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\AtBeginDocument{{\noindent\small
{\em Electronic Journal of Differential Equations},
Vol. 2004(2004), No. 84, pp. 1--9.\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 2004 Texas State University - San Marcos.}
\vspace{9mm}}
\begin{document}
\title[\hfilneg EJDE-2004/84\hfil First order impulsive differential
inclusions]
{On periodic boundary value problems of first-order
perturbed impulsive differential inclusions}
\author[B. C. Dhage, A. Boucherif, S. K. Ntouyas\hfil EJDE-2004/84\hfilneg]
{Bapurao C. Dhage, Abdelkader Boucherif, Sotiris K. Ntouyas}
\address{Bapurao C. Dhage \hfill\break
Kasubai, Gurukul Colony,
Ahmedpur-413 515, Dist: Latur, Maharashtra, India}
\email{bcd20012001@yahoo.co.in}
\address{Abdelkader Boucherif \hfill\break
Department of Mathematical sciences,
King Fahd University of Petroleum and Minerals, P. O. Box 5046,
Dhahran 31261, Saudi Arabia}
\email{aboucher@kfupm.edu.sa}
\address{Sotiris K. Ntouyas \hfill\break
Department of Mathematics, University of Ioannina,
451 10 Ioannina, Greece}
\email{sntouyas@cc.uoi.gr}
\date{}
\thanks{Submitted April 19, 2004. Published June 13, 2004.}
\subjclass[2000]{34A60, 34A37}
\keywords{Impulsive differential inclusion, existence theorem}
\begin{abstract}
In this paper we present an existence result for a first
order impulsive differential inclusion with periodic boundary
conditions and impulses at the fixed times under the convex
condition of multi-functions.
\end{abstract}
\maketitle
\newtheorem{theorem}{Theorem}[section]
\newtheorem{lemma}{Lemma}[section]
\newtheorem{definition}{Definition}[section]
\newtheorem{remark}{Remark}[section]
\numberwithin{equation}{section}
\section{Introduction}
In this paper, we study the existence of
solutions to a periodic nonlinear boundary value problems for
first order Carath\'eodory impulsive ordinary differential inclusions
with convex multi-functions. Given a closed and bounded interval
$ J:=[0,T] $ in $\mathbb{R}$, the set of real numbers, and given the
impulsive moments $t_1 , t_2 , \dots , t_p$
with $0 =t_0 < t_1 < t_2 < \dots < t_p < t_{p+1} = T$,
$J'=J\setminus\{t_1 , t_2 , \dots , t_p\}$, $J_j =(t_j , t_{j+1})$,
consider the
following periodic boundary-value problem for impulsive differential
inclusions (in short IDI):
\begin{gather} \label{e11}
x'(t) \in F(t, x(t))+G(t, x(t)) \mbox{ a.e. } t \in J', \\
\label{e12}
x(t_j^+ ) = x(t_j^-) + I_j (x(t_j^-)), \\
\label{e13}
x(0) = x(T),
\end{gather}
where $F, G :
J \times \mathbb{R} \to P_{f}(\mathbb{R})$ are impulsive
multi-functions, $I_j : \mathbb{R}\to \mathbb{R}$,
$j=1,2, \dots, p $ are the impulse functions and
$x(t_j^+ )$ and $x(t_j^- )$ are respectively the right and the
left limit of $x$ at $t = t_j$.
Let $C(J, \mathbb{R})$ and $L^1(J, \mathbb{R})$ denote the
space of continuous and Lebesgue integrable real-valued functions on
$J$.
Consider the Banach space
\[%\begin{align*}
X := \bigl\{ x: J \to \mathbb{R} : x \in C(J', \mathbb{R}),
x(t_j^+ ) , x(t_j^- ) \mbox{ exist},
\; x(t_j^- ) = x(t_j),\; j=1,2,\dots, p \bigr\}
\]%\end{align*}
equipped with the norm
$\| x \| = \max\{|x(t)|: t \in J\}$,
and the space
\[
Y := \{ x \in X : x \mbox{ is differentiable a.e. on } (0,T) ,
x' \in L^1(J , \mathbb{R}) \} \,.
\]
By a solution of
(\ref{e11})--(\ref{e13}), we mean a function
$x$ in $Y_T := \{ v \in Y : v(0) = v(T)\}$
that satisfies the differential inclusion (\ref{e11}), and the
impulsive
conditions (\ref{e12}).
Several papers have been devoted to the study of initial and boundary
value
problems for impulsive differential inclusions
(see for example \cite{BB,BBN}). Some basic results in the theory of
periodic
boundary value problems for
first order impulsive differential equations may be found in
\cite{N1,N2,N3} and the references therein. Also, for a general theory
on
impulsive differential equations we refer the interested reader to
\cite{R}
and the monographs \cite{LBS} and \cite{SP}. Our aim is to provide
sufficient conditions on the multifunctions $F$, $G$ and the impulsive
functions $I_{j}$, that insure the existence of solutions of problem
IDI
(\ref{e11})--(\ref{e13}).
\section{Preliminaries}
Let $(E, \| \cdot \|)$ be a Banach space and let $P_{f}(E)$
denote the class of all non-empty subsets of $E$ with the property $f$.
Thus $P_{cl}(E),
P_{bd}(E), P_{cv}(E)$ and $P_{cp}(E)$ denote respectively the classes
of
all closed,
bounded, convex and compact subsets of $E$. Similarly $P_{cl, cv,
bd}(E)$
and $P_{cp,
cv}(E)$ denote the classes of all closed, convex and bounded and
compact and
convex subsets of $E$. For $x\in E$ and $Y,Z\in P_{bd,cl}(E)$ we
denote by
$D(x,Y)=\inf\{\|x-y\|: y\in Y\}$, and
$\rho(Y,Z)=\sup_{a\in Y}D(a,Z)$.
Define a function $H: P_{bd,cl}(E)\times P_{bd,cl}(E)\to
\mathbb{R}^+$ by
$$ H(A,B)= \max \{\rho(A,B,\rho(B,A)\}.
$$
The function $H$ is called a
Hausdorff metric on $E$. Note that $\|Y\|=H(Y,\{0\})$.
A map $F:E\to P(E)$ is called a
{\em multi-valued mapping} on $E$. A point $u\in E$ is called a {\em
fixed
point} of
the multi-valued operator $F:E\to P(E)$ if $u\in F(u)$. The fixed
points
set of $F$ will be denoted by $\mathop{\rm Fix}(F)$.
A multivalued map $F: [a,b] \subset \mathbb{R} \to P_{cl,bd}(E)$
is said to be {\em measurable} if for each $x \in X$, the distance
between
$x$ and $F(t)$ is
a measurable function on $[a,b]$. A function $f : [a,b]\to E$
is called {\em measurable selector} of the
multi-function $F$ if $f$ is measurable and $f(t)\in
F(t)$ for almost everywhere $t\in [a,b]$.
\begin{definition} \label{def2.1} \rm
Let $F:E\to P_{bd,cl}(E)$ be a multi-valued operator. Then $F$
is called a multi-valued contraction if there exists a constant
$\alpha\in
(0,1)$ such
that for each $x,y\in E$ we have
$$ H(F(x),F(y))\le \alpha \|x-y\|. $$
The constant $\alpha$ is
called a contraction constant of $F$.
\end{definition}
A multifunction $F$ is called {\em upper semi-continuous (u.s.c.)}
if for each $x_0 \in E$, the set $F(x_0)$ is a nonempty and closed
subset of $E$, and for each open set $N \subset E$ containing
$F(x_0)$, there exists an open neighborhood $M$ of $x_0$ such that
$F(M) \subset N$. If $F$ is nonempty and compact-valued, then $F$
is u.s.c. if and only if $F$ has a closed graph, i.e., given
sequences $\{ x_n {\}}_{n=1}^{\infty}\to x_0 , \{ y_n
{\}}_{n=1}^{\infty} \to y_0 \,, y_n \in F(x_n)$ for every
$n=1,2,\dots $ imply $y_0 \in F(x_0).$\vskip0.3cm $F$ is {\em bounded on
bounded sets} if $\bigcup F(S)$ is bounded in $E$ for every
bounded set $S \subset E$, i.e., $\sup_{ x \in S} \{ \sup\{|y| :
y\in F(x)\}\}<+ \infty$. Again the operator $F$ is called {\em
compact} if $\overline{\bigcup F(E)}$ is a compact subset of $E$.
$F$ is said to be {\em completely continuous} if it is u.s.c. and
$\bigcup F(S)$ is relatively compact set in $E$ for every bounded
subset $S$ of $E$. Finally a multi-valued operator $F$ is called
{\em convex (resp. compact) valued} if $F(x)$ is a convex (resp.
compact) set in $E$ for each $x\in E$.\vskip0.3cm The following form of a
fixed point theorem of Dhage \cite{D} will be used while proving
our main existence result.
\begin{theorem}[Dhage \cite{D}] \label{t21}
Let $B(0,r)$ and $B[0,r]$ denote respectively the open and closed
balls
in a Banach
space $E$ centered at origin and of radius $r$ and let $A:E\to
P_{cl,cv,bd}(E)$ and
$B:B[0,r]\to P_{cp,cv}(E)$ be two multi-valued operators satisfying
\begin{itemize}
\item[(i)] $A$ is multi-valued contraction , and
\item[(ii)] $B$ is completely continuous.
\end{itemize}
Then either
\begin{itemize}
\item[(a)] the operator
inclusion $x\in Ax+Bx$ has a solution in $B[0,r]$, or
\item[(b)] there exists an $u\in E$ with
$\|u\|=r$ such that
$\lambda u\in Au+Bu$ for some $\lambda>1$.
\end{itemize}
\end{theorem}
In the following section we prove
the main existence results of this paper.
\section{Main Results}
Consider the following linear periodic problem with some given impulses
${\theta}_j \in \mathbb{R}$, $j=1,2,\dots, p $:
\begin{gather} \label{e31}
x'(t) + k x(t) = \sigma (t), \mbox{a.e. } t \in J', \\
\label{e32}
x(t_j^+ ) - x(t_j^-) = {\theta}_j , j=1,2,\dots, p, \\
\label{e33}
x(0) = x(T),
\end{gather}
where $k > 0$, and $\sigma \in L^1(J)$. The solution of
(\ref{e31})--(\ref{e33}) is given by (see \cite[Lemma 2.1]{N1})
\begin{equation} \label{e34}
x(t) = \int_{0}^{T}{g_k(t,s) \sigma (s) \,ds}
+\sum_{j=1}^{p}{g_k(t,t_j) {\theta}_j},
\end{equation}
where
\[
g_k (t,s)=\begin{cases}
\dfrac{e^{-k(t-s)}}{1-e^{-kT}}, & 0\leq s \leq t\leq T \\[5pt]
\dfrac{e^{-k(T+t-s)}}{1-e^{-kT}},& 0\leq t < s\leq T \,.
\end{cases}
\]
Clearly the function $g_k(t, s)$ is discontinuous and nonnegative on
$J\times J$ and
has a jump at $t=s$.
Let $$ M_k := \max{\{ |g_k(t,s)| : t,s \in [0,T] \} }=
\frac{1} {1-e^{-kT}}.
$$
Now $x \in Y_T$ is a solution of(\ref{e11})--(\ref{e13}) if and only
if
\begin{equation}\label{e35}
x(t)\in B_k^1 x(t) + B_k^2 x(t),\quad t\in J
\end{equation}
where the multi-valued operators $B_k^1 $ and $B_k^2 $ are
defined by
\begin{gather}\label{e36}
\mathcal{B}_k^1 x(t)=\int_{0}^{T} g_k(t,s) F(s, x(s))\,ds, \\
\label{e37} \mathcal{B}_k^2 x(t)=\int_{0}^{T} g_k(t,s) [k x(s) +
G(s, x(s))] \,ds + \sum_{j=1}^{p} g(t,t_j)I_j(x(t_j^-)).
\end{gather}
\begin{definition} \rm
A multi-function $\beta:J \times \mathbb{R} \to P_{f}(\mathbb{R})$ is
called
an impulsive Carath\'eodory if
\begin{itemize}
\item [(i)] $\beta(\cdot, x)$ is measurable for every $x \in
\mathbb{R}$ and
\item [(ii)] $\beta(t, \cdot)$ is upper semi-continuous a.e. on $J$.
\end{itemize}
Further the impulsive Carath\'eodory multifunction
$\beta$ is called impulsive $L^{1}$-Carath\'eodory if
\begin{itemize}
\item [(iii)] for every $r>0$ there exists a
function $h_r \in L^1(J)$ such that
$$
\|\beta(t,x)\|=\sup \{|u| : u\in \beta(t,x)\} \leq h_r(t)
\mbox{a.e.\ } t\in J
$$
for all $x \in \mathbb{R}$ with $|x | \leq r$.
\end{itemize}
\end{definition}
Denote
$$
S^{1}_{\beta}(x)= \{v\in L^1(J, \mathbb{R}) : v(t)\in \beta(t, x)
\mbox{ a.e. $t\in J$}\}.
$$
\begin{lemma}[Lasota and Opial \cite{LO}] \label{l31}
Let $E$ be a Banach space. Further if $\mathop{\rm dim}
(E) < \infty$ and $\beta: J\times E\to P_{bd,cl}(E)$ is
$L^{1}$-Carath\'eodory, then $S^{1}_{\beta}(x) \ne \emptyset$
for each $x \in E$.
\end{lemma}
\begin{definition} \rm
A measurable multi-valued function $F: J\to P_{cp}(\mathbb{R})$ is
said to be integrably bounded if there exists a function $h\in
L^1(J,\mathbb{R})$ such that $|v|\le h(t)$ a.e. $t\in J$
for all $v\in F(t)$.
\end{definition}
\begin{remark}\label{r31} \rm
It is known that if $F:J\to \mathbb{R}$ is an integrably bounded
multi-function, then the set $S_F^1$ of all Lebesgue integrable
selections of $F$ is closed and non-empty. See Covitz and Nadler
\cite{CN}.
\end{remark}
We now introduce the following assumptions:
\begin{itemize}
\item[(H1)] The functions $I_j: \mathbb{R} \to \mathbb{R}$,
$j=1,2,\dots, p$ are continuous, and there exist $c_j \in \mathbb{R}$,
$j=1,2,\dots, p$ such that $| I_j (x)| \leq c_j$,
$j=1,2,\dots, p$ for every $x \in \mathbb{R}$.
\item[(H2)] $G:J \times \mathbb{R} \to P_{cp,cv}(\mathbb{R})$ is an
impulsive
Carath\'eodory multi-function.
\item[(H3)] There exist a real number $k>0$ and a
Carath\'eodory function $\omega: J \times \mathbb{R}_+ \to
\mathbb{R}_+$ which
is nondecreasing with respect to its second argument such that
$$
\|G(t,x) + k x \|=\sup\{|v| : v\in G(t,x)+k x\}\leq \omega(t, |x|)
$$
a.e. $t \in J',x \in \mathbb{R}$.
\item[(H4)] The multi-function
$t\mapsto F(t,x)$ is measurable and integrally bounded for each $x\in
\mathbb{R}$.
\item[(H5)] The multi-function $F(t,x)$ is
$F:J\times \mathbb{R} \to P_{cl, cv, bd}(\mathbb{R})$ and
there exists a function
$\ell\in L^1(J,\mathbb{R})$ such that
\[
H(F(t,x),F(t,y)) \le \ell(t)|x-y|\quad \mbox{a.e. }t\in J
\]
for all $x,y\in \mathbb{R}$.
\end{itemize}
Note that the hypotheses (H1)--(H5) are not new, they have
been used extensively in the literature on differential inclusions.
Also
(H3) in the special case $\omega(t,r)= \phi(t)\psi(r)$ has been used
by several authors. See Dhage \cite{D}
and the references therein.
\begin{lemma}\label{l32} Assume that (H2)--(H3) hold.
Then the operator $S_{k+G}^1:Y_T \to P_{f}(L^1(J,\mathbb{R}))$ defined
by
\begin{equation}\label{e38}
S_{k+G}^1(x) := \big\{ v\in L^1(J, \mathbb{R}) : v(t) \in k
x(t)+G(t,x(t)) \mbox{ a.e. } t \in J \big\}
\end{equation}
is well defined, u.s.c., closed and
convex valued, and sends bounded subsets of $Y_T$ into bounded subsets
of
$L^1(J,\mathbb{R})$.
\end{lemma}
\begin{proof} Since (H2) holds, by Lemma \ref{l31}
$S_{k+G}^1(x)\ne\emptyset$ for each $x\in Y_T$. Below we show that
$S_{k+G}^1$ has the desired properties
on $Y_T$.
\noindent {\bf Step I:} First we show that $S_{k+G}^1$ has closed
values on
$Y_T$. Let
$x\in Y_T$ be arbitrary and let $\{\omega_{n}\}$ be a sequence in
$S_{k+G}^1(x)\subset
L^{1}(J,\mathbb{R}) $ such that $\omega_{n} \to \omega$. Then
$\omega_{n} \to
\omega$ in
measure. So there exists a subset $S$ of positive integers such that
$\omega_{n}
\to \omega$ a.e. $n\to \infty$ through $S$. Since the hypothesis
(H2) holds,
we have
$\omega\in S_{k+G}^1 (x)$. Therefore, $S_{k+G}^1(x)$ is a closed set in
$L^{1}(J,\mathbb{R})$.
Thus for each $x\in Y_T$, $S_{k+G}^1 (x)$ is a non-empty, closed
subset of
$L^{1}(J,\mathbb{R})$ and consequently $S_{k+G}^1$ has non-empty and
closed values
on $Y_T$.
\noindent {\bf Step II:}
Next we show that $S_{k+G}^1(x)$ is convex subset of
$L^{1}(J,\mathbb{R})$ for each
$x\in Y_T$.
Let
$v_1, v_2 \in S_{k+G}^1(x)$ and let $\lambda\in [0, 1]$. Then there
exist
functions $f_1, f_2\in S_{k+G}^1(x)$ such that
$$
v_1(t)= k x(t)+f_1(t)\quad\mbox{and}\quad
v_2(t)= k x(t)+f_2(t)
$$
for $t\in J$. Therefore we have
\begin{align*}
\lambda v_1(t)+(1-\lambda)v_2(t)&=
\lambda\bigl[k x(t)+f_1(t)\big]+(1-\lambda)\bigl[ k x(t)+f_2(t)\bigr]\\
&=\lambda k x(t)+(1-\lambda) k x(t)+\lambda f_1(t)+(1-\lambda) f_2(t)\\
&= k x(t)+f_3(t)
\end{align*}
where $f_3(t)= \lambda f_1(t)+(1-\lambda) f_2(t)$ for all $t\in J$.
Since $G(t, x)$ is convex for each $x\in \mathbb{R}$, one has
$f_3(t)\in G(t,
x(t))$ for all $t\in
J$. Therefore,
$$ \lambda v_1(t)+(1-\lambda)v_2(t)\in k x(t)+G(t, x(t))
$$
for all $t\in J$ and consequently $\lambda v_1+(1-\lambda) v_2 \in
S_{k+G}^1(x)$.
As a result $S_{k+G}^1(x)$ is a convex subset of $L^1(J, \mathbb{R})$.
\noindent {\bf Step III:} Next we show that $S_{k+G}^1$ is an
u.s.c. multi-valued operator on $Y_T$. Let $\{x_n\}$ be a sequence in
$Y_T$
such that $x_n\to x_*$ and let $\{y_n\}$ be a sequence such that
$y_n\in S_{k+G}^1(x_n)$ and $y_n\to y_*$. To finish, it suffices to
show
that $y_*\in S_{k+G}^1(x_*)$. Since
$y_n\in S_{k+G}^1(x_n)$, there is a function $f_n\in S_{k+G}^1(x_n)$
such that
$y_n(t)= k x_n(t)+ f_n(t)$ for all $t\in J$ and that
$y_*(t)= k x_*(t)+ f_*(t)$, where $f_n\to f_*$ as $n\to\infty$. Now the
multi-function
$G(t,x)$ is an upper semi-continuous in $x$ for all $t\in J$, one has
$f_*(t)\in G(t, x_*(t))$ for all $t\in J$.
Hence it follows that $y_*\in S_{k+G}^1(x_*)$.
\noindent {\bf Step IV:} Finally we show that $S_{k+G}^1$ maps
bounded sets of $Y_T$ into bounded sets of $L^1(J ,
\mathbb{R})$. Let $M$ be a bounded subset of $Y_T$. Then there
is a real number $r>0$ such that $\|x\|\le r$ for all
$x\in M$. Let $y\in S_{k+G}^1(S)$ be arbitrary. Then
there is an $x\in M$ such that $y\in S_{k+G}^1(x)$ and
therefore $y(t)\in k x(t)+ G(t , x(t))$ a.e. $t\in J$.
Now by (H3),
\begin{align*}
\|y\|_{L^1}&=\int_0^T |y(t)|\,dt\\
&\le \int_0^T\|k x(t) + G(t, x(t))\|\,dt\\
&\le \int_0^T \omega(t, |x(t|)\,dt\\
&\le \int_0^T \omega(t, r)\,dt.
\end{align*}
Hence $S_{k+G}^1(S)$ is a bounded set in $L^1(J ,
\mathbb{R})$.\par
Thus the multi-valued operator
$S_{k+G}^1$ is an upper semi-continuous and has closed, convex values
on
$Y_T$. The proof
is complete.\end{proof}
\begin{lemma}\label{l33}
Assume $(H_1)-(H_3)$. The multivalued
operator $\mathcal{B}_k^2$ defined by (\ref{e37}) is completely
continuous
and has
convex, compact values on $Y_T$.
\end{lemma}
\begin{proof} Since $S_{k+G}^1$ is as
upper semi-continuous and has closed and convex values and since (H1)
holds,
$\mathcal{B}_k^2$ is u.s.c. and has closed-convex values on $Y_T$.
To show
$\mathcal{B}_k^2$ is relatively compact, we use the Arzel\'a-Ascoli
theorem. Let
$M\subset B[0,r]$ be any set. Then $\|x\|\le r$ for all $x\in M$. First
we
show that
$\mathcal{B}_k^2(M)$ is uniformly bounded. Now for any $x\in M$ and
for any $y\in
\mathcal{B}_k^2(x)$ one has
\begin{align*}
|y(t)|&\le \int_0^T
|g_k(t,s)| \|[kx(s)+G(s,x(s))]\|\,ds
+\sum_{j=1}^{p} |g_k(t,t_j)| |I_j(x(t_j^{-}))|\\
&\le \int_0^T M_k \omega(s,|x(s)|)\,ds +M_k\sum_{j=1}^{p} c_j\\
&\le M_k \int_0^T \omega(s, r)\,ds +M_k\sum_{j=1}^{p} c_j,
\end{align*}
where $M_k$ is the bound of $g_k$ on $[0,T]\times [0,T]$.
Taking the
supremum over $t$,
\[ \|\mathcal{B}_k^2 x\|\le M_k \Big[ \int_0^T \omega(s, r)\,ds
+\sum_{j=1}^{p}
c_j\Big]\]for all $x\in M$. Hence $\mathcal{B}_k^2(M)$ is a uniformly
bounded set in
$Y_T$. Next we
prove the equi-continuity of the set $\mathcal{B}_k^2(M)$ in $Y_T$. Let
$y\in
B_k^2(M)$ be arbitrary.
Then there is a $v\in S_{k+G}(x)$ such that
$$ y(t)= \int_{0}^{T} g_k(t,s) v(s) \,ds
+ \sum_{j=1}^{p}g_k(t,t_j) I_j(x(t_j^-)), \quad t\in J,$$
for some $x\in M$.
To finish, it is sufficient to show that $y'$ is bounded
on $[0,T]$. Now for any $t\in [0,T]$,
\begin{align*}
|y'(t)|&\le \Bigl|\int_0^T \frac{\partial}{\partial
t} g_k(t,s) v(s)\,ds +\sum_{j=1}^{p} \frac{\partial}{\partial t}
g_k(t,t_k)I_j(y_j(t_j^-))\Bigr|\\
&= \Bigl|\int_0^T (-k) g_k(t,s) v(s)\,ds
+\sum_{j=1}^{p} (-k) g_k(t,t_k)I_j(y_j(t_j^-))\Bigr|\\
&\le k M_k \int_0^T \omega(s,r)\,ds + k M_k
\sum_{j=1}^p c_j
= c.
\end{align*}
Hence for any $t, \tau \in [0, T]$ and for all $y\in B_k^2(M)$ one has
$$
|y(t)- y(\tau)|\le c |t-\tau |\to 0\quad \mbox{as}\quad t\to
\tau.
$$
This shows that $\mathcal{B}_k^2(M)$ is a equi-continuous set and
consequently relatively compact in view of Arzel\'a-Ascoli theorem.
Obviously
$\mathcal{B}_k^2 (x)\subset \mathcal{B}_k^2(B[0, r])$ for each $x\in
B[0,
r]$. Since
$\mathcal{B}_k^2(B[0, r])$ is relatively compact, $\mathcal{B}_k^2 (x)$
is
relatively
compact and which is compact in view of hypothesis (H2). Hence
$\mathcal{B}_k^2$ is a
completely continuous multi-valued operator on $Y_T$. The proof of
the
lemma is complete.\end{proof}
\begin{lemma}\label{l34} Assume that the hypotheses
(H4)--(H5) hold. Then the operator $B_k^1$ defined by \eqref{e36} is a
multi-valued contraction operator on $Y_T$, provided
$M_k\|\ell\|_{L^1}<1$.
\end{lemma}
\begin{proof} Define a mapping $\mathcal{B}_k^1 : Y_T\to Y_T$ by
\eqref{e36}. We show that $\mathcal{B}_k^1$ is a multi-valued
contraction on $Y_T$.
Let $x,y\in Y_T$ be arbitrary and let $u_1\in \mathcal{B}_k^1(x)$.
Then $u_1\in Y_T$ and
\[ u_1(t)=\int_{0}^{T}g_k(t,s) v_1(s) \,ds \]
for some $v_1\in S_F^1(x)$. Since
$H(F(t,x(t)),F(t,y(t))\le\ell(t)|x(t)-y(t)|$, one obtains that there
exists a $w\in F(t,y(t))$ such that
$$|v_1(t)-w|\le \ell(t)|x(t)-y(t)|.$$
Thus the multi-valued operator
$U$ defined by $U(t)=S_F^1(y)(t)\cap K(t)$,where
$$K(t)=\{w\mid |v_1(t)-w|\le\ell(t)|x(t)-y(t)|\}$$
has nonempty values and is measurable. Let $v_2$ be a
measurable selection for $U$ (which exists by
Kuratowski-Ryll-Nardzewski's
selection
theorem. See [3]). Then $v_2\in F(t,y(t))$ and
$$|v_1(t)-v_2(t)|\le\ell(t)|x(t)-y(t)| \quad \mbox{a.e. } t\in J.$$
Define
\[ u_2(t) =\int_{0}^{T}{g_k(t,s) v_2(s) \,ds}.
\]
It follows that $u_2\in \mathcal{B}_k^1(y)$ and
\begin{align*}
|u_1(t)-u_2(t)|&\le\Big|\int_0^{T}g_k(t,s) v_1(s)\,ds-\int_0^{T}
g_k(t,s)
v_2(s)\,ds\Big|\\
&\le\int_0^{T}M_k |v_1(s)-v_2(s)|\,ds\\
&\le\int_0^{T}M_k\ell(s)|x(s)-y(s)|\,ds\\
&\le M_k\|\ell\|_{L^1}\|x-y\|.
\end{align*}
Taking the supremum over $t$, we
obtain
$$
\|u_1-u_2\|\le M_k \|\ell\|_{L^1}\|x-y\|.
$$
From this and the analogous inequality
obtained by interchanging the roles of $x$ and $y$ we get that
$$
H(\mathcal{B}_k^1(x),\mathcal{B}_k^1(y))\le\mu \|x-y\|,
$$
for all $x,y\in Y_T$. This shows that $\mathcal{B}_k^1$
is a multi-valued contraction, since $\mu = M_k\|\ell\|_{L^1}<1$.
\end{proof}
\begin{theorem}\label{t31}
Assume (H1)--(H5) are satisfied. Further if there exists a real number
$r>0$ such that
\begin{equation}\label{e39}
r > \frac{M_k \int_0^T \omega(s, r)\,ds+ M_k
F_0+M_k\sum_{j=1}^{p}c_j}{1- M_k \|\ell\|_{L^1} }
\end{equation}
where $M_k \|\ell\|_{L^1} < 1 $ and $F_0= \int_0^T \|F(s ,
0)\|\,ds$, then the problem IDI (\ref{e11})--(\ref{e13}) has at
least one solution on $J$.
\end{theorem}
\begin{proof} Define an open ball $B(0,r)$ in $Y_T$, where the real
number $r$ satisfies the inequality given in condition (\ref{e39}).
Define the
multi-valued operators $\mathcal{B}_k^1$ and $\mathcal{B}_k^2$ on $Y_T$
by
\eqref{e36} and \eqref{e37}. We shall show that the operators
$\mathcal{B}_k^1$
and $\mathcal{B}_k^2$ satisfy all the conditions of Theorem \ref{t21}.
\noindent {\bf Step I:} The assumptions
(H2)--(H3) imply by Lemma \ref{l33} that \, $\mathcal{B}_k^2$ is
completely continuous multi-valued operator on $B[0, r]$. Again
since (H4)--(H5) hold, by Lemma \ref{l34}, $\mathcal{B}_k^1$
is a multi-valued contraction on $Y_T$ with a contraction constant
$\mu = M_k\|\ell\|_{L^1}$.
% $\mu=M_k \sum_{j=1}^{p}{c_j}$.
Now an application of Theorem \ref{t21}
yields that either the operator
inclusion $x\in\mathcal{B}_k^1 x+\mathcal{B}_k^2 x$ has a solution
in $B[0,r]$, or, there exists an $u\in Y_T$ with $\|u\|=r$
satisfying $\lambda u\in B_k^1 u+B_k^2 u $ for some $\lambda >1$.
\noindent {\bf Step II: } Now we show that the second assertion of
Theorem \ref{t21} is not true.
Let $u \in Y_T$ be a possible solution of $\lambda u\in B_k^1 u+ B_k^2
u $
for some real
number $\lambda >1$ with $\|u\|= r$. Then we have,
\begin{align*}u(t) &\in \lambda^{-1}\int_{0}^{T}{g_k(t,s)
F(s,u(s))\,ds}
+\lambda^{-1}\int_{0}^{T}{g_k(t,s)[k u(s) + G(s,u(s))]\,ds} \\
&+\lambda^{-1}\sum_{j=1}^{p}{g_k(t,t_j)I_j(u(t_j^-))}\,.
\end{align*}
Hence by (H3)-(H5),
\begin{align*}
|u(t)|&\le \int_0^T
|g_k(t,s)| \omega(s,|u(s)|)\,ds+\int_0^T |g_k(t,s)| |\ell(s)|
|u(s)|\,ds\\
&\quad +\int_0^T
|g_k(t,s)| \|F(s,0)\|\,ds+ \sum_{j=1}^{p} |g_k(t,s)|
|I_j(u(t_j^{-})|\\
&\le M_k \int_0^T \omega(s, \|u\|)\,ds+M_k \int_0^T |\ell(s)| \|u\|\,ds
+M_k F_0+M_k \sum_{j=1}^{p}c_j\\&\le M_k \int_0^T \omega(s,\|u\|)\,ds
+M_k \|\ell\|_{L^1}\|u\|
+M_k F_0 +M_k \sum_{j=1}^{p}c_j.
\end{align*}
Taking the supremum over $t$ we get
\[
\|u\|\le M_k\int_0^T \omega(s, \|u\|)\,ds+M_k \|\ell\|_{L^1}\|u\|
+M_k F_0 +M_k \sum_{j=1}^{p}c_j.
\]
Substituting $\|u\|= r$ in the above inequality yields
$$
r \le \frac{M_k \int_0^T \omega(s, r)\,ds+ M_k
F_0+M_k\sum_{j=1}^{p}c_j}{1- M_k \|\ell\|_{L^1} }
$$
which is a contradiction to (\ref{e39}). Hence the operator inclusion
$x\in\mathcal{B}_k^1 x+\mathcal{B}_k^2 x$ has a solution in $B[0,r]$.
This further implies that the IDI (\ref{e11})--(\ref{e13}) has a
solution on $J$.
The proof is complete.
\end{proof}
\begin{remark} \rm
On taking $F(t, x)\equiv 0$ on $J'\times \mathbb{R}$ in
Theorem \ref{t31} we obtain as a special case the existence result in
\cite{D2} for the impulsive differential inclusion
(\ref{e11})--(\ref{e13})
with $F(t,x)\equiv 0$.
\end{remark}
\begin{remark}
In this paper we have dealt with the perturbed impulsive
differential inclusions involving convex multi-functions. Note that
the
continuity of the multi-function is important here, however in a
forthcoming
paper we will relax the continuity condition of one of the
multi-functions
and discuss the existence results for mild discontinuous perturbed
impulsive
differential inclusions.
\end{remark}
\subsection*{Acknowledgement} A. Boucherif is grateful to KFUPM for its
constant support.
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\end{document}