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

\AtBeginDocument{{\noindent\small
\emph{Electronic Journal of Differential Equations},
Vol. 2011 (2011), No. 97, 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}
\thanks{\copyright 2011 Texas State University - San Marcos.}
\vspace{9mm}}

\begin{document}
\title[\hfilneg EJDE-2011/97\hfil Multiple periodic solutions]
{Multiple periodic solutions for impulsive  functional 
 differential equations with feedback control}

\author[P. Guo, Y. Liu\hfil EJDE-2011/97\hfilneg]
{Peilian Guo, Yansheng Liu}  % in alphabetical order

\address{Peilian Guo \newline
Department of Mathematics, Shandong Normal University, 
Jinan, 250014, China}
\email{guopeilian0127@163.com}

\address{Yansheng Liu \newline
Department of Mathematics, Shandong Normal University,
Jinan, 250014,  China}
\email{yanshliu@gmail.com}

\thanks{Submitted April 8, 2011. Published July 30, 2011.}
\subjclass[2000]{34K13, 34K45}
\keywords{Impulse; functional differential equations;
 feedback control; \hfill\break\indent periodic solutions}

\begin{abstract}
 Using the well known Leggett-Williams fixed point theorem, we
 study the existence of periodic solutions for a class of
 impulsive functional equations with feedback control.
 The main results are illustrated with two examples.
\end{abstract}

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

\section{Introduction}

Consider the  impulsive functional differential equation with
feedback control
\begin{equation} \label{e1.1}
\begin{gathered}
x'(t)= -r(t)x(t)+F(t,x(t-\tau_{1}(t)),x(t-\tau_{2}(t)),\dots,
x(t-\tau_{n}(t)),u(t-\zeta(t)));\\
u'(t)= -\eta(t)u(t)+h(t)x(t-\sigma(t));\\
I_{k}(x(t_{k}))=\Delta x|_{t=t_{k}},\quad k=1,2,\dots,m,
\end{gathered}
\end{equation}
 where $\Delta x\big|_{t=t_k}=x(t_{k}+0)-x(t_{k}-0)$,
$0\leq t_1<t_2<\dots<t_m<\omega$, and $F\in C(\mathbb{R}^{n+2},[0,\infty))$,
$r,\eta,h\in C(\mathbb{R},(0,\infty))$, $\sigma,\zeta,\tau_{i}\in C(\mathbb{R},\mathbb{R})$,
$i=1,2,\dots,n$, $I_{k}\in C(\mathbb{R},[0,+\infty))$,
$k=1,2,\dots,m$. All of the above functions are $\omega$-periodic
 in $t$, $\omega>0$ is a constant.

The existence of periodic solutions of functional differential equations
with feed back has been studied extensively by many authors and  the
methods used are coincidence degree theory, Schauder fixed point
theorems,  Krasnoselskii's fixed point theorem, and upper-lower
solutions method (see \cite{c1,l1,l2,l3,l5,l6,l8,w1}, and the
references therein).

Using the Krasnoselskii's fixed point theorem, Li and Wang \cite{l1}
investigated the existence of positive periodic solutions of the
delay differential system with feedback control
\begin{equation} \label{e1.2}
\begin{gathered}
\frac{dx}{dt}= -b(t)x(t)+F(t,x(t-\tau_{1}(t)),x(t-\tau_{2}(t)),
 \dots,x(t-\tau_{n}(t)),u(t-\delta(t)));\\
\frac{du}{dt}= -\eta(t)u(t)+a(t)x(t-\sigma(t)).
\end{gathered}
\end{equation}
They obtained the existence of at least one positive
$\omega$-periodic solution.

 Liu and Li \cite{l5} employed Avery-Henderson fixed point theorem to
study the existence of positive periodic solutions to the following
nonlinear nonautonomous functional differential system with feed back
\begin{equation} \label{e1.3}
\begin{gathered}
\frac{dx}{dt}= -r(t)x(t)+F(t,x_{t},u(t-\delta(t)));\\
\frac{du}{dt}= -h(t)u(t)+g(t)x(t-\sigma(t)).
\end{gathered}
\end{equation}
They obtained the existence of at least two positive solutions
under some complicated assumptions such as
\begin{itemize}
\item[(A)]  $F(t,\phi_{t},(\Phi\phi)(t-\delta(t)))
 >\frac{c}{\lambda_{\eta}}$ for
 $\beta ce^{-\int^{\omega}_{0}r(s)ds}\leq\phi_{t}(\theta)
 \leq\frac{c}{\beta}e^{-\int^{\eta}_{0}r(s)ds}$,
 $n\beta\omega g_{*}ce^{-\int^{\omega}_{0}r(s)ds}
 \leq(\Phi\phi)(t-\delta(t)))\leq\frac{m\omega g^{*}c}{\beta}
 e^{-\int^{\eta}_{0}r(s)ds}$, $t\in[\eta,\omega]$, $\theta\in\mathbb{R}$;

\item[(B)]  $F(t,\phi_{t},(\Phi\phi)(t-\delta(t)))>\frac{b}{\xi_{\eta}}$
 for $\beta be^{-\int^{\eta}_{0}r(s)ds}\leq\phi_{t}(\theta)
 \leq\frac{b}{\beta}e^{-\int^{\eta}_{0}r(s)ds}$,
 $n\beta\omega g_{*}ce^{-\int^{\eta}_{0}r(s)ds}\leq(\Phi\phi)
 (t-\delta(t)))\leq\frac{m\omega g^{*}b}{\beta}
 e^{-\int^{\eta}_{0}r(s)ds}$, $t\in[0,\omega]$, $\theta\in\mathbb{R}$;

\item[(C)]  $F(t,\phi_{t},(\Phi\phi)(t-\delta(t)))
 >\frac{a}{\lambda_{l}}$ for
$\beta ae^{-\int^{\omega}_{0}r(s)ds}\leq\phi_{t}(\theta)
 \leq\frac{a}{\beta}e^{-\int^{l}_{0}r(s)ds}$,
$n\beta\omega g_{*}ae^{-\int^{\omega}_{0}r(s)ds}\leq(\Phi\phi)
(t-\delta(t)))\leq\frac{m\omega g^{*}a}{\beta}
e^{-\int^{l}_{0}r(s)ds}$, $t\in[l,\omega]$, $\theta\in\mathbb{R}$.

\end{itemize}
We notice that the above conditions are not applicable since
they are too complicated to confirm, and there was no example
in \cite{l6} to  demonstrate their conclusions.

On the other hand, many physical systems undergo abrupt changes
at certain moments due to instantaneous perturbations which lead
to impulse effects, and a lot of such equations  arise in
many mathematical models of real processes and phenomena,
for example, physics, population dynamics, biotechnology,
 and economics (see \cite{c2,g1,l4,l7,s1}, and the references therein).
So, in recent years, impulsive differential equations have received
a lot of attention. As far as we know, there is no paper to study
the existence of triple solutions for impulsive functional equations
with feedback control.
The goal of present paper is to attempt to fill this gap.
And we shall show the impulsive effect plays a crucial role
in some cases (see Remark 3.1).

Comparing with \eqref{e1.2} and \eqref{e1.3}, we note that
\eqref{e1.1} has the impulsive effects. The following are
the main features of present paper. First, the result we obtain
is the existence of three nonnegative $\omega$-periodic solutions.
Second, the method used here is Leggett-Williams fixed point theorem.
Furthermore, the assumptions here are easily checked. Finally,
two examples illustrate the applications of the main result.

The organization of this paper is as follows.
In the next section, some lemmas are presented.
In section 3, we state and prove our main result about the
existence of triple periodic solutions of \eqref{e1.1}.
At last, two examples are given to show the applications
of our main result in section 4.

At the end of this section, we state the Leggett-Williams fixed
point theorem which will be used in section 3.

Let $E$ be a real Banach space with norm $\|\cdot\|$ and $P\subset E$
 be a cone of $E$, $P_r=\{x\in P :\| x\|<r\}(r>0)$.
Consider a nonnnegative continuous and concave functional $\alpha(x)$
defined on $P$, i.e. $\alpha: P\to \mathbb{R}^{+}=[0,\infty)$ is
continuous and satisfies
$\alpha(tx+(1-t)y)\geq t\alpha(x)+(1-t)\alpha(y)$, for $x,y\in P$
and $t\in[0,\omega]$.

Let
$$
P(\alpha;a,b)=\{x\in P: a\leq \alpha(x),\| x\|\leq b\},
$$
where $0<a<b$. It is not difficult to see that $P(\alpha;a,b)$
is a bounded closed convex subset of $P$.

\begin{lemma}[Leggett-Williams Fixed Point Theorem] \label{lem1.1}
 Let operator $A:\overline{P_c}\to P_c$ be completely
continuous and let $\alpha$ be a nonnegative concave functional
on $P$ such that $\alpha(x)\leq\| x\|$ for every $x\in \overline{P_c}$.
Suppose that there exist $0<d<a<b\leq c$ such that
\begin{itemize}
\item[(A1)] $\{x: x\in P(\alpha;a,b),\alpha(x)>a\}\neq\emptyset$
 and $\alpha(Ax)> a$ for each $x\in P(\alpha,a,b)$;
\item[(A2)] $\| Ax\|<d$ for $x\in \overline{P_d}$;
\item[(A3)] $\alpha(Ax)>a$ for $x\in P(\alpha;a,c)$
with $\| Ax\|>b$.
\end{itemize}
Then $A$ has at least three fixed points $x_1,x_2,x_3$ satisfying
$x_1\in P_{d}$, $x_2\in U$,
$x_3\in \overline{P_c}\backslash(P_{d}\cup U)$,
where
$U=\{x: x\in P(\alpha;a,c),\alpha(x)>a\}$.
\end{lemma}


\section{Preliminaries}

Let $PC(\mathbb{R})=\{x: x$ is a map from $[0, \omega]$ into $\mathbb{R}$ such that
$x(t)$ is continuous at $t\neq t_k$, left continuous at $t=t_k$,
and the right limit $x(t_k+0)$ exists for $k=1, 2, \dots, m \}$.

Evidently, $PC_{\omega}(\mathbb{R})=\{x\in PC(\mathbb{R}): x(t)=x(t+\omega),
\forall t\in \mathbb{R} \}$ is a Banach space
with the norm
$\|x\|=\sup_{t\in[0,\omega]}{|x(t)|}, \forall x\in PC_{\omega}(\mathbb{R})$.

Let $P:=\{x\in PC_{\omega}(\mathbb{R}): x(t)\geq \lambda\|x\|,t\in[0,\omega]\} $, where $\lambda=\exp\{-\int_{0}^{\omega}r(\upsilon)d\upsilon\}$.
Obviously, $P$ is a cone of Banach space $PC_{\omega}(\mathbb{R})$.

First, we transform \eqref{e1.1} into another form.
 Suppose $(x, u)$ is a solution of \eqref{e1.1} and $x$
is $\omega$-periodic. By integrating the second equation
of \eqref{e1.1} from $t$ to $t+\omega$, we obtain that
\begin{equation} \label{e2.1}
u(t)=\int_{t}^{t+\omega}g(t,s)h(s)x(s-\sigma(s))ds:=(\Phi x)(t),
\end{equation}
where
$$
g(t,s)=\frac{\exp\{\int_{t}^{s}\eta(\xi)d\xi\}}
{\exp\{\int_{0}^{\omega}\eta(\xi)d\xi\}-1}.
$$
From this, we know
\begin{align*}
u(t+\omega)&= \int_{t+\omega}^{t+2\omega}g(t+\omega,s)h(s)x
 (s-\sigma(s))ds\\
&= \int_{t}^{t+\omega}g(t+\omega,\upsilon+\omega)h
(\upsilon+\omega)x(\upsilon+\omega-\sigma(\upsilon+\omega))d\nu\\
&= \int_{t}^{t+\omega}g(t+\omega,\upsilon+\omega)h(\upsilon)x
(\upsilon-\sigma(\upsilon))d\nu\\
&= \int_{t}^{t+\omega}g(t,\upsilon)h(\upsilon)x(\upsilon
 -\sigma(\upsilon))d\nu\\
&= u(t).
\end{align*}
Therefore, the existence of $\omega$-periodic solution of \eqref{e1.1}
is equivalent to that of the equation
\begin{align*}
\frac{dx}{dt}
&= -r(t)x(t)+F(t,x(t-\tau_{1}(t)),x(t-\tau_{2}(t)),\dots,x(t-\tau_{n}(t)),(\Phi x)(t-\zeta(t)))\\
&= -r(t)x(t)+F(t,Ux(t)),
\end{align*}
where
\begin{equation} \label{e2.2}
Ux(t)=(x(t-\tau_{1}(t)),x(t-\tau_{2}(t)),\dots,x(t-\tau_{n}(t)),
(\Phi x)(t-\zeta(t))).
\end{equation}
To solve the above equation, we transform it into
$$
(\frac{dx}{dt}+r(t)x(t))e^{\int_{0}^{t}r(\nu)d\nu}
=(F(t,Ux(t))e^{\int_{0}^{t}r(\nu)d\nu};
$$
that is,
$$
(x(t)e^{\int_{0}^{t}r(\nu)d\nu})'
=(F(t,Ux(t))e^{\int_{0}^{t}r(\nu)d\nu}.
$$
So, integrating the above equality from $t$ to $t+\omega$,
 and noticing that $x(t)=x(t+\omega)$, we have
$$
x(t)=\int_{t}^{t+\omega}G(t,s)F(s,Ux(s))ds+\sum_{j=1}^{m}
G(t,t_{k_j})I_j(x(t_j)),
$$
where
$$
G(t,s)=\frac{\exp\{\int_{t}^{s}r(\nu)d\nu\}}
{\exp\{\int_{0}^{\omega}r(\nu)d\nu\}-1}
$$
for $(t,s)\in \mathbb{R}^{2}$ and $t_{k_j}$ satisfies
$t_{k_j}=t_j+k\omega$, $t_{k_j}\in [t,t+\omega]$,
$I_{k_j}(x(t_{k_j}))=I_j(x(t_j))$, $j=1,2,\dots,m$.
It is clear that $G(t,s)>0$ and $G(t,s)=G(t+\omega,s+\omega)$
for all $(t,s)\in \mathbb{R}^{2}$.

Now we define an operator $\Psi$ on $P$ as
\begin{equation} \label{e2.3}
\begin{aligned}
&(\Psi x)(t)\\
&=\int_{t}^{t+\omega}G(t,s)F(s,x(s-\tau_{1}(s)),\dots,
 x(s-\tau_{n}(s)),(\Phi x)(s-\zeta(s)))ds \\
&\quad +\sum_{j=1}^{m} G(t,t_{k_j})I_j(x(t_j)),\quad
 \forall x\in P,\; t\in \mathbb{R}.
\end{aligned}
\end{equation}
It is obvious that $x$ is an $\omega$-periodic solution
for \eqref{e1.1} if and only if $x$ is a fixed point of the
operator $\Psi$.

\begin{lemma}\label{lem2.1}
$\Psi :P \to P$ is well defined.
\end{lemma}

\begin{proof}
 First, it is easy to see $\Psi:PC(\mathbb{R})\to PC(\mathbb{R})$. Next, since
\begin{align*}
&(\Psi x)(t+\omega)\\
&= \int_{t+\omega}^{t+2\omega}G(t+\omega,s)F(s,Ux(s))ds
 +\sum_{j=1}^{m} G(t+\omega,t_{k_j}+\omega)I_j(x(t_j))\\
&= \int_{t}^{t+\omega}G(t+\omega,\upsilon+\omega)F(\upsilon+\omega,
Ux(\upsilon+\omega))d\upsilon+\sum_{j=1}^{m}
 G(t,t_{k_j})I_j(x(t_j))\\
&= \int_{t}^{t+\omega}G(t,\upsilon)F(\upsilon,Ux(\upsilon))d\upsilon
 +\sum_{j=1}^{m} G(t,t_{k_j})I_j(x(t_j))
= (\Psi x)(t),
\end{align*}
we have $\Psi\in PC_{\omega}(\mathbb{R})$.
Observe that
 \begin{equation} \label{e2.4}
\begin{aligned}
p:=\frac{1}{\exp\{\int_{0}^{\omega}r(\nu)d\nu\}-1}
&\leq G(t,s)
=\frac{\exp\{\int_{t}^{s}r(\nu)d\nu\}}
 {\exp\{\int_{0}^{\omega}r(\nu)d\nu\}-1}\\
&\leq\frac{\exp\{\int_{0}^{\omega}r(\nu)d\nu\}}
 {\exp\{\int_{0}^{\omega}r(\nu)d\nu\}-1}:=q
\end{aligned}
\end{equation}
for all $s\in[t,t+\omega]$. Hence, we obtain that, for $x\in P$,
\begin{equation} \label{e2.5}
\|\Psi x\|\leq q\int_{0}^{\omega}F(s,Ux(s))ds
+q\sum_{j=1}^{m}I_j(x(t_j))
\end{equation}
and
$$
(\Psi x)(t)\geq p\int_{0}^{\omega}F(s,Ux(s))ds
+p\sum_{j=1}^{m}I_j(x(t_j))\geq\frac{p}{q}\| \Psi x\|
=\lambda\| \Psi x\|.
$$
Thus, $\Psi x\in P$.
\end{proof}


\begin{lemma} \label{lem2.2}
$\Psi:P \to P$ is completely continuous.
\end{lemma}

\begin{proof}
First we show that $\Psi$ is continuous. According to our assumptions
and \eqref{e2.1}, we know that for any $\varepsilon>0$,
there exists $\delta>0$ such that for $x,y \in PC_{\omega}(\mathbb{R})$
with $\|x-y\|<\delta$,
\begin{gather*}
\sup_{0\leq s\leq\omega}|F(s,Ux(s))-F(s,Uy(s))|
<\frac{\varepsilon}{2q\omega}, \\
\max_{0\leq t\leq\omega}|I_{k}(x(t_{k}))
-I_{k}(y(t_{k}))|<\frac{\varepsilon}{2qm},\quad
 k=1,2,\dots,m,
\end{gather*}
where $q$ is defined in \eqref{e2.4}. So for each $t\in \mathbb{R}$,
we have
\begin{align*}
&|(\Psi x)(t)-(\Psi y)(t)|\\
&\leq  q\int_{t}^{t+\omega}|F(s,Ux(s))-F(s,Uy(s))|ds
 +q\sum_{k=1}^{m}|I_{k}(x(t_{k}))-I_{k}(y(t_{k}))|\\
&\leq  q \frac{\varepsilon} {2q\omega}\omega
 +q\sum_{k=1}^{m}\frac{\varepsilon}{2qm}
 =\frac{\varepsilon}{2}+\frac{\varepsilon}{2}=\varepsilon.
\end{align*}
This yields $\|\Psi x-\Psi y\|<\varepsilon$ when $\|x-y\|<\delta$.
 Hence $\Psi$ is continuous.

Next, we show that $\Psi$ is compact. Let $S\subset PC_{\omega}(\mathbb{R})$
 be a bounded subset, that is, there exists $M>0$ such that
$\|x\|\leq M$ for $\forall x\in S$. Therefore,
\begin{align*}
|(\Phi x)(t-\zeta(t))|
&= |\int_{t-\zeta(t)}^{t-\zeta(t)+\omega}
 g(t-\zeta(t),s)h(s)x(s-\sigma(s))ds|\\
&\leq  M \frac{\exp(\int_{0}^{\omega}\eta(\xi)d\xi)}
 {\exp(\int_{0}^{\omega}\eta(\xi)d\xi)-1}
 \int_{t}^{t+\omega}h(s)ds<\infty
\end{align*}
and
$$
|x(t-\tau_{i}(t))|\leq \|x\|\leq M,\ i=1,2,\dots,m,\quad
 \forall t\in \mathbb{R}.
$$
From the continuity of $F$ and $I_{k}$, there exist $M_{1}>0$ and
$M_{2}>0$ such that
\begin{gather*}
|F(s,Ux(s))|\leq M_{1},\quad s\in[0,\omega],\quad \forall x\in S,\\
|I_{k}(x(t_{k}))|\leq M_{2},\quad \forall x\in S.
\end{gather*}
Then,
$$
\|\Psi x\|\leq q\int_{0}^{\omega}|F(s,Ux(s))|ds
+q\sum_{j=1}^{m}I_j(x(t_j))\leq q\omega M_{1}+qm M_{2},
$$
which implies $\Psi(S)$ is uniformly bounded.
Finally, notice that
\begin{align*}
\frac{d}{dt}(\Psi x)(t)&= -r(t)(\Psi x)(t)+G(t,t+\omega)
 F(t+\omega,Ux(t+\omega))-G(t,t)F(t,Ux(t))\\
&= -r(t)(\Psi x)(t)+[G(t,t+\omega)-G(t,t)]F(t,Ux(t))\\
&= -r(t)(\Psi x)(t)+F(t,Ux(t)).
\end{align*}
This guarantees that, for each $x\in S$, we have
\begin{align*}
|\frac{d}{dt}(\Psi x)(t)|
&\leq \| r(t)\| (q\int_{0}^{\omega}|F(s,Ux(s))|ds
 +q\sum_{j=1}^{m}I_j(x(t_j)))
 +|F(t,Ux(t))|\\&\leq \| r(t)\|(q\omega M_{1}+qmM_{2})+M_{1}
 <\infty.
\end{align*}
Consequently, $\Psi(S)$ is equi-continuous.
By Ascoli-Arzela theorem, the function $\Psi$ is completely
continuous from $P$ to $P$.
\end{proof}


 \section{Main Results}

For convenience, first let us list the following assumptions.
\begin{itemize}
\item[(H1)] $\lim_{|v|\to +\infty}
 \max_{0\leq t\leq \omega}\frac{F(t,v)}{|v|}=0$,
 $\lim_{x\to +\infty}\frac{I_{i}(x)}{x}=0$, $i=1,2,\dots,m$,
 where $v=(v_{1},v_{2},\dots,v_{n+1})\in\mathbb{R}^{n+1}$,
 $|v|=\max_{1\leq j\leq n+1}|v_j|$;

\item[(H2)] There exist two positive numbers $a$ and $b$ with
 $ab^{-1}<\lambda$ such that one of the following two conditions holds:
\begin{itemize}
 \item[(i)] for some $j$ ($j=1,2,\dots,n$),
 $F(t,v)>\frac{1}{p\omega}v_j$ as
 $a\leq\min_{1\leq i\leq n}\{v_{i}\}\leq\max_{1\leq i\leq n}
 \{v_{i}\}\leq b$ and
 $ap_{1}\overline{h}\leq v_{n+1} \leq bq_{1}\overline{h}$,
 where
 $$
 q_{1}=\frac{\exp(\int_{0}^{\omega}\eta(\xi)d\xi)}
 {\exp(\int_{0}^{\omega}\eta(\xi)d\xi)-1},\quad
 p_{1}=\frac{1}{\exp(\int_{0}^{\omega}\eta(\xi)d\xi)-1},\quad
 \overline{h}=\int_{0}^{\omega}h(s)ds;
 $$

 \item[(ii)] there exists $1\leq j\leq m$ such that $I_j(x)>lx$
 for $x\in[a,b]$, where $l=1/p$;
\end{itemize}

\item[(H3)] $\lim_{|v|\to 0}\max_{0\leq t\leq \omega}
 F(t,v)/|v|=0$,
$\lim_{x\to 0} I_{i}(x)/x=0$, $i=1,2,\dots,m$.

\end{itemize}
Now we state our main result on the existence of three nonnegative
$\omega$-periodic solutions for \eqref{e1.1}.

\begin{theorem}\label{3.1}
Suppose {\rm (H1)--(H3)} hold. Then \eqref{e1.1} has at least
three nonnegative periodic solutions.
\end{theorem}

\begin{proof}
Define a functional $\alpha(x)$ on cone $P$ by
 \begin{equation} \label{e3.1}
\alpha(x)=\min\{x(t): t\in[0,\omega]\},\; \forall x\in P.
\end{equation}
Evidently, $\alpha : P\to \mathbb{R}^{+}=[0,\infty)$ is nonnegative
continuous and concave. Moreover, $\alpha(x)\leq\|x\|$ for each
$x\in P$. Notice that
\begin{equation} \label{e3.2}
P(\alpha;a,b)=\{x\in P: a\leq \alpha(x), \|x\| \leq b\}
=\{x\in P: a\leq x(t)\leq b,\; \forall t\in[0,\omega]\}.
\end{equation}

For using Lemma 1.1, we first prove that there
exists a positive number $c$ with $c\geq b$ such that
$\Psi:\overline{P}_c\to{P}_c$.
By (H1), we know there exist $\varepsilon$ with
$0<\varepsilon <(q\omega+qm+qq_{1}\omega\int_{0}^{\omega}h(s)ds)^{-1}$
and $L>0$ such that
$$
\frac{F(t,v)}{|v|}<\varepsilon\quad\text{and}\quad
\frac{I_{k}(x)}{x}<\varepsilon \quad\text{for } |v|>L,\; x>L,\;
\forall t\in[0,\omega].
$$
Notice that $F$ is continuous on $[0,\omega]\times \mathbb{R}^{+}$
and $I_{k}(x)$ is also continuous on $\mathbb{R}^{+}$. Thus, there exists
$M>0$ such that
\begin{gather*}
F(t,v)<\varepsilon|v|+M,\\
I_{k}(x)<\varepsilon x+M,\quad k=1,2,\dots,m,\; \forall
 t\in[0,\omega].
\end{gather*}
Choose
$$
c=\max\{b,\frac{qM\omega+qmM}{1-q\varepsilon(\omega+m+q_{1}
\omega\int_{0}^{\omega}h(s)ds)}\}.
$$
 So for each $x\in\overline{P}_c$, $t\in[0,\omega]$, we have
\[
|\Phi x(t-\zeta(t))|
\leq  c \frac{\exp(\int_{0}^{\omega}\eta(\xi)d\xi)}
 {\exp(\int_{0}^{\omega}\eta(\xi)d\xi)-1}
 \int_{t-\zeta(t)}^{t-\zeta(t)+\omega}h(s)ds
=  c q_{1}\int_{0}^{\omega}h(s)ds,
\]
where
$$
q_{1}=\frac{\exp(\int_{0}^{\omega}\eta(\xi)d\xi)}
 {\exp(\int_{0}^{\omega}\eta(\xi)d\xi)-1}.
$$
Then,
\begin{equation} \label{e3.3}
|Ux(t)|=\max\{\max_{1\leq i\leq n}|x(t-\tau_{i}(t)|,
|\Phi x(t-\zeta(t))|\}
\leq c+cq_{1}\int_0^{\omega}h(s)ds.
\end{equation}
This together with \eqref{e2.3} guarantees that
\begin{align*}
 |(\Psi x)(t)|
&\leq  q\int_{t}^{t+\omega}| F(s,Ux(s))| ds
 +q\sum_{k=1}^{m} I_{k}(x(t_{k}))\\
&\leq  q\omega(\varepsilon c+\varepsilon cq_{1}
\int_0^{\omega}h(s)ds+M)+qm(\varepsilon c+M)
< c ,\quad \forall t\in[0,\omega].
\end{align*}
Combining this inequality with Lemma 2.2, we obtain
that $\Psi:\overline{P}_c\to{P}_c$ is completely continuous.

Next we show that the condition (A1) of Lemma 1.1 holds.
From \eqref{e3.1}, \eqref{e3.2}, (H2), we know that
$b\in \{x: x\in P(\alpha;a,b),\alpha(x)>a\}$.
Suppose first (i) of (H2) holds. Then, for each
$x\in P(\alpha;a,b)=\{x\in P: a\leq x(t)\leq b,\;
\forall t\in[0,\omega]\}$, we have
\begin{align*}
(\Psi x)(t)&\geq  p\int_{t}^{t+\omega}F(s,Ux(s))ds
+p\sum_{k=1}^{m} I_{k}(x(t_{k}))\\
&\geq  p \int_{t}^{t+\omega}F(s,Ux(s))ds
>p\frac{1}{p\omega}\int_{t}^{t+\omega}x(t-\tau_j(t))ds\\
 &\geq  p\frac{1}{p\omega}a\omega=a,\quad \forall t\in[0,\omega].
\end{align*}
It is easy to see that (A1) of Lemma 1.1 holds.

Next, assume (ii) of (H2) is satisfied. It follows that
\begin{align*}
(\Psi x)(t)
&\geq  p\int_{t}^{t+\omega}F(s,Ux(s))ds
 +p\sum_{k=1}^{m} I_{k}(x(t_{k}))\\
&\geq  p I_j(x(t_j))>pl x(t_j)\\
&\geq  pla=a,\quad \forall t\in[0,\omega],
\end{align*}
which also means that (A1) of Lemma 1.1 holds.

In addition, for each $x\in P(\alpha;a,c)$ with
$\|\Psi x\|\geq b$, by \eqref{e2.3} and (H2), we know that
\begin{equation} \label{e3.4}
(\Psi x)(t)\geq\lambda\|\Psi x\|\geq\lambda b>a,\quad \forall
 t\in[0,\omega].
\end{equation}
This guarantees that (A3) of Lemma 1.1 is satisfied.

Finally, from (H3), there exist $\varepsilon$
satisfying $0<\varepsilon
<(q\omega+qm+qq_{1}\omega\int_{0}^{\omega}h(s)ds)^{-1}$ and
$\delta<a$ such that
$$
\frac{F(t,v)}{|v|}<\varepsilon\quad\text{and}\quad
\frac{I_{k}(x)}{x}<\varepsilon\quad\text{for }
|v|<\delta,\; 0<x<\delta,\; k=1,2,\dots,m,\; \forall t\in[0,\omega].
$$
Choose $d=\min\{a,\delta,\delta/(1+q_{1}\int_0^{\omega}h(s)ds)\}$.
 This together with \eqref{e2.2} guarantees that, for each
$x\in\overline{P}_d$,
$$
|Ux(t)|\leq d+dq_{1}\int_0^{\omega}h(s)ds<\delta.
$$
Thus,
 \begin{align*}
|(\Psi x)(t)|
&\leq  q\int_{t}^{t+\omega} | F(s,Ux(s))| ds
 +q\sum_{j=1}^{m} I_{k}(x(t_{k}))\\
&\leq  q\omega\varepsilon\Big(d+dq_{1} \int_0^{\omega}h(s)ds\Big)
 +qm\varepsilon d\\
&= \varepsilon d\Big(q\omega+qq_{1}\omega \int_0^{\omega}h(s)ds+qm\Big)
<d,
\end{align*}
which implies $\|\Psi x\|\leq d$, that is, (A2) of Lemma 1.1 holds.

 In conclusion, by Lemma 1.1, the operator $\Psi$ has at least
three fixed points $x_1,x_2$ and $x_3$, that is, \eqref{e1.1}
has at least three nonnegative periodic solutions $x_1,x_2$
and $x_3$ satisfying $x_1\in P_{d}$,
$x_2\in U=\{x: x\in P(\alpha;a,c),\alpha(x)>a\}$,
and $x_3\in \overline{P_c}\backslash(P_{d}\cup U)$.
\end{proof}

We remark that Condition (H2) indicates that the impulse plays
an important role.

\section{Examples}

In this section, two examples illustrate the application of our
main result obtained in section 3.

\begin{example} \label{examp4.1} \rm
Consider the  impulsive functional differential equations
with feedback control
\begin{equation} \label{e4.1}
\begin{gathered}
\begin{aligned}
x'(t)
 &= 9\ln(1+x^{2}(t-\sin(2\pi t)))+\sin(2\pi t)
 \sqrt{x(t-\sin(2\pi t))} \\
 &\quad\times \ln(1+u(t-\sin(2\pi t))
 -(\ln(\frac{5}{4}e)+\cos(2\pi t))x(t);
\end{aligned}\\
u'(t)= -(\frac{3}{2}+\cos(2\pi t))u(t)+(\frac{3}{2}
 +\cos(2\pi t))x(t-\sin(2\pi t));\\
\Delta x|_{t=1/2}=I_{1}(x(\frac{1}{2})),
\end{gathered}
\end{equation}
where $I_{1}(x)=x^{2}e^{-x}$, $x\in[0,+\infty)$.
Then, \eqref{e4.1} has three nonnegative periodic solutions.
\end{example}

\begin{proof}
 Equation \eqref{e4.1} can be regarded as of the form \eqref{e1.1},
where
\begin{equation} \label{e4.2}
F(t,v_{1},v_{2},v_{3})=9\ln(1+v_{1}^{2})
+\sin(2\pi t)\sqrt{v_{2}}\ln(1+v_{3}),
\end{equation}
$\tau_{1}(t)=\tau_{2}(t)=\sigma(t)=\zeta(t)=\sin(2\pi t)$,
$\eta(t)=h(t)=\frac{3}{2}+\cos(2\pi t)$,
$r(t)=\ln(\frac{5}{4}e)+\cos(2\pi t)$, $\omega=1$,
$t_{1}=1/2$, $p=1/(\frac{5}{4}e-1)$,
$\frac{1}{p \omega}=\frac{5}{4}e-1$.

Now we prove that (H1)--(H3) hold. From \eqref{e4.2}, we have
\begin{equation} \label{e4.3}
|F(t,v_{1},v_{2},v_{3}|\leq 9\ln(1+|v|^{2})+\sqrt{|v|}\ln(1+2|v|),\quad
 \forall t\in[0,1],\; v\in \mathbb{R}^{3},
\end{equation}
where $|v|=\max_{1\leq j\leq 3}|v_j|$. obviously, (H1) and (H3)
are satisfied.

Next, we show that (i) of (H2) holds. Choose $a=\sqrt{e^{2}-1}$
and $b=e^{3}$. Then $a^{-1}b=\frac{e^{3}}{\sqrt{e^{2}-1}}>\frac{5}{4}e$.
Notice that the function $\frac{\ln(1+x^{2})}{x}$ defined
on $[\sqrt{e^{2}-1},e^{3}]$ takes its minimum at $x=e^{3}$.
Therefore, when
$a\leq \min\{v_{1},v_{2}\}\leq \max\{v_{1},v_{2}\}\leq b$, we have
$$
F(t,v_{1},v_{2},v_{3})\geq9\ln(1+v_{1}^{2})\geq
9\frac{\ln(1+e^{6})}{e^{3}}v_{1}>(\frac{5}{4}e-1)v_{1}
 =\frac{1}{p \omega}v_{1}.
$$
This means (i) of (H2) is satisfied.
Consequently, by Theorem 3.1, \eqref{e4.1} has three nonnegative
periodic solutions.
\end{proof}

\begin{example} \label{examp4.2}\rm
 Consider the  system
\begin{equation} \label{e4.4}
\begin{gathered}
\begin{aligned}
x'(t)&=8\sin(2\pi t)\sqrt{x(t-2\sin(2\pi t))
 +2u(t-2\sin(2\pi t))}\\
&\quad\times \ln(1+ x(t-2\sin(2\pi t)))
 -(\ln(\frac{3}{2}e)+\cos(2\pi t))x(t);
\end{aligned}\\
u'(t)= -(2+\sin(2\pi t))u(t)+(2+\sin(2\pi t))x(t-2\sin(2\pi t));\\
\Delta x|_{t=1/2}=I_{1}(x(\frac{1}{2})),
\end{gathered}
\end{equation}
where $I_{1}(x)=150x^{2}e^{-x}$, $x\in[0,+\infty)$.
Then, \eqref{e4.2} has three nonnegative periodic solutions.
\end{example}

\begin{proof} Equation \eqref{e4.4} can be regarded as of the form
\eqref{e1.1}, where
\begin{equation} \label{e4.5}
F(t,v_{1},v_{2},v_{3})=8\sin(2\pi t)\sqrt{v_{1}+2v_{3}}\ln(1+ v_{2}),
\end{equation}
$\tau_{1}(t)=\tau_{2}(t)=\sigma(t)=\zeta(t)=2\sin(2\pi t)$,
$\eta(t)=h(t)=2+\sin(2\pi t)$,
$r(t)=\ln(\frac{3}{2}e)+\cos(2\pi t)$, $\omega=1$,
$t_{1}=1/2$, $p=1/(\frac{3}{2}e-1)$,
$l=\frac{3}{2}e-1$.

As in the proof of Example 4.1, it is easy to see that (H1) and (H3)
are satisfied. Now it remains to show that (ii) of (H2) holds.
Choose $a=1$ and $b=2e$. Then $a^{-1}b=2e>\frac{3}{2}e$.
Notice the function $xe^{-x}$ defined
on $[1,2e]$ takes its minimum at $x=2e$. So, it is not difficult
to see
$$
I_{1}(x)=150x^{2}e^{-x}\geq150(2e)e^{-(2e)}x
=\frac{150(2e)}{e^{2e}}x>({\frac{3}{2}e-1})x=lx,
\quad \forall x\in[1,2e],
$$
which implies (ii) of (H2) is satisfied.
Consequently, by Theorem 3.1, \eqref{e4.4} has three nonnegative
periodic solutions.
\end{proof}

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\end{document}