% Trinh Tuan Anh (family name, middle name, given name, respectively)
\documentclass[reqno]{amsart}
\usepackage{hyperref}
\usepackage{amssymb}

\AtBeginDocument{{\noindent\small
\emph{Electronic Journal of Differential Equations},
Vol. 2013 (2013), No. 261, pp. 1--16.\newline
ISSN: 1072-6691. URL: http://ejde.math.txstate.edu or http://ejde.math.unt.edu
\newline ftp ejde.math.txstate.edu}
\thanks{\copyright 2013 Texas State University - San Marcos.}
\vspace{9mm}}

\begin{document}
\title[\hfilneg EJDE-2013/261\hfil Existence and global asymptotic stability]
{Existence and global asymptotic stability of positive periodic solutions
of a Lotka-Volterra type competition systems with delays and feedback controls}

\author[A. T. Trinh \hfil EJDE-2013/261\hfilneg]
{Anh Tuan Trinh}  % in alphabetical order

\address{Anh Tuan Trinh \newline
Department of Mathematics\\
Hanoi National University of Education\\
136 Xuan Thuy Road, Hanoi, Vietnam}
\email{anhtt@hnue.edu.vn}

\thanks{Submitted January 8, 2013. Published November 26, 2013.}
\subjclass[2000]{34K13, 92B05, 92D25, 93B52}
\keywords{Lotka-Volterra  competition system; time delay;
 feedback control; \hfill\break\indent 
positive periodic solution; global asymptotic stability; 
 continuation theorem}


\begin{abstract}
 The existence of positive periodic solutions of a periodic
 Lotka-Volterra type competition system with delays
 and feedback controls is studied by applying the continuation theorem
 of coincidence degree theory.
 By contracting a suitable Liapunov functional, a set of sufficient
 conditions for the global asymptotic stability of the positive periodic
 solution of the system is given.
 A counterexample is given to show that the result on the existence
 of positive periodic solution in \cite{tk2} is incorrect.
\end{abstract}

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


\section{Introduction}\label{sec.int}

In this paper, we consider the following non-autonomous
Lotka-Volterra $n$-species competition system with delays and feedback controls
\begin{equation}\label{1.1}
\begin{gathered}
\begin{aligned}
\dot x_i(t)
&= g_i(x_i(t))\Big[r_i(t)-\sum_{j=1}^na_{ij}(t)x_j(t)-\sum_{j=1}^n
 \sum_{k=1}^mb_{ijk}(t)x_j(t-\tau_{ijk}(t))\\
&\quad-\sum_{j=1}^n\sum_{k=1}^m\int_{-\infty}^tc_{ijk}(t,s)x_j(s) d s
 -d_{i}(t)u_i(t)\\
&\quad -\sum_{k=1}^me_{ik}(t)u_ i(t-\sigma_{ik}(t))
 -\int_{-\infty}^tf_{i}(t,s)u_i(s)ds \Big],
\end{aligned}\\
\begin{aligned}
\dot u_i(t)&=-\alpha_i(t)u_i(t)+\beta _i(t)x_i(t)
 +\sum_{k=1}^mp_{ik}(t)x_i(t-\gamma_{ik}(t))\\
&\quad +\int_{-\infty}^tv_{i}(t,s)x_i(s)ds,
\end{aligned}
\end{gathered}
\end{equation}
where $i\in\{1,2,\dots,n\}$, $u_i$ denote indirect feedback control variables.
For system \eqref{1.1}, we introduce the following hypotheses
\begin{itemize}
\item[(H1)]  $r_i, \alpha _i\in C(\mathbb{R},\mathbb{R})$,
$a_{ij}, b_{ijk}, d_i, e_{ik}, p_{ik}, \beta_i\in C(\mathbb{R},\mathbb{R}_+)$ are
 $\omega$-periodic ($\omega$ is a fixed positive number) with
 $\int_0^\omega r_i(t)dt>0, \int_0^\omega \alpha_i(t)dt>0$,
 $i,j=1,2,\dots,n$; $k=1,2,\dots,m$.

\item[(H2)]
$c_{ijk},f_{i},v_{i}:\mathbb{R}\times \mathbb{R} \to \mathbb{R}_+$ are
$\omega$-periodic functions; i.e.,
$$
c_{ijk}(t+\omega,s+\omega)=c_{ijk}(t,s), f_{i}(t+\omega,s+\omega)
=f_{i}(t,s), v_{i}(t+\omega,s+\omega)=v_{i}(t,s)
$$
and $\int_{-\infty}^tc_{ijk}(t,s)ds$,
$\int_{-\infty}^tf_{i}(t,s)ds$, $\int_{-\infty}^tv_{i}(t,s)ds $
are continuous with respect to $t$. Moreover
\begin{gather*}
\int_0^{+\infty}\int_{-t}^0c_{ijk}(s+t,s)\,ds\,dt<+\infty, \quad
\int_0^{+\infty}\int_{-t}^0f_{i}(s+t,s)\,ds\,dt<+\infty,\\
\int_0^{+\infty}\int_{-t}^0v_{i}(s+t,s)\,ds\,dt
<+\infty, \quad i,j=1,2,\dots,n; k=1,2,\dots,m.
\end{gather*}

\item[(H3)] $g_i\in C(\mathbb{R}_+,\mathbb{R}_+)$ is strictly increasing,
$g_i(0)=0$ and $\lim_{v\to 0^+}\frac{g_i(v)}{v}$ is a positive constant.
Moreover, there are positive constants $l$ and $L$ such that
$l\leq \frac{g_i(v)}{v}\leq L$ for all $v>0$, $i=1,2,\dots,n$.

\item[(H4)] $\tau_{ijk}$, $\sigma_{ik}$,
$\gamma _{ik}\in C(\mathbb{R},\mathbb{R}_+)$ are
$\omega$-periodic for $i,j=1,2,\dots$ $n, k=1,2,\dots,m$.

\item[(H5)] $\tau_{ijk}$, $\sigma_{ik}$,
$\gamma _{ik} \in C^1(\mathbb{R},\mathbb{R}_+)$ and
$\dot \tau_{ijk}(t)<1$, $\dot \sigma_{ik}(t)<1$,
$\dot \gamma _{ik}(t)<1$ for all $t\in \mathbb{R}$, $i,j=1,2,\dots,n$,
$k=1,2,\dots,m$.
\end{itemize}

We consider \eqref{1.1}  with the  initial conditions
\begin{equation}\label{1.2}
\begin{gathered}
x_i(s)=\phi_i(s),  s\in (-\infty, 0],\quad
 \phi_i\in C((-\infty, 0], \mathbb{R}_+),  \phi_i(0)>0 \\
u_i(s)=\psi_i (s), s\in (-\infty, 0], \quad
 \psi_i\in C((-\infty, 0], \mathbb{R}_+),\quad \psi_i(0)>0,
\end{gathered}
\end{equation}
for $ i=1,2,\dots,n$.
Throughout this paper, we use the following symbols:
for an $\omega$-periodic function $f\in C(\mathbb{R},\mathbb{R})$, we 
define
\begin{equation}\label{1.3}
\begin{gathered}
 \bar f = \frac{1}{\omega}\int_0^\omega f(t)dt,\quad
c^*_{ijk}(t)=\int_{-\infty}^tc_{ijk}(t,s)ds,\quad
f^*_{i}(t)=\int_{-\infty}^tf_{i}(t,s)ds,\\
v^*_{i}(t)=\int_{-\infty}^tv_{i}(t,s)ds,\quad
G_i(t,s)=\frac{\exp \{ \int_t^s\alpha_i(v)dv \}}
 {\exp\{\int_0^\omega \alpha_i(v)dv\}-1 }, s\geq t,\\
P_i(t)=d_i(t)\int_t^{t+\omega}G_i(t,s)\Big[\beta_i (s)
 +\sum_{k=1}^mp_{ik}(s)+v^*_{i}(s)\Big]ds,\\
Q_i(t)=\sum_{j=1}^me_{ij}(t)\int_t^{t+\omega}G_i(t,s)\Big[\beta_i (s)
+\sum_{k=1}^mp_{ik}(s)+v^*_{i}(s)\Big]ds,\\
R_i(t)=\int_{-\infty}^t f_{i}(t,s)\int_s^{s+\omega}G_i(s,\tau)
\Big[\beta_i (\tau)+\sum_{k=1}^mp_{ik}(\tau)+v^*_{i}(\tau)\Big]d\tau ds.
\end{gathered}
\end{equation}

Fan et al.~\cite{tk4} studied system \eqref{1.1} in the case
$k=1$, $g_i(v_i)=v_i$, $ f_{i}(t,s)=v_{i}(t,s)=0$ for $ i=1,2,\dots,n$
and obtained several results for the existence and
global asymptotically stability of positive periodic solutions of the system.
Recently, Yan and Liu \cite{tk2} considered system \eqref{1.1} in the case
$e_{ik}(t)=p_{ik}(t)=0$ and $ f_{i}(t,s)=v_{i}(t,s)=0$ for
$k=2,3,\dots,m$, $i=1,2,\dots,n$,
\begin{equation}\label{1.4}
\begin{gathered}
\begin{aligned}
\dot x_i(t)&=g_i(x_i(t))\Big[r_i(t)-\sum_{j=1}^na_{ij}(t)x_j(t)
 -\sum_{j=1}^n\sum_{k=1}^mb_{ijk}(t)x_j(t-\tau_{ijk}(t))\\
&\quad -\sum_{j=1}^n\sum_{k=1}^m\int_{-\infty}^tc_{ijk}(t,s)x_j(s) d s
 -d_{i}(t)u_i(t)-e_{i1}(t)u_ i(t-\sigma_{i1}(t)) \Big],
\end{aligned}\\
\dot u_i(t)=-\alpha_i(t)u_i(t)+\beta _i(t)x_i(t)+p_{i1}(t)x_i(t-\gamma_{i1}(t)),
\quad i=1,2,\dots,n.
\end{gathered}
\end{equation}
By employing  fixed point index theory on cones,  Yan and Liu \cite{tk2}
established the following result.

\begin{theorem}[\cite{tk2}] \label{thm1.1}
Assume that {\rm (H1)--(H4)} hold.  For system \eqref{1.4}, to have at
least one positive $\omega$-periodic solution, a necessary and sufficient
condition is
\begin{equation}\label{1.5}
\min_{1\leq i\leqslant n}\Big\{ \sum_{j=1}^n\Big[\bar a_{ij}
+\sum_{k=1}^m(\bar b_{ijk}+\bar c^*_{ijk})\Big]+\bar P_i+\bar Q_i\Big\}>0
\end{equation}
and
\begin{equation}\label{1.6}
\min_{1\leq i\leq n}\{ \bar \beta _i+\bar p_{i1} \}>0.
\end{equation}
\end{theorem}

Unfortunately, the sufficient condition in the above theorem is incorrect,
as shown by the following example.

\noindent \textbf{Example.}  Consider the system
\begin{equation}\label{1.7}
\begin{gathered}
\dot x_1(t)=x_1(t)\Big[1-x_1(t)-3x_2(t)-2x_2(t-\tau_{1})-u_1(t)
 -u_ 1(t-\sigma_{1}) \Big],\\
\dot x_2(t)=x_1(t)\Big[2-3x_1(t)-x_2(t)-2x_1(t-\tau_{2})-u_2(t)
 -u_ 2(t-\sigma_{2}) \Big],\\
\dot u_1(t)=-u_1(t)+x_1(t)+x_1(t-\gamma_{1}),\ \dot u_2(t)
 =-u_2(t)+x_2(t)+x_2(t-\gamma_{2}),
\end{gathered}
\end{equation}
where $\tau_{i}, \sigma_i, \gamma_i$, $i=1,2,$ are positive constants.
It is easy to see that system \eqref{1.7} satisfies all hypotheses
of Theorem \ref{thm1.1} with $\omega =1$.
On the other hand, if $(x^*_1(t),x^*_2(t),u^*_1(t),u^*_2(t))$
is a positive 1-periodic solution of system \eqref{1.7}, then
\begin{equation}\label{1.8}
\begin{gathered}
\frac{d}{dt}\ln  x^*_1(t)
 =\Big[1-x^*_1(t)-3x^*_2(t)-2x^*_2(t-\tau_{1})-u^*_1(t)-u^*_ 1(t-\sigma_{1})
 \Big],\\
\frac{d}{dt}\ln  x^*_2(t)
 =\Big[2-3x^*_1(t)-x^*_2(t)-2x^*_1(t-\tau_{2})-u^*_1(t)-u^*_ 2(t-\sigma_{2})
 \Big],\\
\frac{d}{dt} u^*_1(t)=-u^*_1(t)+x^*_1(t)+x^*_1(t-\gamma_{1}),\\
 \frac{d}{dt} u^*_2(t)=-u^*_2(t)+x^*_2(t)+x^*_2(t-\gamma_{2}),
\end{gathered}
\end{equation}
Integrating \eqref{1.8} from 0 to 1 and simplifying, we obtain
\begin{equation}\label{1.9}
\bar x^*_1+5 \bar x^*_2+2\bar u^*_1=1,\quad 
5 \bar x^*_1  + \bar x^*_2+2\bar u^*_2=2,\quad
\bar u^*_1=2\bar x^*_1,\quad  
\bar u^*_2=2x^*_2.
\end{equation}
This implies
\[
5 \bar x^*_1+5 \bar x^*_2=1,\quad
5 \bar x^*_1+5 \bar x^*_2=2,
\]
which is impossible.  Thus, system \eqref{1.7} has no positive 1-periodic
solution and then the sufficient condition in Theorem \ref{thm1.1}
(given in \cite{tk2}) is incorrect.

In the proof of Theorem \ref{thm1.1} (see \cite{tk2}), authors considered a map
$\Phi: K \to K,$ where
$$
K=\{(x_1,\dots,x_n) \in E:x_i(t)\geqslant
\delta _i \|x_i\|_0,\; i=1,2,\dots,n,\;
  t\in [0,\omega ]\}
$$
(with $\delta_i=\exp\{-\bar r_i\omega\}$)
is a cone of the Banach space
$E=\{x\in C(\mathbb{R},\mathbb{R}^n): x(t+\omega)=x(t) \text{ for all }
t\in \mathbb{R}\}$
with the norm $\|x\|_0=\sum_{i=1}^n\|x_i\|_0$
(where  $\|x_i\|_0=\max_{t\in [0,\omega]}|x_i(t)|$).
 By employing  fixed point index theory on cone,  it was proved
in \cite{tk2} that there exist positive constants
$r$ and $R$ $(r<R)$ such that $\Phi $ has at least one fixed point
$x^*$ in $K_{r,R}=\{x\in K:r< \|x\|_0\leq R\}$,
and then it was concluded that $(x^*(t),u^*(t))$ is positive
$\omega$-periodic solution of system \eqref{1.1}.
The mistake in the proof of Theorem \ref{thm1.1}
(see \cite{tk2}) is that  $x^*\in K_{r,R}$ does not imply that $x^*(t)$
is positive for $n\geq 2$.

Our purpose of this paper is by using the technique of coincidence degree theory
developed by Gains and Mawhin in \cite{tk3} to study the existence of positive
periodic solutions of system \eqref{1.1}. Moreover, by contracting a
suitable Liapunov functional,
we study the global asymptotic stability of the positive periodic solution
of system \eqref{1.1}.
The remainder of this paper is organized as follows. Section 2 is preliminaries,
in which we introduce the continuation theorem and some lemmas.
Section 3 contains our main results on the existence and the global asymptotic
stability of positive periodic solutions of system \eqref{1.1}.

\section{Preliminaries}

Let $Y$ and $Z$ be two normed vector spaces, $L:\operatorname{Dom}L\subset Y\to Z$
be a linear mapping,
and $N:Y\to Z$ be a continuous mapping. The mapping $L$ will be
called a Fredholm mapping of index zero if
$\dim\ker L=\operatorname{codim\, Im}L<+\infty$ and $\operatorname{Im}L$
is closed in $Z$.
If $L$ is a Fredholm mapping of index zero and there exist continuous projectors
$P:Y\to Y$ and $Q:Z\to Z$ such that $\operatorname{Im}P = \ker  L$,
 $\operatorname{Im}L=\ker  Q=\operatorname{Im}(I-Q)$,
it follows that $L|_{\operatorname{Dom}L\cap \ker P}:(I-P)X\to
\operatorname{Im}L$ is invertible.
We denote the inverse of that map by $K_P$. If $\Omega$ is an open bounded
subset of $Y$, the mapping $N$ will be called $L$-compact on $\bar \Omega$
if $QN(\bar \Omega)$ is bounded and $K_P(I-Q)N:\bar \Omega \to Y$ is compact.
Since $\operatorname{Im}Q$ is isomorphic to $\ker L$,
there exists an isomorphism $J:\operatorname{Im}Q\to \ker L$.

\begin{lemma}[Continuation theorem \cite{tk3}] \label{lem2.1}
Let $L$ be a Fredholm mapping of index zero and $N$ be $L$-compact
on $\bar \Omega$. Suppose that
\begin{itemize}
\item[(i)] for each $\lambda \in (0,1)$, every solution $x$
of $Lx=\lambda Nx$ is such that $x\not \in \partial \Omega$;
\item[(ii)] $QNx\ne  0$ for each $x\in \partial \Omega \cap \ker L$ and
$\deg (JQN,\Omega \cap \ker L,0)\ne 0$.
\end{itemize}
Then the operator equation $Lx=Nx$ has at least one solution
lying in $\operatorname{Dom}L\cap \bar \Omega$.
\end{lemma}

Let us consider the  system
\begin{equation}\label{2.1}
\begin{gathered}
\begin{aligned}
\dot x_i(t)&=g_i(x_i(t))\Big[r_i(t)-\sum_{j=1}^na_{ij}(t)x_j(t)
 -\sum_{j=1}^n\sum_{k=1}^mb_{ijk}(t)x_j(t-\tau_{ijk}(t))\\
&\quad -\sum_{j=1}^n\sum_{k=1}^m\int_{-\infty}^tc_{ijk}(t,s)x_j(s) d s
 -d_{i}(t)(V_ix_i)(t)\\
&\quad -\sum_{k=1}^me_{ik}(t)(V_ix_ i)(t-\sigma_{ik}(t))
 -\int_{-\infty}^tf_{i}(t,s)(V_ix_i)(s)ds \Big],
\end{aligned}\\
u_i(t)=(V_ix_i)(t), \quad i=1,2,\dots,n,
\end{gathered}
\end{equation}
where
\begin{equation}\label{2.2}
\begin{aligned}
(V_ix_i)(t)&=\int_t^{t+\omega}G_i(t,s)\Big[\beta_i(s)x_i(s)
 +\sum_{k=1}^mp_{ik}(s)x_i(s-\gamma_{ik}(s))\\
&\quad +\int_{-\infty}^sv_i(s,\tau)x_i(\tau)d\tau\Big]ds, \quad i=1,2,\dots,n.
\end{aligned}
\end{equation}

\begin{lemma}\label{lem2.2}
Assume  {\rm (H1)--(H4)} hold.
Then $(x_1(t),\dots,x_n(t),u_1(t),\dots,u_n(t))$ is an $\omega$-periodic
solution of system \eqref{1.1}
if and only if it is an $\omega$-periodic solution of system \eqref{2.1}.
\end{lemma}

\begin{proof}
$(\Rightarrow)$ Let $(x(t),u(t))=(x_1(t),\dots,x_n(t),u_1(t),\dots,u_n(t))$
be an $\omega$-periodic solution of system \eqref{1.1}.
It is easy to see from \eqref{1.1} that
\begin{align*}
u_i(\bar t)&=\exp\{-\int_t^{\bar t}\alpha_i(\tau)d\tau\}
\Big[u_i(t)+\int_t^{\bar t}\exp\{\int_t^{\tau}\alpha_i(\tau)d\tau\}
\Big\{\beta _i(\tau)x_i(\tau)\\
&\quad +\sum_{k=1}^mp_{ik}(\tau)x_i(t-\gamma_{ik}(\tau))
+\int_{-\infty}^\tau v_{i}(\tau,v)x_i(v)dv\Big\}d\tau\Big],
\end{align*}
for $\bar t\geqslant t$, $i=1,2,\dots,n$.
Thus, since $u_i(t)=u_i(t+\omega)$ for $i=1,2,\dots,n$, it follows that
\begin{align*}
u_i(t)
&=u_i(t+\omega)\\
&=\exp\{-\int_t^{t+\omega}\alpha_i(\tau)d\tau\}
\Big[u_i(t)+\int_t^{t+\omega} \exp\{\int_t^{\tau}\alpha_i(\tau)d\tau\}\Big\{\beta _i(\tau)x_i(\tau)\\
&+\sum_{k=1}^mp_{ik}(\tau)x_i(t-\gamma_{ik}(\tau))
+\int_{-\infty}^\tau v_{i}(\tau,v)x_i(v)dv\Big\}d\tau\Big],
\end{align*}
for $t\in \mathbb{R}$, $i=1,2,\dots,n$.
Hence,
\begin{align*}
u_i(t)&=\int_t^{t+\omega}G_i(t,\tau)\Big[\beta_i(\tau)x_i(\tau)
 +\sum_{k=1}^mp_{ik}(\tau)x_i(\tau-\gamma_{ik}(\tau))\\
&\quad +\int_{-\infty}^sv_i(\tau,v)x_i(v)dv\Big]d\tau=(V_ix_i)(t), \quad
i=1,2,\dots,n,
\end{align*}
which implies that $(x(t),u(t))$ is an $\omega$-periodic solution of
system \eqref{2.1}.

$(\Leftarrow)$ Let $(x_1(t),\dots,x_n(t),u_1(t),\dots,u_n(t))$
be an $\omega$-periodic solution of system \eqref{2.1}.
It is easy to see from \eqref{2.2} that $(u_1(t),\dots,u_n(t))$
satisfies the  system
$$
\dot u_i(t)=-\alpha_i(t)u_i(t)+\beta _i(t)x_i(t)
+\sum_{k=1}^mp_{ik}(t)x_i(t-\gamma_{ik}(t))
+\int_{-\infty}^tv_{i}(t,s)x_i(s)ds,
$$
for $i=1,2,\dots,n$.
Thus, $(x(t),u(t))$ is an $\omega$-periodic solution of system \eqref{1.1}.
The proof is complete.
\end{proof}

\begin{lemma}[\cite{tk1}] \label{lem2.3}
Suppose that $\nu \in C^1(\mathbb{R},\mathbb{R}_+)$ is $\omega$-periodic
and $\nu '(t)<1$ for all $t\in [0,\omega]$.
Then the function $t-\nu (t)$ has a unique inverse $\eta (t)$
satisfying $\eta \in C(\mathbb{R},\mathbb{R})$
with $\eta (t+\omega)=\eta (t)$ for all $t\in \mathbb{R}$.
\end{lemma}

\begin{lemma}\label{lem2.4}
Suppose that $q_{ij}>0, \delta_i>0$ for $i,j=1,2,\dots,n$. If
\begin{equation}\label{2.3}
\delta_i-\sum_{j\ne i}^nq_{ij}\frac{\delta_j}{q_{jj}}>0, i=1,2,\dots,n,
\end{equation}
then the  system of algebraic equations
\begin{equation}\label{2.4}
\delta_i-\sum_{j=1}^nq_{ij}x_j=0, \quad i=1,2,\dots,n
\end{equation}
has a unique solution $x^*=(x_1^*,\dots,x_n^*)\in \mathbb{R}^n$.
Moreover, $x^*_i>0$ for $i=1,2,\dots,n$.
\end{lemma}

\begin{proof}
Let $y_i=q_{ii}x_i$ for $i=1,2,\dots,n$, so that system \eqref{2.4} becomes
\begin{equation}\label{2.5}
\delta_i-\sum_{j=1}^nq^*_{ij}y_j=0, i=1,2,\dots,n,
\end{equation}
where $q^*_{ij}=q_{ij}/q_{jj}$, $i,j=1,2,\dots,n$. Clearly,
 $q^*_{ii}=1$ for $i=1,2,\dots,n$. By \eqref{2.3},
\begin{equation}\label{2.6}
\delta_i-\sum_{j\ne i}^nq^*_{ij}\delta_j>0, \quad i=1,2,\dots,n.
\end{equation}
Let $\epsilon$ be a positive number such that
$\epsilon <\min_{1\leq i \leq n}
\big[\delta_i-\sum_{j\ne i}^nq^*_{ij}\delta_j\big]$. Denote
\[
{\mathcal D}=\{y=(y_1,\dots,y_n)\in \mathbb{R}^n:
\epsilon \leq y_i \leq \delta_i, i=1,2,\dots,n\},
\]
$F=(F_1,\dots,F_n):\mathbb{R}^n\to \mathbb{R}^n$,
where 
\[
F_i(y)=\delta_i-\sum_{j=1}^nq^*_{ij}y_j, \quad i=1,2,\dots,n;
\]
and $H=F+I$, where $I$ is the identity operator on
$\mathbb{R}^n$. It is easy to see that $\epsilon < H_i(y)<\delta_i$ for
$i=1,2,\dots,n,$ $y \in {\mathcal D}$; i.e.,
$H(y)\in \operatorname{int}({\mathcal D})$-the interior of $\mathcal D$ for all
$y\in {\mathcal D}$.
Thus, $\deg (H-I,\operatorname{int}{\mathcal D},0)=\deg (F,\operatorname{int}{\mathcal D},0)=1$.
This implies that the equation $Fy=0$ has at least one solution $y^*$ in
$\operatorname{int}{\mathcal D}$.
Since $Fy\neq 0$ for all $y \in \partial {\mathcal D}$ and $Fy=0$
is linear equation, it follows that $y^*$ is the unique solution
in $\mathbb{R}^n$ of the equation $Fy=0$.
Thus, equation \eqref{2.4} has  a unique solution
 $x^*=(x_1^*,\dots,x_n^*)\in \mathbb{R}^n$.
Moreover, $x^*_i>0$ for $i=1,2,\dots,n.$ The proof is complete.
\end{proof}

\begin{definition}\label{def2.1} \rm
Let $(\tilde x(t),\tilde u(t))=(\tilde x_1(t),\dots,\tilde x_n(t),
\tilde u_1(t),\dots, \tilde u_n(t))$ be a positive
$\omega$-periodic solution of system \eqref{1.1}.
It is said to be globally asymptotically stable if any positive solution
$(x(t),u(t))=(x_1(t),\dots,x_n(t),u_1(t),\dots,u_n(t))$ of
\eqref{1.1}-\eqref{1.2} satisfies
$$
\lim_{t\to +\infty}\sum_{i=1}^n\Big\{|x_i(t)-\tilde x_i(t)|
+ |u_i(t)-\tilde u_i(t)|\Big\}=0.
$$
\end{definition}


\begin{remark} \label{rmk2.1} \rm
Let us put $y_i=h_i(x_i):=\int_1^{x_i}\frac{ds}{g_i(s)}$, $i=1,2,\dots,n$.
By (H3), it is easy to see that
$h_i:(0,+\infty)\to \mathbb{R}$, $x_i\mapsto y_i=h_i(x_i)$ has a unique inverse
$\varphi_i:\mathbb{R}\to (0,+\infty )$, $y_i\mapsto x_i=\varphi _i(y_i)$.
Moreover, $\varphi_i \in C^1(\mathbb{R},(0,+\infty))$ and
 $\varphi_i$ is strictly monotone increasing.
\end{remark}

\begin{remark} \label{rmk2.2}\rm
  By (H5), Lemma \ref{lem2.3} implies that the functions
$t-\tau_{ijk}(t), t-\sigma_{ik}(t)$
and $t-\gamma_{ik}(t)$ have the unique inverses, respectively.
Let $\mu_{ijk}(t),\ \zeta_{ik}(t)$ and $\xi_{ik}(t)$
represent the inverses of functions $t-\tau_{ijk}(t), t-\sigma_{ik}(t)$ and
$t-\gamma_{ik}(t)$, respectively.
Obviously, $\mu_{ijk}, \zeta_{ik}, \xi_{ik}\in C(\mathbb{R},\mathbb{R})$ and
$\mu_{ijk}(t+\omega)=\mu_{ijk}(t), \zeta_{ik}(t+\omega)
=\zeta_{ik}(t), \xi_{ik}(t+\omega)=\xi_{ik}(t)$ for all $t\in \mathbb{R}$.
\end{remark}


\begin{remark} \label{rmk2.3}\rm
 It is easy from (H1)--(H4) to show that solutions of \eqref{1.1}-\eqref{1.2}
are well defined for all $t\geq 0$ and satisfy $x_i(t)>0$ and
$u_i(t)>0$ for all $t\geq 0$ and $i=1,2,\dots,n$.
\end{remark}

\section{Main Results}


\begin{theorem}\label{thm3.1}
 Assume that {\rm (H1)--(H4)} hold. Let
\begin{gather}\label{3.1}
\int_0^\omega\Big[\beta_i(s)+\sum_{k=1}^mp_{ik}(s)+v^*_i(s) \Big]ds>0 , 
\quad i=1,2,\dots,n,\\
\label{3.2}
A_i:=\bar a_{ii}+\sum_{k=1}^m\bar b_{iik}+\sum_{k=1}^m\bar c^*_{iik}
+\bar P_i +\bar Q_i +\bar R_i >0, \quad i=1,2,\dots,n, \\
\label{3.3}
\bar r_i>\sum_{j\ne i}^n\Big(\bar a_{ij}+\sum_{k=1}^m\bar b_{ijk}
+\sum_{k=1}^m\bar c^*_{ijk}
+\bar P_j+\bar Q_j+\bar R_j \Big)\varphi_j(B_j), \quad i=1,2,\dots,n,
\end{gather}
where
$B_i=h_i(\bar r_i/A_i)+(\bar r_i +\overline{|r_i|})\omega$, 
$i=1,2,\dots,n$.
Then system \eqref{1.1} has at least one positive $\omega$-periodic solution.
\end{theorem}

\begin{proof}
Consider the  system
\begin{equation}\label{3.4}
\begin{aligned}
\dot y_i(t)
&=r_i(t)-\sum_{j=1}^na_{ij}(t)\varphi_j(y_j(t))
 -\sum_{j=1}^n\sum_{k=1}^mb_{ijk}(t)\varphi_j(y_j(t-\tau_{ijk}(t)))\\
&-\sum_{j=1}^n\sum_{k=1}^m\int_{-\infty}^tc_{ijk}(t,s)\varphi_j(y_j(s) )d s 
-d_{i}(t)(V_i\varphi_i(y_i))(t)\\
&-\sum_{k=1}^me_{ik}(t)(V_ i\varphi_i(y_i))(t-\sigma_{ik}(t))
-\int_{-\infty}^tf_{i}(t,s)(V_i\varphi_i(y_i))(s)ds.
\end{aligned}
\end{equation}
By \eqref{1.3}, \eqref{2.2} and \eqref{3.1},  if system \eqref{3.4} 
has an $\omega$-periodic solution $(y_1^*(t),\dots,y^*_n(t))$, then
\begin{align*}
u^*_i(t)&=(V_i\varphi_i(y^*_i))(t) \\
&\geqslant \int_t^{t+\omega}
\Big(\frac{1}{\exp\{\bar \alpha_i\omega\}-1}\min_{\tau \in [0,\omega ]}
\varphi_i(y^*_i(\tau))\Big)\Big[\beta_i(s)\\
&\quad +\sum_{k=1}^mp_{ik}(s)+v^*_i(s) \Big]ds>0,\quad
 t\in \mathbb{R},\; i=1,2,\dots,n.
\end{align*}
Thus, $(x^*_1(t),\dots,x^*_n(t),u^*_1(t),\dots,u_n^*(t))$ with values
$x_i^*(t)=\varphi_i(y^*_i(t))$ and $u^*_i(t)=(V_i\varphi_i(y^*_i))(t)$ 
for $i=1,2,\dots,n$ is a positive
$\omega$-periodic solution of system \eqref{2.1}. 
So, by Lemma \ref{lem2.2}, we only need to show that
system \eqref{3.4} has at least one $\omega$-periodic solution in 
order to complete the proof.
To apply the continuation theorem of coincidence degree theory to the existence
of an $\omega$-periodic solution of system \eqref{3.4}, we take
$$
Y=Z=\Big\{y(t)=(y_1(t),\dots,y_n(t)) \in C(\mathbb{R},\mathbb{R}^n): 
y(t+\omega)=y(t)\text{ for all } t \in \mathbb{R}\Big\}.
$$
Denote $\|y\|_0=\sum_{i=1}^n\|y_i\|_0$, where 
$\|y_i\|_0=\max_{t\in [0,\omega]}|y_i(t)|$.
Then $Y$ and $Z$ are Banach spaces when they endowed with the norm $\|\cdot\|_0$.

We define $L:\operatorname{Dom}L\subset Y\to Z$ and $N:Y \to Z$ 
by setting $Ly=\dot y$ and $Ny=Fy=( F_1y,\dots,F_ny)$, where
\begin{equation}
\begin{aligned}
F_iy(t)=&r_i(t)-\sum_{j=1}^na_{ij}(t)\varphi_j(y_j(t))
 -\sum_{j=1}^n\sum_{k=1}^mb_{ijk}(t)\varphi_j(y_j(t-\tau_{ijk}(t)))\\
&-\sum_{j=1}^n\sum_{k=1}^m\int_{-\infty}^tc_{ijk}(t,s)\varphi_j(y_j(s) )d s 
 -d_{i}(t)(V_i\varphi_i(y_i))(t)\\
&-\sum_{k=1}^me_{ik}(t)(V_ i\varphi_i(y_i))(t-\sigma_{ik}(t))
 -\int_{-\infty}^tf_{i}(t,s)(V_i\varphi_i(y_i))(s)ds.
\end{aligned}
\end{equation}
Further, we define continuous projectors $P:Y\to Y$ and $Q:Z\to Z$ as follows
$$
Py = \frac{1}{\omega}\int_0^\omega y(s)ds,\quad
Qz=\frac{1}{\omega}\int_0^\omega z(s)ds.
$$
We easily see that $ \operatorname{Im}L=\{z\in Z:\int_0^\omega z(s)ds=0\}$ and
$\ker L=\mathbb{R}^n$. So, $\operatorname{Im}L$ is closed in $Z$ and
$\dim\ker L =n = \operatorname{codim\,Im}L$. Hence, $L$ is a Fredholm 
mapping of index zero.
Clearly that $\operatorname{Im}P = \ker  L$, 
$\operatorname{Im}L=\ker  Q=\operatorname{Im}(I-Q)$/
Furthermore, the generalized inverse (to $L$) 
$K_P:\operatorname{Im}L\to \ker P\cap \operatorname{Dom}L$
has the form
$$
K_Pz(t)=\int_0^tz(s)ds-\frac{1}{\omega}\int_0^\omega\int_0^tz(s)\,ds\,dt.
$$
We know that
$$
QNy(t)=\frac{1}{\omega}\int_0^\omega Fy(t)dt.
$$
Thus,
\begin{align*}
K_P(I-Q)Ny(t)
&=(K_PN-K_PQN)y(t)\\
&=\int_0^tFy(s)ds-\frac{1}{\omega}\int_0^\omega\int_0^tFy(s)\,ds\,dt
+\Big(\frac{1}{2}-\frac{t}{\omega} \Big)\int_0^\omega Fy(s)ds.
\end{align*}
It is easy to see that $QN$ and $K_p(I-Q)N$ are continuous. 
Furthermore, it can be verified that
$\overline{K_P(I-Q)N(\bar{\Omega})}$ is compact for any open bounded 
set $\Omega \subset Y$
by using Arzela-Ascoli theorem and $QN(\bar{\Omega})$ is bounded. 
Therefore, $N$ is $L$-compact on
$\bar \Omega$ for any open bounded subset $\Omega \subset Y$.
Now we are in a position to search for an appropriate open bounded subset
$\Omega$ for the application of the continuation theorem 
(Lemma \ref{lem2.1}) to system \eqref{3.4}.

Corresponding to the operator equation $Ly=\lambda Ny$ ($\lambda \in (0,1)$), 
 we have
\begin{equation}\label{3.6}
\begin{aligned}
\dot y_i(t)
&=\lambda \Big[r_i(t)-\sum_{j=1}^na_{ij}(t)\varphi_j(y_j(t))
 -\sum_{j=1}^n\sum_{k=1}^mb_{ijk}(t)\varphi_j(y_j(t-\tau_{ijk}(t)))\\
&\quad -\sum_{j=1}^n\sum_{k=1}^m\int_{-\infty}^tc_{ijk}(t,s)\varphi_j(y_j(s) )d s -d_{i}(t)(V_i\varphi_i(y_i))(t)\\
&\quad -\sum_{k=1}^me_{ik}(t)(V_ i\varphi_i(y_i))(t-\sigma_{ik}(t))
-\int_{-\infty}^tf_{i}(t,s)(V_i\varphi_i(y_i))(s)ds\Big].
\end{aligned}
\end{equation}
Integrating \eqref{3.6} from 0 to $\omega$ and simplifying, we obtain
\begin{equation}\label{3.7}
\begin{aligned}
\bar r_i \omega
&= \sum_{j=1}^n\int_0^\omega a_{ij}(t)\varphi_j(y_j(t))dt
 +\sum_{j=1}^n\sum_{k=1}^m\int_0^\omega b_{ijk}(t)
 \varphi_j(y_j(t-\tau_{ijk}(t)))dt\\
&\quad +\sum_{j=1}^n\sum_{k=1}^m\int_0^\omega 
 \int_{-\infty}^tc_{ijk}(t,s)\varphi_j(y_j(s) )\,ds\,dt
+\int_0^\omega d_{i}(t)(V_i\varphi_i(y_i))(t)dt\\
&\quad +\sum_{k=1}^m\int_0^\omega e_{ik}(t)(V_ i\varphi_i(y_i))
 (t-\sigma_{ik}(t))dt\\
&\quad +\int_0^\omega \int_{-\infty}^tf_{i}(t,s)(V_i\varphi_i(y_i))(s)\,ds\,dt.
\end{aligned}
\end{equation}
Let
\begin{equation}\label{3.8}
y_i(\eta _i)=\max_{t\in [0,\omega ]}y_i(t),
y_i(\theta _i)=\min_{t\in [0,\omega ]}y_i(t),\quad 
\eta _i,\theta _i\in [0,\omega ],\; i=1,2,\dots,n.
\end{equation}
It is easy to see from \eqref{1.3}, \eqref{2.2} and \eqref{3.8}  that
\begin{equation}\label{3.9}
\begin{gathered}
\int_0^\omega d_{i}(t)(V_i\varphi_i(y_i))(t)dt 
 \geq \bar P_i \omega \varphi_i(y_i(\theta_i)),\\
\sum_{k=1}^m\int_0^\omega e_{ik}(t)(V_ i\varphi_i(y_i))(t-\sigma_{ik}(t))dt
 \geq \bar Q_i \omega \varphi_i(y_i(\theta_i)),\\
\int_0^\omega \int_{-\infty}^tf_{i}(t,s)(V_i\varphi_i(y_i))(s)\,ds\,dt 
 \geq \bar R_i\omega  \varphi_i(y_i(\theta_i))
\end{gathered}
\end{equation}
and
\begin{equation}\label{3.10}
\begin{gathered}
\int_0^\omega d_{i}(t)(V_i\varphi_i(y_i))(t)dt 
 \leq \bar P_i \omega \varphi_i(y_i(\eta_i)),\\
\sum_{k=1}^m\int_0^\omega e_{ik}(t)(V_ i\varphi_i(y_i))(t-\sigma_{ik}(t))dt
 \leq \bar Q_i \omega \varphi_i(y_i(\eta_i)),\\
\int_0^\omega \int_{-\infty}^tf_{i}(t,s)(V_i\varphi_i(y_i))(s)\,ds\,dt 
 \leq \bar R_i\omega  \varphi_i(y_i(\eta_i)).
\end{gathered}
\end{equation}
It follows from \eqref{1.3}, \eqref{3.7}, \eqref{3.8} and \eqref{3.9} that
\begin{equation}\label{3.11}
\bar r_i \geq \Big(\bar a_{ii}+\sum_{k=1}^m\bar b_{iik}
 +\sum_{k=1}^m\bar c^*_{iik}
+\bar P_i +\bar Q_i +\bar R_i\Big)\varphi_i(y_i(\theta_i))
=A_i \varphi_i(y_i(\theta_i)), 
\end{equation}
for $i=1,2,\dots,n$.
Thus, by \eqref{3.2} and Remark \ref{rmk2.1}, we have
\begin{equation}\label{3.12}
y_i(\theta _i)\leq h_i(\frac{\bar r_i}{A_i}), \quad i=1,2,\dots,n.
\end{equation}
From \eqref{3.6} and \eqref{3.7}, we know that
\begin{equation}\label{3.13}
\int_0^\omega |\dot y_i(t)|dt \leq (\bar r_i +\overline{|r_i|})\omega, 
\quad  i=1,2,\dots,n,
\end{equation}
and thus, by \eqref{3.12},
\begin{equation}\label{3.14}
y_i(t)\leq y_i(\theta_i)+ \int_0^\omega |\dot y_i(t)|dt 
\leq h_i(\frac{\bar r_i}{A_i})
+ (\bar r_i +\overline{|r_i|})\omega =B_i,
\quad t\in [0,\omega ],
\end{equation}
for $i=1,2,\dots,n$.
It is easy to see from \eqref{3.7}, \eqref{3.8},  \eqref{3.10} and 
\eqref{3.14} that
\begin{equation}\label{3.15}
\begin{aligned}
&\Big(\bar a_{ii}+\sum_{k=1}^m\bar b_{iik}+\sum_{k=1}^m\bar c^*_{iik}
+\bar P_i +\bar Q_i +\bar R_i  \Big) \varphi_i(y_i(\eta_i))\\
&\geqslant \bar r_i  -\sum_{j\ne i}^n \Big(\bar a_{ij}
 +\sum_{k=1}^m\bar b_{ijk}+\sum_{k=1}^m\bar c^*_{ijk}
+\bar P_j+\bar Q_j+\bar R_j \Big)\varphi_j(y_j(\eta_j)) ,
\end{aligned}
\end{equation}
for $ i=1,2,\dots,n$.
Thus, by \eqref{3.3} and \eqref{3.14}, we have
\[
\varphi_i(y_i(\eta_i))\geq \frac{\bar r_i
 -\sum_{j\ne i}^n \Big(\bar a_{ij}+\sum_{k=1}^m[\bar b_{ijk}+\bar c^*_{ijk}]
+\bar P_j+\bar Q_j+\bar R_j \Big)\varphi_j(B_j)}{ \bar a_{ii}
+\sum_{k=1}^m[\bar b_{iik}+\bar c^*_{iik}]+\bar P_i +\bar Q_i +\bar R_i }=:C_i,
\]
for $ i=1,2,\dots,n$;
or 
\begin{equation}\label{3.16}
 y_i(\eta_i)\geq h_i(C_i), \quad i=1,2,\dots,n,
\end{equation}
From \eqref{3.13} and \eqref{3.16}, it follows that
\begin{equation}\label{3.17}
y(t) \geq y_i(\eta_i)-\int_0^\omega |\dot y_i(t)|dt
\geq h_i(C_i)-(\bar r_i +\overline{|r_i|})\omega =:D_i, t\in [0,\omega ],
\end{equation}
for $i=1,2,\dots,n$.
From \eqref{3.12} and \eqref{3.16} we  see  that
\begin{equation}\label{3.18}
\|y\|_0\leq M:= \sum_{i=1}^n (|B_i|+|D_i|).
\end{equation}
By \eqref{3.3}, Lemma \ref{lem2.3} implies that the following system of 
algebraic equations
\begin{equation}\label{3.19}
\bar r_i=\sum_{j=1}^n \Big[ \bar a_{ij}+\sum_{k=1}^m\bar b_{ijk}
+\sum_{k=1}^m\bar c^*_{ijk}\Big]\varphi_j(y_j)
+(\bar P_i+\bar Q_i+\bar R_i)\varphi_i(y_i), 
\end{equation}
for $i=1,2,\dots,n$,
has a unique solution $y^*=(y^*_1,\dots,y^*_n)\in \mathbb{R}^n$.
Let  $S=\|y^*\|_0+M$. Evidently, $S$ is independent of the choice of
$\lambda$. Let $\Omega :=\{ y \in Y: \|y\|_1<S\}$. It is clear that $\Omega$
satisfies the requirement (i) in Lemma \ref{lem2.1}. Moreover,
$QNy\neq 0$ for any $y\in \partial \Omega \cap \ker L$.
Let us take $J=I$, where $I$ is the identity operator on $\mathbb{R}^n$. 
By straightforward computation, we have
$$
\deg (JQN,\Omega \cap \ker L,0)=\operatorname{sgn}
\Big\{ (-1)^n\Big[\det (w_{ij}) \Big]\varphi_1(y^*_1)
\dots\varphi_n(y^*_n)\Big\}\ne 0,
$$
where $w_{ii}= A_i$, 
\[
w_{ij}= \bar a_{ij}+\sum_{k=1}^m\bar b_{ijk}
+\sum_{k=1}^m\bar c^*_{ijk}\ (j\ne i), \quad i,j=1,2,\dots,n.
\]
By Lemma \ref{lem2.1}, we  conclude that the equation $Ly=Ny$ has at 
least one solution in $Y$.
Therefore, system \eqref{1.1} has at least one positive $\omega$-periodic 
solution. The proof is complete.
\end{proof}

\begin{theorem}\label{thm3.2} 
Assume that {\rm (H1)--(H5)}, \eqref{3.1} and \eqref{3.3}  hold.
If there exist positive constants $\nu_1,\nu _2,\dots,\nu_n$ such that
\begin{equation}\label{3.20}
\begin{gathered}
\begin{aligned}
&\min_{t\in [0,\omega]}\Big\{  \nu_i\Big[ a_{ii}(t)-\beta_i(t)-\sum_{k=1}^m
\frac{p_{ik}(\xi_{ik}(t))}{1-\dot \gamma _{ik}(\xi_{ik}(t))}-\int_0^{+\infty}v_i(t+\tau,t)d\tau \Big]  \\
&- \sum_{j\ne i}^n\nu_ja_{ji}(t)-\sum_{j=1}^n\nu_j\sum_{k=1}^m
\frac{b_{jik}(\mu_{jik}(t))}{1-\dot \tau_{jik}(\mu_{jik}(t))}\\
&- \sum_{j=1}^n\nu_j \sum_{k=1}^m\int_0^{+\infty}c_{jik}(t+\tau,t)d\tau\Big\}>0,
\end{aligned}\\
\min_{t\in [0,\omega]}\Big\{\alpha_i(t)-d_i(t)-\sum_{k=1}^m
\frac{e_{ik}(\zeta_{ik}(t))}{1-\dot \sigma_{ik}(\zeta_{ik}(t))}
- \int_0^{+\infty}f_i(t+\tau,t)d\tau \Big\}>0, 
\end{gathered}
\end{equation}
for $i=1,2,\dots,n$,
then system \eqref{1.1} has a unique positive $\omega$-periodic 
solution which is globally asymptotically stable.
\end{theorem}

\begin{proof} 
From \eqref{3.20}, we conclude that $\bar a_{ii}>0$ for $i=1,2,\dots,n$,
which means that \eqref{3.2} holds. By Theorem \ref{thm3.1},
system \eqref{1.1} has a positive $\omega$-periodic solution 
$(\tilde x(t),\tilde u(t))$. Let $( x(t),u(t))$
be any other positive solution of  \eqref{1.1} with initial 
condition \eqref{1.2}.
Consider the Liapunov functional $V(t)=V(\tilde x(t), \tilde u(t)),(x(t),u(t)))$ 
defined by
\begin{equation}\label{3.21}
\begin{aligned}
V(t)&=\sum_{i=1}^n\nu_i\Big\{V_i^{(1)}(t)+V_i^{(2)}(t)
 + \sum_{j=1}^n\sum_{k=1}^m\Big[V_{ijk}^{(3)}(t)+V_{ijk}^{(4)}(t)\Big]\\
&\quad +\sum_{k=1}^m\Big[V_{ik}^{(5)}(t)+V_{ik}^{(6)}(t)\Big]
 +V_i^{(7)}(t)+V_i^{(8)}(t) \Big\},
\end{aligned}
\end{equation}
where, for  $i,j=1,2,\dots,n$, $k=1,2,\dots,m$, we have
\begin{gather*}
V_i^{(1)}(t)=\Big|\int_{\tilde x_i(t)}^{x_i(t)}\frac{ds}{g_i(s)}\Big|,\quad
  V_i^{(2)}(t)= |u_i(t)-\tilde u_i(t)|,\\
V_{ijk}^{(3)}(t)=\int_{t-\tau_{ijk}(t)}^t \frac{b_{ijk}
 (\mu_{ijk}(s))}{1-\dot \tau_{ijk}(\mu_{ijk}(s))}|x_j(s)-\tilde x_j(s)|\,ds,\\
V_{ijk}^{(4)}(t)=\int_0^{+\infty}\int_{t-\tau}^t c_{ijk}
 (s+\tau,s)|x_j(s)-\tilde x_j(s)|\,ds\,d\tau,\\
V_{ik}^{(5)}(t)=\int_{t-\sigma_{ik}(s)}^t\frac{e_{ik}
 (\zeta_{ik}(s))}{1-\dot \sigma_{ik}(\zeta_{ik}(s))}|u_i(s)-\tilde u_i(s)|\,ds,\\
V_{ik}^{(6)}(t)=\int_{t-\gamma_{ik}(t)}^t\frac{p_{ik}
 (\xi_{ik}(s))}{1-\dot \gamma _{ik}(\xi_{ik}(s))}|x_i(s)-\tilde x_i(s)|\,ds,\\
V_i^{(7)}(t)=\int_0^{+\infty}\int_{t-\tau}^t f_i
 (s+\tau ,s) |u_i(s)-\tilde u_i(s)|\,ds\,d\tau, \\
V_i^{(8)}(t)=\int_0^{+\infty}\int_{t-\tau}^t v_i(s+\tau,s)|u_i(s)
 -\tilde u_i(s)|\,ds\,d\tau
\end{gather*}

Clearly  $V(t)$ is continuous on $[0,+\infty)$.
Calculating the upper right derivative  of $V_i^{(1)}(t),\dots,V_i^{(8)}(t)$
along the solutions of system \eqref{1.1} for $t>0$, we obtain
\begin{align*}
&D^+V_i^{(1)}\\
&=\Big[\frac{\dot x_i(t)}{g_i(x_i(t))}
-\frac{\dot {\tilde x}_i(t)}{g_i(\tilde x_i(t))}\Big] 
\operatorname{sgn}[x_i(t)-\tilde x_i(t)]\\
&=\operatorname{sgn}[x_i(t)-\tilde x_i(t)]
 \Big[-\sum_{j=1}^na_{ij}(x_j(t)-\tilde x_j(t))\\
&\quad -\sum_{j=1}^n\sum_{k=1}^mb_{ijk}(t)(x_j(t-\tau_{ijk}(t))
 -\tilde x_j(t-\tau_{ijk}(t)))\\
&\quad -\sum_{j=1}^n\sum_{k=1}^m\int_{-\infty}^tc_{ijk}(t,\tau)(x_j(\tau)
 -\tilde x_j(t))d\tau -d_i(t)(u_i(t)-\tilde u_i(t))\\
&\quad  -\sum_{k=1}^me_{ik}(t)(u_i(t-\sigma_{ik}(t))
 -\tilde u_i(t-\sigma_{ik}(t)))
-\int_{-\infty}^t f_i(t,\tau)(u_i(\tau)-\tilde u_i(\tau))d\tau\Big];
\end{align*}
and thus,
\begin{equation}\label{3.22}
\begin{aligned}
D^+ V_i^{(1)}&\leqslant  -a_{ii}(t)|x_i(t)-\tilde x_i(t)|
 +\sum_{j\ne i}^na_{ij}(t)|x_j(t)-\tilde x_j(t)|\\
&\quad +\sum_{j=1}^n\sum_{k=1}^mb_{ijk}(t)|x_j(t-\tau_{ijk}(t))
 -\tilde x_j(t-\tau_{ijk}(t))|\\
&\quad + \sum_{j=1}^n\sum_{k=1}^m\int_{-\infty}^tc_{ijk}(t,\tau)|x_j(\tau)
 -\tilde x_j(\tau)|d\tau -d_i(t)|u_i(t)-\tilde u_i(t)|\\
&\quad +\sum_{k=1}^me_{ik}(t)|u_i(t-\sigma_{ik}(t))
 -\tilde u_i(t-\sigma_{ik}(t))|\\
&\quad +\int_{-\infty}^t f_i(t,\tau)|u_i(\tau) -\tilde u_i(\tau)|d\tau,
\end{aligned}
\end{equation}
\begin{equation}\label{3.23}
\begin{aligned}
D^+ V_i^{(2)}
&=[\dot u_i(t)-\dot{\tilde u}_i(t)]\operatorname{sgn}[u_i(t)
 -\tilde u_i(t)]\\
&=\operatorname{sgn}[u_i(t)-\tilde u_i(t)]\Big[-\alpha_i(t)(u_i(t)
 -\tilde u_i(t))+\beta_i(t)(x_i(t)-\tilde x_i(t))\\
&\quad +\sum_{k=1}^mp_{ik}(t)(x_i(t-\gamma_{ik}(t))
 -\tilde x_i(t-\gamma_{ik}(t)))\\
&\quad +\int_{-\infty}^t v_i(t,\tau)(x_i(\tau)- \tilde x_i(\tau))d\tau\Big]\\
&\leq -\alpha_i(t)|u_i(t)-\tilde u_i(t)|+\beta_i(t)|x_i(t)-\tilde x_i(t)|\\
&\quad +\sum_{k=1}^mp_{ik}(t)|x_i(t-\gamma_{ik}(t))
 -\tilde x_i(t-\gamma_{ik}(t))|\\
&\quad +\int_{-\infty}^t v_i(t,\tau)|x_i(\tau)- \tilde x_i(\tau)|d\tau;
\end{aligned}
\end{equation}
\begin{gather}\label{3.24}
\begin{aligned}
\dot V_{ijk}^{(3)}
&=\frac{b_{ijk}(\mu_{ijk}(t))}{1-\dot \tau_{ijk}(\mu_{ijk}(t)))}
 |x_j(t)-\tilde x_j(t)|\\
&\quad -b_{ijk}(t)|x_j(t-\tau_{ijk}(t))-\tilde x_j(t-\tau_{ijk}(t))|,
\end{aligned} \\
\label{3.25}
\begin{aligned}
\dot V_{ijk}^{(4)}
&= \int_0^{+\infty}c_{ijk}(t+\tau,t)|x_j(t)-\tilde x_j(t)|d\tau \\
&\quad -\int_0^{+\infty}c_{ijk}(t,t-\tau)|x_j(t-\tau)-\tilde x_j(t-\tau)|d\tau;
\end{aligned}\\
\label{e3.26}
\dot V_{ik}^{(5)}= \frac{e_{ik}(\zeta_{ik}(t))}{1-\dot \sigma_{ik}
 (\zeta_{ik}(t)))}|u_i(t)-\tilde u_i(t)|
-e_{ik}(t)|u_i(t-\sigma_{ik}(t))-\tilde u_i(t-\sigma_{ik}(t))|; \\
\label{e3.27}
\dot V_{ik}^{(6)}= \frac{p_{ik}(\xi_{ik}(t))}{1-\dot \gamma_{ik}(\xi_{ik}(t)))}
 |x_i(t)-\tilde x_i(t)|
-p_{ik}(t)|x_i(t-\gamma_{ik}(t))-\tilde x_i(t-\gamma_{ik}(t))|; \\
\label{e3.28}
\begin{aligned}
\dot V_{i}^{(7)}
&= \int_0^{+\infty}f_i(t+\tau,t)|u_i(t)-\tilde u_i(t)|d\tau\\
&\quad -\int_0^{+\infty}f_i(t,t-\tau)|u_i(t-\tau)-\tilde u_i(t-\tau)|d\tau; 
\end{aligned}\\
\label{e3.29}
\begin{aligned}
\dot V_{i}^{(8)}&= \int_0^{+\infty}v_i(t+\tau,t)|x_i(t)-\tilde x_i(t)|d\tau\\
&\quad -\int_0^{+\infty}v_i(t,t-\tau)|x_i(t-\tau)-\tilde x_i(t-\tau)|d\tau.
\end{aligned}
\end{gather}

From \eqref{3.21}-\eqref{e3.29} it follows that
\begin{align}
D^+V  &\leq \sum_{i=1}^n\nu_i\Big\{ \Big[-a_{ii}(t)+\beta_i(t)
+\sum_{k=1}^m\frac{p_{ik}(\xi_{ik}(t))}{1-\dot \gamma_{ik}(\xi_{ik}(t)))}\notag\\
&\quad +\int_0^{+\infty}v_i(t+\tau,t)d\tau\Big]|x_i(t)-\tilde x_i(t)| \notag\\
&\quad +\sum_{j\ne i}^na_{ij}(t)|x_j(t)-\tilde x_j(t)|+\sum_{j=1}^n
\sum_{k=1}^m\frac{b_{ijk}(\mu_{ijk}(t))}{1-\dot \tau_{ijk}
 (\mu_{ijk}(t)))}|x_j(t)-\tilde x_j(t)|  \notag\\
&\quad +\sum_{j=1}^n\sum_{k=1}^m\Big[\int_0^{+\infty}c_{ijk}
 (t+\tau,t)d\tau \Big]|x_j(t)-\tilde x_j(t)|\Big\}
+\sum_{i=1}^n\nu_i\Big\{-\alpha_i(t)+d_i(t) \notag\\
&\quad +\sum_{k=1}^m \frac{e_{ik}(\zeta_{ik}(t))}{1-\dot \sigma_{ik}
 (\zeta_{ik}(t)))}
+\int_0^{+\infty}f_i(t+\tau,t)d\tau\Big\}|u_i(t)-\tilde u_i(t)| \notag\\
&=\sum_{i=1}^n\Big\{  \nu_i\Big[- a_{ii}(t)+\beta_i(t)
 +\sum_{k=1}^m\frac{p_{ik}(\xi_{ik}(t))}{1-\dot \gamma _{ik}(\xi_{ik}(t))}
+\int_0^{+\infty}v_i(t+\tau,t)d\tau \Big]  \notag\\
&\quad + \sum_{j\ne i}^n\nu_ja_{ji}(t) 
 +\sum_{j=1}^n\nu_j\sum_{k=1}^m\frac{b_{jik}
 (\mu_{jik}(t))}{1-\dot \tau_{jik}(\mu_{jik}(t))} \notag\\
&\quad +\sum_{j=1}^n\nu_j\sum_{k=1}^m\int_0^{+\infty}c_{jik}
 (t+\tau,t)d\tau\Big\}|x_i(t)-\tilde x_i(t)| \notag \\
&\quad +\sum_{i=1}^n\Big\{\nu_i\Big[ -\alpha_i(t)+d_i(t) 
 +\sum_{k=1}^m \frac{e_{ik}(\zeta_{ik}(t))}{1-\dot \sigma_{ik}(\zeta_{ik}(t))} \notag\\
&\quad + \int_0^{+\infty}f_i(t+\tau,t)d\tau \Big]\Big\}|u_i(t)-\tilde u_i(t)|.
\label{3.30}
\end{align}
By \eqref{3.20}, there exists positive number $\delta$ such that
\begin{equation}\label{3.31}
\begin{gathered}
\begin{aligned}
&\min_{t\in [0,\omega]}\Big\{  \nu_i\Big[ a_{ii}(t)-\beta_i(t)
-\sum_{k=1}^m\frac{p_{ik}(\xi_{ik}(t))}{1-\dot \gamma _{ik}(\xi_{ik}(t))}
 -\int_0^{+\infty}v_i(t+\tau,t)d\tau \Big]  \\
&- \sum_{j\ne i}^n\nu_ja_{ji}(t)-\sum_{j=1}^n\nu_j\sum_{k=1}^m\frac{b_{jik}
 (\mu_{jik}(t))}{1-\dot \tau_{jik}(\mu_{jik}(t))}\\
&-\sum_{j=1}^n\nu_j\sum_{k=1}^m\int_0^{+\infty}c_{jik}(t+\tau,t)d\tau\Big\}>\delta,
\end{aligned}\\
\min_{t\in [0,\omega]}\Big\{\nu_i\Big[ \alpha_i(t)-d_i(t)
 -\sum_{k=1}^m \frac{e_{ik}(\zeta_{ik}(t))}{1-\dot \sigma_{ik}(\zeta_{ik}(t))}
- \int_0^{+\infty}f_i(t+\tau,t)d\tau \Big]\Big\}>\delta, 
\end{gathered}
\end{equation}
for $i=1,2,\dots,n$.
In the view of \eqref{3.30} and \eqref{3.31} we have
\begin{equation}\label{3.32}
D^+V (t)\leq - \delta \sum_{i=1}^n\Big\{|x_i(t)-\tilde x_i(t)|
+ |u_i(t)-\tilde u_i(t)|\Big\},\quad t\geq 0.
\end{equation}
Integrating both sides of \eqref{3.32} from 0 to $t$, we obtain
$$
V(t)-V(0)\leq -\delta \int_0^t\sum_{i=1}^n\Big\{|x_i(s)
-\tilde x_i(s)|+ |u_i(s)-\tilde u_i(s)|\Big\} ds, t\geq 0.
$$
Thus,
$$
 \int_0^t\sum_{i=1}^n\Big\{|x_i(s)-\tilde x_i(s)|
+ |u_i(s)-\tilde u_i(s)|\Big\} ds\leq \frac{V(0)}{\delta}, \quad t\geq 0,
$$
which implies 
\begin{equation}\label{3.33}
\int_0^{+\infty} \sum_{i=1}^n\Big\{|x_i(s)-\tilde x_i(s)|
+ |u_i(s)-\tilde u_i(s)|\Big\}ds\leq \frac{V(0)}{\delta}.
\end{equation}
Since
$$
\nu_i\Big\{\Big|\int_{\tilde x_i(t)}^{x_i(t)}\frac{ds}{g_i(s)}\Big|
+|u_i(t)-\tilde u_i(t)|\Big\}\leq V(t)\leq V(0), \quad t \geq 0,
$$
it follows from (H3) that $x_i(t)$ and $u_i(t)$ are  bounded on $[0,+\infty)$,
and hence from \eqref{1.1} we can conclude that $\dot x_i(t)$ and
 $\dot u_i(t)$ are also bounded
on $[0,+\infty)$. This implies that $x_i(t)$ and $u_i(t)$ are uniformly 
continuous on $[0,+\infty)$.
Therefore, $\sum_{i=1}^n|x_i(t)-\tilde x_i(t)|+ |u_i(t)-\tilde u_i(t)|$ 
is uniformly continuous on $[0,+\infty)$. Thus, \eqref{3.33} implies that
$$
\lim_{t\to +\infty}\sum_{i=1}^n\Big\{|x_i(t)-\tilde x_i(t)|
+ |u_i(t)-\tilde u_i(t)|\Big\}=0
$$
and $(\tilde x(t),\tilde u(t))$ is the unique positive $\omega$-periodic 
solution of system \eqref{1.1}. The proof is complete.
\end{proof}

As an example we consider  system \eqref{1.1} with
\begin{gather*}
n=2, \quad m=1,\quad r_1(t)=r_2(t)\equiv \frac{\ln 5}{2},\quad
\alpha _1(t)=\alpha_2(t)\equiv 6,\\
a_{11}(t)=a_{22}(t)=21 + \sin 2\pi t,\quad
 a_{12}(t)=a_{21}(t)=1+\sin 2\pi t,\\
b_{111}(t)=b_{121}(t)=b_{211}(t)=b_{221}(t)=1+2\sin \pi t,\\
 c_{111}(t,s)=c_{121}(t,s)=c_{211}(t,s)=c_{221}(t,s)=v_1(t,s)=v_2(t,s)
  =\exp\{-(t-s)\},\\
e_{11}(t)= e_{21}(t)=1+\cos 2\pi t, \quad d_1(t)=d_2(t)=1-\sin 2\pi t,\\
\beta_1(t)=\beta_2(t)= 1+\cos 2\pi t,\quad p_{11}(t)=p_{21}(t)=1-\cos 2\pi t,\\
\tau_{111}(t)=\tau_{121}(t)=\tau_{211}(t)=\tau_{221}(t)\equiv \tau^*>0,\\
\sigma_{11}(t)=\sigma_{21}(t)\equiv \sigma^* >0,\quad
 \gamma_{11}(t)=\gamma_{21}(t)\equiv \gamma^* >0,\ g_1(v)=g_2(v)=v.
\end{gather*}
It is easy to see that (H1)--(H5) hold. By straightforward computation, we have
\begin{gather*}
c^*_{ij1}(t)=f^*_i(t)=v^*_i(t)=1,\quad \bar P_i=\bar Q_i=\bar R_i=\frac{3}{5},\\
A_i= 24.8,\quad  B_i=\ln \frac{\ln 5}{49.6}+\ln 5, \quad
 \varphi_i(B_i)=\frac{5\ln 5}{49.6}, \quad i,j=1,2.
\end{gather*}
Let $\nu_1=\nu_2=1$. We can easily see that
\begin{gather*}
\int_0^1\Big[\beta_i(s)+p_{i1}(s)+v^*_i(s) \Big]ds= 3 >0 ,\\
\sum_{j\ne i}^2\Big(\bar a_{ij}+\bar b_{ij1}+\bar c^*_{ij1}+\bar P_j
+\bar Q_j+\bar R_j \Big)\varphi_j(B_j)=\frac{30}{31}\times \frac{\ln 5}{2} 
< \frac{\ln 5}{2}=\bar r_i,\\
\begin{aligned}
&\min_{t\in [0,2 \pi ]}\Big\{  \nu_i\Big[ a_{ii}(t)-\beta_i(t)
-\frac{p_{i1}(\xi_{i1}(t))}{1-\dot \gamma _{i1}(\xi_{i1}(t))}
-\int_0^{+\infty}v_i(t+\tau,t)d\tau \Big] - \sum_{j\ne i}^2\nu_ja_{ji}(t)\\
&-\sum_{j=1}^2\nu_j\frac{b_{ji1}(\mu_{ji1}(t))}{1-\dot \tau_{ji1}(\mu_{ji1}(t))}
-\sum_{j=1}^2\nu_j\int_0^{+\infty}c_{ji1}(t+\tau,t)d\tau\Big\}\\
&=\min_{t\in[0,2\pi ]}[14-\cos 2\pi (t+\gamma^*)-2\sin 2\pi (t+\tau^*)]>0,
\end{aligned} \\
\begin{aligned}
&\min_{t\in [0,2\pi ]}\Big\{\alpha_i(t)-d_i(t)
- \frac{e_{i1}(\zeta_{i1}(t))}{1-\dot \sigma_{i1}(\zeta_{i1}(t))}
- \int_0^{+\infty}f_i(t+\tau,t)d\tau \Big\}\\
&=\min_{t\in [0,2\pi ]}[3+\sin 2\pi t - \cos 2\pi (t+\sigma^* )]>0 , \quad i=1,2
\end{aligned}
\end{gather*}
Thuerefore, conditions \eqref{3.1}, \eqref{3.3} and \eqref{3.30} hold. 
Hence, by Theorem \ref{thm3.2},
the system has a unique positive 1-periodic solution which 
is globally asymptotically stable.

\subsection*{Acknowledgements}
The authors would like to thank the editors and the anonymous reviewers 
for their constructive comments.
This work was supported by Hanoi National University of Education 
and the Minister of Education and Training of Vietnam.

\begin{thebibliography}{00}

\bibitem{tk4} Fan, M.; Wang, K.; Wong, P.J.Y.;  Agarwal, R.P.; 
\emph{Periodicity and stability in periodic $n$-species Lotka-Volterra 
competition systems with feedback controls and deviating arguments}. 
Acta Mathematica Sinica, Vol. 19, 2003, pp. 801 - 822.

\bibitem{tk3} Gains, R.E.; Mawhin, J.K.; 
\emph{Coincidence Degree and Nonlinear Differential Equations}. 
Springer-Verlag, Berlin. 1977.

\bibitem{tk1} Liu, Z.J.; Tan, R.H.;  Chen, L.S.; 
\emph{Global stability in a periodic delayed predator-prey system}.
Applied Mathematics and Computation, Vol. 186, 2007, pp. 389 - 403.

\bibitem{tk2} Yan, J.; Liu, G.; 
\emph{Periodicity and stability for a Lotka-Volterra type competition
 with feedback controls and deviating arguments}.
Nonlinear Analysis, Vol. 74, 2011, pp. 2916 - 2928.
\end{thebibliography}

\end{document}