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

\AtBeginDocument{{\noindent\small
\emph{Electronic Journal of Differential Equations},
Vol. 2008(2008), No. 38, pp. 1--8.\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 2008 Texas State University - San Marcos.}
\vspace{9mm}}

\begin{document}
\title[\hfilneg EJDE-2008/38\hfil Positive periodic solutions]
{Positive periodic solutions of neutral functional
differential equations with a parameter and impulse}

\author[X. Fan, Y. Li\hfil EJDE-2008/38\hfilneg]
{Xuanlong Fan, Yongkun Li}

\address{Xuanlong Fan \newline
Department of Mathematics\\
Yunnan University\\
Kunming, Yunnan 650091, China} 
\email{fanxuanlong@126.com}

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

\thanks{Submitted December 16, 2007. Published March 14, 2008.}
\thanks{Supported by grants 10361006 and 2003A0001M from the
National Natural Sciences \hfill\break\indent
 Foundation of China, and of Yunnan Province}
\subjclass[2000]{34K13, 34K40} 
\keywords{Periodic solution; functional differential
 equation; fixed point; cone}

\begin{abstract}
 In this paper, we consider first-order neutral differential
 equations with a parameter and impulse in the form of
 \begin{gather*}
 \frac{d}{dt}[x(t)-c x(t-\gamma)]=-a(t)g(x(h_1(t)))x(t)+\lambda
 b(t) f\big(x(h_2(t))\big),\quad t\neq t_j;\\
 \Delta \big[x(t)-c x(t-\gamma)\big]=I_j\big(x(t)\big),\quad
 t=t_j,\; j\in\mathbb{Z}^+.
 \end{gather*}
 Leggett-Williams fixed point
 theorem, we prove the existence of three positive periodic solutions.
\end{abstract}

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


\section{Introduction}

The  existence of periodic solutions of delay
differential equations with or without impulses has been a focus
of theoretical and practical importance because time-delay
occurs  areas such as  mechanics, physics, biology,
economy, population dynamic models, large-scale systems, automatic
control systems, neural networks, chaotic systems, and so on.
The existence of periodic solutions of time-delay systems
with or without impulses has been extensively studied by
Many researchers have studied this problem; se for example
\cite{b1,d1,d2,k1,l1, l2,l3,l4,l5,l6,l7,l8, m1, s1,w1,w2,z1,z2}
and references therein.
However, relatively few papers have been
published on the existence of periodic solutions for neutral
functional differential equations.


In this paper, we are concerned with the  neutral
differential equation
\begin{equation}  \label{e1.1}
\begin{gathered}
\frac{d}{dt}\big[x(t)-c x(t-\gamma)\big]
=-a(t)g\big(x(h_1(t))\big)x(t)+\lambda b(t) f\big(x(h_2(t))\big),\quad
t\neq t_j,\\
\Delta \big[x(t)-c x(t-\gamma)\big]=I_j\big(x(t)\big),\quad
t=t_j,\; j\in\mathbb{Z}^+,
\end{gathered}
\end{equation}
where $\lambda>0$ is a positive parameter and there exists a
positive constant $q$ such that $t_{j+q}=t_j+\omega$,
$I_{j+q}(x(t_{j+q}))=I_j(x(t_j))$, $j\in \mathbb{Z}^+$. Without loss of
generality, we assume that $[0,\omega]\cap\{t_j, j\in
\mathbb{Z}^+\}=\{t_1,t_2,\cdots,t_q\}$.

In this paper, we will use the following assumptions:
\begin{itemize}
  \item [(H1)]
$a\in C(\mathbb{R},[0,+\infty))$ is  $\omega$-periodic and there
exists $t_1^*\in (0,\omega)$ such that $a(t_1^*)> 0$;
  \item [(H2)]
$b\in C(\mathbb{R},[0,+\infty))$ is $\omega$-periodic and there
exists $t_2^*\in(0,\omega)$ such that $b(t_2^*)>0$;
  \item [(H3)]
$h_1(t),\,h_2(t)\in C(\mathbb{R},\mathbb{R})$ are $\omega$-periodic;
   \item [(H4)]
$g\in C([0,\infty),[0,\infty))$, $I_j\in C([0,\infty),[0,\infty))$
and $f\in C([0,\infty),[0,\infty))$ are continuous, $0<l\leq
g(u)<L<\infty$ for all $u>0$, $l,L$ are two positive constants.
\end{itemize}

Using Leggett-Williams fixed point
theorem \cite{l1} to show the existence of at least three positive
periodic solutions of \eqref{e1.1}. To the best of the authors' knowledge,
few authors discuss at least three positive periodic solutions for
neutral functional differential equations with impulses and
parameters.


\section{Preliminaries}

To obtain the existence of  periodic solutions of system
\eqref{e1.1}, we first make the following preparations.

Let $\beta=\int_0^\omega a(s)\,ds$, where $a$ is a continuous
$\omega$-periodic function. In what follows, we set
$$
\mathbb{X}=\big\{x\in C(\mathbb{R},\mathbb{R}): x(t+\omega)=x(t)\big\}
$$
and define $\|x\|=\max\{|x(t)|:\, t\in[0,\omega]\}$. Then
$(\mathbb{X},\|\cdot\|)$ is a Banach space. Let
 $A: \mathbb{X}\to \mathbb{X}$ be defined by
$$
(Ax)(t)= x(t)-c x(t-\gamma).
$$

\begin{lemma}[\cite{l9}] \label{lem2.1}
If $|c|\neq 1$, then $A$ has continuous bounded inverse $A^{-1}$ and
for all $x\in \mathbb{X}$,
\begin{equation} \label{e2.1}
(A^{-1}x)(t)=\begin{cases}
                     \sum_{j\geq 0}c^jx(t-j\gamma), &\text{if  }|c|<1, \\
                    -\sum_{j\geq 1}c^{-j}x(t+j\gamma), &\text{if }
                    |c|>1\,.
                   \end{cases}
\end{equation}
Then
\[
\|A^{-1}x\|\leq \frac{\|x\|}{|1-|c||}.
\]
\end{lemma}
To establish the existence of periodic solutions of \eqref{e1.1}, we
first consider the  system
\begin{equation} \label{e2.2}
\begin{gathered}
\frac{d}{dt}u(t)=-a(t)g\big((A^{-1}u)(h_1(t))\big)(A^{-1}u)(t)
+\lambda b(t) f\big((A^{-1}u)(h_2(t))\big),\quad t\neq t_j,\\
\Delta u(t)=I_j\big((A^{-1}u)(t)\big),\quad t=t_j,\, j\in\mathbb{Z}^+,
\end{gathered}
\end{equation}
where $A^{-1}$ is defined by \eqref{e2.1}. By Lemma \ref{lem2.1},
we conclude the following result.

\begin{lemma} \label{lem2.2}
$u(t)$ is an $\omega$-periodic solution of  \eqref{e2.2} if and only if
$(A^{-1}u)(t)$ is an $\omega$-periodic solution of \eqref{e1.1}.
\end{lemma}

Let $\mathbb{X}$ be a Banach space and $K$ is a closed, nonempty
subset of $\mathbb{X}$. $K$ is a cone provided that
\begin{itemize}
  \item [(i)]
$\alpha_1 u+\beta_1 y\in K$ for all $u,v\in K$ and
$\alpha_1,\beta_1\geq 0;$
  \item [(ii)]
  $u,-u\in K$ imply $u=0$.
\end{itemize}

Define $K_r=\{x\in K:\|x\|\leq r\}$. Let $\alpha(x)$ denote the
positive continuous concave functional on $K$, that is
$\alpha: K\to [0,\infty)$ is continuous and satisfies
\[
\alpha(\lambda x+(1-\lambda)y)\geq
\lambda\alpha(x)+(1-\lambda)\alpha(y)\quad\text{for all }
 x,y\in K, 0\leq \lambda\leq 1,
 \]
and we denote the set $K(\alpha,a_1,b_1)=\{x\in K: a_1\leq
\alpha(x), \|x\|\leq b_1\}$.

\begin{lemma}[\cite{l1}] \label{lem2.3}
Let $K$ be a cone of the real Banach space $\mathbb{X}$ and
$\Phi: K_{c_3} \to K_{c_3}$ be a completely continuous operator, and
suppose that there exists a concave positive functional $\alpha$
with that $\alpha(x)\leq \|x\|$ for $x\in K$ and numbers $c_1,c_2,c_3,c_4$
with $0<c_4<c_1<c_2\leq c_3$, satisfying the following conditions:
\begin{itemize}
  \item [(1)]
$\{x\in K(\alpha,c_1,c_2): \alpha(x)>c_1\}\neq \emptyset$ and
$\alpha(\Phi x)>c_1$ if $x\in K(\alpha,c_1,c_2)$;
  \item [(2)]
  $\|\Phi x\|<c_4$ if $x\in K_{c_4}$;
\item [(3)]
  $\alpha(\Phi x)>c_1$ for all $x\in K(\alpha,c_1,c_2)$ with
$\|\Phi x\|>c_2$.
\end{itemize}
Then $\Phi$ has at least three fixed points in $K_{c_3}$.
\end{lemma}

Aiming to apply Lemma \ref{lem2.2} to \eqref{e2.2}, we rewrite \eqref{e2.2} as
\begin{equation} \label{e2.3}
\begin{gathered}
\begin{aligned}
\frac{d}{dt}u(t)&=-a(t)g\big((A^{-1}u)(h_1(t))\big)u
(t)+\big[a(t)\widehat{G}\big(u(t),u(h_1(t))\big)\\
&\quad +\lambda b(t) f\big((A^{-1}u)(h_2(t))\big)\big],\quad t\neq t_j,
\end{aligned}\\
\Delta u(t)=I_j\big((A^{-1}u)(t)\big),\quad  t=t_j,\, j\in\mathbb{Z}^+,
\end{gathered}
\end{equation}
where
\begin{align*}
\widehat{G}(u(t),u(h_1(t)))
&= g\big((A^{-1}u)(h_1(t))\big)\big[u(t)-(A^{-1}u)(t)\big]\\
&= -c g\big((A^{-1}u)(h_1(t))\big)(A^{-1}u)(t-\gamma).
\end{align*}

The following lemma is fundamental in our discussion. Since the
method is similar to that in the literature \cite{n1}, we omit the proof.

\begin{lemma}
$x(t)$ is an $\omega$-periodic solution of  \eqref{e1.1} is equivalent
to $u(t)$ is an $\omega$-periodic solution of
\begin{equation} \label{e2.4}
\begin{aligned}
u(t)&=\int_t^{t+\omega}G(t,s)\Big[a(t)\widehat{G}\big(u(t),u(h_1(t))\big)
+\lambda b(t) f\big((A^{-1}u)(h_2(t))\big)\Big]\,ds \\
&\quad +\sum_{j:t_j\in[0,\omega]}G(t,t_j)I_j\big((A^{-1}u)(t_j)\big),
\end{aligned}
\end{equation}
where
\[
G(t,s)=\frac{e^{\int_{t}^{s}a(\theta)g((A^{-1}x)(h_1(\theta))\,d\theta}}
{e^{\int_{0}^{\omega}a(\theta)g((A^{-1}x)(h_1(\theta))\,d\theta}-1},\quad
s\in [t,t+\omega].
\]
\end{lemma}

For $u\in \mathbb{X}$ and $t\in \mathbb{R}$, let
 the map $\Phi $ be defined  by
\begin{equation} \label{e2.5}
\begin{aligned}
 (\Phi u)(t)&=\int_t^{t+\omega}G(t,s)\Big[a(t)\widehat{G}
 \big(u(t),u(h_1(t))\big)+\lambda b(t)
f\big((A^{-1}u)(h_2(t))\big)\Big]\,ds \\
&\quad +\sum_{j:t_j\in[0,\omega]} G(t,t_j)I_j\big((A^{-1}u)(t_j)\big)
\end{aligned}
\end{equation}
It is easy to see that
$G(t+\omega,s+\omega)=G(t,s)$ and
\begin{equation} \label{e2.6}
\frac{1}{\sigma^L-1} \leq G(t,s)
\leq \frac{\sigma^l}{\sigma^l-1},\quad s\in [t,t+\omega],
\end{equation}
where $\sigma=\exp\big(-\int_{0}^{\omega}a(\theta)\,d\theta\big)$.
Define the cone $K$ in $\mathbb{X}$ by
\begin{align*}
K=\Big\{ u\in \mathbb{X}: u(t)\geq \delta\|u\|,\ t\in
[0,\omega]\Big\},
\end{align*}
where $0<\delta=\frac{(\sigma^l-1)}{\sigma^l(\sigma^L-1)}<1$.

The following lemma is useful in the proofs of our main results.
Since the method is similar to that in the literature \cite{l5},
we omit the proof.

\begin{lemma} \label{lem2.5}
If $c\in(-\delta,0]$ and $u\in K$. Then
\[
l|c|\frac{\delta-|c|}{1-c^2}\|u\|\leq \widehat{G}\big(u(t),
u(h_1(t))\big)\leq L\frac{|c|}{1-|c|}\|u\|.
\]
\end{lemma}

\begin{lemma}
Assume that {\rm (H1)--(H4)} and $c\in(-\delta,0]$ hold, then
$\Phi$ maps $K$ into $K$.
\end{lemma}

\begin{proof}
 For any $u\in K$, it is clear that $\Phi u\in C(\mathbb{R},\mathbb{R})$,
we have
\begin{align*}
(\Phi u)(t+\omega)
&= \int_t^{t+\omega}G(t+\omega,s+\omega)
 \Big[a(t+\omega)\widehat{G}\big(u(t+\omega),u(h_1(t+\omega))\big)\\
&\quad + \lambda b(t+\omega) f\big((A^{-1}u)(h_2(t+\omega))\big)\Big]\,ds
\\
&\quad + \sum_{j:t_j\in[0,\omega]}G(t+\omega,t_j+\omega)I_j
 \big((A^{-1}u)(t_j+\omega))\big)\\
&= \int_t^{t+\omega}G(t,s)\Big[a(t)\widehat{G}\big(u(t),u(h_1(t))\big)
+\lambda b(t) f\big((A^{-1}u)(h_2(t))\big)\Big]\,ds\\
&\quad + \sum_{j:t_j\in[0,\omega]}G(t,t_j)I_j\big((A^{-1}u)(t_j)\big)\\
&= (\Phi u)(t).
\end{align*}
Thus, $(\Phi u)(t+\omega)=(\Phi u)(t)$, $t\in \mathbb{R}$. So that
$\Phi u \in \mathbb{X}$. Since $c\in(-\delta,0]$, it follows that
$G(u(t),u(h_1(t)))\geq 0$ for $t\in\mathbb{R}$. In view of \eqref{e2.5}, \eqref{e2.6},
for $u \in K$, $t\in [0,\omega]$, we have
\begin{align*}
\|(\Phi u)\|
&\leq \frac{\sigma^l} {\sigma^l-1}\Big(\int_t^{t+\omega}
 \Big[a(t)\widehat{G}\big(u(t),u(h_1(t))\big)+\lambda b(t)
 f\big((A^{-1}u)(h_2(t))\big)\Big]\,ds\\
&\quad + \sum_{j:t_j\in[0,\omega]}I_j\big((A^{-1}u)(t_j)\big)\Big)
\end{align*}
and
\begin{align*}
|(\Phi u)(t)|
&\geq \frac{1}{\sigma^L-1}\Big(\int_t^{t+\omega}
 \Big[a(t)\widehat{G}\big(u(t),u(h_1(t))\big)
 +\lambda b(t) f\big((A^{-1}u)(h_2(t))\big)\Big]\,ds\\
&\quad + \sum_{j:t_j\in[0,\omega]}I_j\big((A^{-1}u)(t_j)\big)\Big)\\
&\geq \delta\|\Phi u\|.
\end{align*}
Hence, $\Phi  x \in K$. The proof  is complete.
\end{proof}

It is easy to see that $\Phi$ is continuous and bounded.
From Lemma \ref{lem2.5}, we know that $\Phi$ maps bounded
sets into relatively compact sets. Furthermore,
 by the theorem of Ascoli-Arzela \cite{r1}, it is easy
to prove that the function $\Phi$ is completely continuous.

For convenience in the following discussion, we introduce the
following notation:
\begin{gather*}
f^0=\limsup_{v\to0}\frac{f(v)}{v},\quad
I^0=\limsup_{v\to0}\sum_{j=1}^q\frac{I_j(v)}{v}, \\
f^\infty=\limsup_{v\to\infty}\frac{f(v)}{v}, \quad
I^\infty=\limsup_{v\to\infty}\sum_{j=1}^q\frac{I_j(v)}{v}
\end{gather*}
and for $c_2>0$,
\[
I_{(c_2)}=\min_{\delta c_2\leq v\leq c_2}\sum_{j=1}^qI_j(v).
\]

\section{Main Result}

Our main result of this paper is stated as follows.

\begin{theorem} \label{thm3.1}
 Assume that {\rm (H1)-(H4)} and $c\in(-\eta,0 ]$, where
\[
\eta:=\min\big\{\delta,\, 1-\frac{l\sigma^l
\beta}{(\sigma^l-1)+L\sigma^l\beta}\big\}.
\]
Then there exists a number $c_2>0$ such that
\begin{itemize}
  \item [(i)] For $\delta c_2\leq u\leq c_2$, $t\in R$,
\[
f\big((A^{-1}u)(h_2(s))\big)>
\frac{\sigma^l(\sigma^L-1)}{\sigma^l-1}u
-\frac{\sigma^l}{\sigma^l-1}I_{(c_2)}
- \frac{\sigma^l}{\sigma^l-1}l|c|\frac{\delta-|c|}{1-c^2}\beta u\,;
\]

\item [(ii)]
\[
f^0+I^0<\frac{(1-|c|)(\sigma^l-1)}{L\sigma^l|c|}-\beta, \quad
f^\infty+I^\infty<\frac{(1-|c|)(\sigma^l-1)}{L\sigma^l|c|}-\beta\,.
\]
\end{itemize}
Then system \eqref{e1.1} has at least three positive
$\omega$-periodic solutions for
\[
\frac{\sigma^l-1}{\sigma^l\int_{t}^{t+\omega}b(s)\,\mathrm{d}s}
<\lambda<\frac{1}{\int_{t}^{t+\omega}b(s)\,\mathrm{d}s}.
\]
\end{theorem}

\begin{proof}
 By the condition
$f^\infty+I^\infty<\frac{(1-|c|)(\sigma^l-1)}{L\sigma^l|c|}-\beta$
of (ii),  one can find that for
\[
\frac{\frac{(1-|c|)(\sigma^l-1)}{L\sigma^l|c|}
 -\beta-(f^\infty+I^\infty)}{2}>\varepsilon>0,
\]
 there exists a $C_0>c_2$ such that
\[
\limsup_{v\to\infty}f(v)\leq(f^\infty+\varepsilon)v,
\quad
 \limsup_{v\to\infty}\sum_{j=1}^qI_j(v)\leq (I^\infty+\varepsilon)v,
\]
 where $u> C_0$.
Let $C_1=C_0/\delta$, if $u\in K,\, \|u\|> C_1$, thus $u>C_0$ and we
have
\begin{equation} \label{e3.1}
\begin{aligned}
(\Phi u)(t)
&= \int_t^{t+\omega}G(t,s)\Big[a(t)\widehat{G}\big(u(t),u(h_1(t))\big)
 +\lambda b(t) f\big((A^{-1}u)(h_2(t))\big)\Big]\,ds \\
&\quad + \sum_{j:t_j\in[0,\omega]}G(t,t_j)I_j\big((A^{-1}u)(t_j)\big) \\
&\leq \frac{\sigma^l}{\sigma^l-1}\Big\{L\frac{|c|}{1-|c|}\|u\|\int_{t}^{t+\omega}a(t)ds \\
&\quad + (f^\infty+\varepsilon)L\frac{|c|}{1-|c|}\|u\|
 \int_{t}^{t+\omega}\lambda b(s)\,\mathrm{d}s
 +(I^\infty+\varepsilon)L\frac{|c|}{1-|c|}\|u\|\Big\}
 \\
&= \frac{L\sigma^l|c|}{(1-|c|)(\sigma^l-1)}\Big\{\beta
+(f^\infty+\varepsilon)\int_{t}^{t+\omega}\lambda
b(s)\,\mathrm{d}s+(I^\infty+\varepsilon)\Big\}\|u\| \\
&< \|u\|.
\end{aligned}
\end{equation}
Take  $K_{C_1}=\{u\in K:\|u\|\leq C_1\}$, then the set $K_{C_1}$
is a bounded set.  Since $\Phi$ is completely continuous, $\Phi$
maps bounded sets into bounded sets and there exists a number $C_2$
such that
 \[
\|\Phi u\|\leq C_2\quad\text{for all } u\in K_{C_1}.
\]
If $C_2\leq C_1$, we obtain that $\Phi: K_{C_1}\to K_{C_1}$ is
completely continuous. If $C_1< C_2$, then from (3.1),
we know that for any $u\in K_{C_2}\backslash K_{C_1}$, $\|u\|>C_1$
and $\|\Phi u\|<\|u\|<C_2$ hold. Then we obtain
$\Phi: K_{C_2}\to K_{C_2}$ is completely continuous.
Now, take $c_3=\max\{C_1,C_2\}$,
obviously $c_3>c_2$ and $\Phi: K_{c_3}\to K_{c_3}$ is completely
continuous.

Define the positive continuous concave functional
$\alpha(u)=\min_{t\in[0,\omega]}\big\{|u(t)|\}$. First, we let
$c_1=\delta c_2$ and take $u\equiv\frac{c_1+c_2}{2}$,
$u\in K(\alpha,c_1,c_2)$, $\alpha(u)>c_1,$ then the set
$\{u\in K(\alpha,c_1,c_2)\}\neq \emptyset$. And  by
(i), if $u\in K(\alpha,c_1,c_2),$ then $\alpha(u)\geq c_1,$ and we have
\begin{align*}
&\alpha(\Phi u) \\
&= \min_{t\in[0,\omega]}\Big\{
\int_t^{t+\omega}G(t,s)\Big[a(t)\widehat{G}\big(u(t),u(h_1(t))\big)+\lambda
b(t) f\big((A^{-1}u)(h_2(t))\big)\Big]\,ds \\
&\quad + \sum_{j:t_j\in[0,\omega]}G(t,t_j)I_j\big((A^{-1}u)(t_j)\big)\Big\}
 \\
&\geq \frac{1}{\sigma^L-1}\Big\{l|c|\frac{\delta-|c|}{1-c^2}u\beta
+\min_{t\in[0,\omega]}\Big\{\int_t^{t+\omega}\lambda b(s)
f\big((A^{-1}u)(h_2(s))\big)\,ds\Big\}+I_{(c_2)}\Big\}
 \\
&> \frac{1}{\sigma^L-1}\Big\{l|c|\frac{\delta-|c|}{1-c^2}u\beta
+\Big[\frac{\sigma^l(\sigma^L-1)}{\sigma^l-1}a(u)
-\frac{\sigma^l}{\sigma^l-1}I_{(c_2)}\\
&\quad -\frac{\sigma^l}{\sigma^l-1}l|c|\frac{\delta-|c|}{1-c^2}\beta
u\Big]\lambda\int_t^{t+\omega}b(s)ds+I_{(c_2)}\Big\} \\
&= \alpha(x)\geq c_1.
\end{align*}
Thus condition (1) of Lemma \ref{lem2.3} holds.

Secondly, by the inequality
$f^0+I^0<\frac{(1-|c|)(\sigma^l-1)}{L\sigma^l|c|}-\beta$ in condition
of (ii),  one can find that for
\[
\frac{\frac{(1-|c|)(\sigma^l-1)}{L\sigma^l|c|}-\beta-(f^0+I^0)}{2}
>\varepsilon>0,
\]
 there exists $c_4$, with $0<c_4<c_1$ such that
\[
\limsup_{v\to0}f(v)\leq(f^0+\varepsilon)v,
\, \, \limsup_{v\to0}\sum_{j=1}^qI_j(v)\leq
(I^0+\varepsilon)v,
\]
where $0\leq v\leq c_4$. If $u\in K_{c_4}=\Big\{u\Big|\|u\|\leq c_4\Big\}$,
then we have
\begin{align*}
(\Phi u)(t)
&= \int_t^{t+\omega}G(t,s)\Big[a(t)\widehat{G}\big(u(t),u(h_1(t))\big)
 +\lambda b(t) f\big((A^{-1}u)(h_2(t))\big)\big]\,ds \\
&\quad + \sum_{j:t_j\in[0,\omega]}G(t,t_j)I_j\big((A^{-1}u)(t_j)\big) \\
&\leq \frac{\sigma^l}{\sigma^l-1}\Big\{L\frac{|c|}{1-|c|}\|u\|
 \int_{t}^{t+\omega}a(t)ds \\
&\quad + (f^0+\varepsilon)L\frac{|c|}{1-|c|}\|u\|
 \int_{t}^{t+\omega}\lambda b(s)\,\mathrm{d}s
 +(I^0+\varepsilon)L\frac{|c|}{1-|c|}\|u\|\Big\}
 \\
&= \frac{L\sigma^l|c|}{(1-|c|)(\sigma^l-1)}\Big\{\beta
+(f^0+\varepsilon)\int_{t}^{t+\omega}\lambda
b(s)\,\mathrm{d}s+(I^0+\varepsilon)\Big\}\|u\| \\
&< \|u\|\leq c_4.
\end{align*}
That is, condition (2) of Lemma \ref{lem2.3} holds.

Finally, if $x\in K(\alpha,c_1,c_3)$ with $\|\Phi u\|>c_2,$ by the
definition of the cone $K$, we have
\[
\Phi u\geq \delta\|\Phi u\|>\delta c_2=c_1.
\]
Thus condition (3) of Lemma \ref{lem2.3} holds. Therefore, by
Lemma \ref{lem2.3}, we
obtain that the operator $\Phi$ has at least three fixed points in
$K_{c_3}$. From Lemma \ref{lem2.2}, we know that \eqref{e1.1} has at least three
fixed points in $K_{c_3}$. The proof of Theorem \ref{thm3.1} is complete.
\end{proof}

\begin{corollary} \label{coro3.1}
The conclusion in Theorem \ref{thm3.1}, sis still true when (ii) is replaced by
\begin{itemize}
  \item [(ii$^*$)] $f^0=0$, $\widehat{f}^0=0$, $f^\infty=0$,
$\widehat{f}^\infty=0$.
\end{itemize}
\end{corollary}

\section{An example}

   Consider the problem
\begin{equation} \label{e4.1}
\begin{gathered}
\frac{d}{dt}\big[x(t)-\frac{1}{3}
x(t-\frac{\pi}{2})\big]=-\frac{1}{2\pi}(\frac{1}{3}+e^{-x(t)})x(t)
+\lambda (1-\sin t)x^2(t) e^{-x(t)},\quad t\neq t_j,\\
\Delta \big[x(t)-\frac{1}{3}
x(t-\frac{\pi}{2})\big]=0.1x^3(t_j)e^{-3x(t_j)},\quad
t=t_j,\;j\in\mathbb{Z}^+,
\end{gathered}
\end{equation}
where $\lambda$ is nonnegative parameter. Take
$\gamma=\frac{\pi}{2}$, $c=\frac{1}{3}$, $a(t)=\frac{1}{2\pi}$,
$b(t)=1-\sin t$, $j=2k$, $k=1,2,\ldots$,
$g(x(h_1(t)))=\frac{1}{3}+e^{-x(t)}$, and
$f(x(h_2(t)))=x^2(t) e^{-x(t)}$. Clearly, $L=\frac{4}{3}$,
$l=\frac{1}{3}$ and $\beta=1$.
According to Corollary \ref{coro3.1}, Equation \eqref{e4.1} has at least
three positive periodic solutions.

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\end{document}