}_{ji}(t)))\\
&\quad +\sum_{i=1}^n\sum_{r=1}^nh_{jir}(t)q_i(x_i(t-\hat{\sigma}_{jir}(t)))
q_r(x_r(t-\hat{\upsilon}_{jir}(t)))+J_j(t),\\
&\quad  t\in[\bar{k},\bar{k}+\omega],\; t\neq t_k,
\end{aligned}\\
\Delta y_j(t_k)= y_j(t_k^+)-y_j(t_k^-)=\beta_{jk}(y_j(t_k)),
y_j(\bar{k})=y_j(\bar{k}+\omega),\\
 j=1,2,\dots,m, k=1,2,\dots,q.
\end{gathered}
\end{equation}
To apply Lemma \ref{l2.4} to system \eqref{e2.1}, we first make the following preparations.
For any non-negative integer $q$, let $t_q<\omega<t_{q+1}=\omega+t_1$ and
$C[\bar{k},\bar{k}+\omega;t_1,t_2,\dots,t_q]
=\{z:[\bar{k},\bar{k}+\omega]\cap\mathbb{T}\to\mathbb{R}^{n+m}|z(t)$
exists for $t\neq t_1,\dots,t_q$; $z(t_k^+)$ and $z(t_k^-)$ exists at
$t\neq t_1,\dots,t_q$; and $z(t_k)=z(t_k^-)$, $k=1,\dots,q\}$. Let
$$
\mathbb{X}=\{z\in C[\bar{k},\bar{k}+\omega;t_1,t_2,\dots,t_q] : z(t+\omega)=z(t),
\; t\in\mathbb{T}\},\quad
\mathbb{Z}=\mathbb{X}\times\mathbb{R}^{(n+m)\times(q+1)}
$$
be endowed with the norm
$$
\|z\|=\sum_{i=1}^n\max_{t\in[\bar{k},\bar{k}+\omega]\cap\mathbb{T}}|x_i(t)|
+\sum_{j=1}^m\max_{t\in[\bar{k},\bar{k}+\omega]\cap\mathbb{T}}|y_j(t)|,
$$
for $z(t)=(x_1(t),\dots,x_n(t),y_1(t),\dots,y_m(t))^T$.
Then $\mathbb{X}$ and $\mathbb{Z}$ are Banach spaces with the norm $\|\cdot\|$. Let
\begin{eqnarray}\label{e2.2}
L : \operatorname{Dom}L\cap \mathbb{X}\to \mathbb{Z}, \quad
z\mapsto (z^\Delta,\Delta z(t_1),\Delta z(t_2),\dots,\Delta z(t_q),0),
\end{eqnarray}
\begin{equation} \label{e2.3}
N: \mathbb{X}\to \mathbb{Z},\quad
Nz=  \begin{pmatrix}
    A_1(t) & \Delta x_1(t_1) &\Delta x_1(t_2) & \cdots &\Delta x_1(t_q) & 0 \\
     \vdots& \vdots & \vdots & \vdots&\vdots&\vdots \\
    A_n(t) & \Delta x_n(t_1)& \Delta x_n(t_1) & \cdots & \Delta x_n(t_q) & 0 \\
    B_1(t) &\Delta y_1(t_1) &\Delta y_1(t_2) & \cdots &\Delta y_1(t_q) & 0 \\
   \vdots& \vdots & \vdots & \vdots&\vdots&\vdots  \\
    B_m(t) &\Delta y_n(t_1)& \Delta y_n(t_1) & \cdots & \Delta y_m(t_q) & 0\\
  \end{pmatrix},
\end{equation}
where $\operatorname{Dom}L=\{z\in C^1[\bar{k},\bar{k}+\omega;t_1,t_2,\dots,t_q]:
 z(t+\omega)=z(t)\}$,
\begin{align*}
A_i(t) &= -c_i(t)x_i(t)+\sum_{j=1}^na_{ij}(t)f_j(x_j(t-\tau_{ij}))\\
&\quad+\sum_{j=1}^n\sum_{l=1}^nb_{ijl}(t)g_j(x_j(t-\sigma_{ijl}))g_l
(x_l(t-\upsilon_{ijl}))+I_i(t)
\end{align*}
for $i=1,2,\dots,n$;
\begin{align*}
B_j(t)&= -d_j(t)y_j(t)+\sum_{i=1}^ne_{ji}(t)p_i(x_i(t-\hat{\tau}_{ji}(t)))\\
&\quad+\sum_{i=1}^n\sum_{r=1}^nh_{jir}(t)q_i(x_i(t-\hat{\sigma}_{jir}(t)))
q_r(x_r(t-\hat{\upsilon}_{jir}(t)))+J_j(t)
\end{align*}
for $j=1,2,\dots,m$.
Taking $z=(f,C_1,C_2,\dots,C_q,d)\in \operatorname{Im} L\subset\mathbb{Z}$, then
\begin{gather*}
\ker  L=\{z\in\mathbb{X}: z=h\in\mathbb{R}^{n+m}\},\\
\operatorname{Im} L=\Big\{(f,C_1,C_2,\dots,C_q,d)\in\mathbb{Z}:
\int_0^\omega f(s)ds+\sum_{k=1}^qC_k+d=0\Big\},
\end{gather*}
and $\dim \ker  L=\operatorname{codim} \operatorname{Im} L=n+m$. Define the two projectors
\begin{gather*}
Pz=\frac{1}{\omega}\int_{\bar{k}}^{\bar{k}+\omega}z(t)\Delta t,\\
QNz=Q(f,C_1,\dots,C_q,d)=\Big(\frac{1}{\omega}
\Big(\int_{\bar{k}}^{\bar{k}+\omega}f(s)\Delta s+\sum_{k=1}^qC_k+d\Big),0,\dots,0,0\Big).
\end{gather*}
It is not difficult to show that $P$ and $Q$ are continuous and satisfy
$$
\operatorname{Im} P=\ker  L, \quad \operatorname{Im} L=\ker  Q=\operatorname{Im}(I-Q).
$$
It is easy to see that $\operatorname{Im L}$ is closed in $\mathbb{Z}$, which leads
to the following lemma.


\begin{lemma} \label{l2.6}
 Let $L$ and $N$ be defined by \eqref{e2.2} and \eqref{e2.3}, respectively, then
$L$ is a Fredholm operator of index zero.
\end{lemma}

\begin{lemma} \label{l2.7}
 Let $L$ and $N$ be defined by $\eqref{e2.2}$ and \eqref{e2.3}, respectively,
suppose that $\Omega$ is an open bounded subset of  $\operatorname{Dom} L$,
then $N$ is $L$-compact on $\overline{\Omega}$.
\end{lemma}

\begin{proof} Through an easy computation, we can find that the inverse
$K_q : \operatorname{Im} L\to\ker  P\cap\operatorname{Dom} L$ of $L_q$ has the form
$$
(K_qz)(t)=\int_{\bar{k}}^{\bar{k}+\omega}z(s)\Delta s+\sum_{t>t_k}C_k-
\frac{1}{\omega}\int_{\bar{k}}^{\bar{k}+\omega}\int_{\bar{k}}^tz(s)
\Delta s\Delta t-\sum_{k=1}^qC_k.
$$
Therefore,
$$
QNz=\begin{pmatrix}
    \begin{pmatrix}
     \frac{1}{\omega}\int_0^\omega A_i(t)dt-\frac{1}{\omega}
      \sum_{k=1}^q\alpha_{ik}(x_i(t_k))\\
    \frac{1}{\omega}\int_0^\omega B_j(t)dt-\frac{1}{\omega}
      \sum_{k=1}^q\beta_{jk}(y_j(t_k))
     \end{pmatrix}
          & 0& \cdots & 0
    \end{pmatrix}
    _{(n+m)\times(q+1)},
$$
and then
\begin{align*}
K_q(I-Q)Nz
&=  \begin{pmatrix}
            \int_0^t A_i(t)dt-\sum_{t>t_k}\alpha_{ik}(x_i(t_k))\\
            \int_0^t B_j(t)dt-\sum_{t>t_k}\beta_{jk}(y_j(t_k))
     \end{pmatrix}_{(n+m)\times1}\\
&\quad- \begin{pmatrix}
              \frac{1}{\omega}\int_0^\omega \int_0^tA_i(s)\,ds\,dt\\
              \frac{1}{\omega}\int_0^\omega \int_0^tB_j(s)\,ds\,dt
   \end{pmatrix}_{(n+m)\times1}\\
&\quad-\Big(\frac{t-\bar{k}}{\omega}-\frac{1}{\omega^2}\int_{\bar{k}}^{\bar{k}+\omega}
       (t-\bar{k})\Delta t\Big)
            \begin{pmatrix}
            \int_0^\omega A_i(t)dt\\
            \int_0^\omega B_j(t)dt
            \end{pmatrix}_{(n+m)\times1} \\
&\quad    -\begin{pmatrix}
\sum_{k=1}^q \alpha_{ik}(x_i(t_k))\\
\sum_{k=1}^q \beta_{jk}(y_j(t_k))
\end{pmatrix}_{(n+m)\times1}.
\end{align*}
Therefore, $QN$ and $K_q(I-Q)N$ are both continuous. Using the Arzela-Ascoli Theorem,
it is easy to show that $\overline{K_p(I-Q)N(\overline{\Omega})}$ is relatively compact.
Moreover, $QN(\overline{\Omega})$ is bounded. Thus, $N$ is $L-$compact on
 $\overline{\Omega}$ for any open bounded set $\Omega\subset \mathbb{X}$.
The proof is completed.
\end{proof}


\section{Existence of periodic solutions}

In this section, we study the existence of periodic solution of \eqref{e1.1} based
on Mawhins' continuation theorem.

\begin{theorem} \label{t3.1}
Assume that {\rm (H1)--(H4)} hold, then system \eqref{e1.1} has at least one
$\omega$-periodic solution.
\end{theorem}

\begin{proof} Based on the Lemma \ref{l2.6} and Lemma \ref{l2.7},  what we need to
do is just to search for an appropriate open, bounded subset $\Omega$ for the
application of the continuation theorem.
Corresponding to the operator equation $Lz=\lambda Nz$, $\lambda\in(0,1)$, we have
\begin{equation} \label{e3.1}
\begin{gathered}
\begin{aligned}
x_i^\Delta(t)
&= \lambda\Big\{-c_i(t)x_i(t)+\sum_{j=1}^ma_{ij}(t)f_j(y_j(t-\tau_{ij}(t))) \\
&\quad+\sum_{j=1}^m\sum_{l=1}^mb_{ijl}(t)g_j(y_j(t-\sigma_{ijl}(t)))
g_l(y_l(t-\upsilon_{ijl}(t)))+I_i(t)\Big\}, \\
&\quad  t\in[\bar{k},\bar{k}+\omega], t\neq t_k,
\end{aligned}
\\
\Delta x_i(t_k)= x_i(t_k^+)-x_i(t_k^-)=\alpha_{ik}(x_i(t_k)),\quad
x_i(\bar{k})=x_i(\bar{k}+\omega),\\
 i=1,2,\dots,n,\; k=1,2,\dots,q,
\\
\begin{aligned}
y_j^\Delta(t)
&= \lambda\Big\{-d_j(t)y_j(t)+\sum_{i=1}^ne_{ji}(t)p_i(x_i(t-\hat{\tau}_{ji}(t)))\\
&\quad +\sum_{i=1}^n\sum_{r=1}^nh_{jir}(t)q_i(x_i(t-\hat{\sigma}_{jir}(t)))
q_r(x_r(t-\hat{\upsilon}_{jir}(t)))+J_j(t)\Big\},\\
&\quad t\in[\bar{k},\bar{k}+\omega],\; t\neq t_k,
\end{aligned}
\\
\Delta y_j(t_k)= y_j(t_k^+)-y_j(t_k^-)=\beta_{jk}(y_j(t_k)),\quad
y_j(\bar{k})=y_j(\bar{k}+\omega),\\
j=1,2,\dots,m,\; k=1,2,\dots,q.
\end{gathered}
\end{equation}
For the sake of convenience, we define
$$
\|f\|_2=\Big(\int_{\bar{k}}^{\bar{k}+\omega}|f(t)|^2\Delta t\Big)^{1/2},
\quad \text{for } f\in(\mathbb{T},\mathbb{R}).
$$
Suppose that $(x_1(t),x_2(t),\dots,x_n(t),y_1(t),y_2(t),\dots,y_m(t))^T\in X$ is a
solution of \eqref{e3.1} for a certain $\lambda\in(0,1)$.
Integrating \eqref{e3.1} over the interval $[\bar{k},\bar{k}+\omega]$, we obtain
\begin{gather*}
\int_{\bar{k}}^{\bar{k}+\omega}A_i(t)\Delta t+\sum_{k=1}^q\alpha_{ik}(x_i(t_k))=0,
 \quad i=1,2,\dots,n,\\
\int_{\bar{k}}^{\bar{k}+\omega}B_j(t)\Delta t+\sum_{k=1}^q\beta_{jk}(y_j(t_k))=0,
 \quad j=1,2,\dots,m.
\end{gather*}
Hence
\begin{equation} \label{e3.2}
\begin{split}
&\int_{\bar{k}}^{\bar{k}+\omega}c_i(s)x_i(s)\Delta s \\
&= \int_{\overline{k}}^{\overline{k}+\omega}
 \Big(\sum_{j=1}^ma_{ij}(t)f_j(y_j(t-\tau_{ij}(t)))\\
&\quad +\sum_{j=1}^m\sum_{l=1}^mb_{ijl}(t)g_j(y_j(t-\sigma_{ijl}(t)))
 g_l(y_l(t-\upsilon_{ijl}(t)))+I_i(t)\Big)\Delta t\\
&\quad +\sum_{k=1}^q\alpha_{ik}(x_i(t_k)),
\end{split}
\end{equation}
\begin{equation} \label{e3.3}
\begin{split}
&\int_{\bar{k}}^{\bar{k}+\omega}d_j(s)y_j(s)\Delta s\\
&= \int_{\overline{k}}^{\overline{k}+\omega}
 \Big(\sum_{i=1}^ne_{ji}(t)p_i(x_i(t-\hat{\tau}_{ji}(t))) \\
&\quad +\sum_{i=1}^n\sum_{r=1}^nh_{jir}(t)
 q_i(x_i(t-\hat{\sigma}_{jir}(t)))q_r(x_r(t-\hat{\upsilon}_{jir}(t)))+J_j(t)\Big)\Delta t\\
&\quad+\sum_{k=1}^q\beta_{jk}(y_j(t_k)),
\end{split}
\end{equation}
where $i=1,2,\dots,n$,  $j=1,2,\dots,m$.
Let $\xi_i,\eta_i,\hat{\xi}_j,\hat{\eta}_j (\neq t_k)\in[\bar{k},\bar{k}+\omega]
 \cap\mathbb{T}$, $k=1,2,\dots,q$, such that
$x_i(\xi_i)=\inf_{t\in[\bar{k},\bar{k}+\omega]\cap\mathbb{T}}x_i(t)$,
$x_i(\eta_i)=\sup_{t\in[\bar{k},\bar{k}+\omega]\cap\mathbb{T}}x_i(t)$,
$y_j(\hat{\xi}_j)=\inf_{t\in[\bar{k},\bar{k}+\omega]\cap\mathbb{T}}y_j(t)$,
$y_j(\hat{\eta}_j)=\sup_{t\in[\bar{k},\bar{k}+\omega]\cap\mathbb{T}}y_j(t)$,
$i=1,2,\dots,n$, $j=1,2,\dots,m$. Then by \eqref{e3.2} and the Lemma \ref{l2.1},
then we have
\begin{align*}
\omega\bar{c}_ix_i(\xi_i)
&\leq \int_{\bar{k}}^{\bar{k}+\omega}\Big|\sum_{j=1}^ma_{ij}(t)f_j(y_j(t-\tau_{ij}(t)))\\
&\quad +\sum_{j=1}^m\sum_{l=1}^mb_{ijl}(t)g_j(y_j(t-\sigma_{ijl}(t)))
g_l(y_l(t-\upsilon_{ijl}(t)))+I_i(t)\Big|\Delta t\\
&\quad+\sum_{k=1}^q|\alpha_{ik}(x_i(t_k))|\\
&\leq \int_{\bar{k}}^{\bar{k}+\omega}
 \sum_{j=1}^m|a_{ij}(t)||f_j(y_j(t-\tau_{ij}(t)))|\Delta t\\
&\quad+\int_{\bar{k}}^{\bar{k}+\omega}\sum_{j=1}^m
 \sum_{l=1}^m|b_{ijl}(t)||g_j(y_j(t-\sigma_{ijl}(t)))||
 g_l(y_l(t-\upsilon_{ijl}(t)))|\Delta t\\
&\quad+\int_{\bar{k}}^{\bar{k}+\omega}|I_i(t)|\Delta t
 +\sum_{k=1}^q|\alpha_{ik}(x_i(t_k))|\\
&\leq \omega\Big(\sum_{j=1}^ma_{ij}^+M_j+\sum_{j=1}^m
 \sum_{l=1}^mb_{ijl}^+N_jN_l+I_i^+\Big)
+\sum_{k=1}^q|\alpha_{ik}(x_i(t_k))|
\end{align*}
for $i=1,2,\dots,n$.
Hence
\begin{equation} \label{e3.4}
\begin{split}
x_i(\xi_i)&\leq \frac{1}{\bar{c}_i}\Big(\Big(\sum_{j=1}^ma_{ij}^+M_j
+\sum_{j=1}^m\sum_{l=1}^mb_{ijl}^+N_jN_l+I_i^+\Big)
+\frac{1}{\omega}\sum_{k=1}^q|\alpha_{ik}(x_i(t_k))|\Big)\\
&= B_i, \quad i=1,2,\dots,n.
\end{split}
\end{equation}
Similarly, by \eqref{e3.3} and Lemma \ref{l2.1}, we obtain
\begin{equation} \label{e3.5}
\begin{split}
y_j(\hat{\xi}_j)&\leq \frac{1}{\bar{d}_j}\Big(\Big(\sum_{i=1}^ne_{ji}^+\hat{M}_i
+\sum_{i=1}^n\sum_{r=1}^nh_{jir}^+\hat{N}_i\hat{N}_r+J_j^+\Big)
+\frac{1}{\omega}\sum_{k=1}^q|\beta_{jk}(y_j(t_k))|\Big)\\
&= \hat{B}_j, \quad j=1,2,\dots,m.
\end{split}
\end{equation}
By \eqref{e3.2}, we  have
\begin{align*}
\omega\bar{c}_ix_i(\eta_i)
&\geq -\int_{\bar{k}}^{\bar{k}+\omega}
 \Big|\sum_{j=1}^ma_{ij}(t)f_j(y_j(t-\tau_{ij}(t)))\\
&\quad -\sum_{j=1}^m\sum_{l=1}^mb_{ijl}(t)
 g_j(y_j(t-\sigma_{ijl}(t)))
 g_l(y_l(t-\upsilon_{ijl}(t)))\\
&\quad +I_i(t)\Big|\Delta t -\sum_{k=1}^q|\alpha_{ik}(x_i(t_k))|\\
&\geq -\int_{\bar{k}}^{\bar{k}+\omega}\sum_{j=1}^m|a_{ij}(t)||f_j(y_j(t-\tau_{ij}(t)))|\Delta t\\
&\quad -\int_{\bar{k}}^{\bar{k}+\omega}\sum_{j=1}^m\sum_{l=1}^m|b_{ijl}(t)||g_j(y_j(t-\sigma_{ijl}(t)))||g_l(y_l(t-\upsilon_{ijl}(t)))|\Delta t\\
&\quad -\int_{\bar{k}}^{\bar{k}+\omega}|I_i(t)|\Delta t-\sum_{k=1}^q|\alpha_{ik}(x_i(t_k))|\\
&\geq -\omega\Big(\sum_{j=1}^ma_{ij}^+M_j
+\sum_{j=1}^m\sum_{l=1}^mb_{ijl}^+N_jN_l+I_i^+\Big)
-\sum_{k=1}^q|\alpha_{ik}(x_i(t_k))|
\end{align*}
 for $i=1,2,\dots,n$. Hence
\begin{equation} \label{e3.6}
\begin{split}
x_i(\eta_i)&\geq -\frac{1}{\bar{c}_i}\Big(\Big(\sum_{j=1}^ma_{ij}^+M_j
+\sum_{j=1}^m\sum_{l=1}^mb_{ijl}^+N_jN_l+I_i^+\Big)
+\frac{1}{\omega}\sum_{k=1}^q|\alpha_{ik}(x_i(t_k))|\Big)\\
&= -B_i, \quad i=1,2,\dots,n.
\end{split}
\end{equation}
Similarly, by \eqref{e3.3}, we obtain
\begin{equation} \label{e3.7}
\begin{split}
y_j(\hat{\eta}_j)
&\geq -\frac{1}{\bar{d}_j}\Big(\Big(\sum_{i=1}^ne_{ji}^+\hat{M}_i
+\sum_{i=1}^n\sum_{r=1}^nh_{jir}^+\hat{N}_i\hat{N}_r+J_j^+\Big)
+\frac{1}{\omega}\sum_{k=1}^q|\beta_{jk}(y_j(t_k))|\Big)\\
&= -\hat{B}_j, \quad j=1,2,\dots,m.
\end{split}
\end{equation}
Set $t_0=t_0^+=\bar{k}$, $t_{q+1}=\bar{k}+\omega$. From \eqref{e3.4}, \eqref{e3.6}
and Lemma \ref{l2.3}, we have
\begin{align} 
&\int_{\bar{k}}^{\bar{k}+\omega}|x_i^\Delta(t)|\Delta t \nonumber\\
&=\sum_{k=1}^q\int_{t_{k-1}^+}^{t_k}|x^\Delta(t)|\Delta t
 +\sum_{k=1}^q|x_i(t_k^+)-x_i(t_k)| \nonumber\\
&\leq \int_{\bar{k}}^{\bar{k}+\omega}|c_i(t)||x_i(t)|\Delta t
 +\int_{\bar{k}}^{\bar{k}+\omega}\sum_{j=1}^m|a_{ij}(t)||f_j(y_j(t-\tau_{ij}(t)))|\Delta t
 \nonumber\\
&\quad+\int_{\bar{k}}^{\bar{k}+\omega}
\sum_{j=1}^m\sum_{l=1}^m|b_{ijl}(t)||g_j(y_j(t-\sigma_{ijl}(t)))||
 g_l(y_l(t-\upsilon_{ijl}(t)))|\Delta t \nonumber\\
&\quad+\int_{\bar{k}}^{\bar{k}+\omega}|I_i(t)|\Delta t+\sum_{k=1}^q|e_{ik}(x_i(t_k))|
 \nonumber\\
&\leq \Big(\int_{\bar{k}}^{\bar{k}+\omega}|c_i(t)|^2\Delta t\Big)^{1/2}
\Big(\int_{\bar{k}}^{\bar{k}+\omega}|x_i(t)|^2\Delta t\Big)^{1/2} 
\label{e3.8} \\
&\quad+\sum_{j=1}^m\Big(\int_{\bar{k}}^{\bar{k}+\omega}|a_{ij}(t)|^2\Delta t\Big)^{1/2}
\Big(\int_{\bar{k}}^{\bar{k}+\omega}|f_j(y_j(t-\tau_{ij}(t)))|^2\Delta t\Big)^{1/2} 
 \nonumber\\
&\quad+\sum_{j=1}^m\sum_{l=1}^mb_{ijl}^+\Big(\int_{\bar{k}}^{\bar{k}+\omega}
|g_j(y_j(t-\sigma_{ijl}(t)))|^2\Delta t\Big)^{1/2} \nonumber \\
&\quad \times\Big(\int_{\bar{k}}^{\bar{k}+\omega}|g_l(y_l(t-\upsilon_{ijl}(t)))|^2
 \Delta t\Big)^{1/2}
+I_i^+\omega+\sum_{k=1}^q|\alpha_{ik}(x_i(t_k))| \nonumber\\
&\leq \sqrt{\omega}c_i^+\|x_i\|_2+\sum_{j=1}^m\omega a_{ij}^+M_j
+\sum_{j=1}^m\sum_{l=1}^m\omega b_{ijl}^+N_jN_l+I_i^+\omega
+\sum_{k=1}^q|\alpha_{ik}(x_i(t_k))|. \nonumber
\end{align}
Similarly, from \eqref{e3.5} and \eqref{e3.7}, and the Lemma \ref{l2.3}, we obtain
\begin{equation} \label{e3.9}
\begin{split}
\int_{\bar{k}}^{\bar{k}+\omega}|y_j^\Delta(t)|\Delta t
&\leq \sqrt{\omega}d_j^+\|y_j\|_2+\sum_{i=1}^n\omega e_{ji}^+\hat{M}_i
+\sum_{i=1}^n\sum_{r=1}^n\omega b_{jir}^+\hat{N}_i\hat{N}_r\\
&\quad +J_j^+\omega+\sum_{k=1}^q|\beta_{jk}(y_j(t_k))|.
\end{split}
\end{equation}
From Lemma \ref{l2.5} and \ref{e3.1}, for $i=1,2,\dots,n$, $j=1,2,\dots,m$, we
obtain
\begin{align*}
x_i(t)&= e_{-\lambda c_i(t)}(t,\bar{k})x_i(\bar{k})+
\int_{\bar{k}}^t\lambda e_{-\lambda c_i(t)}(t,\sigma(s))
\Big(\sum_{j=1}^ma_{ij}(s)f_j(y_j(s-\tau_{ij}(s)))\\
&\quad+\sum_{j=1}^m\sum_{l=1}^mb_{ijl}(s)g_j(y_j(s-\sigma_{ijl}(s)))
 g_l(y_l(s-\upsilon_{ijl}(s)))+I_i(s)\Big)\Delta s\\
&\quad+\sum_{k : t_k\in[\bar{k},t)_\mathbb{T}}e_{-\lambda c_i(t)}(t,t_k)
\alpha_{ik}(x(t_k)),
\end{align*}
\begin{align*}
y_j(t)&= e_{-\lambda d_j(t)}(t,\bar{k})y_j(\bar{k})+
\int_{\bar{k}}^t\lambda e_{-\lambda d_j(t)}(t,\sigma(s))
 \Big(\sum_{i=1}^ne_{ji}(s)p_i(x_i(s-\hat{\tau}_{ji}(s))) \\
&\quad+\sum_{i=1}^n\sum_{r=1}^nh_{jir}(s)q_r(x_i(s-\hat{\sigma}_{jir}(s)))
 q_r(x_r(s-\hat{\upsilon}_{jir}(s)))+J_j(s)\Big)\Delta s\\
&\quad+\sum_{k : t_k\in[\bar{k},t)_\mathbb{T}}e_{-\lambda d_j(t)}(t,t_k)
 \beta_{jk}(y(t_k)).
\end{align*}
Hence
\begin{align*}
|x_i(t)|
&\leq |x_i(\bar{k})|+\sum_{j=1}^m\int_{\bar{k}}^{\bar{k}+\omega}
|a_{ij}(s)||f_j(y_j(s-\tau_{ij}(s)))|\Delta s\\
&\quad+\sum_{j=1}^m\sum_{l=1}^m\int_{\bar{k}}^{\bar{k}+\omega}
|b_{ijl}(s)||g_j(y_j(s-\sigma_{ijl}(s)))||g_l(y_l(s-\upsilon_{ijl}(s)))|\Delta s\\
&\quad+\int_{\bar{k}}^{\bar{k}+\omega}|I_i(s)|\Delta s+\sum_{k : t_k\in[\bar{k},t)_\mathbb{T}}|\alpha_{ik}(x(t_k))|\\
&\leq |x_i(\bar{k})|+\sum_{j=1}^m\bar{a}_{ij}N_j\omega+
\sum_{j=1}^m\sum_{l=1}^m\bar{b}_{ijl}N_jN_l\omega+\bar{I}_i\omega
+\sum_{k=1}^q|\alpha_{ik}(x_i(t_k))|\\
&= u_i,\\
|y_j(t)|&\leq |y_j(\bar{k})|+\sum_{i=1}^n\bar{e}_{ji}\hat{N}_i\omega+
\sum_{i=1}^n\sum_{r=1}^n\bar{h}_{jir}\hat{N}_i\hat{N}_r\omega+\bar{J}_j\omega
+\sum_{k=1}^q|\beta_{jk}(y_j(t_k))|\\
&=  \hat{u}_j,
\end{align*}
where $i=1,2,\dots,n$, $j=1,2,\dots,m$; that is,
\begin{equation}\label{e3.10}
\begin{gathered}
\|x_i\|_2=\Big(\int_{\bar{k}}^{\bar{k}+\omega}|x_i(t)|^2\Delta t\Big)^{1/2}
 \leq u_i\sqrt{\omega},\\
\|y_j\|_2=\Big(\int_{\bar{k}}^{\bar{k}+\omega}|y_j(t)|^2\Delta t\Big)^{1/2}
 \leq \hat{u}_j\sqrt{\omega}.
\end{gathered}
\end{equation}
Substituting \eqref{e3.10} in \eqref{e3.8} and \eqref{e3.9}, we obtain
\begin{equation} \label{e3.11}
\begin{split}
\int_{\bar{k}}^{\bar{k}+\omega}|x_i^\Delta(t)|\Delta t
&\leq  \omega c_i^+u_i+\sum_{j=1}^m\omega a_{ij}^+M_j
+\sum_{j=1}^m\sum_{l=1}^m\omega b_{ijl}^+N_jN_l\\
&\quad+I_i^+\omega+\sum_{k=1}^q|\alpha_{ik}(x_i(t_k))|,
\quad i=1,2,\dots,n,
\end{split}
\end{equation}
and
\begin{equation} \label{e3.12}
\begin{split}
\int_{\bar{k}}^{\bar{k}+\omega}|y_j^\Delta(t)|\Delta t
&\leq  \omega d_j^+\hat{u}_j+\sum_{i=1}^n\omega e_{ji}^+\hat{M}_i
+\sum_{j=1}^n\sum_{r=1}^n\omega h_{jir}^+\hat{N}_i\hat{N}_r\\
&\quad+J_j^+\omega+\sum_{k=1}^q|\beta_{jk}(y_j(t_k))|,
\quad j=1,2,\dots,m.
\end{split}
\end{equation}
From \eqref{e3.6}-\eqref{e3.9} and \eqref{e3.11}-\eqref{e3.12} and Lemma \ref{l2.2},
there exist positive constants
$\zeta_i, \hat{\zeta}_j$ $(i=1,2,\dots,n, j=1,2,\dots,m)$ such that for
$t\in[\bar{k},\bar{k}+\omega]\cap\mathbb{T}$,
$|x_i(t)|\leq\zeta_i$, $|y_j(t)|\leq\hat{\zeta}_j$,  $i=1,2,\dots,n$, $j=1,2,\dots,m$.
 Clearly, $\zeta_i$ and $\hat{\zeta}_j$ ($i=1,2,\dots,n, j=1,2,\dots,m$) are independent of
$\lambda$. Denote $H^\ast=\sum_{i=1}^n\zeta_i+\sum_{j=1}^m\hat{\zeta}_j+C$, where $C>0$
is taken sufficiently large such that
\begin{align*}
&\min_{1\leq i\leq n,1\leq j\leq m}\{\bar{c}_i,\bar{d}_j\}H^*\\
&> n\max\Big(|\bar{I}_i|+\sum_{j=1}^m|\bar{a}_{ij}|M_j
+\sum_{j=1}^m\sum_{l=1}^m\bar{b}_{ijl}M_jN_l
 -\frac{1}{\omega}\sum_{k=1}^q|\alpha_{ik}(x_i(t_k))|\Big)\\
&\quad+m\max\Big(|\bar{J}_j|+\sum_{i=1}^n|\bar{e}_{ji}|\hat{M}_i
+\sum_{i=1}^n\sum_{r=1}^n\bar{h}_{jir}\hat{M}_i\hat{N}_r
 -\frac{1}{\omega}\sum_{k=1}^q|\beta_{jk}(y_j(t_k))|\Big).
\end{align*}
Now we take $\Omega=\{(x_1(t),\dots,x_n(t),y_1(t),\dots,y_m(t))^T
: \|(x_1(t),\dots,x_n(t),y_1(t),\\\dots,y_m(t))^T\|<H^*\}$.
Thus $(a)$ of Lemma \ref{l2.4} is satisfied.
When $(x_1(t),\dots,x_n(t),\\y_1(t),\dots,y_m(t))^T\in\partial\Omega\cap\mathbb{R}^n$,
$(x_1(t),\dots,x_n(t),y_1(t),\dots,y_m(t))^T$
is a constant vector in $\mathbb{R}^n$ with $|x_1|+\cdots+|x_n|+|y_1|+\cdots+|y_m|=H^\ast$,
then
$$
QN \begin{pmatrix}
      (x_i)_{n\times1} \\
      (y_j)_{m\times1}
    \end{pmatrix}
=  \begin{pmatrix}
    \begin{aligned}
     &\Big(-\bar{c}_ix_i(t)+\sum_{j=1}^m \bar{a}_{ij}f_j(y_j(t-\tau_{ij}(t)))\\
     &+\sum_{j=1}^m\sum_{l=1}^m\bar{b}_{ijl}
      g_j(y_j(t-\sigma_{ijl}(t)))g_l(y_l(t-\upsilon_{ijl}(t)))\\
     &+\bar{I}_i-\frac{1}{\omega}\sum_{k=1}^q\alpha_{ik}(x_i(t_k))\Big)_{n\times1}
    \end{aligned} \\
\begin{aligned}
     &\Big(-\bar{d}_jy_j(t)+\sum_{i=1}^n \bar{e}_{ji}p_i(x_i(t-\hat{\tau}_{ji}(t))) \\
     & +\sum_{i=1}^n\sum_{r=1}^n\bar{h}_{jir}
     q_i(x_i(t-\hat{\sigma}_{jir}(t)))q_r(x_r(t-\hat{\upsilon}_{jir}(t)))\\
     & +\bar{J}_j-\frac{1}{\omega}\sum_{k=1}^q\beta_{jk}(y_j(t_k))\Big)_{m\times1}
    \end{aligned}
\end{pmatrix}_{(n+m)\times1}.
$$
Therefore,
\begin{align*}
&\Big\|QN\begin{pmatrix}
      (x_i)_{n\times1}\\
      (y_j)_{m\times1}\\
    \end{pmatrix}\Big\|\\
 &=  \sum_{i=1}^n\Big|\bar{c}_ix_i(t)-\sum_{j=1}^m
\bar{a}_{ij}f_j(y_j(t-\tau_{ij}(t))) \\
&\quad-\sum_{j=1}^m\sum_{l=1}^m\bar{b}_{ijl}
g_j(y_j(t-\sigma_{ijl}(t)))g_l(y_l(t-\upsilon_{ijl}(t)))
-\bar{I}_i+\frac{1}{\omega}\sum_{k=1}^q\alpha_{ik}(x_i(t_k))\Big|\\
&\quad+\sum_{j=1}^m\Big|\bar{d}_jy_j(t)-\sum_{i=1}^n
\bar{e}_{ji}p_i(x_i(t-\hat{\tau}_{ji}(t))) \\
&\quad -\sum_{i=1}^n\sum_{r=1}^n\bar{b}_{jir}
q_i(x_i(t-\hat{\sigma}_{jir}(t)))q_r(x_r(t-\hat{\upsilon}_{jir}(t)))
-\bar{J}_j+\frac{1}{\omega}\sum_{k=1}^q\beta_{jk}(y_j(t_k))\Big|\\
&\geq \sum_{i=1}^n\bar{c}_i|x_i(t)|+\frac{n}{\omega}\sum_{k=1}^q|\alpha_{ik}(x_i(t_k))|
-\sum_{i=1}^n\sum_{j=1}^m|\bar{a}_{ij}|M_j-\sum_{i=1}^n\sum_{j=1}^m\sum_{l=1}^m\bar{b}_{ijl}
M_jN_l\\
&\quad-\sum_{i=1}^n|\bar{I}_i|+\sum_{j=1}^m\bar{d}_j|y_j(t)|+\frac{m}{\omega}\sum_{k=1}^q|\beta_{jk}(y_j(t_k))|
-\sum_{j=1}^m\sum_{i=1}^n|\bar{e}_{ji}|\hat{M}_i\\
&\quad-\sum_{j=1}^m\sum_{i=1}^n\sum_{r=1}^n\bar{b}_{jir}
\hat{M}_i\hat{N}_r-\sum_{j=1}^m|\bar{J}_j|\\
&\geq \sum_{i=1}^n(\bar{c}_i|x_i(t)|)+\sum_{j=1}^m(\bar{d}_j|y_j(t)|)\\
&\quad-\sum_{i=1}^n\Big(|\bar{I}_i|+\sum_{j=1}^m|\bar{a}_{ij}|M_j
+\sum_{j=1}^m\sum_{l=1}^m\bar{b}_{ijl}M_jN_l-\frac{1}{\omega}\sum_{k=1}^q|
\alpha_{ik}(x_i(t_k))|\Big)\\
&\quad-\sum_{j=1}^m\Big(|\bar{J}_j|+\sum_{i=1}^n|\bar{e}_{ji}|\hat{M}_i
+\sum_{i=1}^n\sum_{r=1}^n\bar{h}_{jir}\hat{M}_i\hat{N}_r-\frac{1}{\omega}
 \sum_{k=1}^q|\beta_{jk}(y_j(t_k))|\Big)\\
&\geq \min(\bar{c}_i)\sum_{i=1}^n|x_i(t)|+\min(\bar{d}_j)\sum_{j=1}^m|y_j(t)|\\
&\quad-n\max\Big(|\bar{I}_i|+\sum_{j=1}^m|\bar{a}_{ij}|M_j
+\sum_{j=1}^m\sum_{l=1}^m\bar{b}_{ijl}M_jN_l-\frac{1}{\omega}
 \sum_{k=1}^q|\alpha_{ik}(x_i(t_k))|\Big)\\
&\quad-m\max\Big(|\bar{J}_j|+\sum_{i=1}^n|\bar{e}_{ji}|\hat{M}_i
+\sum_{i=1}^n\sum_{r=1}^n\bar{h}_{jir}\hat{M}_i\hat{N}_r-\frac{1}{\omega}
\sum_{k=1}^q|\beta_{jk}(y_j(t_k))|\Big)>0.
\end{align*}
Consequently,
$QN(x_1,\dots,x_n,y_1,\dots,y_m)^T\neq(0,\dots,0,0,\dots,0)^T$ for
$(x_1,\dots,x_n,\\ y_1,\dots,y_m)^T\in\partial\Omega\cap\ker L$.
This satisfies condition (b) of Lemma \ref{l2.4}.
 Define $\Psi : \ker L\times[0,1]\to X$ by
\begin{align*}
\Psi(x_1,\dots,x_n,y_1,\dots,y_m,\chi)
&=  -\chi(x_1,\dots,x_n, y_1,\dots,y_m)^T\\
&\quad+(1-\chi)QN(x_1,\dots,x_n,y_1,\dots,y_m)^T.
\end{align*}
When $(x_1,\dots,x_n,y_1,\dots,y_m)^T\in\partial\Omega\cap\ker L$,
 $(x_1,\dots,x_n,y_1,\dots,y_m)^T$ is a constant vector in
 $\mathbb{R}^{n+m}$ with $\sum_{i=1}^n|x_i|+\sum_{j=1}^m|y_j|=H^\ast$,
we have
$$
\Psi(x_1,\dots,x_n,y_1,\dots,y_m,\chi)\neq(0,\dots,0,0,\dots,0)^T.
$$
Therefore,
\begin{align*}
&\deg \Big(QN(x_1(t),\dots,x_n(t),y_1(t),\dots,y_m(t))^T, \Omega\cap\ker L,
 (0,\dots,0,0,\dots,0)^T\Big)\\
&=  \deg \Big(QN(-x_1(t),\dots,-x_n(t),-y_1(t),\dots,-y_m(t))^T, \Omega\cap\ker L,\\
& (0,\dots,0,0,\dots,0)^T\Big)\neq0.
\end{align*}
Condition (c) of Lemma \ref{l2.4} is also satisfied. Thus, by Lemma \ref{l2.4}
we obtain that $Lx=Nx$ has at least one solution in $\mathbb{X}$.
 That is, system \eqref{e1.1} has at least one $\omega$-periodic solution.
The proof is complete.
\end{proof}

\section{Global exponential asymptotic stability of periodic solutions}

In this section, we will construct some suitable Lyapunov functions to study
the global exponential asymptotic stability of the periodic solution of \eqref{e1.1}.

\begin{theorem}\label{t4.1}
Assume that {\rm (H1)--(H4)} hold. Suppose further that
\begin{itemize}
\item[(H5)] there exists $n+m$ positive constants
$\varepsilon_i>0$ and $\varepsilon_{n+j}>0$, $i=1,2\dots,n$, $j=1,2\dots,m$  such that
\begin{gather*}
-c_i^-\varepsilon_i+\sum_{j=1}^ma_{ij}^+L_j\varepsilon_{n+j}
+\sum_{j=1}^m\sum_{l=1}^mb_{ijl}^+\left(H_jN_l\varepsilon_{n+j}
+H_lN_j\varepsilon_{n+l}\right)<0,\quad
i=1,2,\dots,n,\\
-d_j^-\varepsilon_{n+j}+\sum_{i=1}^ne_{ji}^+\hat{L}_i\varepsilon_i
+\sum_{i=1}^n\sum_{r=1}^nh_{jir}^+\left(\hat{H}_i\hat{N}_r\varepsilon_i
+\hat{H}_r\hat{N}_i\varepsilon_r\right)<0,\quad
j=1,2,\dots,m;
\end{gather*}


\item[(H6)]  the impulse operators $e_i(x_i(t))$, $i=1,2,\dots,n$ satisfy
\begin{gather*}
\alpha_i(x_i(t_k))=-\gamma_{ik}(x_i(t_k)-x_i^\ast(t)),\quad
0<\gamma_{ik}<2,\quad i=1,2,\dots,n,\quad k\in\mathbb{Z}^+,\\
\beta_j(y_j(t_k))=-\delta_{jk}(y_j(t_k)-y_j^\ast(t)),\quad
0<\delta_{jk}<2,\quad j=1,2,\dots,m,\quad k\in\mathbb{Z}^+.
\end{gather*}
\end{itemize}
Then the $\omega$-periodic solution of \eqref{e1.1} is globally exponentially stable.
\end{theorem}

\begin{proof}
According to Theorem \ref{t3.1}, we know that \eqref{e1.1} has an $\omega$-periodic
solution
$z^\ast(t)=(x^\ast_1(t),\dots,x^\ast_n(t),y^\ast_1(t),\\\dots,y^\ast_m(t))^T$.
Suppose that $x(t)=(x_1(t),\dots,x_n(t),y_1(t),\dots,y_m(t))^T$ is an arbitrary solution
of \eqref{e1.1}. Then it follows from system \eqref{e1.1} that
\begin{align*}
(x_i(t)-x_i^\ast(t))^\Delta
&= -c_i(t)(x_i(t)-x_i^\ast(t))\\
&\quad +\sum_{j=1}^ma_{ij}(t)\Big(f_j(y_j(t-\tau_{ij}(t)))-f_j(y_j^\ast(t-\tau_{ij}(t)))
 \Big)\\
&\quad +\sum_{j=1}^m\sum_{l=1}^mb_{ijl}(t)
 \Big(g_j(y_j(t-\sigma_{ijl}(t)))g_l(y_l(t-\upsilon_{ijl}(t)))\\
&\quad -g_j(y_j^\ast(t-\sigma_{ijl}(t)))g_l(y_l^\ast(t-\upsilon_{ijl}(t)))\Big),
\end{align*}
\begin{align*}
(y_j(t)-y_j^\ast(t))^\Delta
&= -d_j(t)(y_j(t)-y_j^\ast(t))\\
&\quad +\sum_{i=1}^ne_{ji}(t)\Big(p_i(x_i(t-\hat{\tau}_{ji}(t)))
 -p_i(x_i^\ast(t-\hat{\tau}_{ji}(t)))\Big)\\
&\quad +\sum_{i=1}^n\sum_{r=1}^nh_{jir}(t)
 \Big(q_i(x_i(t-\hat{\sigma}_{jir}(t)))q_r(x_r(t-\hat{\upsilon}_{jir}(t))) \\
&\quad -q_i(x_i^\ast(t-\hat{\sigma}_{jir}(t)))q_r(x_r^\ast(t-\hat{\upsilon}_{jir}(t)))\Big),
\end{align*}
for $i=1,2,\dots,n$, $t>0$, $t\neq t_k$, $k\in\mathbb{Z}^+$, with initial values given by
\begin{gather*}
x_i(t)=\phi_{x_i}(s),\quad s\in[-\theta,0]\cap\mathbb{T},\quad i=1,2,\dots,n,\\
y_j(t)=\phi_{y_j}(s),\quad s\in[-\hat{\theta},0]\cap\mathbb{T},\quad j=1,2,\dots,m,
\end{gather*}
where $\theta$ and $\hat{\theta}$ are defined as before. By condition (H3),  we obtain
\begin{equation} \label{e4.1}
\begin{split}
&[(x_i(t)-x_i^\ast(t))^\Delta]^+\\
&\leq -c_i^-|x_i(t)-x_i^\ast(t)|
 +\sum_{j=1}^ma_{ij}^+L_j|y_j(t-\tau_{ij}(t))-y_j^\ast(t-\tau_{ij}(t))|\\
&\quad +\sum_{j=1}^m\sum_{l=1}^mb_{ijl}^+
\Big(H_jN_l|y_j(t-\sigma_{ijl}(t))-y_j^\ast(t-\sigma_{ijl}(t))|\\
&\quad +H_lN_j|y_l(t-\upsilon_{ijl}(t))-y_l^\ast(t-\upsilon_{ijl}(t))|\Big),
\end{split}
\end{equation}
\begin{align*}
[(y_j(t)-y_j^\ast(t))^\Delta]^+
&\leq -d_j^-|y_j(t)-y_j^\ast(t)|
 +\sum_{i=1}^ne_{ji}^+\hat{L}_i|x_i(t-\hat{\tau}_{ji}(t))-x_i^\ast(t-\hat{\tau}_{ji}(t))|\\
&\quad +\sum_{i=1}^n\sum_{r=1}^nh_{jir}^+\Big(\hat{H}_i\hat{N}_r|x_i(t-\hat{\sigma}_{jir}
 (t))-x_i^\ast(t-\hat{\sigma}_{jir}(t))| \\
&\quad +\hat{H}_r\hat{N}_i|x_r(t-\hat{\upsilon}_{jir}(t))
 -x_r^\ast(t-\hat{\upsilon}_{jir}(t))|\Big),
\end{align*}
for $i=1,2,\dots,n$, $j=1,2,\dots,m$, $t>0$, $t\neq t_k$, $k\in\mathbb{Z}^+$,
 where $[x_i^\Delta(t)]^+$ and $[y_j^\Delta(t)]^+$ denote the upper right derivative.
 Also, in view of condition (H6), one has
\begin{align*}
x_i(t_k+0)-x_i^\ast(t_k+0)
&=x_i(t_k)+\alpha_i(x_i(t_k))-x_i^\ast(t_k)-\alpha_i(x_i^\ast(t_k))\\
&=(1-\gamma_{ik})(x_i(t_k)-x_i^\ast(t_k)),\\
y_j(t_k+0)-y_j^\ast(t_k+0)&=y_j(t_k)+\beta_j(y_j(t_k))-y_j^\ast(t_k)-\beta_i(y_j^\ast(t_k))\\
&=(1-\beta_{jk})(y_j(t_k)-y_j^\ast(t_k)),
\end{align*}
thus
\begin{gather*}
|x_i(t_k+0)-x_i^\ast(t_k+0)|=|1-\gamma_{ik}||x_i(t_k)-x_i^\ast(t_k)|
 \leq|x_i(t_k)-x_i^\ast(t_k)|,\\
|y_j(t_k+0)-y_j^\ast(t_k+0)|=|1-\delta_{jk}||y_j(t_k)-y_j^\ast(t_k)|
 \leq|y_j(t_k)-y_j^\ast(t_k)|,
\end{gather*}
for $i=1,2,\dots,n$,  $j=1,2,\dots,m$, $k\in\mathbb{Z}^+$. According to condition (H5),
 it can always find a small enough constant $\eta>0$ satisfying
 $1-\mu(t)\eta>0$ for all $t\in\mathbb{T}$, namely, $-\eta\in\mathfrak{R}^+$ such that
\begin{equation} \label{e4.2}
\begin{aligned}
&(-c_i^-+\eta)\varepsilon_i+\sum_{j=1}^ma_{ij}^+L_je_\eta(t,t-\tau_{ij}(t))
 \varepsilon_{n+j} \\
&+\sum_{j=1}^m\sum_{l=1}^mb_{ijl}^+\big(H_jN_l\varepsilon_{n+j}e_\eta(t,t-\sigma_{ijl}(t))
+H_lN_j\varepsilon_{n+l}e_\eta(t,t-\upsilon_{ijl}(t))\big)<0
\end{aligned}
\end{equation}
for $i=1,2,\dots,n$; 
\begin{align*}
&(-d_j^-+\eta)\varepsilon_{n+j}+\sum_{i=1}^ne_{ji}^+\hat{L}_ie_\eta(t,t-\hat{\tau}_{ji}(t))
 \varepsilon_i \\
&+\sum_{i=1}^n\sum_{r=1}^nh_{jir}^+
 \big(\hat{H}_i\hat{N}_r\varepsilon_ie_\eta(t,t-\hat{\sigma}_{jir}(t))
 +\hat{H}_r\hat{N}_i\varepsilon_re_\eta(t,t-\hat{\upsilon}_{jir}(t))\big)<0
\end{align*}
for $j=1,2,\dots,m$.

Now we define a Lyapunov function $V=(\varphi_1,\dots,\varphi_n,\psi_1,\dots,\psi_m)^T$,
with
\begin{gather*}
\varphi_i(t)=e_\eta(t,\varrho)|x_i(t)-x_i^\ast(t)|,\quad
 t\in[-\theta,\infty]\cap\mathbb{T}, i=1,2,\dots,n,\\
\psi_j(t)=e_\eta(t,\varrho)|y_j(t)-y_j^\ast(t)|,\quad
 t\in[-\hat{\theta},\infty]\cap\mathbb{T},j=1,2,\dots,m,
\end{gather*}
where $\varrho\in[-\max\{\theta,\hat{\theta}\},0]\cap\mathbb{T}$.
In view of \eqref{e4.1}, we obtain
\begin{align} 
&[\varphi_i^\Delta(t)]^+ \nonumber \\
&=\eta e_\eta(t,\varrho)|x_i(t)-x^\ast_i(t)|+e_\eta\sigma(t),\varrho)
\operatorname{sign}(x_i(t)-x^\ast_i(t)) \nonumber\\
&\quad\times\Big(-c_i(t)(x_i(t)-x_i^\ast(t))
+\sum_{j=1}^ma_{ij}(t)\big(f_j(y_j(t-\tau_{ij}(t)))-f_j(y_j^\ast(t-\tau_{ij}(t)))\big)
\nonumber\\
&\quad +\sum_{j=1}^m\sum_{l=1}^mb_{ijl}(t)
\big(g_j(y_j(t-\sigma_{ijl}(t)))g_l(y_l(t-\upsilon_{ijl}(t)))\nonumber\\
&\quad -g_j(y_j^\ast(t-\sigma_{ijl}(t)))g_l(y_l^\ast(t-\upsilon_{ijl}(t)))\big)\Big)
\nonumber\\
&\leq\eta e_\eta(t,\varrho)|x_i(t)-x^\ast_i(t)|+e_\eta(\sigma(t),\varrho) \label{e4.3}  \\
&\quad \times\Big(-c_i^-|x(t)-x^\ast(t)|
+\sum_{j=1}^ma_{ij}^+L_j|y_j(t-\tau_{ij}(t))-y_j^\ast(t-\tau_{ij}(t))| \nonumber\\
&\quad +\sum_{j=1}^m\sum_{l=1}^mb_{ijl}^+
 \big(H_jN_l|y_j(t-\sigma_{ijl}(t))-y_j^\ast(t-\sigma_{ijl}(t))| \nonumber\\
&\quad +H_lN_j|y_l(t-\upsilon_{ijl}(t))-y_l^\ast(t-\upsilon_{ijl}(t))|\big)\Big)
\nonumber\\
&\leq e_\eta(\sigma(t),\varrho)\Big((-c_i^-+\eta)|x_i(t)-x_i^\ast(t)|
+\sum_{j=1}^ma_{ij}^+L_j|y_j(t-\tau_{ij}(t))-y_j^\ast(t-\tau_{ij}(t))| \nonumber\\
&\quad +\sum_{j=1}^m\sum_{l=1}^mb_{ijl}^+
 \big(H_jN_l|y_j(t-\sigma_{ijl}(t))-y_j^\ast(t-\sigma_{ijl}(t))| \nonumber\\
&\quad +H_lN_j|y_l(t-\upsilon_{ijl}(t))-y_l^\ast(t-\upsilon_{ijl}(t))|\big)\Big) \nonumber\\
&\leq (1+\mu(t)\eta)\Big((-c_i^-+\eta)\varphi_i(t)
+\sum_{j=1}^ma_{ij}^+L_je_\eta(t,t-\tau_{ij}(t))\psi_j(t-\tau_{ij}(t)) \nonumber\\
&\quad +\sum_{j=1}^m\sum_{l=1}^mb_{ijl}^+
 \big(H_jN_le_\eta(t,t-\sigma_{ijl}(t))\psi_j(t-\sigma_{ijl}(t)) \nonumber\\
&\quad +H_lN_je_\eta(t,t-\upsilon_{ijl}(t))\psi_l(t-\upsilon_{ijl}(t))\big)\Big), \nonumber
\end{align}

\begin{equation} \label{e4.4}
\begin{split}
&[\psi_i^\Delta(t)]^+ \\
&\leq(1+\mu(t)\eta)\Big((-d_j^-+\eta)\psi_j(t)
+\sum_{i=1}^ne_{ji}^+\hat{L}_ie_\eta(t,t-\hat{\tau}_{ji}(t))
\varphi_i(t-\hat{\tau}_{ji}(t)) \\
&\quad +\sum_{i=1}^n\sum_{r=1}^nh_{jir}^+
\big(\hat{H}_i\hat{N}_re_\eta(t,t-\hat{\sigma}_{jir}(t))
\varphi_i(t-\hat{\sigma}_{jir}(t))  \\
&\quad +\hat{H}_r\hat{N}_ie_\eta(t,t-\hat{\upsilon}_{jir}(t))
 \varphi_r(t-\hat{\upsilon}_{jir}(t))\big)\Big),
\end{split}
\end{equation}
for $i=1,2,\dots,n$, $j=1,2,\dots,m$.
Also,
\begin{gather*}
\varphi_i(t_k+0)=|1-\gamma_{ik}|\varphi_i(t_k)\leq\varphi_i(t_k),\quad
 i=1,2,\dots,n,\; k\in\mathbb{Z}^+,\\
\psi_j(t_k+0)=|1-\delta_{jk}|\psi_j(t_k)\leq\psi_j(t_k),\quad
 j=1,2,\dots,m,\; k\in\mathbb{Z}^+.
\end{gather*}
Let $\varepsilon_M=\max_{1\leq i\leq n,1\leq j\leq m}\{\varepsilon_i,\varepsilon_{n+j}\}$,
$\varepsilon_L=\min_{1\leq i\leq n,1\leq j\leq m}\{\varepsilon_i,\varepsilon_{n+j}\}$,
$l_0=(1-\varrho)(\sum_{i=1}^n|\phi_{x_i}-x_i^\ast|_0+\sum_{j=1}^m|\phi_{y_j}-y_j^\ast|_0)
/{\varepsilon_L}$, where $-\varrho\geq0$ is a constant,
$|\phi_{x_i}-x_i^\ast|_0=\sup_{\varrho\in[-\theta,0]\cap\mathbb{T}}|\phi_{x_i}
(\varrho)-x_i^\ast(\varrho)|$,
$|\phi_{y_j}-y_j^\ast|_0=\sup_{\varrho\in[-\hat{\theta},0]
\cap\mathbb{T}}|\phi_{y_j}(\varrho)-y_j^\ast(\varrho)|$.
Then
\begin{gather*}
|\varphi_i(\varrho)|=e_\eta(t,\varrho)|\phi_{x_i}(\varrho)-x_i^\ast(\varrho)|
 <\varepsilon_il_0,
\quad \varrho\in[-\theta,0]\cap\mathbb{T},\; i=1,2,\dots,n,\\
|\psi_i(\varrho)|=e_\eta(t,\varrho)|\phi_{y_j}(\sigma)-y_j^\ast(\varrho)|
 <\varepsilon_{n+j}l_0,
\quad \varrho\in[-\hat{\theta},0]\cap\mathbb{T},\; j=1,2,\dots,m.
\end{gather*}
In the following, we will prove that $|\varphi_i(t)|<\varepsilon_il_0$,
$|\psi_j(t)|<\varepsilon_{n+j}l_0$,
for $t>0$, $i=1,2,\dots,n$, $j=1,2,\dots,m$.
If it is not true, no loss of generality, then there exist some $i_0$ and
$t_1$ $(t_1>0)$ such that
$|\varphi_{i_0}(t_1)|=\varepsilon_{i_0}l_0$, $[\varphi_{i_0}^\Delta(t_1)]^+\geq0$
and  $|\varphi_i(t)|<\varepsilon_il_0$, $t\in[-\theta,t_1]\cap\mathbb{T}$,
 $|\psi_j(t)|<\varepsilon_{n+j}l_0$, $t\in[-\hat{\theta},t_1]\cap\mathbb{T}$,
for  $i=1,2,\dots,n$, $j=1,2,\dots,m$.
However, from \eqref{e4.2} and \eqref{e4.3}, we obtain
\begin{align*}
&[\varphi_{i_0}^\Delta(t_1)]^+\\
&\leq(1+\mu(t_1)\eta)\Big((-c_{i_0}^-+\eta)\varepsilon_{i_0}
+\sum_{j=1}^ma_{i_0j}^+L_je_\eta(t_1,t_1-\tau_{{i_0}j}(t_1))\varepsilon_j\\
&\quad +\sum_{j=1}^m\sum_{l=1}^mb_{i_0jl}^+
\big(H_jN_le_\eta(t_1,t_1-\sigma_{i_0jl}(t_1))\varepsilon_j
+H_lN_je_\eta(t_1,t_1-\upsilon_{i_0jl}(t_1))\varepsilon_l\big)\Big)l_0\\
&<0,
\end{align*}
this is a contradiction. So $|\varphi_i(t)|<\varepsilon_il_0$,
$|\psi_j(t)|<\varepsilon_{n+j}l_0$,
for $t>0$, $i=1,2,\dots,n$, $j=1,2,\dots,m$. That is,
\[
e_\eta(t,\varrho)|x_i(t)-x_i^\ast(t)|<\varepsilon_il_0
\leq\frac{\varepsilon_i}{\varepsilon_L}(1-\varrho)
\Big(\sum_{i=1}^n|\phi_{x_i}-x_i^\ast|_0+\sum_{j=1}^m|\phi_{y_j}-y_j^\ast|_0\Big)
\]
for $t>0$, $i=1,2,\dots,n$, and
\[
e_\eta(t,\varrho)|y_j(t)-y_j^\ast(t)|<\varepsilon_{n+j}l_0
\leq\frac{\varepsilon_{n+j}}{\varepsilon_L}
(1-\varrho)\Big(\sum_{i=1}^n|\phi_{x_i}-x_i^\ast|_0+\sum_{j=1}^m|\phi_{y_j}-y_j^\ast|_0
\Big)
\]
for $t>0$, $j=1,2,\dots,m$,
which means that
\begin{equation} \label{e4.5}
\begin{split}
&\sum_{i=1}^n|x_i(t)-x_i^\ast(t)|_0+\sum_{j=1}^m|y_j(t)-y_j^\ast(t)|_0\\
&\leq\frac{\varepsilon_M}{\varepsilon_m}e_{\ominus\eta}(t,\varrho)
 (1-\varrho)(n+m)\Big(\sum_{i=1}^n|\phi_{x_i}(\varrho)-x_i^\ast(\varrho)
|_0\\
&\quad +\sum_{j=1}^m|\phi_{y_j}(\varrho)-y_j^\ast(\varrho)|_0\Big).
\end{split}
\end{equation}
Denote $-\vartheta=\ominus\eta=-\eta/(1+\mu\eta)\in\mathbb{R}$,
$N=N(\varrho)=\varepsilon_M/\varepsilon_m(1-\varrho)(n+m)>1$, in
view of \eqref{e4.5}, we have
\[
\|z(t)-z^\ast(t)\|\leq Ne_{-\vartheta}(t,\varrho)
\Big(\sum_{i=1}^n|\phi_{x_i}(\varrho)-x_i^\ast(\varrho)|_0
+\sum_{j=1}^m|\phi_{y_j}(\varrho)-y_j^\ast(\varrho)|_0\Big),
\]
and we can conclude that the $\omega$-periodic solution of \eqref{e1.1}
is globally exponentially stable and this completes the proof.
\end{proof}

\section{An illustrative example}

In this section, we give an example to illustrate the results in this article.
 Let $\mathbb{T}=\bigcup_{k=0}^\infty[2k,2k+1]$. We will apply our main results 
to the BAM HHNNs with impulses and delays on time scales
\begin{equation} \label{e5.1}
\begin{gathered}
\begin{aligned}
x_i^\Delta(t)
&= -c_i(t)x_i(t)+\sum_{j=1}^2a_{ij}(t)f_j(y_j(t-\tau_{ij}(t)))\\
&\quad +\sum_{j=1}^2\sum_{l=1}^2b_{ijl}(t)
 g_j(y_j(t-\sigma_{ijl}(t)))g_l(y_l(t-\upsilon_{ijl}(t)))+I_i(t),\;
 t\neq t_k,
\end{aligned}\\
\Delta x_i(t_k)
= \alpha_{ik}(x_i(t_k))=-\gamma_{ik}(x_i(t_k)-x_i^\ast(t)),\quad i=1,2,\; k=1,2,\dots,
\\
\begin{aligned}
y_j^\Delta(t) &= -d_j(t)y_j(t)+\sum_{i=1}^2e_{ji}(t)p_i(x_i(t-\hat{\tau}_{ji}(t)))\\
&\quad +\sum_{i=1}^2\sum_{r=1}^2h_{jir}(t)q_i(x_i(t-\hat{\sigma}_{jir}(t)))
q_r(x_r(t-\hat{\upsilon}_{jir}(t)))+J_j(t),\; t\neq t_k,
\end{aligned}
\\
\Delta y_j(t_k)= \beta_{jk}(y_j(t_k))=-\delta_{jk}(y_j(t_k)-y_j^\ast(t)),\quad
j=1,2, \quad k=1,2,\dots,
\end{gathered}
\end{equation}
where
\begin{gather*}
f_1(x)=\sin\big(\frac{1}{\sqrt{2}}x\big),\quad
f_2(x)=\sin\big(\frac{1}{2\sqrt{2}}x\big),\quad
g_1(x)=\arctan\big(\frac{1}{\sqrt{2}}x\big),\\
g_2(x)=\arctan\big(\frac{1}{2\sqrt{2}}x\big),\quad
p_1(x)=\cos\big(\frac{1}{\sqrt{3}}x\big),\quad
p_2(x)=\cos\big(\frac{1}{3\sqrt{3}}x\big),\\
q_1(x)=\operatorname{arccot}\big(\frac{1}{\sqrt{3}}x\big),\quad
q_2(x)=\operatorname{arccot}\big(\frac{1}{3\sqrt{3}}x\big).
\end{gather*}
Obviously, $f_j(x),g_j(x) (j=1,2)$, $p_i(x),q_i(x) (i=1,2)$ satisfy (H2) and (H3), and
\begin{gather*}
M_1=M_2=L_1=L_2=H_1=H_2=1,\quad N_1=N_2=\frac{\pi}{2},\\
\hat{M}_1=\hat{M}_2=\hat{L}_1=\hat{L}_2=\hat{H}_1=\hat{H}_2=1,\quad
\hat{N}_1=\hat{N}_2=\pi.
\end{gather*}
Take
\begin{gather*}
a_{11}(t)=1+\cos(2\pi t),\quad a_{12}(t)=2+\cos(2\pi t),\quad a_{21}(t)=2+\cos(2\pi t),\\
a_{22}(t)=3+\cos(2\pi t),\quad c_1(t)=20+5\sin(2\pi t),\quad c_2(t)=33+16\sin(2\pi t),\\
I_1(t)=1+\sin(2\pi t),\quad I_2(t)=1+\cos(2\pi t),\quad b_{111}(t)=b_{222}(t)=\frac{1}{4}+\frac{1}{4}\sin(2\pi t),\\
b_{112}(t)=b_{212}(t)=\frac{1}{3}+\frac{1}{3}\cos(2\pi t),\quad
b_{121}(t)=b_{221}(t)=\frac{1}{5}+\frac{1}{5}\cos(2\pi t),\\
 b_{122}(t)=b_{211}(t)=\frac{1}{6}+\frac{1}{6}\sin(2\pi t),\quad
\gamma_{1k}=1+\frac{1}{2}\sin(2+k),\\
\gamma_{2k}=1+\frac{6}{7}\cos(5+k^2), k\in\mathbb{Z}^+,\quad
e_{11}(t)=1+\sin(2\pi t),\quad e_{12}(t)=1+2\sin(2\pi t),\\
e_{21}(t)=1+2\cos(2\pi t),\quad e_{22}(t)=3+\sin(2\pi t),\quad d_1(t)=21+6\cos(2\pi t),\\
d_2(t)=31+14\cos(2\pi t),\quad J_1(t)=2+3\sin(2\pi t), J_2(t)=3+2\cos(2\pi t),\\
h_{111}(t)=h_{222}(t)=\frac{1}{8}+\frac{1}{8}\sin(2\pi t),\quad
h_{112}(t)=h_{212}(t)=\frac{1}{6}+\frac{1}{6}\cos(2\pi t),\\
h_{121}(t)=h_{221}(t)=\frac{1}{10}+\frac{1}{10}\sin(2\pi t),\quad
 h_{122}(t)=h_{211}(t)=\frac{1}{12}+\frac{1}{12}\cos(2\pi t),\\
\delta_{1k}=1+\frac{2}{5}\cos(3+k),\quad
\delta_{2k}=1+\frac{4}{7}\sin(9+k^2), k\in\mathbb{Z}^+.
\end{gather*}
One can verify that (H1) is satisfied, and $\omega=1$, $c_1^-=15$, $c_2^-=17$,
$a_{11}^+=2$, $a_{12}^+=3$, $a_{21}^+=3$, $a_{22}^+=4$, $b_{111}^+=b_{222}^+=1/2$,
$b_{112}^+=b_{212}^+=2/3$, $b_{121}^+=b_{221}^+=2/5$, $b_{122}^+=b_{211}^+=1/3$,
$0<\gamma_{ik}<2 (i=1,2)$, $d_1^-=15$, $d_2^-=17$,
$e_{11}^+=2$, $e_{12}^+=3$, $e_{21}^+=3$, $e_{22}^+=4$, $h_{111}^+=h_{222}^+=1/4$,
$h_{112}^+=h_{212}^+=1/3$, $h_{121}^+=h_{221}^+=1/5$, $h_{122}^+=h_{211}^+=1/6$,
$0<\delta_{ik}<2 (i=1,2)$, so, if we
take $\varepsilon_1=\varepsilon_2=\varepsilon_3=\varepsilon_4=1$, we can obtain
\begin{gather*}
\begin{aligned}
&-c_1^-\varepsilon_1+\sum_{j=1}^2a_{1j}^+L_j\varepsilon_{2+j}
+\sum_{j=1}^2\sum_{l=1}^2b_{1jl}^+\big(H_jN_l\varepsilon_{2+j}
+H_lN_j\varepsilon_{2+l}\big)\\
& =-15+5+\frac{19}{10}\pi<0,
\end{aligned}\\
\begin{aligned}
&-c_2^-\varepsilon_2+\sum_{j=1}^2a_{2j}^+L_j\varepsilon_{2+j}
+\sum_{j=1}^2\sum_{l=1}^2b_{2jl}^+\big(H_jN_l\varepsilon_{2+j}
+H_lN_j\varepsilon_{2+l}\big)\\
&=-17+7+\frac{19}{10}\pi<0,
\end{aligned}\\
-d_3^-\varepsilon_3+\sum_{i=1}^2e_{1i}^+\hat{L}_i\varepsilon_i
+\sum_{i=1}^2\sum_{r=1}^2h_{1ir}^+\big(\hat{H}_i\hat{N}_r\varepsilon_i
+\hat{H}_r\hat{N}_i\varepsilon_r\big)=-15+5+\frac{19}{10}\pi<0,\\
-d_4^-\varepsilon_4+\sum_{i=1}^2e_{2i}^+\hat{L}_i\varepsilon_i
+\sum_{i=1}^2\sum_{r=1}^2h_{2ir}^+\big(\hat{H}_i\hat{N}_r\varepsilon_i
+\hat{H}_r\hat{N}_i\varepsilon_r\big)=-17+7+\frac{19}{10}\pi<0.
\end{gather*}
Conditions (H5) and (H6) are satisfied. From Theorem \ref{t3.1} and \ref{t4.1},
we know that \eqref{e5.1} has at least one $1$-periodic solution and this solution
is exponential stable.


\subsection*{Conclusion}

By using the continuation theorem of coincidence degree theory and constructing
some suitable Lyapunov functions, sufficient conditions are derived to guarantee the 
stability and existence of periodic solutions for a class of BAM HHNNs with impulses
and delays on time scales. In fact, both continuous and discrete systems, are very 
important in implementing and applications.
But it is troublesome to study the existence and stability of periodic solutions for 
continuous and discrete systems respectively. Therefore, it is meaningful to study 
that on time scales which can unify the continuous and discrete situations. 
The system we study here gives an affirmative exemplum for this problem.

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\end{document}