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

\AtBeginDocument{{\noindent\small
\emph{Electronic Journal of Differential Equations},
Vol. 2009(2009), No. 100, pp. 1--11.\newline
ISSN: 1072-6691. URL: http://ejde.math.txstate.edu or http://ejde.math.unt.edu
\newline ftp ejde.math.txstate.edu}
\thanks{\copyright 2009 Texas State University - San Marcos.}
\vspace{8mm}}

\begin{document}
\title[\hfilneg EJDE-2009/100\hfil Positive almost periodic solutions]
{Positive almost periodic solutions of  non-autonomous
delay competitive systems with weak Allee effect}

\author[Y. Li, K. Zhao\hfil EJDE-2009/100\hfilneg]
{Yongkun Li, Kaihong Zhao}  % in alphabetical order

\address{Yongkun Li \newline
Department of Mathematics, Yunnan University\\
Kunming, Yunnan 650091, China}
\email{yklie@ynu.edu.cn}

\address{Kaihong Zhao \newline
Department of Mathematics, Yuxi Normal University\\
Yuxi, Yunnan 653100, China} 
\email{zhaokaihongs@126.com}

\thanks{Submitted April 6, 2009. Published August 19, 2009.}
\thanks{Supported by grant 04Y239A from the Natural
Sciences Foundation of Yunnan Province.} 
\subjclass[2000]{34K14, 92D25} 
\keywords{Positive almost periodic solution; coincidence  degree; 
delay; \hfill\break\indent non-autonomous competitive systems}

\begin{abstract}
 By using Mawhin's continuation theorem of coincidence degree theory,
 we obtain sufficient conditions for the existence of positive
 almost periodic  solutions for a non-autonomous delay
 competitive system with weak Allee effect.
\end{abstract}

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

\section{Introduction}

 The Lotka-Volterra type systems have been studied in
various fields of epidemiology, chemistry, economics and
biological science. In the past few years, there has been
increasing interest in studying dynamical characteristics such as
stability, persistence and periodicity of Lotka-Volterra type
systems. There have been considerable works on the qualitative
analysis of Lotka-Volterra type systems with delays;
see \cite{l1,l2, l3,l4, l6,y1,x1}.
Naturally, the study of almost periodic solutions for
Lotka-Volterra type systems is important and of great interest.

There are two main approaches to obtain sufficient
conditions for the existence and stability of the almost periodic
solutions of biological models: One is using the fixed point
theorem, Lyapunov functional method and differential inequality
techniques \cite{c1,l5,y2};
the other is using functional hull theory
and Lyapunov functional method \cite{m1,m2,m3}. However, to the best
of our knowledge, there are very few published papers considering the
almost periodic solutions for non-autonomous Lotka-Volterra type
systems by applying the method of coincidence degree theory.
Motivated by this, in this paper, we apply the coincidence theory
to study the existence of positive almost periodic solutions for
the following non-autonomous delay competitive systems with weak
Allee effect
\begin{equation} \label{e1.1}
\begin{aligned}
\dot{u}_{i}(t)
&=u_{i}(t)\Big[F_{i}(t,u_{i}(t-\tau_{ii}(t)))
 -\sum_{j=1}^{n}b_{ij}(t)u_{j}(t)\\
&\quad -\sum_{j=1,j\neq i}^{n}c_{ij}(t)u_{j}(t-\tau_{ij}(t))
 -\sum_{j=1}^{n}\int_{-\sigma_{ij}}^{0}\mu_{ij}(t,s)u_{j}(t+s)\,
\mathrm{d}s\Big],
\end{aligned}
\end{equation}
where $i=1,2,\dots ,n$, $u_{i}(t)$ stands for the $i$th species
population density at
time $t\in\mathbb{R}, b_{ij}(t)\geq0, c_{ij}(t)\geq0, \tau_{ij}(t)$
are continuous almost periodic functions on $R$, $\mu_{ij}(t,s)$ are
positive almost periodic functions on $\mathbb{R}\times[-\sigma_{ij},0]$,
continuous with respect to $t\in\mathbb{R}$ and integrable with respect to
$s\in [-\sigma_{ij},0]$, where $\sigma_{ij}$ are nonnegative
constants, moreover
$\int_{-\sigma_{ij}}^{0}\mu_{ij}(t,s)\,\mathrm{d}s=1,
i,j=1,2,\dots ,n$. The per capita growth rate $F_{i}\in C(\mathbb{R}^{2},\mathbb{R})$
is defined by the form for each $i=1,2,\dots ,n$,
\begin{equation} \label{e1.2}
 F_{i}(t,x)=r_{i}(t)-f_{i}(t,x)x.
\end{equation}
In this definition, $r_{i}$ is the natural reproduction rate and
$f_{i}$ represents the inner-specific competition, $c_{ij}$  in
\eqref{e1.1} represents the interspecific competition.
In addition, $f_{i}$
satisfies the following condition for each $i=1,2,\dots ,n$,
\begin{equation} \label{e1.3}
\frac{\partial f_{i}(t,x)}{\partial x}>0  \quad
\text{and $f_{i}(t,x)$ are  almost  periodic  in $t$,}
\end{equation}
for  each $t\in\mathbb{R}$, there  exists a constant $\alpha_{i}>0$
such  that
\begin{equation} \label{e1.4}
f_{i}(t,\alpha_{i})=0.
\end{equation}
 The situations formulated by $\frac{\partial f_{i}}{\partial x}>0$
and $\frac{\partial f_{i}}{\partial x}<0$ are called the weak
Allee effect and the strong Allee effect respectively.
Details about the Allee effect can be found in \cite{s1,t1,w1}.
The initial condition for \eqref{e1.1} is
\begin{equation} \label{e1.5}
u_{i}(s)=\phi_{i}(s),\quad i=1,2,\dots ,n,
\end{equation}
where $\phi_{i}(s)$ are positive bounded continuous function on
$[-\tau,0]$, $i=1,2,\dots ,n$ and
$\tau=\max_{1\leq i,j\leq n} \{\max_{t\in\mathbb{R}}|\tau_{ij}(t)|,
\sigma_{ij}\}$.

The organization of the rest of this paper is as follows. In
Section 2, we introduce some preliminary results which are needed
in later sections. In Section 3, we establish our main results for
the existence of almost periodic solutions of \eqref{e1.1}. Finally, we
make the conclusion in Section 4.

\section{Preliminaries}

To obtain the existence of an almost periodic solution of
 \eqref{e1.1}, we firstly make the following preparations.

\begin{definition}[\cite{f1}] \label{def2.1} \rm
 Let $u(t):\mathbb{R}\to \mathbb{R}$ be continuous in $t$.
$u(t)$ is said to be almost periodic on $\mathbb{R}$, if, for
any $\epsilon>0$, the set
$K(u,\epsilon)=\{\delta:|u(t+\delta)-u(t)|<\epsilon,
\text{ for any } t\in\mathbb{R}\}$ is relatively dense,
that is for any $\epsilon>0$,
it is possible to find a real number $l(\epsilon)>0$, for any
interval with length $l(\epsilon)$, there exists a number
$\delta=\delta(\epsilon)$ in this interval such that
$|u(t+\delta)-u(t)|<\epsilon$, for any $t\in\mathbb{R}$.
\end{definition}

\begin{definition} \label{def2.2} \rm
A solution $u(t)=(u_{1}(t),u_{2}(t),\dots ,u_{n}(t))^{T}$ of \eqref{e1.1}
is called an almost periodic solution if and only if for each
 $i=1,2,\dots ,n,u_{i}(t)$ is almost periodic.
\end{definition}

   For convenience, we denote $AP(\mathbb{R})$ is the set of all
real valued, almost periodic functions on $\mathbb{R}$ and let
\begin{gather*}
\wedge(f_{j})=\Big\{\widetilde{\lambda}\in\mathbb{R}:
\lim_{T\to\infty}\frac{1}{T}\int_{0}^{T}f_{j}(s)
e^{-i\widetilde{\lambda}s}\,\mathrm{d}s\neq0\Big\},\quad j=1,2,\dots ,n,
\\
\mathop{\rm mod}(f_{j})=\Big\{\sum^{N}_{i=1}n_{i}\widetilde{\lambda_{i}}:n_{i}\in
Z,N\in N^+,\widetilde{\lambda_{i}}\in
\wedge(f_{j})\Big\},\quad j=1,2,\dots ,n
\end{gather*}
be the set of Fourier exponents and the module of $f_{j}$,
respectively, where $f_{j}(\cdot)$ is almost periodic. Suppose
$f_{j}(t,\phi_{j})$ is almost periodic in $t$, uniformly with
respect to $\phi_{j}\in C([-\tau,0],\mathbb{R})$.
$K_{j}(f_{j},\epsilon,\phi_{j}(s))
=\{\delta:|f_{j}(t+\delta,\phi_{j}(s))-f_{j}(t,\phi_{j}(s))|
<\epsilon,\forall t\in\mathbb{R}$\}
 denote the set of $\epsilon$-almost periods
uniformly with respect to $\Phi_{j}(s)\in C([-\tau,0],\mathbb{R})$.
$l_{j}(\epsilon)$ denote the length of inclusion interval.
$m(f_{j})=\frac{1}{T}\int^{T}_{0}f_{j}(s)\,\mathrm{d}s$ be the mean
value of $f_{j}$ on interval $[0,T]$, where $T>0$ is a constant.
Clearly, $m(f_{j})$ depends on $T$.
$m[f_{j}]=\lim_{T\to\infty}\frac{1}{T}\int^{T}_{0}f_{j}(s)\,\mathrm{d}s$.

\begin{lemma}[\cite{f1}] \label{lem2.1}
Suppose that $f$ and $g$ are almost periodic. Then the following
statements are equivalent
\begin{itemize}
\item[(i)]    $\mathop{\rm mod}(f)\supset \mathop{\rm mod}(g)$,
\item[(ii)]  for any sequence $\{t_{n}^{*}\}$, if
 $\lim_{n\to\infty}f(t+t_{n}^{*})=f(t)$ for each $t\in\mathbb{R}$,
then there exists a subsequence $\{t_{n}\}\subseteq\{t_{n}^{*}\}$
such that $\lim_{n\to\infty}g(t+t_{n})=f(t)$ for each $t\in\mathbb{R}$.
\end{itemize}
\end{lemma}

\begin{lemma}[\cite{e1}] \label{lem2.2}
 Let $u\in AP(\mathbb{R})$. Then
$\int_{t-\tau}^{t}u(s)\,\mathrm{d}s$ is almost periodic.
\end{lemma}

Let $X$ and $Z$ be Banach spaces. A linear mapping $L: \mathop{\rm
dom}(L)\subset X\to Z$ is called Fredholm mapping if its kernel,
denoted by $\ker(L)=\{X\in \mathop{\rm dom}(L):Lx=0\}$, has finite
dimension and its range,  denoted by $\mathop{\rm Im}(L)=\{Lx:x\in
\mathop{\rm dom}(L)\}$, is closed and has finite  codimension. The
index of  $L$  is defined by the integer $\dim K(L)-\mathop{\rm
codim} \mathop{\rm dom}(L)$. If $L$ is a Fredholm mapping with index
0, then there exists continuous projections $P:X\to X$ and $Q:Z\to
Z$ such that $\mathop{\rm Im}(P)=\ker(L)$ and $\ker(Q)=\mathop{\rm
Im}(L)$. Then $L|_{\mathop{\rm dom}(L)\cap \ker(P)}:\mathop{\rm
Im}(L) \cap \ker(P)\to \mathop{\rm Im}(L)$ is bijective, and its
inverse mapping is denoted by $K_{P}:\mathop{\rm Im}(L)\to
\mathop{\rm dom}(L)\cap \ker(P)$. Since $\ker(L)$ is isomorphic to
$\mathop{\rm Im}(Q)$, there exists a bijection $J:\ker(L)\to
\mathop{\rm Im}(Q)$. Let $\Omega$ be a bounded open subset of $X$
and let $N:X\to Z$ be a continuous mapping. If
$QN(\overline{\Omega})$ is bounded and
$K_{P}(I-Q)N:\overline{\Omega}\to X$ is compact, then $N$ is called
L-$compact$ on $\Omega$, where $I$ is the identity.

Let $L$ be a Fredholm linear mapping with index 0 and let $N$ be a
$L$-compact mapping on $\overline{\Omega}$. Define mapping
$F:\mathop{\rm dom}(L)\cap \overline{\Omega}\to Z$ by $F=L-N$. If $Lx\neq
Nx$ for all $x\in \mathop{\rm dom}(L)\cap\partial\Omega$, then by using
$P,Q,K_{P},J$ defined above, the coincidence degree of $F$ in
$\Omega$ with respect to $L$ is defined by
\[
\deg_{L}(F,\Omega)=\deg(I-P-(J^{-1}Q+K_{P}(I-Q))N,\Omega,0),
\]
where $\deg(g,\Gamma,p)$ is the Leray-Schauder degree of $g$ at $p$
relative to $\Gamma$.

Then The Mawhin's continuous theorem is given as follows:

\begin{lemma}[\cite{g1}] \label{lem2.3}
Let $\Omega\subset X$ be an open bounded set and let
$N:X\to Z$ be a continuous operator which is $L$-compact on
$\overline{\Omega}$. Assume
\begin{itemize}
\item[(a)] for each $\lambda\in (0,1),x\in\partial\Omega\cap
\mathop{\rm dom}(L),Lx\neq\lambda Nx$;
\item[(b)] for each $x\in\partial\Omega\cap L,QNx\neq 0$;
\item[(c)] $\deg(JNQ,\Omega\cap \ker(L),0)\neq 0$.
\end{itemize}
Then $Lx=Nx$ has at least one solution in
$\overline{\Omega}\cap \mathop{\rm dom}(L)$.
\end{lemma}

\section{Main Result}

In this section, we state and prove the main results of
this paper.
By making the substitution
$u_{i}(t)=\exp\{y_{i}(t)\}$, $i=1,2,\dots ,n$,
\eqref{e1.1} can be reformulated as
\begin{equation} \label{e3.1}
\begin{aligned}
\dot{y}_{i}(t)
&=r_{i}(t)-f_{i}\big(t,\exp\{y_{i}(t-\tau_{ii}(t))\}\big)
\exp\big\{y_{i}(t-\tau_{ii}(t))\big\} \\
&\quad -\sum_{j=1}^{n}b_{ij}(t)\exp\big\{y_{j}(t)\big\}
 -\sum_{j=1,i\neq j}^{n}c_{ij}(t)\exp\big\{y_{j}(t-\tau_{ij}(t))\big\}\\
&\quad -\sum_{j=1}^{n}\int_{-\sigma_{ij}}^{0}\exp\big\{y_{j}(t+s)\big\}ds,
\quad i=1,2,\dots ,n.
\end{aligned}
\end{equation}
The initial condition \eqref{e1.5} can be rewritten as
\begin{equation} \label{e3.2}
y_{i}(s)=\ln\phi_{i}(s)=:\psi_{i}(s),\quad i=1,2,\dots ,n
\end{equation}
Set
$X=Z=V_{1}\oplus V_{2}$,
where
\begin{gather*}
\begin{aligned}
V_{1}&=\Big\{y(t)=(y_{1}(t),y_{2}(t),\dots ,y_{n}(t))^{T}\in
C(\mathbb{R},\mathbb{R}^{n}):y_{i} (t)\in AP(\mathbb{R}),\\
&\mathop{\rm mod}(y_{i}(t))\subset
\mathop{\rm mod}(H_{i}(t)),
 \forall\widetilde{\lambda_{i}}\in\wedge(y_{i}(t))
\text{ satisfies } |\widetilde{\lambda_{i}}|>\beta,\;
i=1,2,\dots ,n\Big\},
\end{aligned}
\\
V_{2}=\{y(t)\equiv(h_{1},h_{2},\dots ,h_{n})^{T}\in\mathbb{R}^{n}\},
\\
\begin{aligned}
H_{i}(t)
&=r_{i}(t)-f_{i}\big(t,\exp\{\psi_{i}(-\tau_{ii}(t))\}\big)
\exp\big\{\psi_{i}(-\tau_{ii}(t))\big\} \\
&\quad -\sum_{j=1}^{n}b_{ij}(t)\exp\big\{\psi_{j}(0)\big\}
 -\sum_{j=1,i\neq j}^{n}c_{ij}(t)\exp\big\{\psi_{j}(-\tau_{ij}(0))\big\}\\
&\quad -\sum_{j=1}^{n}\int_{-\sigma_{ij}}^{0}\mu_{ij}(t,s)
\exp\big\{\psi_{j}(s)\big\}\,\mathrm{d}s
\end{aligned}
\end{gather*}
and $\psi_i(\cdot)$ is defined as \eqref{e3.2}, $i=1,2,\dots ,n$.
$\beta$ is a given constant.
For $y=(y_{1},y_{2},\dots ,y_{n})^{T}\in Z$, define
$\|y\|=\max_{1\leq i\leq n}\sup_{t\in\mathbb{R}}|y_{i}(t)|$.

\begin{lemma} \label{lem3.1}
$Z$ is a Banach space equipped with the norm $\|\cdot\|$.
\end{lemma}

\begin{proof}
If $y^{\{k\}}\subset V_{1}$ and
$y^{\{k\}}=(y_{1}^{\{k\}},y_{2}^{\{k\}},\dots ,y_{n}^{\{k\}})^T$
converges to \\
$\overline{y}=(\overline{y}_{1},\overline{y}_{2},\dots ,
\overline{y}_{n})^T$,
that is $y^{\{k\}}_{j}\to \overline{y}_{j}$, as
$k\to \infty$, $j=1,2,\dots ,n$. Then it is easy to show that
$\overline{y}_{j}\in AP(\mathbb{R})$ and
$\mathop{\rm mod}(\overline{y}_{j})\in
\mathop{\rm mod}(H_{j})$.
For any $\widetilde{\lambda}_{j}\leq\beta$, we have
\[
\lim_{T\to\infty}\frac{1}{T}\int^{T}_{0}y^{\{k\}}_{j}(t)
e^{-i\widetilde{\lambda}_{j}t}dt=0,\quad j=1,2,\dots ,n;
\]
therefore,
\[
\lim_{T\to\infty}\frac{1}{T}\int^{T}_{0}\overline{y}_{j}(t)
e^{-i\widetilde{\lambda}_{j}t}dt=0,\quad j=1,2,\dots ,n,
\]
which implies $\overline{y}\in V_{1}$. Then it is not difficult to
see that $V_{1}$ is a Banach space equipped with the norm
$\|\cdot\|$. Thus, we can easily verify that $x$ and $Z$ are Banach
spaces equipped with the norm $\|\cdot\|$. The proof is
complete.
\end{proof}

\begin{lemma} \label{lem3.2}
Let $L:X\to Z$, $Ly=\dot{y}$, then $L$ is a Fredholm mapping
of index $0$.
\end{lemma}

\begin{proof}
Clearly, $L$ is a linear operator and $\ker(L)=V_{2}$. We claim that
$\mathop{\rm Im}(L)=V_{1}$. Firstly, we suppose that
$z(t)=(z_{1}(t),z_{2}(t),\dots ,z_{n}(t))^{T}
\in \mathop{\rm Im}(L)\subset Z$.
Then there exist
$z^{\{1\}}(t)=(z^{\{1\}}_{1}(t),z^{\{1\}}_{2}(t),\dots ,z^{\{1\}}_{n}(t))^{T}\in
V_{1}$ and constant vector
$z^{\{2\}}=(z^{\{2\}}_{1},z^{\{2\}}_{2},\dots ,z^{\{2\}}_{n})^{T}\in
V_{2}$ such that
\[
z(t)=z^{\{1\}}(t)+z^{\{2\}};
\]
that is,
\[
z_{i}(t)=z_{i}^{\{1\}}(t)+z_{i}^{\{2\}},\quad i=1,2,\dots ,n.
\]
 From the definition of $z_{i}(t)$ and $z_{i}^{\{1\}}(t)$, we can
easily see that $\int^{t}_{t-\tau}z_{i}(s)\,\mathrm{d}s$ and
$\int^{t}_{t-\tau}z_{i}^{\{1\}}(s)\,\mathrm{d}s$ are almost periodic
function. So we have $z_{i}^{\{2\}}\equiv 0$, $i=1,2,\dots ,n$, then
$z^{\{2\}}\equiv(0,0,\dots ,0)^T$, which implies $z(t)\in V_{1}$,
that is $\mathop{\rm Im}(L)\subset V_{1}$.

On the other hand, if $u(t)=(u_{1}(t),u_{2}(t),\dots,u_{n}(t))^T\in
 V_{1}\backslash\{0\}$, then we have
$\int^{t}_{0}u_{j}(s)\,\mathrm{d}s\in  AP(\mathbb{R}),j=1,2,\dots ,n$.
If $\widetilde{\lambda}_j\neq0$, then we  obtain
\[
\lim_{T\to\infty}\frac{1}{T}\int^{T}_{0}\Big(\int^{t}_{0}u_{j}(s)ds\Big)
e^{-i\widetilde{\lambda}_{j}t}\,\mathrm{d}t
=\frac{1}{i\widetilde{\lambda}_{j}}
\lim_{T\to\infty}\frac{1}{T}\int^{T}_{0}u_{j}(t)
e^{-i\widetilde{\lambda}_{j}t}\,\mathrm{d}t,
\]
$j=1,2,\dots ,n$. It follows that
\[
\wedge\Big[\int_{0}^{t}u_{j}(s)\,\mathrm{d}s-m\Big(\int_{0}^{t}u_{j}(s)\,\mathrm{d}s\Big)\Big]
=\wedge(u_{j}(t)),\quad j=1,2,\dots ,n,
\]
hence
\[
\int_{0}^{t}u(s)\,\mathrm{d}s-m\Big(\int^{t}_{0}u(s)\,\mathrm{d}s\Big)\in
V_{1}\subset X
\]
Note that
$\int_{0}^{t}u(s)\,\mathrm{d}s-m(\int_{0}^{t}u(s)\,\mathrm{d}s)$ is
the primitive of $u(t)$ in $X$, we have $u(t)\in \mathop{\rm Im}(L)$,
that is $V_{1}\subset \mathop{\rm Im}(L)$.
Therefore, $\mathop{\rm Im}(L)=V_{1}$.

Furthermore, one can easily show that $\mathop{\rm Im}(L)$ is
closed in $Z$ and
\[
\dim \ker(L)=n=\mathop{\rm codim} \mathop{\rm Im}(L);
\]
therefore, $L$ is a Fredholm mapping of index 0. The proof
is complete.
\end{proof}

\begin{lemma} \label{lem3.3}
Let
$N:X\to Z$, $Ny=(G_{1}^{y},G_{2}^{y},\dots ,G_{n}^{y})^T$,
where
\begin{align*}
G_{i}^{y}
&=r_{i}(t)-f_{i}\big(t,\exp\{y_{i}(t-\tau_{ii}(t))\}\big)
 \exp\big\{y_{i}(t-\tau_{ii}(t))\big\} \\
&\quad -\sum_{j=1}^{n}b_{ij}(t)\exp\big\{y_{j}(t)\big\}-\sum_{j=1,i\neq
j}^{n}c_{ij}(t)\exp\big\{y_{j}(t-\tau_{ij}(t))\big\}
 \\
&\quad -\sum_{j=1}^{n}\int_{-\sigma_{ij}}^{0}\exp\big\{y_{j}(t+s)\big\}\,
\mathrm{d}s,\,i=1,2,\dots ,n.
\end{align*}
Set
\begin{gather*}
P:X\to Z,\quad Py=\big(m(y_{1}),m(y_{2}),\dots ,m(y_{n})\big)^T,\\
Q:Z\to Z,\quad Qz=\big(m[z_{1}],m[z_{2}],\dots ,m[z_{n}]\big)^T.
\end{gather*}
Then $N$ is $L$-compact on $\overline{\Omega}$, where $\Omega$ is an
open bounded subset of $X$.
\end{lemma}

\begin{proof}
Obviously, $P$ and $Q$ are continuous projectors such that
\[
\mathop{\rm Im} {P}=\ker(L),\, \mathop{\rm Im}(L)=\ker(Q).
\]
It is clear that
$(I-Q)V_{2}=\{0\}$, $(I-Q)V_{1}=V_{1}$.
Hence
\[
\mathop{\rm Im}(I-Q)=V_{1}=\mathop{\rm Im}(L).
\]
Then in view of
\[
\mathop{\rm Im}(P)=\ker(L), \,\mathop{\rm Im}(L)=\ker(Q)
=\mathop{\rm Im}(I-Q),
\]
we obtain that the inverse $K_{P}:\mathop{\rm Im}(L)\to \ker(P)\cap
\mathop{\rm dom}(L)$ of $L_{P}$ exists and is given by
\[
K_{P}(z)=\int_{0}^{t}z(s)\,\mathrm{d}s
-m\Big[\int_{0}^{t}z(s)\,\mathrm{d}s\Big].
\]
Thus,
\begin{gather*}
QNy=\big(m[G_{1}^{y}],m[G_{2}^{y}],\dots ,m[G_{n}^{y}]\big)^T,\\
K_{P}(I-Q)Ny=\big(f(y_{1})-Q(f(y_{1})),f(y_{2})-Q(f(y_{2})),
\dots ,f(y_{n})-Q(f(y_{n}))\big)^T,
\end{gather*}
where
\[
f(y_{i})=\int_{0}^{t}\big(G_{i}^{y}-m[G_{i}^{y}]\big)\,\mathrm{d}s,\quad
i=1,2,\dots ,n.
\]

Clearly, $QN$ and $(I-Q)N$ are continuous. Now we will show that
$K_{P}$ is also continuous. By assumptions, for any $0<\epsilon<1$
and any compact set $\phi_{i}\subset C\big([-\tau,0],\mathbb{R}\big)$, let
$l_{i}(\epsilon_{i})$ be the length of the inclusion interval of
$K_{i}(H_{i},\epsilon_{i},\phi_{i})$, $i=1,2,\dots ,n$. Suppose that
$\{z^{k}(t)\}\subset \mathop{\rm Im}(L)=V_{1}$ and
$z^{k}(t)=(z_{1}^{k}(t),z_{2}^{k}(t),\dots ,z_{n}^{k}(t))^T$
uniformly converges to
$\overline{z}(t)=(\overline{z}_{1}(t),\overline{z}_{2}(t),\dots ,
\overline{z}_{n}(t))^T$,
that is $z_{i}^{k}\to\overline{z}_{i}$, as
$k\to\infty$, $i=1,2,\dots ,n$. Because of
$\int_{0}^{t}z^{k}(s)\,\mathrm{d}s\in Z$, $k=1,2,\dots ,n$, there
exists $\sigma_{i}(0<\sigma_{i}<\epsilon_{i})$ such that
$K_{i}(H_{i},\sigma_{i},\phi_{i})\subset
K_{i}(\int_{0}^{t}z_{i}^{k}(s)ds,\sigma_{i},\phi_{i})$,
$i=1,2,\dots ,n$.
Let $l_{i}(\sigma_{i})$ be the length of the inclusion interval of
$K_{i}(H_{i},\sigma_{i},\phi_{i})$ and
\[
l_{i}=\max\big\{l_{i}(\epsilon_{i}),l_{i}(\sigma_{i})\big\},\quad
i=1,2,\dots ,n.
\]
It is easy to see that $l_{i}$ is the length of the inclusion
interval of $K_{i}(H_{i},\sigma_{i},\phi_{i})$ and
$K_{i}(H_{i},\epsilon_{i},\phi_{i})$, $i=1,2,\dots ,n$. Hence, for any
$t\notin[0,l_{i}]$, there exists
$\xi_{t}\in K_{i}(H_{i},\sigma_{i},\phi_{i})\subset
K_{i}(\int_{0}^{t}z_{i}^{k}(s)\,\mathrm{d}s,\sigma_{i},\phi_{i})$
such that $t+\xi_{t}\in [0,l_{i}]$, $i=1,2,\dots ,n$. Hence, by the
definition of almost periodic function we have
\begin{equation} \label{e3.3}
\begin{aligned}
&\big\|\int_{0}^{t}z^{k}(s)\,\mathrm{d}s\big\|\\
&=\max_{1\leq i\leq n}\sup_{t\in\mathbb{R}}
 \Big|\int_{0}^{t}z_{i}^{k}(s)\,\mathrm{d}s\Big| \\
&\leq\max_{1\leq i\leq n}\sup_{t\in [0,l_{i}]}
 \Big|\int_{0}^{t}z_{i}^{k}(s)\,\mathrm{d}s\Big|
 +\max_{1\leq i\leq n}\sup_{t\notin
 [0,l_{i}]}\Big|\int_{0}^{t}z_{i}^{k}(s)\,\mathrm{d}s
 -\int_{0}^{t+\xi_{t}}z_{i}^{k}(s)\,\mathrm{d}s\\
&\quad +\int_{0}^{t+\xi_{t}}z_{i}^{k}(s)\,\mathrm{d}s\Big|\\
&\leq2\max_{1\leq i\leq n}\sup_{t\in[0,l_{i}]}\Big|
 \int_{0}^{t}z_{i}^{k}(s)\,\mathrm{d}s\Big|
 +\max_{1\leq i\leq n}\sup_{t\notin[0,l_{i}]}\Big|
 \int_{0}^{t}z_{i}^{k}(s)\,\mathrm{d}s
 -\int_{0}^{t+\xi_{t}}z_{i}^{k}(s)\,\mathrm{d}s\Big| \\
&\leq2\max_{1\leq i\leq n}\Big|\int_{0}^{l_{i}}z_{i}^{k}(s)\,\mathrm{d}s
 \Big|+\max_{1\leq i\leq n}\epsilon_{i}.
\end{aligned}
\end{equation}
 From this inequality, we can conclude that $\int^{t}_{0}z(s)ds$ is
continuous, where
$z(t)=(z_{1}(t),z_{2}(t),\dots ,z_{n}(t))^T\in \mathop{\rm Im}(L)$.
Consequently, $K_{P}$ and $K_{P}(I-Q)Ny$ are continuous.

 From \eqref{e3.3}, we also have $\int_{0}^{t}z(s)\,\mathrm{d}s$ and
$K_{P}(I-Q)Ny$ also are uniformly bounded in $\overline{\Omega}$.
Further, it is not difficult to verify that $QN(\overline{\Omega})$
is bounded and $K_{P}(I-Q)Ny$ is equicontinuous in
$\overline{\Omega}$. By the Arzela-Ascoli theorm, we have
immediately conclude that $K_{P}(I-Q)N(\overline{\Omega})$ is
compact. Thus $N$ is $L$-compact on $\overline{\Omega}$. The proof
is complete.
\end{proof}

 By \eqref{e1.3}, $f_{i}(t,x)$ can be represented as a
power-series at $\alpha_{i}$ of $x$, in form of
\[
f_{i}(t,x)=f_{i}(t,\alpha_{i})+\frac{\partial f_{i}}{\partial
x}\Big|_{(t,\alpha_{i})}x+o(x),\quad i=1,2,\dots ,n,
\]
where $o(x)$ is a higher-order infinitely small quantity of $x$.
By \eqref{e1.4}, we  conclude that
$f_{i}(t,\alpha_{i})=0$, $i=1,2,\dots ,n$.
For convenience, we denote
$\frac{\partial f_{i}}{\partial x}\big|_{(t,\alpha_{i})}
:=c_{ii}(t)$, $i=1,2,\dots ,n$. By \eqref{e1.3},
$c_{ii}(t)>0$.

\begin{theorem} \label{thm3.1}
Assume that
\begin{gather*}
m[r_{i}(t)]=\lim_{T\to\infty}\frac{1}{T}\int_{0}^{T}r_{i}(t)
\,\mathrm{d}t>0, \\
m\Big[\sum_{j=1}^{n}(b_{ij}(t)+c_{ij}(t))\Big]=\lim_{T\to\infty}
\frac{1}{T}\int_{0}^{T}\sum_{j=1}^{n}(b_{ij}(t)+c_{ij}(t))
\,\mathrm{d}t>0 \,.
\end{gather*}
Then \eqref{e1.1} has at least one positive almost periodic
solution.
\end{theorem}

\begin{proof}
To use the continuation theorem of coincidence degree
theorem to establish the existence of a solution of \eqref{e3.1}, we set
Banach space $X$ and $Z$ the same as those in Lemma \ref{lem3.1} and set
mappings $L,N,P,Q$ the same as those in Lemma \ref{lem3.2} and
Lemma \ref{lem3.3},
respectively. Then we can obtain that $L$ is a Fredholm mapping of
index 0 and $N$ is a continuous operator which is $L$-compact on
$\overline{\Omega}$.

Now, we are in the position of searching for an appropriate open,
bounded subset $\Omega$ for the application of the continuation
theorem. Corresponding to the operator equation
\[
Ly=\lambda Ny,\lambda\in(0,1),
\]
we obtain
\begin{equation} \label{e3.4}
\begin{aligned}
\dot{y}_{i}(t)
&=\lambda\Big[r_{i}(t)-c_{ii}(t)\exp\{y_{i}(t-\tau_{ii}(t))\}
-o(\exp\{2y_{i}(t-\tau_{ii}(t))\}) \\
&\quad -\sum_{j=1}^{n}b_{ij}(t)\exp\{y_{j}(t)\}-\sum_{j=1,i\neq
j}^{n}c_{ij}(t)\exp\{y_{j}(t-\tau_{ij}(t))\}\\
&\quad -\sum_{j=1}^{n}\int_{-\sigma_{ij}}^{0}\mu_{ij}(t,s)
\exp\{y_{j}(t+s)\}\,\mathrm{d}s\Big],\quad i=1,2,\dots ,n.
\end{aligned}
\end{equation}
Assume that $y(t)=(y_{1}(t),y_{2}(t),\dots ,y_{n}(t))^T\in X$ is a
solution of \eqref{e3.4} for some $\lambda\in(0,1)$.
 Denote
\[
M_{1}=\max_{1\leq i\leq n}\sup_{t\in\mathbb{R}}\{y_{i}(t)\},\quad
M_{2}=\min_{1\leq i\leq n}\inf_{t\in\mathbb{R}}\{y_{i}(t)\},
\]
by \eqref{e3.4}, we derive
\begin{align*}
m[r_{i}(t)]
&=m\Big[c_{ii}(t)\exp\{y_{i}(t-\tau_{ii}(t))\}
 +o(\exp\{2y_{i}(t-\tau_{ii}(t))\}) \\
&\quad +\sum_{j=1}^{n}b_{ij}(t)\exp\{y_{j}(t)\}+\sum_{j=1,i\neq
j}^{n}c_{ij}(t)\exp\{y_{j}(t-\tau_{ij}(t))\} \\
&\quad + \sum_{j=1}^{n}\int_{-\sigma_{ij}}^{0}\mu_{ij}(t,s)
\exp\{y_{j}(t+s)\}\,\mathrm{d}s\Big],\quad i=1,2,\dots ,n
\end{align*}
and consequently
\[
m[r_{i}(t)]\leq\exp\{M_{1}\}\Big\{m\Big[\sum_{j=1}^{n}(b_{ij}(t)
+c_{ij}(t))\Big]+n+1\Big\},\quad i=1,2,\dots ,n;
\]
that is,
\begin{equation} \label{e3.5}
M_{1}\geq\ln\frac{m[r_{i}(t)]}{m[\sum_{j=1}^{n}(b_{ij}(t)
+c_{ij}(t))]+n+1},\quad i=1,2,\dots ,n.
\end{equation}
Similarly, we can get
\begin{equation} \label{e3.6}
M_{2}\leq\ln\frac{m[r_{i}(t)]}{m[\sum_{j=1}^{n}(b_{ij}(t)
+c_{ij}(t))]+n-1},\quad i=1,2,\dots ,n.
\end{equation}
By \eqref{e3.5} and \eqref{e3.6}, we find that there exist
$t_{1}^{i}\in\mathbb{R}$, $i=1,2,\dots ,n$ such that
$y_{i}(t_{1}^{i})\leq R_{1}$,
where
\[
R_{1}=\max_{1\leq i\leq n}\Big|
\ln\frac{m[r_{i}(t)]}{m[\sum_{j=1}^{n}(b_{ij}(t)+c_{ij}(t))]+n-1}\Big|+1.
\]
Furthermore, we have
\begin{equation} \label{e3.7}
\begin{aligned}
\big\|y\big\|
&\leq\max_{1\leq i\leq n}|y_{i}(t_{1}^{i})|
 +\max_{1\leq i\leq n}\sup_{t\in\mathbb{R}}
 \Big|\int_{t_{1}^{i}}^{t}\dot{y}_{i}(s)\,\mathrm{d}s\Big| \\
&\leq R_{1} +2\max_{1\leq i\leq n}\sup_{t\in[t_{1}^{i},t_{1}^{i}+l_{i}]
}\Big|\int_{t_{1}^{i}}^{t}\dot{y}_{i}(s)\,\mathrm{d}s\Big|+\max_{1\leq
i\leq n}\epsilon_{i} \\
&\leq R_{1}+2\max_{1\leq i\leq n}
\Big|\int_{t_{1}^{i}}^{t_{1}^{i}+l_{i}}\dot{y}_{i}(s)\,\mathrm{d}s\Big|+1.
\end{aligned}
\end{equation}
Choose a point $\widetilde{\tau}_{i}$ such that
$\widetilde{\tau_{i}}-t_{1}^{i}\in[l_{i},2l_{i}]\cap
K_{i}(H_{i},\sigma_{i},\phi_{i})$, where
$\sigma_{i}(0<\sigma_{i}<\epsilon_{i})$ satisfies
$K_{i}(H_{i},\sigma_{i},\phi_{i})\subset
K_{i}(y_{i},\epsilon_{i},\phi_{i})$, $i=1,2,\dots ,n$. Integrating
\eqref{e3.4} from $t_{1}^{i}$ to $\widetilde{\tau}_{i}$, we get
\begin{align*}
&\lambda\int_{t_{1}^{i}}^{\widetilde{\tau}_{i}}\Big[c_{ii}(t)
\exp\{y_{i}(t-\tau_{ii}(t))\}+o(\exp\{2y_{i}(t-\tau_{ii}(t))\})
+\sum_{j=1}^{n}b_{ij}(t)\exp\{y_{j}(t)\}
 \\
&+\sum_{j=1,i\neq j}^{n}c_{ij}(t)\exp\{y_{j}(t-\tau_{ij}(t))\}
+\sum_{j=1}^{n}\int_{-\sigma_{ij}}^{0}\mu_{ij}(t,s)\exp\{y_{j}(t+s)\}\,\mathrm{d}s\Big]\,\mathrm{d}t
 \\
&=\lambda\int_{t_{1}^{i}}^{\widetilde{\tau}_{i}}r_{i}(t)\,\mathrm{d}t-
\int_{t_{1}^{i}}^{\widetilde{\tau}_{i}}\dot{y}_{i}(t)\,\mathrm{d}t
\\
&\leq \lambda\int_{t_{1}^{i}}^{\widetilde{\tau}_{i}}|r_{i}(t)|
 \,\mathrm{d}t+\epsilon_{i},\quad i=1,2,\dots ,n.
\end{align*}
 From the above inequality and \eqref{e3.4}, we obtain
\begin{align*}
&\int_{t_{1}^{i}}^{\widetilde{\tau}_{i}}|\dot{y}_{i}(t)|\,\mathrm{d}t\\
&\leq \lambda\int_{t_{1}^{i}}^{\widetilde{\tau}_{i}}|r_{i}(t)|\,\mathrm{d}t+
\lambda\int_{t_{1}^{i}}^{\widetilde{\tau}_{i}}
 \Big[c_{ii}(t)\exp\{y_{i}(t-\tau_{ii}(t))\}
+o(\exp\{2y_{i}(t-\tau_{ii}(t))\}) \\
&\quad + \sum_{j=1}^{n}b_{ij}(t)\exp\{y_{j}(t)\} +\sum_{j=1,i\neq
j}^{n}c_{ij}(t)\exp\{y_{j}(t-\tau_{ij}(t))\} \\
&\quad +\sum_{j=1}^{n}\int_{-\sigma_{ij}}^{0}\mu_{ij}(t,s)
 \exp\{y_{j}(t+s)\}\,\mathrm{d}s\Big]\,\mathrm{d}t
 \\
&\leq 2\int_{t_{1}^{i}}^{\widetilde{\tau}_{i}}|r_{i}(t)|dt+\epsilon_{i}
 \\
&\leq 2\int_{t_{1}^{i}}^{\widetilde{\tau}_{i}}|r_{i}(t)|\,\mathrm{d}t+1,
\quad i=1,2,\dots ,n,
\end{align*}
which together with \eqref{e3.7} and
$\widetilde{\tau}\geq t_{1}^{i}+l_{i}$, $i=1,2,\dots ,n$, we have
$\|y\|\leq\overline{R}$,
where
\[
\overline{R}=R_{1}+4\max_{1\leq i\leq
n}\int_{0}^{\widetilde{\tau}}|r_{i}(t)|dt+3.
\]
Clearly, $\overline{R}$ is independent of $\lambda$. Take
\[
\Omega=\{y=(y_{1},y_{2},\dots ,y_{n})^T\in X:\|y\|<\overline{R}+1\}.
\]
It is clear that $\Omega$ satisfies the requirement (a)
in Lemma \ref{lem2.3}. when $y\in \partial\Omega\cap \ker(L)$,
$y=(y_{1},y_{2},\dots ,y_{n})^T$ is a constant vector in
$\mathbb{R}^{n}$ with $\|y\|=\overline{R}+1$. Then
\[
QNy=\big(m[G_{1}],m[G_{2}],\dots ,m[G_{n}]\big)^T,\quad y\in X
\]
where
\begin{align*}
G_{i}&=r_{i}(t)-f_{i}(t,\exp\{y_{i}\})\exp\{y_{i}\}\\
&\quad -\sum_{j=1}^{n}b_{ij}(t)\exp\{y_{j}\}-\sum_{j=1,i\neq
j}^{n}c_{ij}(t)\exp\{y_{j}\}\\
&\quad -\sum_{j=1}^{n}\int_{-\sigma_{ij}}^{0}\mu_{ij}(t,s)
\exp\{y_{j}\}\,\mathrm{d}s,\quad i=1,2,\dots ,n,
\end{align*}
thus $QNy\neq0$, which implies that the requirement (b)
in Lemma \ref{lem2.3}
is satisfied. Furthermore, take the isomorphism
$J:\mathop{\rm Im}(Q)\to \ker(L),Jz\equiv z$ and let
$\Phi(\gamma;y)=-\gamma y+(1-\gamma)JQNy$, then for any
$y\in\partial\Omega\cap \ker(L),y^T\Phi(\gamma;y)<0$, we have
\[
\deg\{JQN,\Omega\cap \ker(L),0\}=\deg\{-y,\Omega\cap \ker(L),0\}\neq0.
\]
So, the requirement (c) in  Lemma \ref{lem2.3}  is satisfied.
Hence, \eqref{e3.1} has at least one almost periodic solution
in $\overline{\Omega}$, that is \eqref{e1.1} has at least one
positive almost periodic solution. The
proof is complete.
\end{proof}

We remark that when $n=1$ in \eqref{e1.1},  if
$F_{1}(t,x)$ is a linear function and $\mu_{11}(t)\equiv0$, then
Theorem \ref{thm3.1} is the same as \cite[Theorem 3.1]{m3}.

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\end{document}