\documentclass[reqno]{amsart}
\usepackage{hyperref}
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\AtBeginDocument{{\noindent\small
{\em Electronic Journal of Differential Equations},
Vol. 2007(2007), No. 13, pp. 1--10.\newline
ISSN: 1072-6691. URL: http://ejde.math.txstate.edu or http://ejde.math.unt.edu
\newline ftp ejde.math.txstate.edu  (login: ftp)}
\thanks{\copyright 2006 Texas State University - San Marcos.}
\vspace{9mm}}

\begin{document}
\title[\hfilneg EJDE-2007/13\hfil Positive periodic solutions]
{Positive periodic solutions of  neutral
logistic equations with distributed delays}

\author[Y. Li, G. Wang,  H. Wang\hfil EJDE-2007/13\hfilneg]
{Yongkun Li, Guoqiao Wang, Huimei Wang }

\address{Department of Mathematics,
Yunnan University, Kunming, Yunnan 650091, China}
\email[Yongkun Li]{yklie@ynu.edu.cn}
\email[Guoqiao Wang]{wgq81@126.com}
\email[Huimei Wang]{wanghmei@163.com}


\date{}
\thanks{Submitted July 14, 2006. Published January 8, 2007.}
\thanks{Supported by grants 10361006 from the National Natural
 Sciences Foundation China, \hfill\break\indent
 and 2003A0001M from the the Natural
 Sciences Foundation of Yunnan Province.}
\subjclass[2000]{34K13, 34K40}
\keywords{Positive periodic solution;  neutral delay logistic equation;
 \hfill\break\indent strict-set-contraction}

\begin{abstract}
  Using a fixed point theorem of strict-set-contraction,
  we establish criteria for the existence of positive periodic
  solutions for the periodic neutral logistic equation,
  with distributed delays,
 $$
 x'(t)= x(t)\Big[a(t)-\sum_{i=1}^n a_i(t)\int_{-T_i}^0  x(t+\theta)\,
 \mathrm{d}\mu_i(\theta)- \sum_{j=1}^m b_j(t)
 \int_{-\hat{T}_j}^0 x'(t+\theta)\,\mathrm{d}\nu_j(\theta)\Big],
 $$
 where the coefficients $a, a_i ,b_j$ are continuous and periodic functions,
 with the same period. The values $T_i, \hat{T}_j$ are positive,
 and the functions  $\mu_i, \nu_j$   are nondecreasing   with
 $\int_{-T_i}^0\,{\rm d} \mu_i=1$ and
 $\int_{-\hat{T}_j}^0\,{\rm d} \nu_j=1$.
\end{abstract}

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

\section{Introduction}

 Consider the  single species neutral logistic model, with discrete
 delays,
\begin{equation} \label{e1.1}
\frac{{\rm d} N(t)}{{\rm d} t}=N(t)\Big[a(t)-\beta(t)N(t)
-\sum_{i=1}^n b_i(t) N_i(t-\tau_i(t))
-\sum_{j=1}^m c_j(t) N'_j(t-\sigma_j(t))\Big],
\end{equation}
where the functions $a(t), \beta(t), b_j(t), c_j(t), \tau_i(t),
\sigma_j(t)$ are continuous $\omega$-periodic, and
$a(t)\geq 0$, $\beta(t)\geq 0$, $b_i(t)\geq 0$, $c_j(t)\geq 0$
($i=1,2,\dots,n$, $j=1,2,\dots,m$).
An ecological justification of model \eqref{e1.1} can be found in
 \cite{g2,g3,k1,p1}.
Using  continuation theory for $k$-set-contractions, Lu
 \cite{l1},  Lu and Ge \cite{l2}
  studied the existence of positive periodic solutions of \eqref{e1.1}.
Yang and Cao \cite{y1} used Mawhin's
continuation theorem \cite{g1} to investigated the existence of positive
periodic solutions of \eqref{e1.1}. The main results obtained in
\cite{k1,l2}  required  $c_j\in C^1,\sigma_j\in C^2$ and
$\sigma_j'<1$ ($j=1,2,\dots,n$).
To the best of our knowledge, this is
  the first paper to study the existence of periodic solutions of neutral
  logistic equations with distributed delays.

The main purpose of this paper  is by using a fixed point theorem of
 strict-set-contraction \cite{c1,g4} to
 establish  the  existence  of positive periodic solutions
 of the following neutral functional differential equation, with
distributed delays,
\begin{equation} \label{e1.2}
x'(t)=x(t)\Big
[a(t)-\sum_{i=1}^n a_i(t)\int_{-T_i}^0x(t+\theta)\,\mathrm{d}\mu_i(\theta)-
\sum_{j=1}^n b_j(t)\int_{-\hat{T}_j}^0x'(t+\theta)\,\mathrm{d}\nu_j(\theta)
\Big]\,,
\end{equation}
where $a,a_i ,b_j \in  C(\mathbb{R},\mathbb{R}^+)$  are $\omega$-periodic
  functions, $T_i, \hat{T}_j$ are postive constants,
$\mu_i, \nu_j:[-T_i,0]\to[0,\infty)$ are nondecreasing functions and
$\int_{-T_i}^0\,{\rm d} \mu_i=1$, $\int_{-\hat{T}_j}^0\,{\rm
d}\nu_j=1$, for $i=1,2,\dots,n$, $j=1,2,\dots,m$.  Note that
\eqref{e1.1} is a special case of  \eqref{e1.2}. For an ecological
justification of \eqref{e1.1}, we refer the reader to \cite{k2}.

 For convenience, we introduce the following notation:
\begin{gather*}
\lambda=e^{-\int_{0}^{\omega}a(s)\,\mathrm{d}s},\quad
\Delta=\int_{0}^{\omega}\Big[\lambda \sum_{i=1}^n a_i(s)-\sum_{j=1}^m b_j(s)\Big]\,\mathrm{d}s,
 \\
\Pi=\int_{0}^{\omega}\Big[\sum_{i=1}^n a_i(s)+\sum_{j=1}^m b_j(s)\Big]\,
\mathrm{d}s, \quad
f^M=\max_{t\in [0,\omega]}\{f(t)\},\\
 f^m=\min_{t\in [0,\omega]}\{f(t)\},
\end{gather*}
where $f$ is a continuous $\omega$-periodic function.
Also we introduce the following assumptions:
\begin{itemize}
  \item [(H1)]
   $\lambda:=\exp\big(-\int_{0}^{\omega}a(s)\,\mathrm{d}s\big)<1$.

  \item [(H2)] $\lambda \sum_{i=1}^n a_i(t)-\sum_{j=1}^m b_j(t)\geq0$.

  \item [(H3)]
 $(1+a^m) \frac{\lambda^2\Delta}{1-\lambda}\geq\max_{t\in [0,\omega]}
\big\{\sum_{i=1}^n a_i(t)+\sum_{j=1}^m b_j(t)\big\}$.

  \item [(H4)]
$\frac{\Pi(a^M-1)}{\lambda(1-\lambda)}\leq\min_{t\in
[0,\omega]}\big\{\lambda
\sum_{i=1}^n a_i(t)-\sum_{j=1}^m b_j(t)\big\}$.

  \item [(H5)]
 $\frac{1-\lambda}{\lambda^2\Delta}\big(\sum_{j=1}^m b_j^M\big)<1$.
\end{itemize}

\section{Preliminaries}

To obtain the existence of periodic solutions to \eqref{e1.2},
we make the following preparations:

Let $E$ be a Banach space and $K$ be a cone in $E$. The semi-order
induced by the cone $K$ is denoted by ``$\leq$''. That is, $x\leq y$
if and only if $y-x\in K$. In addition, for a bounded subset
$A\subset E$, let the Kuratowski measure of
non-compactness be defined by
\begin{align*}
\alpha_{E}(A)=\inf\Big\{&\delta>0: \text{ there
is a finite number of subsets }A_i\subset A\\
&\text{such that }A=\cup_iA_i \text{ and }
\mathop{\rm diam} (A_i)\leq \delta\Big\},
\end{align*}
where $\mathop{\rm diam}(A_i)$ denotes the diameter of the set $A_i$.

Let $E, F$ be two Banach spaces and  $D\subset E$, a continuous and
bounded mapping  $\Phi : \bar{\Omega} \to F$  is called
 $k$-set contractive if for every bounded set $S\subset D$ we have
 \[
 \alpha_F(\Phi(S))\leq k\alpha_E(S).
 \]
The mapping $\Phi$  is called strict-set-contractive if it is
$k$-set-contractive for some $0 \leq k < 1$.
 From \cite{c1,g1}, we cite the following lemma which is useful for the
proof of our main result.

\begin{lemma}[\cite{c1,g4}] \label{lem2.1}
Let $K$ be a cone of the real Banach space $X$ and $K_{r,R}=\{x\in
K:r\leq ||x||\leq R\}$ with $R>r>0$. Suppose that $\Phi:K_{r,R}\to
K$ is  strict-set-contractive such that one of the following two
conditions is satisfied:
\begin{itemize}
  \item [(i)]
($\Phi  x \nleq x$ for all $x \in K$, $\|x\|= r$) and
($\Phi x\ngeq x$ for all $x\in K$, $\|x\|=R$).
  \item [(ii)]
($\Phi  x \ngeq x$ for all $x \in K$, $\|x\|= r$) and
($\Phi x\nleq x$ for all $x\in K$, $\|x\|=R$).
\end{itemize}
Then $\Phi$ has at least one  fixed point in $K_{r,R}$.
\end{lemma}

To apply Lemma \ref{lem2.1} to \eqref{e1.2}, we set
\[
C^0_\omega=\{x\in C^0(\mathbb{R},\mathbb{R}): x(t+\omega)=x(t)\}
\]
with the norm $|x|_0=\max_{t\in [0,\omega]}\{|x(t)|\}$,
and
\[
C^1_\omega=\{x\in C^1(\mathbb{R},\mathbb{R}): x(t+\omega)=x(t)\}
\]
 with the norm $|x|_1=\max\{|x|_0, |x'|_0\}$. Then
 $C^0_\omega$ and  $C^1_\omega$ are all Banach spaces.

 Define the cone $K$ in $C^1_{\omega}$ by
 \begin{equation} \label{e2.1}
K=\{x\in C^1_{\omega}: x(t)\geq \lambda|x|_1, t\in [0,\omega]\}.
 \end{equation}
Let the mapping $\Phi $ be defined  by
\begin{equation} \label{e2.2}
\begin{aligned}
 (\Phi x)(t)&= \int_{t}^{t+\omega}G(t,s)x(s)\Big[\sum_{i=1}^n a_i(s)
 \int_{-T_i}^0x(s+\theta)\,\mathrm{d}\mu_i(\theta)\\
&\quad  +\sum_{j=1}^m b_j(s)\int_{-\hat{T}_j}^0x'(s+\theta)\,\mathrm{d}\nu_j(\theta)
\Big]\,\mathrm{d}s,
\end{aligned}
\end{equation}
where $x\in K, t\in \mathbb{R}$,  and
\[
G(t,s)=\frac{e^{-\int_{t}^{s}a(\theta)\,{\rm d}\theta}}
{1-e^{-\int_{0}^{\omega}a(\theta)\,{\rm d}\theta}},\quad s\in
[t,t+\omega].
\]
It is easy to see that $G(t+\omega,s+\omega)=G(t,s)$ and
\[
\frac{\lambda}{1-\lambda}\leq G(t,s)\leq\frac{1}{1-\lambda},\quad
s\in [t,t+\omega].
\]
Next,  we give some lemmas concerning the $K$ and $\Phi$
defined above.

\begin{lemma} \label{lem2.2}
Assume that (H1)--(H3) hold.
\begin{itemize}
  \item [(i)] If $a^M\leq 1$, then $\Phi: K\to K$ is well defined.
  \item [(ii)] If (H4) holds and $a^M>1$, then $\Phi: K\to K$ is well defined.
\end{itemize}
\end{lemma}

\begin{proof}
For any $x\in K$, it is clear that
$\Phi x\in C^1(\mathbb{R},\mathbb{R})$. In view of \eqref{e2.2}, for
$t\in \mathbb{R}$, we obtain
\begin{align*}
  (\Phi x)(t+\omega)&= \int_{t+\omega}^{t+2\omega}G(t+\omega,s)x(s)
  \Big[\sum_{i=1}^n a_i(s)\int_{-T_i}^0x(s+\theta)\,\mathrm{d}\mu_i(\theta)\\
  &\quad +
\sum_{j=1}^m b_j(s)\int_{-\hat{T}_j}^0x'(s+\theta)\,\mathrm{d}\nu_j(\theta)
\Big]\,\mathrm{d}s\\
&= \int_{t}^{t+\omega}G(t+\omega,u+\omega)x(u+\omega)\Big[
a(u+\omega)\int_{-T_i}^0x(u+\omega+\theta)\,\mathrm{d}\mu_i(\theta)\\
&\quad +
b(u+\omega)\int_{-\hat{T}_j}^0x'(u+\omega+\theta)\,\mathrm{d}\nu_j(\theta)\Big]\,\mathrm{d}u\\
&= \int_{t}^{t+\omega}G(t,u)x(u)\Big[
a(u)\int_{-T_i}^0x(u+\theta)\,\mathrm{d}\mu_i(\theta)\\
&\quad +
 b(u)\int_{-\hat{T}_j}^0x'(u+\theta)\,\mathrm{d}\nu_j(\theta)\Big]\,\mathrm{d}u\\
&= (\Phi x)(t).
\end{align*}
 That is, $(\Phi x)(t+\omega)=(\Phi x)(t)$,
$t\in \mathbb{R}$. So $\Phi x \in C^1_{\omega}$. In view of (H2), for
$x \in K$, $t\in [0,\omega]$, we have
\begin{equation} \label{e2.3}
\begin{aligned}
& \sum_{i=1}^n a_i(t)\int_{-T_i}^0x(t+\theta)\,\mathrm{d}\mu_i(\theta)
 +
\sum_{j=1}^m b_j(t)\int_{-\hat{T}_j}^0x'(t+\theta)\,\mathrm{d}\nu_j(\theta)\\
&\geq
\sum_{i=1}^n a_i(t)\int_{-T_i}^0x(t+\theta)\,\mathrm{d}\mu_i(\theta)
 -
\sum_{j=1}^m b_j(t)\int_{-\hat{T}_j}^0|x'(t+\theta)|\,\mathrm{d}\nu_j(\theta)\\
&\geq \sum_{i=1}^n a_i(t)\int_{-T_i}^0\lambda|x|_1\,\mathrm{d}\mu_i(\theta)
 -
\sum_{j=1}^m b_j(t)\int_{-\hat{T}_j}^0|x|_1\,\mathrm{d}\nu_j(\theta)\\
&\geq \lambda \sum_{i=1}^n a_i(t)
-\sum_{j=1}^m b_j(t)]|x|_1\,\mathrm{d}\nu_j(\theta)\geq0.
\end{aligned}
\end{equation}
Therefore, for $x\in K$, $t\in [0,\omega]$, we find
\begin{align*}
   |\Phi x|_0 &\leq \frac{1}{1-\lambda}\int_{0}^{\omega}x(s)
   \Big[\sum_{i=1}^n a_i(s)\int_{-T_i}^0x(s+\theta)\,\mathrm{d}\mu_i(\theta)\\
&\quad +\sum_{j=1}^m b_j(s)\int_{-\hat{T}_j}^0x'(s+\theta)\,\mathrm{d}\nu_j(\theta)\Big
]\,\mathrm{d}s
\end{align*}
and
\begin{equation} \label{e2.4}
\begin{aligned}
(\Phi x)(t)
&\geq \frac{\lambda}{1-\lambda}\int_{t}^{t+\omega}x(s)\Big[
\sum_{i=1}^n a_i(s)\int_{-T_i}^0x(s+\theta)\,\mathrm{d}\mu_i(\theta)\\
&\quad
+ \sum_{j=1}^m b_j(s)\int_{-\hat{T}_j}^0x'(s+\theta)\,\mathrm{d}\nu_j(\theta)
\Big]\,\mathrm{d}s\\
&= \frac{\lambda}{1-\lambda}\int_{0}^{\omega}x(s)\Big[
\sum_{i=1}^n a_i(s)\int_{-T_i}^0x(s+\theta)\,\mathrm{d}\mu_i(\theta)\\
&\quad
+ \sum_{j=1}^m b_j(s)\int_{-\hat{T}_j}^0x'(s+\theta)\,\mathrm{d}\nu_j(\theta)\Big]\,\mathrm{d}s\\
&\geq \lambda|\Phi x|_0.
\end{aligned}
\end{equation}
Now, we show that $(\Phi x)'(t)\geq \lambda |(\Phi x)'|_0, t\in [0,\omega]$.
 From \eqref{e2.2}, we have
\begin{align}
  (\Phi x)'(t)
&= G(t,t+\omega)x(t+\omega)\Big[
a(t+\omega)\int_{-T_i}^0x(t+\omega+\theta)\,\mathrm{d}\mu_i(\theta) \nonumber\\
&\quad +
b(t+\omega)\int_{-\hat{T}_j}^0x'(t+\omega+\theta)\,\mathrm{d}\nu_j(\theta)
\Big] \nonumber\\
&\quad -G(t,t)x(t)\Big[
\sum_{i=1}^n a_i(t)\int_{-T_i}^0x(t+\theta)\,\mathrm{d}\mu_i(\theta)\nonumber \\
&\quad +\sum_{j=1}^m b_j(t)\int_{-\hat{T}_j}^0x'(t+\theta)
  \,\mathrm{d}\nu_j(\theta)\Big]
 +a(t)(\Phi x)(t) \label{e2.5}\\
&= a(t)(\Phi x)(t)-x(t)\Big[
\sum_{i=1}^n a_i(t)\int_{-T_i}^0x(t+\theta)\,\mathrm{d}\mu_i(\theta) \nonumber\\
&\quad +
\sum_{j=1}^m b_j(t)\int_{-\hat{T}_j}^0x'(t+\theta)\,\mathrm{d}\nu_j
(\theta)\Big]. \nonumber
\end{align}
It follows from \eqref{e2.3} and \eqref{e2.5} that if $(\Phi x)'(t)\geq 0$,
then
\begin{equation} \label{e2.6}
    (\Phi x)'(t)\leq a(t)(\Phi x)(t)\leq a^M (\Phi x)(t)\leq(\Phi x)(t).
\end{equation}
On the other hand, from \eqref{e2.4}, \eqref{e2.5} and (H3), if
$(\Phi x)'(t) < 0$, then
\begin{align*} %2.7
&-(\Phi x)'(t) \\
&= x(t)\Big[
\sum_{i=1}^n a_i(t)\int_{-T_i}^0x(t+\theta)\,\mathrm{d}\mu_i(\theta)
+ \sum_{j=1}^m b_j(t)\int_{-\hat{T}_j}^0x'(t+\theta)\,\mathrm{d}
\nu_j(\theta)\Big]\\
&\quad -a(t)(\Phi x)(t)\\
&\leq |x|_1^2\Big[\sum_{i=1}^n a_i(t)+\sum_{j=1}^m b_j(t)\Big]-a^m(\Phi x)(t)\\
&\leq (1+a^m)\frac{\lambda^2}{1-\lambda}|x|_1^2\int_{0}^{\omega}\Big[\lambda
\sum_{i=1}^n a_i(s)+  \sum_{j=1}^m b_j(s)\Big]\,\mathrm{d}s-a^m(\Phi x)(t)\\
&= (1+a^m)\int_{0}^{\omega}\frac{\lambda}{1-\lambda}\lambda|x|_1\Big[\lambda|x|_1
\sum_{i=1}^n a_i(s)-|x|_1 \sum_{j=1}^m b_j(s)\Big]\,\mathrm{d}s-a^m(\Phi x)(t)\\
&\leq (1+a^m)\int_{t}^{t+\omega}G(t,s)x(s)\Big[\sum_{i=1}^n a_i(s)
\int_{-T_{1}}^0x(s+\theta)\,\mathrm{d}\mu_i(\theta)\\
&\quad -\sum_{j=1}^m b_j(s)
\int_{-T_{2}}^0 |x'(\theta+s)|\,\mathrm{d}\nu_j(\theta)\Big]\,\mathrm{d}s-a^m(\Phi x)(t)\\
&\leq (1+a^m)\int_{t}^{t+\omega}G(t,s)x(s)
\Big[\sum_{i=1}^n a_i(s)\int_{-T_{1}}^0x(s+\theta)\,\mathrm{d}\mu_i(\theta)\\
&\quad +\sum_{j=1}^m b_j(s)\int_{-T_{2}}^0
 x'(\theta+s)\,\mathrm{d}\nu_j(\theta)\Big]\,\mathrm{d}s-a^m(\Phi x)(t)\\
&= (1+a^m)(\Phi_ix)(t)-a^m(\Phi x)(t)\\
&= (\Phi x)(t).
\end{align*}
It follows from the above inequality and \eqref{e2.6} that
$|(\Phi x)'|_0\leq |\Phi x|_0$.
So $|\Phi x|_1 = |\Phi x|_0$. By \eqref{e2.2} we have
$(\Phi x)(t) \geq \lambda |\Phi x|_1$. Hence, $\Phi  x \in K$.
The proof of (i) is complete.

(ii) In view of the proof of (i), we only need to prove that
$(\Phi x)'(t) \geq 0$ implies
\[
(\Phi x)'(t) \leq (\Phi x)(t).
\]
 From \eqref{e2.3}, \eqref{e2.5}, (H2) and (H4), we
obtain
\begin{align*}
(\Phi_ix)'(t)
&\leq a(t)(\Phi x)(t)-\lambda|x|_1\Big[
\sum_{i=1}^n a_i(t)\int_{-T_i}^0x(t+\theta)\,\mathrm{d}\mu_i(\theta)\\
&\quad -\sum_{j=1}^m b_j(t)\int_{-\hat{T}_j}^0|x'(t+\theta)|\,\mathrm{d}\nu_j(\theta)\Big]\\
&\leq a(t)(\Phi x)(t)-\lambda|x|_1^2\Big[\lambda
\sum_{i=1}^n a_i(t)
- \sum_{j=1}^m b_j(t)\Big]\\
&\leq a^M(\Phi
x)(t)-\lambda|x|_1^2\frac{a^M-1}{\lambda(1-\lambda)}\int_{0}^{\omega}\Big[\sum_{i=1}^n a_i(s)
+\sum_{j=1}^m b_j(s)\Big]\,\mathrm{d}s\\
&\leq a^M(\Phi
x)(t)-(a^M-1)\int_{t}^{t+\omega}\frac{1}{1-\lambda}|x|_1\Big[\sum_{i=1}^n a_i(s)|x|_1
 +\sum_{j=1}^m b_j(s)|x|_1\Big]\,\mathrm{d}s \\
&\leq a^M(\Phi x)(t)-(a^M-1)\int_{t}^{t+\omega}G(t,s)x(s)
\Big[\sum_{i=1}^n a_i(t)\int_{-T_i}^0x(t+\theta)\,\mathrm{d}\mu_i(\theta)\\
&\quad  +\sum_{j=1}^m b_j(t)\int_{-\hat{T}_j}^0|x'(t+\theta)|\,\mathrm{d}\nu_j(\theta)\Big]\,\mathrm{d}s \\
&\leq a^M(\Phi x)(t)-(a^M-1)\int_{t}^{t+\omega}G(t,s)x(s)
\Big[\sum_{i=1}^n a_i(t)\int_{-T_i}^0x(t+\theta)\,\mathrm{d}\mu_i(\theta)\\
&\quad  + \sum_{j=1}^m b_j(t)\int_{-\hat{T}_j}^0x'(t+\theta)\,\mathrm{d}\nu_j(\theta)\Big]\,\mathrm{d}s\\
&= a^M(\Phi x)(t)-(a^M-1)(\Phi x)(t)\\
&= (\Phi x)(t).
\end{align*}
The proof of (ii) is complete.
\end{proof}


\begin{lemma} \label{lem2.3}
Assume that (H1)--(H3) hold and $R\sum_{j=1}^m b_j^M <1$.
\begin{itemize}
  \item [(i)] If $a^M\leq 1$, then $\Phi : K \bigcap\bar{\Omega}_R \to K$
  is strict-set-contractive,
  \item [(ii)] If (H4) holds and $a^M> 1$, then $\Phi : K
\bigcap\bar{\Omega}_R \to K$ is strict-set-contractive,
\end{itemize}
where $\Omega_R=\{x\in C^1_{\omega}: |x|_1<R\}$.
\end{lemma}

\begin{proof}
 We only need to prove (i), since the proof of
(ii) is similar. It is easy to see that $\Phi $ is continuous and
bounded. Now we prove that $\alpha _{C_\omega^1} (\Phi(S)) \leq
\big(R\sum_{j=1}^m b_j^M\big)\alpha_{C_{\omega}^1}(S)$ for
any bounded set $S\subset\bar{\Omega}_R$. Let $\eta = \alpha
_{C_\omega ^1 } (S)$. Then, for any positive number $\varepsilon
<\big(R\sum_{j=1}^m b_j^M\big)\eta$, there is a finite
family of subsets $\{S_i\}$ satisfying $S = \bigcup_ iS_i$ with
diam($S_i) \leq \eta + \varepsilon$. Therefore,
\begin{equation} \label{e2.8}
| x - y|_1 \leq \eta + \varepsilon \quad
\text{for all }x,y \in S_i.
\end{equation}
As $S$ and $S_i$ are precompact in $C_\omega ^0$, it follows that
there is a finite family of subsets $\{S_{ij}\}$ of $S_i$ such that
$S_i = \bigcup_j S_{ij}$ and
\begin{equation} \label{e2.9}
|x - y|_0 \leq \varepsilon \quad \text{for all } x,y \in S_{ij}.
\end{equation}
In addition, for any $x \in S$ and $t \in [0,\omega]$, we have
\begin{align*}
  |(\Phi x)(t)|&=  \int_{t}^{t+\omega}G(t,s)x(s)\Big[
\sum_{i=1}^n a_i(t)\int_{-T_i}^0x(t+\theta)\,\mathrm{d}\mu_i(\theta)\\
&\quad +\sum_{j=1}^m b_j(t)\int_{-\hat{T}_j}^0x'(t+\theta)\,\mathrm{d}\nu_j(\theta)
\Big]\,\mathrm{d}s\\
&\leq \frac{R^2}{1-\lambda}\int_{t}^{t+\omega}\Big[\sum_{i=1}^n a_i(s)
+\sum_{j=1}^m b_j(s)\Big]\,\mathrm{d}s:=H
\end{align*}
and
\begin{align*}
|(\Phi x)'(t)|&= \Big|a(t)(\Phi x)(t)-x(t)\Big[\sum_{i=1}^n a_i(s)
\int_{-T_i}^0x(s+\theta)\,\mathrm{d}\mu_i(\theta)\\
&\quad +\sum_{j=1}^m b_j(s)\int_{-\hat{T}_j}^0x'(s+\theta)\,
\mathrm{d}\nu_j(\theta)\Big]\Big|\\
&\leq a^MH+R^2\sum_{j=1}^m (a^M+b^M).
\end{align*}
Applying the Arzela-Ascoli Theorem, we know that $\Phi(S)$ is
precompact in $C_{\omega}^0$. Then, there is a finite family of
subsets $\{S_{ijk}\}$ of $S_{ij}$ such that
$S_{ij} =\bigcup_kS_{ijk}$ and
\begin{equation} \label{e2.10}
|\Phi x - \Phi y|_0\leq \varepsilon \quad \text{for all} x, y \in S_{ijk}.
\end{equation}
  From \eqref{e2.3}, \eqref{e2.5} and \eqref{e2.8}-\eqref{e2.10} and (H2), for any $x, y \in
S_{ijk}$, we obtain
\begin{align*}
&|(\Phi x)'-(\Phi y)'|_0 \\
&= \max_{t\in [0,\omega]}\Big\{\Big|a(t)(\Phi x)(t)-a(t)(\Phi y)(t)\\
&\quad -x(t)\Big[\sum_{i=1}^n a_i(t)\int_{-T_i}^0x(t+\theta)\,\mathrm{d}\mu_i(\theta)+
\sum_{j=1}^m b_j(t)\int_{-\hat{T}_j}^0x'(t+\theta)\,\mathrm{d}\nu_j(\theta)
\Big]\\
&\quad +y(t)\Big[\sum_{i=1}^n a_i(t)\int_{-T_i}^0y(t+\theta)\,\mathrm{d}\mu_i(\theta)+
\sum_{j=1}^m b_j(t)\int_{-\hat{T}_j}^0y'(t+\theta)\,\mathrm{d}\nu_j(\theta)
\Big]\Big|\Big\}\\
&\leq \max_{t\in
[0,\omega]}\{|a(t)[(\Phi x)(t)-(\Phi y)(t)]|\}\\
&\quad +\max_{t\in [0,\omega]}\Big\{\Big|x(t)\Big[
\sum_{i=1}^n a_i(t)\int_{-T_i}^0x(t+\theta)\,\mathrm{d}\mu_i(\theta)+
\sum_{j=1}^m b_j(t)\int_{-\hat{T}_j}^0x'(t+\theta)\,\mathrm{d}\nu_j(\theta)
\Big]\\
&\quad -y(t)\Big[\sum_{i=1}^n a_i(t)\int_{-T_i}^0y(t+\theta)\,\mathrm{d}\mu_i(\theta)+
\sum_{j=1}^m b_j(t)\int_{-\hat{T}_j}^0y'(t+\theta)\,\mathrm{d}\nu_j(\theta)\Big]\Big|\Big\}\\
&\leq a^M|(\Phi x)-(\Phi y)|_0\\
&\quad +\max_{t\in [0,\omega]}\Big\{\Big|x(t)\Big[\Big(
\sum_{i=1}^n a_i(t)\int_{-T_i}^0x(t+\theta)\,\mathrm{d}\mu_i(\theta)\\
&\quad +
\sum_{j=1}^m b_j(t)\int_{-\hat{T}_j}^0x'(t+\theta)\,\mathrm{d}\nu_j(\theta)
\Big)\\
&\quad -\Big(\sum_{i=1}^n a_i(t)\int_{-T_i}^0y(t+\theta)\,\mathrm{d}\mu_i(\theta)+
\sum_{j=1}^m b_j(t)\int_{-\hat{T}_j}^0y'(t+\theta)\,\mathrm{d}\nu_j(\theta)\Big)\Big]\Big|\Big\}\\
&\quad +\max_{t\in [0,\omega]}\Big\{\Big|y(t)\Big[
\sum_{i=1}^n a_i(t)\int_{-T_i}^0y(t+\theta)\,\mathrm{d}\mu_i(\theta)\\
&\quad +
\sum_{j=1}^m b_j(t)\int_{-\hat{T}_j}^0y'(t+\theta)\,\mathrm{d}\nu_j(\theta) \Big][x(t)-y(t)]\Big|\Big\}\\
&\leq a^M\varepsilon +R\max_{t\in
[0,\omega]}\Big\{\sum_{i=1}^n a_i(t)\int_{T_i}^0|x(s+\theta)-y(s+\omega)|\,\mathrm{d}\mu_i(s)\\
&\quad
+\sum_{j=1}^m b_j(t)\int_{\hat{T}_j}^0|x'(s+\theta)-y'(s+\omega)|\,\mathrm{d}\nu_j(s)
\\
&\quad +\varepsilon\max_{t\in
[0,\omega]}\Big\{\sum_{i=1}^n a_i(t)\int_{-T_i}^0y(t+\theta)\,\mathrm{d}\mu_i(\theta)+
\sum_{j=1}^m b_j(t)\int_{-\hat{T}_j}^0y'(t+\theta)\,\mathrm{d}\nu_j(\theta)\Big\}\\
&\leq a^M\varepsilon +R\varepsilon\sum_{i=1}^n  a_i^M
+R(\eta+\varepsilon)\sum_{j=1}^m b_j^M +R\varepsilon
\sum_{i=1}^n a_i^M
+R\varepsilon\sum_{j=1}^m b_j^M\\
&= R\eta\sum_{j=1}^m  b_j^M+\hat{H}\varepsilon,
\end{align*}
where $\hat{H}=a^M+2R\sum_{i=1}^n a_i^M+2R\sum_{j=1}^n b_j^M$.
 From the above equation and \eqref{e2.10},  we have
\begin{align*}
|\Phi  x - \Phi y|_1 &\leq
\Big(R\sum_{j=1}^m b_j^M\Big)\eta+\hat{H}\varepsilon\quad
\text{for all } x, y \in S_{ijk}.
\end{align*}
 Since $\varepsilon$ is arbitrary small, it follows that
\[
 \alpha_{C_{\omega}^1}(\Phi  (S))\leq
 \Big(R\sum_{j=1}^m b_j^M\Big)\alpha_{C_{\omega}^1}(S).
\]
 Therefore, $\Phi $ is strict-set-contractive. The proof of
 Lemma \ref{lem2.3} is complete.
\end{proof}


\section{Main Result}

Our main result of this paper is as follows.

\begin{theorem} \label{thm3.1}
Assume that
(H1)--(H3), (H5) hold.
\begin{itemize}
  \item [(i)] If $a^M \leq 1$, then system \eqref{e1.2} has at
least one positive $\omega$-periodic solution.
  \item [(ii)] If (H4) holds and $a^M > 1$, then system \eqref{e1.2}
has at least one positive $\omega$-periodic solution.
\end{itemize}
\end{theorem}

\begin{proof}
We only need to prove (i), since the proof of (ii) is similar.
Let $R=\frac{1-\lambda}{\lambda^2\Delta}$ and $0 < r <
\frac{\lambda(1-\lambda)}{\Pi}$. Then we have $0< r< R$.
 From Lemma \ref{lem2.2} and \ref{lem2.3}, we know that $\Phi$ is
strict-set-contractive on $K_{r,R}$.
In view of \eqref{e2.5}, we see that if there exists $x^*\in K$
such that $\Phi x^* = x^*$, then $x^*$ is one positive
$\omega$-periodic solution of \eqref{e1.2}. Now, we shall prove that
condition (ii) of Lemma \ref{lem2.1} hold.

First, we prove that $\Phi  x \ngeq x$, for all $x \in K$,
$|x|_1=r$. Otherwise, there exists $x \in K$, $|x|_1= r$
 such that $\Phi x\geq x$. So $|x| > 0$ and $\Phi  x - x \in K$, which
implies that
\begin{equation} \label{e3.1}
(\Phi  x)(t) - x(t) \geq \lambda |\Phi x - x |_1 \geq 0 \quad
\text{for all } t \in [0,\omega].
\end{equation}
Moreover, for $t \in [0,\omega]$, we
have
\begin{equation} \label{e3.2}
\begin{aligned}
  (\Phi x)(t)
&=  \int_{t}^{t+\omega}G(t,s)x(s)\Big[
\sum_{i=1}^n a_i(s)\int_{-T_i}^0x(s+\theta)\,\mathrm{d}\mu_i(\theta)\\
&\quad +
\sum_{j=1}^m b_j(s)\int_{-\hat{T}_j}^0x'(s+\theta)\,\mathrm{d}\nu_j(\theta)
\Big]\,\mathrm{d}s\\
&\leq \frac{1}{1-\lambda}r|x|_0\int_{0}^{\omega}\Big[\sum_{i=1}^n a_i(s)
+\sum_{j=1}^m b_j(s)\Big]\,\mathrm{d}s\\
&= \frac{1}{1-\lambda}\Pi|x|_0
< \lambda|x|_0.
\end{aligned}
\end{equation}
In view of \eqref{e3.1} and \eqref{e3.2}, we have
\[
|x|_0\leq |\Phi x|_0<\lambda|x|_0<|x|_0,
\]
 which is a contradiction. Finally, we prove that
 $\Phi x\nleq x$ for all $x\in K$, $|x|_1=R$ also holds.
For this case, we only need to prove that
\[
\Phi  x \nless x \quad   x \in K,\, |x|_1= R.
\]
Suppose, for the sake of contradiction, that there exists $x \in K$
and $|x|_1= R$ such that $\Phi  x < x$.
Thus $x - \Phi x \in K \setminus \{0\}$. Furthermore,
for any $t \in [0,\omega]$, we have
\begin{equation} \label{e3.3}
    x(t)-(\Phi x)(t)\geq \lambda|x-\Phi x|_1>0.
\end{equation}
In addition, for any $t \in [0,\omega]$, we find
\begin{equation} \label{e3.4}
\begin{aligned}
(\Phi  x)(t)&= \int_{t}^{t+\omega}G(t,s)x(s)\Big[
\sum_{i=1}^n a_i(s)\int_{-T_i}^0x(s+\theta)\,\mathrm{d}\mu_i(\theta)\\
&\quad + \sum_{j=1}^m b_j(s)\int_{-\hat{T}_j}^0x'(s+\theta)\,
\mathrm{d}\nu_j(\theta)\Big]\,\mathrm{d}s\\
&\geq \frac{\lambda}{1-\lambda}|x|_0^2\int_{0}^{\omega}\Big[\lambda
\sum_{i=1}^n a_i(s) -
\sum_{j=1}^m b_j(s)\Big]\,\mathrm{d}s\\
&= \frac{\lambda^2}{1-\lambda} \Delta R^2
= R.
\end{aligned}
\end{equation}
 From \eqref{e3.3} and \eqref{e3.4}, we obtain
$|x|>|\Phi  x|_0 \geq R$,
which is a contradiction. Therefore, conditions (i) and (ii)
hold. By Lemma \ref{lem2.1}, we see that $\Phi$ has at least one nonzero
fixed point in $K$. Therefore, \eqref{e1.2} has at least one positive
$\omega$-periodic solution. The proof of Theorem \ref{thm3.1} is complete.
\end{proof}

We remark that from the proof of our results, if some (or all)
$\hat{T}_j (j=1,2,\dots,n)$ are replaced by $\infty$ the conclusion
of Theorem \ref{thm3.1} remains valid.

As an example of \eqref{e1.1}, consider the  equation
\begin{equation} \label{e3.5}
 x'(t)= x(t)\Big[\frac{1+\cos
t}{4\pi}-(5-2\sin t)\int_{-1}^0 x(t+\theta){\rm d}
\theta-\frac{1-\sin
t}{20}\int_{-1}^0x'(t+\theta)\,\mathrm{d}\theta\Big].
\end{equation}
 Obviously,
\[
a(t)=\frac{1+\cos t}{4\pi},\quad
a_1(t)=5-2\sin t,\quad  b_1(t)=\frac{1-\sin t}{20}.
\]
Furthermore, we have
\begin{gather*}
\lambda=e^{-\frac{1}{2}},\,\max_{0\leq t\leq 2\pi}\{\lambda
a_1(t)-b_1(t)\}>1.7>0,\\
 \Delta=\int_{0}^{2\pi}[\lambda a_1(s)-b_1(s)]\,\mathrm{d}s
=10\pi e^{-\frac{1}{2}}-\frac{1}{10}\pi > 18,
\\
\max_{0\leq t\leq 2\pi}\{ a_1(t)+b_1(t)\}=\frac{71}{10},
\\
(1+a^m)\frac{\lambda^2}{1-\lambda}\Delta>16>\frac{71}{10}>\max_{0\leq
t\leq 2\pi}\{ a_1(t)+b_1(t)\},
\\
\frac{1-\lambda}{\lambda^2\Delta}b_1^M<5.95\times 10^{-3}<1.
\end{gather*}
Hence, (H1)--(H3), (H5) hold and $a^M\leq 1$. According to
Theorem \ref{thm3.1}, system \eqref{e3.5} has at least one positive $2\pi$-periodic
solution.

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\end{document}