\documentclass[reqno]{amsart}

\AtBeginDocument{{\noindent\small
{\em Electronic Journal of Differential Equations},
Vol. 2003(2003), No. 52, pp. 1--10.\newline
ISSN: 1072-6691. URL: http://ejde.math.swt.edu or http://ejde.math.unt.edu
\newline ftp ejde.math.swt.edu  (login: ftp)}
\thanks{\copyright 2003 Southwest Texas State University.}
\vspace{9mm}}

\begin{document}

\title[\hfilneg EJDE--2003/52\hfil Multiple solutions for nonresonance]
{Multiple solutions for nonresonance impulsive
functional differential equations}

\author[Mouffak Benchohra \&  Abdelghani Ouahab\hfil EJDE--2003/52\hfilneg]
{Mouffak Benchohra \&  Abdelghani Ouahab}

\address{Mouffak Benchohra \hfill\break
Laboratoire de Math\'ematiques, Universit\'e de Sidi Bel Abbes \\
BP 89 2000 Sidi Bel Abbes, Alg\'erie}
\email{benchohra@univ-sba.dz}

\address{Abdelghani Ouahab \hfill\break
Laboratoire de Math\'ematiques, Universit\'e de Sidi Bel Abbes \\ 
BP 89 2000 Sidi Bel Abbes, Alg\'erie} 
\email{ouahabi\_abdelghani@yahoo.fr}

\date{}
\thanks{Submitted January 8, 2003. Published May 3, 2003.}
\subjclass[2000]{34A37, 34K25}
\keywords{Nonresonance impulsive functional differential equations, \hfill\break\indent
 boundary conditions, fixed point, multiple solutions, cone, concave functional}


\begin{abstract}
  In this paper we investigate the existence of multiple solutions for
  first and second order impulsive functional differential equations with
  boundary conditions. Our main tool is the Leggett and Williams fixed
  point theorem.
\end{abstract}

\maketitle

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

\section{Introduction}

This paper is concerned with the existence of three nonnegative solutions
for initial value problems for first and second order
impulsive functional differential equations with boundary conditions.
Initially we consider
the first order impulsive functional differential equation,
\begin{gather}\label{e1}
y'(t)-\lambda y(t)=f(t,y_{t}), \quad\mbox{a.e. }t\in [0,T], \;
t\neq t_k, \; k=1,\dots,m, \\  \label{e2}
\Delta y|_{t=t_{k}}=I_{k}(y(t_k^{-})), \quad  k=1,\dots,m, \\ \label{e3}
y(t)=\phi(t),\quad  t\in [-r,0], \;  y(0)=y(T),
\end{gather}
where $\lambda >0$,  $f:[0,T]\times D\to \mathbb{R}^{+}$, $I_k\in
C(\mathbb{R},\mathbb{R}^{+})$, $0<r<\infty$,
$0=t_{0}<t_{1}<\dots<t_{m}<t_{m+1}=T$,
$\Delta y|_{t=t_{k}}=y(t_{k}^{+})- y(t_{k}^{-})$, $y(t_{k}^{-})$ and
$y(t_{k}^{+})$ represent the left and right limits of $y(t)$ at
$t=t_{k}$, respectively. $D=\{\Psi:[-r,0]\to\mathbb{R}^{+}; \Psi\
$is continuous everywhere except for a finite number of points \
$\bar{t}$ \ at which \ $\Psi(\bar{t})$\ and\ $\Psi(\bar{t}^+)$\
exist  and \ $\Psi(\bar{t}^-)=\Psi(\bar{t})\}$,  $\phi\in  D$.
For any  function $y$ defined on $[-r,T]$ and
any $t\in J$, we denote by $y_{t}$ the
element of $D$ defined by
$$
y_{t}(\theta)=y(t+\theta), \quad  \theta\in [-r,0].
$$
Here $y_{t}(\cdot)$ represents the history of
the state from time $t-r$, to the present time $t$.

Later, we study the second order impulsive functional
differential equations  with boundary conditions and fixed
moments of the form
\begin{gather}\label{e4}
y''(t)-\lambda y(t)=f(t,y_{t}), \quad\mbox{a.e. }t\in [0,T], \;
t\neq t_k,\;  k=1,\dots,m, \\ \label{e5}
\Delta y|_{t=t_{k}}=I_{k}(y(t_k^{-})), \quad  k=1,\dots,m, \\ \label{e6}
\Delta y'|_{t=t_{k}}=\overline I_{k}(y(t_k^{-})), \quad  k=1,\dots,m, \\
\label{e7}
y(t)=\phi(t), \quad t\in [-r,0],\quad y(0)-y(T)=\mu_{0},
\quad y'(0)-y'(T)=\mu_{1},
\end{gather}
where  $F$, $I_{k}$, and $\phi$ are as in problem
(\ref{e1})-(\ref{e3}),
$\overline I_{k}\in C(\mathbb{R},\mathbb{R}^{+})$ $k=1,\dots,m$, and
$\mu_0, \mu_1 \in \mathbb{R}$.

Note that when $\mu_{0}=\mu_{1}=0$ we have
periodic boundary conditions. Differential
equations with impulses are a basic tool to
study evolution processes that are subjected to
abrupt changes in their state.  Such equations
arise naturally from a wide variety of
applications, such as space-craft control,
inspection processes in operations research, drug
administration, and threshold theory in biology.
There has been a significant development in
impulse theory, especially in the area of impulsive
differential equations  with fixed moments;
see the monographs of Bainov and
Simeonov \cite{BaSi}, Lakshmikantham {\em et al}.
 \cite{LaBaSi}, and Samoilenko and Perestyuk
\cite{SaPe} and the papers of  Benchohra {\em et al }\cite{BeEl},
 Franco {\em et al} \cite{FrLiNiRo} and the references cited therein.

The existence of multiple solutions for boundary value problems for impulsive differential
equations was studied by Guo and  Liu \cite{GuLi} and  Agarwal and  O'Regan \cite{AgRe1}.
Notice that when the impulses are absent  (i.e.  $I_k=0$, $k=1,\dots,m)$
 the existence of three solutions
and multiple solutions for ordinary differential equations
was studied in \cite{AgRe,AgReWo,AnAvPe,AnRiHe,AvHe,HeTh}.

The main theorems of this note extend some existence results in the
above literature to the impulsive case. Our approach  here  is based on the
Leggett and Williams fixed point theorem in cones \cite{LeWi}.

\section{Preliminaries}

In this section, we introduce notation, definitions, and preliminary facts
 which are used for the rest of this paper.

By $C(J,\mathbb{R})$ we denote the Banach space  of all continuous
functions from $J$ into $\mathbb{R}$ with the norm
$$
\|y\|_{\infty}:=\sup\{|y(t)|: t\in J\}.
$$

$L^{1}(J,\mathbb{R})$ denotes the Banach space of measurable functions $y:J\to\mathbb{R}$ which
are Lebesgue integrable, with $$ \|y\|_{L^{1}}=\int_{0}^{T}|y(t)|dt. $$ Let $(a,b)$ be an open
interval and $AC^i(a,b),\mathbb{R})$ be the space of $i$-times differentiable functions
$y:(a,b) \to \mathbb{R}$, whose $i^{th}$ derivative, $y^{(i)}$, is absolutely continuous.

Let $(E,\|\cdot\|)$ be a Banach space  and $C$ be a cone in
$E$. By concave
nonnegative continuous functional $\psi$ on $C$
we mean a continuous mapping $\psi:\ C\to[0,\infty)$ with
$$
\psi(\lambda x+(1-\lambda)y)\geq\lambda\psi(x)+(1-\lambda)\psi(y)
\quad\hbox{for all } x, y\in C \; \lambda\in[0,1].
$$
To define solutions of (\ref{e1})-(\ref{e3}) we
shall consider the  space
\begin{align*}
&PC=\big\{y: [0,T]\to\mathbb{R}: y_{k}\in
C(J_{k},\mathbb{R}),\; k=0,\dots,m
\hbox{ and there exist } y(t^{-}_{k})  \\
&\hbox{ and }  y(t^{+}_{k})  \hbox{ with }
y(t^{-}_{k})=y(t_{k}), k=1,\dots,m\big\}
\end{align*}
which is a Banach space with the norm
$$
\|y\|_{PC}=\max\{\|y_{k}\|_{J_{k}}, k=0,\dots,m \},
$$
where $y_{k}$ is the restriction of $y$ to $J_{k}=(t_{k},t_{k+1}], \
k=0,\dots,m$.
Set $\Omega=\{y:[-r,T]\to\mathbb{R}: y\in D\cap PC\}$. Then
$\Omega$ is a Banach space with norm
$$
\|y\|_{\Omega}=\sup\{|y(t)|: t\in[-r,T]\}.
$$

\begin{definition} \label{def2.1} \rm
A map $f:J\times D\to \mathbb{R}^{+}$ is said to be
$L^{1}$-Carath\'eodory  if
\begin{itemize}
\item[(i)] $t\mapsto f(t,u)$ is  measurable for each $y\in D;$
\item[(ii)] $u\mapsto f(t,u)$ is continuous for almost all
$t\in J;$
\item[(iii)] For each $q>0$,  there exists $h_{q} \in L^{1}(J,\mathbb{R}_{+})$
such that
$|f(t,u)|\leq h_{q}(t)$ for all
$\|u\|_{D}\leq q$  and for almost all $t\in J$.
\end{itemize}
\end{definition}

Our consideration is based on the following fixed point theorem given by
Leggett and Williams in  1979  \cite{LeWi}(see also
  Guo and  Lakshmikantham \cite{GuLa}).

\begin{theorem}\label{t1}
Let $E$ be a Banach space, $C\subset E$ a
cone of $E$ and $R>0$ a constant. Let
$C_{R}=\{y\in C: \|y\|< R \}$. Suppose a concave nonnegative
continuous functional $\psi$ exists on the cone $C$ with
$\psi(y)\leq\|y\|$ for $y\in\overline C_R$, and let
$N:\overline C_R\to\overline C_R$ be a continuous compact map.
Assume there are numbers $r, L$ and $K$ with $0<r<L<K\leq R$ such that
\begin{itemize}
\item[(A1)] $\{ y\in C(\psi, L, K): \psi(y)>L\}\neq \emptyset$
and $\psi(N(y))>L$ for all $ y\in C(\psi, L, K)$;
\item[(A2)] $\|N(y)\|<r$ for all $y\in \overline C_r;$
\item[(A3)] $\psi(N(y))> L$ for all $y\in C(\psi, L, R)$
with $\|N(y)\|>K$, where
$C_{K}=\{y\in C: \|y\|\leq K\}$ and
$$
C(\psi, L, K)=\{y\in C: \psi(y)\geq L\ \hbox{and}\ \|y\|\leq K\}.
$$
\end{itemize}
Then $N$ has at least three fixed points $y_1, y_2, y_3$ in
$\overline C_R$. Furthermore, we have
$$
y_1\in C_r,\quad y_2\in\{ y\in C(\psi, L, R):\psi(y)>L\}\,\quad
y_3\in \overline C_R-\{C(\psi, L, R)\cup \overline C_r\}.
$$
\end{theorem}

\section{First Order Impulsive FDEs}

Let us start by defining what we mean by a solution of problem
(\ref{e1})--(\ref{e3}).

\begin{definition} \label{def3.1} \rm
A function $y\in \Omega\cap\cup_{k=0}^{m}AC((t_{k},t_{k+1}),\mathbb{R})$ is
said to be a solution of (\ref{e1})--(\ref{e3})  if $y$ satisfies
 $y'(t)-\lambda y(t)=f(t,y_t)$  a.e. on  $J\backslash\{t_{1},\dots,t_{m}\}$,  and
 $\Delta y|_{t=t_{k}}=I_{k}(y(t_{k}^{-}))$, $k=1,\dots,m$,
$y(t)=\phi(t)$, $t\in[-r,0]$,  and $y(0)=y(T)$.
\end{definition}

For the next theorem we need the following assumptions:
\begin{itemize}
\item[(H1)] There exist constants $c_{k}$ such that
$|I_{k}(x)|\leq c_{k}$, $k=1,\dots,m$ for each $x\in\mathbb{R}$

\item[(H2)] There exist a function $g: [0,\infty)\to[0,\infty)$ continuous and
non-decreasing, a function
 $p\in L^{1}(J,\mathbb{R}_{+})$, $r>0$, and a constant $0< M\leq 1$ such that
$$
M\,p(t)g(\|u\|)\leq |H(t,s)f(t,u)|\leq p(t)g(\|u\|)
$$
for each $(t,s,u)\in J\times J\times D$, and
$$
\frac{1}{1-e^{-\lambda T}}\sum_{k=1}^m c_k +g(r)\int_0^Tp(t)dt<r;
$$
\item[(H3)]  There exist $L>r$ and an interval $[a,b]\subset(0,T)$
such that
\begin{gather*}
\min_{t\in[a,b]}\Big(\sum_{k=1}^mH(t,t_k)I_k(y(t_{k}))\Big)\geq M\sum_{k=1}^mc_k, \\
M\Big(\sum_{k=1}^mc_k+g(L)\int_0^Tp(s)ds\Big)>L;
\end{gather*}
\item[(H4)] There exist $R, K, 0<M_1<M,\ M_1^{-1}L<K\leq R$ such that
\begin{align*}
&\min_{t\in[a,b]}\Big(\sum_{k=1}^mH(t,t_k)I_k(y(t_{k}))
+\int_0^TH(t,s)f(s,y_s)ds\Big)\\
&\geq M_1M_*\Bigl(\sum_{k=1}^m
c_k+g(R)\int_0^Tp(s)ds\Bigr)
\end{align*}
where $M_*=\sup_{(t,s)\in[0,T]\times[0,T]}|H(t,s)|$ and
$$
\frac{1}{1-e^{-\lambda T}}\sum_{k=1}^{m}c_k +g(R)\int_0^Tp(t)dt<R\,.
$$
\end{itemize}

\begin{theorem} \label{t2}
Assume (H1)--(H4) are satisfied.
Then problem  (\ref{e1})--(\ref{e3}) has at least three solutions.
\end{theorem}

\begin{proof} We transform the problem into a fixed point problem.
Consider the operator,
$N: \Omega\to \Omega$
defined by
$$
N(y)(t)=\begin{cases}
\phi(t),& \mbox{if } t\in [-r,0];\\
\int_{0}^{T}H(t,s)f(s,y_s)ds+\sum_{k=1}^{m}H(t,t_k)I_k(y(t^-_k)),&
\mbox{if } t\in [0,T],
\end{cases}
$$
where
$$
H(t,s)=(e^{-\lambda T}-1)^{-1}
\begin{cases} e^{-\lambda (T+s-t)},& 0\leq s\leq t\leq T, \\
e^{-\lambda(s-t)},& 0\leq t<s\leq T. \end{cases}
$$

\begin{remark}\label{r1}\rm
 It is easy to show  that the fixed points of
$N$ are solutions to the problem (\ref{e1})-(\ref{e3}); see  \cite{BeHeNt1}.
\end{remark}

We shall show that $N$ satisfies the assumptions of Theorem \ref{t1}.
This will be done in several steps.

\noindent{\bf Step 1:}  $N$ is continuous.
Let $\{y_{n}\}$ be a sequence such that $y_{n}\to y$ in
$\Omega$. Then
\begin{align*}
&| N(y_{n}(t))- N(y(t))|\\
&\leq \int_{0}^{T}|H(t,s)| |f(s,y_{ns})-f(s,y_s)|ds
+\sum_{k=1}^{m}|H(t,t_k)||I_{k}(y_{n}(t_k))-I_{k}(y(t_k))| \\
&\leq \frac{1}{1-e^{-\lambda T}}\int_{0}^{T}|f(s,y_{ns})-f(s,y_s)|ds
+\frac{1}{1-e^{-\lambda T}}\sum_{k=1}^{m}|I_{k}(y_{n}(t_k))-I_{k}(y(t_k))|.
\end{align*}
Since the functions  $H$, $I_k$, $k=1,\dots,m$ are continuous
and $f$ is an $L^{1}$-Carath\'eodory,
\begin{align*}
&\|N(y_{n})-N(y)\|_{\Omega}\\
&\leq \frac{1}{1-e^{-\lambda T}}\|f(.,y_{n}) -f(.,y)\|_{L^1}
+\frac{1}{1-e^{-\lambda T}}\sum_{k=1}^{m} |I_{k}(y_{n}(t_k))-I_{k}(y(t_k))|
\end{align*}
which approaches zero as $n\to\infty$.

\noindent{\bf Step 2:} $ N$ maps bounded sets into bounded sets in $\Omega$.
Indeed, it is sufficient to show that for any $q>0$ there exists a positive
constant $\ell$ such that for each
$y\in B_{q}=\{y\in \Omega: \|y\|_{\Omega}\leq q \}$ one has
$\| N(y)\|_{\Omega}\leq \ell$.
Let $y\in B_{q}$. Then for $t\in [0,T]$ we have
$$
 N(y)(t)=\int_{0}^{T}H(t,s)f(s,y_s)ds
+\sum_{k=1}^{m}H(t,t_{k})I_{k}(y(t_{k})).
$$
By (H2) we have for each $t\in [0,T]$
\begin{align*}
| N(y)(t)|&\leq \int_{0}^{T}|H(t,s)||f(s,y_s)|d\,s
+\sum_{k=1}^{m}|H(t,t_{k})||I_{k}(y(t_k))| \\
&\leq \int_{0}^{T}|H(t,s)|h_q(s)d\,s+
\sum_{k=1}^{m}|H(t,t_{k})|c_{k}.
\end{align*}
Then for each $h\in N(B_{q})$ we have
$$
\| N(y)\|_{\Omega}\leq \frac{1}{1-e^{-\lambda T}}
\Big(\int_{0}^{T}h_q(s)d\,s+\sum_{k=1}^{m}c_{k}\Big):=\ell.
$$
{\bf Step 3:} $N$ maps bounded set into equicontinuous sets of
$\Omega$.
Let $\tau_{1}, \tau_{2}\in [0,T]$, $\tau_{1}<\tau_{2}$ and
$B_{q}$ be a bounded set of $\Omega$ as in Step 2.
Let $y\in B_{q}$ and $t\in [0,T]$ we have
$$
 N(y)(t)=\int_{0}^{T}H(t,s)f(s,y_s)d\,s
+\sum_{k=1}^{m}H(t,t_{k})I_{k}(y(t_{k})).
$$
Then
\begin{align*}
&|N(y)(\tau_{2})-N(y)(\tau_{1})|\\
&\leq \int_{0}^{T}|H(\tau_{2},s)-H(\tau_{1},s)|h_q(s)ds
+ \sum_{k=1}^{m}|H(\tau_{2},t_{k})-H(\tau_{1},t_{k})| c_{k}.
\end{align*}
As $\tau_{2}\to \tau_{1}$ the right-hand side of the
above inequality tends to zero.

As a consequence of Steps 1 to 3 together
with the Arzela-Ascoli theorem we can conclude that
 $N:\Omega\to \Omega$ is  completely continuous.

Let $C=\{ y\in\Omega :y(t)\geq 0\hbox{ for } t\in[-r, T]\}$
be a cone in $\Omega$. Since  $f,  H,  I_k$, $k=1,\dots,m$  are positive functions,
then $N(C)\subset C$ and $N: \overline C_{R}\to\overline C_{R}$
is compact.  By  (H1), (H2), (H4) we can show  that if
$y\in\overline C_{R}$ then $N(y)\subset\overline C_{R}$.
Let $\psi : C\to [0,\infty)$ defined by
$\psi(y)=\min_{t\in[a,b]}y(t)$. It is clear that $\psi$ is a
nonnegative concave continuous functional and $\psi(y)\leq
\|y\|_{\Omega}$ for $y\in\overline C_{R}$. Now it remains to show
that the hypotheses of Theorem \ref{t1} are satisfied. First
notice that condition (A2) of Theorem \ref{t1} holds since for
$y\in\overline C_r$, and from (H1) and (H2) we have
\begin{align*}
|N(y)(t)|&\leq \int_{0}^{T}|H(t,s)||f(s,y_s)|ds
+\sum_{k=1}^{m}|H(t,t_k)|I_{k}(y(t_k))| \\
&\leq \int_{0}^{T}g(\|y_s\|)p(s)ds
+\sum_{k=1}^{m}|H(t,t_k)|I_{k}(y(t_k))| \\
&\leq g(r)\|p\|_{L^{1}}+\frac{1}{1-e^{-\lambda T}}
\sum_{k=1}^{m}c_k<r.
\end{align*}
Let $K\geq L$ and $y(t)=\frac{L+K}{2}$ for $t\in[-r,T]$. By the definition of
$C(\psi, L,K), y$ belongs to  $C(\psi, L,K)$. Then
$y\in \{ y\in C(\psi, L, K): \psi(y)>L\}$. Also if $y\in C(\psi, L,K)$  then
$$
\psi(N(y))=\min_{t\in[a,b]}\Big(
\sum_{k=1}^{m}H(t,t_{k})I_k(y(t_k))+\int_0^TH(t,s)f(s,y_s)ds\Big).
$$
Then from (H3) we have
\begin{align*}
 \psi(N(y))&=\min_{t\in[a,b]}\left( \sum_{k=1}^{m}H(t,t_k)I_k(y(t_k))
+\int_0^TH(t,s)f(s,y_s)ds\right) \\
&\geq \Big(M\sum_{k=1}^mc_k+M\int_0^Tg(\|y_s\|)p(s)ds \Big)\\
&\geq M\Big( \sum_{k=1}^mc_k+g(L)\int_0^Tp(s)ds\Big) > L.
\end{align*}
So the conditions (A1) and (A2) of Theorem \ref{t1} are satisfied.

Finally we will be prove that (A3) of Theorem \ref{t1} holds. Let
$y\in C(\psi, L, R)$   with $\|N(y)\|_{\Omega}>K$
Thus
\begin{align*}
 \psi(N(y))&= \min_{t\in[a,b]}\Big(\sum_{k=1}^{m}H(t,t_k)I_k(y(t_k))
+\int_0^TH(t,s)f(s,y_s)ds\Big)\\
&\geq M_1M_*\Big(\sum_{k=1}^{m}c_k+g(R)\int_0^Tp(s)ds\Big) \\
&\geq M_1\|N(y)\|_{\Omega}>M_1K >L.
\end{align*}
Thus condition (A3) holds. Then  Leggett and Williams
fixed point theorem implies that
$N$ has at least three fixed points $y_1, y_2, y_3$
which are  solutions to problem (\ref{e1})--(\ref{e3}).
Furthermore, we have
$$
y_1\in C_r,\ \ y_2\in\{ y\in C(\psi, L, R):\psi(y)>L\},\quad
y_3\in C_R-\{C(\psi, L, R)\cup( C_r)\}.
$$
\end{proof}

\section{Second Order Impulsive  FDEs}

In this section we give an existence result for the boundary-value problem
(\ref{e4})--(\ref{e7}).

\begin{definition} \label{def4.1} \rm
A function $y\in \Omega\cap\cup_{k=0}^{m}AC^{1}((t_{k},t_{k+1}),\mathbb{R})$
is said to be a solution of (\ref{e4})--(\ref{e7})  if $y$
satisfies
 $y''(t)-\lambda y(t)=f(t,y_{t})$
a.e. on  $J\backslash\{t_{1},\dots,t_{m}\}$\ and the conditions
$\Delta y|_{t=t_{k}}=I_{k}(y(t_{k}^{-})), \
\Delta y'|_{t=t_{k}}=\overline I_{k}(y(t_{k}^{-})), \
k=1,\dots,m$, $y(t) =\phi(t), \ t \in [-r,0], \
 y(0)-y(T)=\mu_0, \ y'(0)-y'(T)=\mu_1$. \end{definition}
We now consider the ``linear problem''
\begin{equation}\label{e8}
y''(t)-\lambda y(t)=g(t), \ \   t\neq t_{k}, \ k=1,\dots, m,
\end{equation}
subjected to the conditions (\ref{e5}), (\ref{e6}), (\ref{e7}),
and where $g\in L^{1}([t_{k},t_{k+1}],\mathbb{R})$.
Note that (\ref{e5})--(\ref{e7}), (\ref{e8}) is not really a linear problem
since the impulsive functions are not necessarily linear. However, if
$I_{k},  \bar I_{k}$, $k=1,\dots,m$ are  linear,
then (\ref{e5})--(\ref{e7}), (\ref{e8}) is a linear impulsive problem.

We need the following auxiliary result:

\begin{lemma}\label{l5}
$y\in \Omega\cap\cup_{k=0}^{m}AC^{1}((t_{k},t_{k+1}),\mathbb{R})$
is a solution of (\ref{e5})--(\ref{e7}), (\ref{e8}), if and only if $y\in\Omega$
is a solution of the impulsive integral functional equation,
\begin{equation}\label{e9}
y(t)=\begin{cases}
\phi(t),   & t\in [-r,0],\\[2pt]
\int_{0}^{T}M(t,s)h(s)ds+M(t,0)\mu_{1}+N(t,0)\mu_{0} \\
 +\sum_{k=1}^{m}[M(t,t_{k})I_{k}(y(t_{k}))
+N(t,t_{k})\bar I_{k}(y(t_{k}))], & t\in [0,T],
\end{cases}
\end{equation}
where
$$
M(t,s)=\frac{-1}{2\sqrt\lambda (e^{\sqrt\lambda T}-1)}
\begin{cases} e^{\sqrt\lambda (T+s-t)}+ e^{\sqrt\lambda (t-s)},&
0\leq s\leq t\leq T,\\
e^{\sqrt\lambda (T+t-s)}+ e^{\sqrt\lambda (s-t)},& 0\leq t<s\leq T,
\end{cases}
$$
and
$$
N(t,s)=\frac{\partial}{\partial
t}M(t,s)=\frac{1}{2(e^{\sqrt\lambda T}-1)}
\begin{cases} e^{\sqrt\lambda (T+s-t)}- e^{\sqrt\lambda (t-s)},
& 0\leq s\leq t\leq T,\\
e^{\sqrt\lambda (s-t)}- e^{\sqrt\lambda (T+t-s)},
& 0\leq t<s\leq T.
\end{cases}
$$
\end{lemma}

We omit the proof of this lemma since it is similar to the proof of results in
\cite{Don}.

We are now in  a position to state and prove our existence result for problem
(\ref{e4})-(\ref{e7}). We first list the following hypotheses:

\begin{itemize}
\item[(H5)]  There exist constants $d_{k}$ such that
$|\overline I_{k}(x)|\leq d_{k}$, $k=1,\dots,m$ for each
$x\in\mathbb{R}$

\item[(H6)] There exist a function $g^*: [0,\infty)\to[0,\infty)$ continuous and
non-decreasing, a function
 $p\in L^{1}(J,\mathbb{R}_{+})$, $r^*>0$, and $0< M^*\leq 1$ such that
$$
M^*p(t)g^*(\|u\|)\leq |M(t,s)f(t,u)|\leq p(t)g^*(\|u\|)
$$
for each $(t,s,u)\in J\times  J\times D$, and
$$
C\sum_{k=1}^{m}(c_k+d_k) +C_*[|\mu_{1}|+|\mu_{0}|] +\sup_{(t,s)\in
[0,T]\times[0,T]}|M(t,s)|g^*(r^*)\int_{0}^{T}p(s)ds<r^*
$$
where
\begin{gather*}
C=\max(\sup_{(t,s)\in [0,T]\times[0,T]}|M(t,s)|,\sup_{(t,s)\in
[0,T]\times[0,T]}|N(t,s)|), \\
C_*=\max(\sup_{t\in [0,T]]}|M(t,0)|,\sup_{t\in [0,T]}|N(t,0)|)
\end{gather*}

\item[(H7)]  There exist\ $L^*>r^*$, $0<M^*\leq1$ and an interval
$[a,b]\subset(0,T)$ such that
\begin{align*}
&\min_{t\in[a,b]}\Big(
\sum_{k=1}^{m}[M(t,t_k)I_k(y(t_{k})) +N(t,t_k)\overline
I_k(y(t_{k}))]+\int_0^TM(t,s)f(s,y_s)ds \Big)\\
&\geq M^*\min_{t\in[a,b]}\Big(
M(t,0)\mu_{1}+N(t,0)\mu_{0}
+g^*(L^*)\int_{0}^{T}p(s)d\,s+\sum_{k=1}^{m}[c_k+d_k]\Big)\\
&>L^*
\end{align*}

\item[(H8)] There exist $R^*, K, 0<M^*_1\leq M^*$\ such that
$M_1^{*-1}L<K\leq R^*$,
$$
C\sum_{k=1}^{m}[c_k+d_k] +C_*(|\mu_{0}|+|\mu_1|)
+ \sup_{(t,s)\in
[0,T]\times[0,T]}|M(t,s)|g^*(R^*)\int_{0}^{T}p(s)ds <R^*
$$
and
\begin{align*}
&\min_{t\in[a,b]}\Big(M(t,0)\mu_1+N(t,0)\mu_0
+\sum_{k=1}^{m}M(t,t_k)I_k(y(t_{k}))\\
&+N(t,t_k)\overline
I_k(y(t_{k}))+\int_0^TM(t,s)f(s,y_s)ds\Big)\\
&\geq
M_1^*\Big( C_*(|\mu_{1}|+|\mu_{0}|) \\
&\quad + \sup_{(t,s)\in
[0,T]\times[0,T]}|M(t,s)|g^*(R^*)\int_{0}^{T}p(s)ds
+C\sum_{k=1}^{m}[c_k+d_k]\Big).
\end{align*}
\end{itemize}

\begin{theorem} \label{thm4.3}
Suppose that hypotheses $(H1), \ (H5)-(H8)$
are satisfied. Then the problem (\ref{e4})--(\ref{e7})
has at least three positive solutions.
\end{theorem}

\begin{proof}
We transform the  problem into a fixed point problem.
Consider the operator  $N_1:\Omega\to \Omega$
defined by
$$
N_1(y)(t)=\begin{cases}
 \phi(t), & t\in [-r,0],\\[2pt]
\int_{0}^{T}M(t,s)f(s,y_s)ds+M(t,0)\mu_{1}+N(t,0)\mu_{0}\\
+\sum_{k=1}^{m}[M(t,t_{k})I_{k}(y(t_{k}))
+N(t,t_{k})\overline I_{k}(y(t_{k}))], & t\in [0,T],
\end{cases}
$$
As in Theorem \ref{t2} we can show that $N_1$
is compact. Now we prove only that the  hypotheses of Theorem \ref{t1}
are satisfied.

Let $C=\{ y\in\Omega :y(t)\geq 0\hbox{ for } t\in[-r, T]\}$
be a cone in $\Omega$. It is clear that $N_1(C)\subset C$ and
$N_1: \overline C_{R^*}\to \overline C_{R^*}$ is  compact. By
(H1), (H5), (H6), (H8) we can show  that if $y\in\overline
C_{R^*}$ then $N_1(y)\in \overline C_{R^*}$.
Let $\psi:C\to [0,\infty)$ defined by $\psi(y)=\min_{t\in[a,b]}y(t)$.
It is clear that $\psi$ is a nonnegative concave continuous functional and
$\psi(y)\leq \|y\|_{\Omega}$ for $y\in\overline C_{R^*}$. Now it
remains to show that the hypotheses of Theorem \ref{t1} are
satisfied. First notice that condition (A2)
holds since for $y\in\overline C_{r^*}$, we have from (H1), (H5),
(H6),
\begin{align*}
|N_1(y)(t)|
&\leq \int_{0}^{T}|M(t,s)||f(s,y_s)|d\,s+|M(t,0)||\mu_{1}|
+|N(t,0)||\mu_{0}| \\
&\quad +\sum_{k=1}^{m}[|M(t,t_{k})|c_{k}+|N(t,t_{k})|d_{k}] \\
&\leq \int_0^Tg^*(\|y_s\|)p(s)d\,s +C_*(|\mu_{1}|+|\mu_{0}|)
+C\sum_{k=1}^{m}(c_{k}+d_k)\\
&\leq \sup_{t\in [0,T]\times[0,T]}|M(t,s)|g^*(L^*)\int_0^Tp(s)ds
+C_*(|\mu_{1}|+|\mu_{0}|) \\
&\quad +C\sum_{k=1}^{m}(c_{k}+d_k)< r^*.
\end{align*}
Let $K^*\geq L^*$ and $y(t)=\frac{L^*+K^*}{2}$ for $t\in[-r,T]$. By the definition of $C(\psi, L^*,K^*), y$ is in  $C(\psi,
L^*,K^*)$. Then $y\in \{ y\in C(\psi, L^*, K^*): \psi(y)>L^*\}$.
Also if $y\in C(\psi, L^*,K^*)$ we have
\begin{align*}
\psi(N_1(y))=\min_{t\in[a,b]}\Big(
&\int_{0}^{T}M(t,s)f(s,y_s)ds+M(t,0)\mu_{1}
+N(t,0)\mu_{0}\\
&+\sum_{k=1}^{m}[M(t,t_{k})I_{k}(y(t_{k})) +N(t,t_{k})\overline
I_{k}(y(t_{k}))\Big).
\end{align*}
Then from  (H7) we get
\begin{align*}
\psi(N_1(y))&\geq M^*\min_{t\in[a,b]}\Big(M(t,0)\mu_1+N(t,0)\mu_0
+\sum_{k=1}^{m}(c_k+d_k)+g^*(L^*)\int_0^Tp(s)ds\Big)\\
&>L^*.
\end{align*}
So conditions (A1) and (A2)  of Theorem \ref{t1} are satisfied.

Finally to see that (A3) holds
let $y\in C(\psi, L^*, R^*)$  with $\|N_{1}(y)\|_{\Omega}>K^*$ then
from (H8) we have
\begin{align*}
&\psi(N_{1}(y))\\
&\geq M_1^*\Big( \sum_{k=1}^{m}(c_k+d_k)
+C_*|\mu_1|+C|\mu_0| +\sup_{t\in [0,T]\times[0,T]}|M(t,s)|
g^*(R^*)\int_0^Tp(s)ds\Big)\\
&\geq M^{*}_1\Big(C_*|\mu_1|+C|\mu_0|+C^*\sum_{k=1}^{m}(c_k+d_k)
+\sup_{t\in [0,T]\times[0,T]}|M(t,s)| g^*(R^*)\int_0^Tp(s)ds\Big)\\
&\geq M_1^{*}\|N_1(y)\|_{\Omega}
> M_1^{*}K>L^*.
\end{align*}
Thus condition (A3) for Theorem \ref{t1} holds.
As consequence of   Leggett and Williams  theorem we deduce that $N_1$
has at least three fixed points $y_1, y_2, y_3$
which are  solutions to problem (\ref{e4})--(\ref{e7}).
Furthermore, we have
$$
y_1\in C_{r^*},\quad y_2\in\{ y\in C(\psi, L^*, R^*):\psi(y)>L^*\},
\quad y_3\in C_{R^*}-\{C(\psi, L^*, R^*)\cup C_{r^*}\}.
$$
\end{proof}

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\end{document}