\documentclass[reqno]{amsart}

\AtBeginDocument{{\noindent\small
{\em Electronic Journal of Differential Equations},
Vol. 2005(2005), No. 30, pp. 1--11.\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 2005 Texas State University - San Marcos.}
\vspace{9mm}}

\begin{document}

\title[\hfilneg EJDE-2005/30\hfil Existence results]
{Existence results for impulsive partial  neutral functional
differential inclusions}

\author[S. K. Ntouyas\hfil EJDE-2005/30\hfilneg]
{Sotiris K. Ntouyas}

\address{Department of Mathematics,
University of Ioannina,
451 10 Ioannina, Greece}
\email{sntouyas@cc.uoi.gr}

\date{}
\thanks{Submitted February 9, 2005. Published March 14, 2005.}
\subjclass[2000]{34A60, 34K05, 34K45}
\keywords{Impulsive neutral functional differential inclusions;
\hfill\break\indent
fixed point theorem; existence theorem}

\begin{abstract}
 In this paper we prove   existence results   for  first
 order  semilinear impulsive neutral functional differential
 inclusions under the mixed Lipschitz and  Carath\'eodory conditions.
\end{abstract}

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

\section{Introduction}

The theory of impulsive differential equations is emerging as an important area
of investigation since it is much richer that the corresponding theory of
differential equations; see the monograph of Lakshmikantham {\em et al}
\cite{LBS}. In this paper, we study the existence of solutions for initial
value problems for first  order impulsive semilinear neutral functional
differential inclusions. More precisely in Section 3 we consider first-order
impulsive semilinear neutral functional differential inclusions of the form
\begin{gather}\label{e11}
\begin{gathered}
\frac{d}{dt}[x(t)- f(t, x_t)]\in Ax(t)+G(t, x_t)\\
\mbox{a.e. } t\in J:=[0,T], \quad
t\neq t_{k} \quad  k=1,\dots,m,
\end{gathered}\\
\label{e12}
x(t_{k}^{+})-x(t_{k}^{-})=I_{k}(x(t_{k}^{-})), \quad  k=1,\dots,m, \\
\label{e13}
x(t)=\phi(t),\quad t\in [-r,0],
\end{gather}
where $A$ is the infinitesimal generator of an analytic semigroup of bounded
linear operators, $S(t), t\ge 0$ on a Banach space $X$,
$ f:J\times \mathcal{D} \to  X$ and $G:J\times \mathcal{D}
\to \mathcal{P}(X)$; $\mathcal{D}$ consists of  functions
$\psi:[-r,0]\to X$ such that $\psi$ is continuous everywhere except for
a finite number of points  $s$  at
which  $\psi(s)$ and the right limit $\psi(s^+)$ exist  and
$\psi(s^-)=\psi(s)$;
$\phi\in \mathcal{D}$,
$(0<r<\infty)$, $0=t_{0}<t_{1}<\dots<t_{m}<t_{m+1}=T$,
$I_{k}: X\to X \ (k=1,2,\dots,m)$, $x(t_{k}^{+})$ and
$x(t_{k}^{-})$ are
respectively the right and the left limit of $x$ at $t=t_{k}$, and
$\mathcal{P}(X)$ denotes the class of all nonempty subsets of
$X$.

For any continuous function $x$ defined on the interval
$[-r,T]\setminus\{t_1,\dots$, $t_m\}$ and any $t\in J$, we denote by
$x_{t}$ the element of $\mathcal{D}$ defined by
$$
x_{t}(\theta)=x(t+\theta), \quad \theta\in [-r,0].
$$
 For $\psi\in \mathcal{D}$   the norm of $\psi$ is defined by
$$
\|\psi\|_\mathcal{D}=\sup\{|\psi(\theta)|, \theta\in[-r,0] \}.
$$

The main tools used in the study is a fixed point
theorem proved by  Dhage \cite{D}. In the following section, we give
some auxiliary results needed in the subsequent part of the paper.

\section{Auxiliary results}

Throughout this paper, $X$ will be a separable Banach space provided
with norm $\|\cdot\|$ and $A: D(A)\to X$ will be the infinitesimal generator
of an analytic semigroup, $S(t),\ t\geq0$, of bounded linear operators on $X$.
For the theory of strongly continuous semigroup, refer to Pazy \cite{Pa}.
If $S(t)$, $t\geq0$, is a uniformly bounded and analytic semigroup such
that $0\in \rho(A)$, then it is  possible to define the fraction  power
$(-A)^{\alpha}$, for $0<\alpha\leq 1$, as closed linear operator on its
domain $D(-A)^{\alpha}$. Furthermore, the subspace $D(-A)^{\alpha}$ is
dense in $X$, and the expression
$$
\|x\|_{\alpha}=\|(-A)^{\alpha}x\|,\quad x\in D(-A)^{\alpha}
$$
defines a norm on $D(-A)^{\alpha}$.
 Hereafter we denote by $X_{\alpha}$ the Banach
space $D(-A)^{\alpha}$ normed with
$\|\cdot\|_{\alpha}$. Then for each $0<\alpha\leq1$, $X_{\alpha}$ is a Banach
space, and $X_{\alpha}\hookrightarrow X_{\beta}$ for
$0<\beta\leq\alpha\leq1$ and
the imbedding is compact whenever the resolvent operator of $A$ is compact.
Also
for every $0<\alpha\le 1$ there exists $C_{\alpha}>0$ such that
\begin{equation}\label{C}
\|(-A)^{\alpha}S(t)\|\le \frac{C_{\alpha}}{t^{\alpha}}, \quad 0<t\le T.
\end{equation}


Let $\mathcal{P}(X)$ denote the class of all nonempty subsets of $X$. Let
$\mathcal{P}_{bd,cl}(X)$  and $\mathcal{P}_{cp,cv}(X)$   denote
respectively the classes  of all  bounded-closed and
compact-convex subsets of $X$. For $x\in X$ and $Y,Z\in
\mathcal{P}_{bd,cl}(X)$ we denote by $D(x,Y)=\inf\{\|x-y\|: y\in Y\}$,
 and $\rho(Y,Z)=\sup_{a\in Y}D(a,Z)$.

Define the function $H: \mathcal{P}_{bd,cl}(X)\times \mathcal{P}_{bd,cl}(X)
\to \mathbb{R}^+$ by
$$
H(A,B)= \max \{\rho(A,B),\rho(B,A)\}.
$$
The function $H$ is called a Hausdorff metric on $X$. Note that
$\|Y\|=H(Y,\{0\})$.

A correspondence $G:X\to \mathcal{P}(X)$ is called a multi-valued
mapping on $X$. A point $x_0\in X$ is called a {\em fixed point of the
multi-valued operator $G:X\to \mathcal{P}(X)$ if $x_0\in
G(x_0)$}. The fixed points set of $G$ will be denoted by $\mathop{\rm Fix}(G)$.

\begin{definition}  \rm
Let $G:X\to \mathcal{P}_{bd,cl}(X)$ be a multi-valued operator. Then
$G$ is called a multi-valued contraction if there exists a constant $k\in
(0,1)$ such that for each $x,y\in X$ we have
$$  H(G(x),G(y))\le k\|x-y\|.
$$
The constant $k$ is called a contraction constant of $G$.
\end{definition}

 A multi-valued mapping $G:X\to \mathcal{P}(X)$ is called {\em
lower  semi-continuous} (shortly l.s.c.) (resp. {\em upper semi-continuous}
(shortly u.s.c.))
if $B$ is any open  subset of $X$ then $\{x\in X : Gx\cap B\ne
 \emptyset\}$(resp. $\{x\in X: Gx\subset B\}$) is an open subset of
 $X$. The multi-valued
 operator $G$ is called {\em compact} if $\overline{G(X)}$ is a compact
subset of $X$. Again $G$ is called {\em totally bounded} if for any
bounded subset $S$ of $X$, $G(S)$ is a totally bounded subset of
$X$. A multi-valued operator $G: X\to \mathcal{P}(X)$ is called
{\em completely continuous} if it is upper semi-continuous and totally
bounded on $X$, for each bounded $B\in \mathcal{P}(X)$. Every
compact
multi-valued operator is totally bounded but the converse may not
be true. However the two notions are equivalent  on a bounded
subset of $X$.

We apply the following form of the fixed point theorem by
Dhage \cite{D} in the sequel.

\begin{theorem}\label{t21}
Let $X$ be a Banach space,  $A:X\to \mathcal{P}_{cl,cv,bd}(X)$ and
$B:X\to \mathcal{P}_{cp,cv}(X)$ two
multi-valued operators satisfying
\begin{itemize}
\item [(a)] $A$ is  contraction with a contraction constant $k$, and
\item [(b)] $B$ is completely continuous.
\end{itemize}
Then either
\begin{itemize}
\item[(i)] The operator inclusion $\lambda x \in Ax+Bx$
has a solution for $\lambda =1$, or
\item[(ii)] The set $\mathcal{E}= \{ u\in X : \lambda u\in Au+Bu
,\;\lambda > 1\}$ is unbounded.
\end{itemize}
\end{theorem}

\section{Existence results}

 Let us state what we mean by a  solution of problem
(\ref{e11})--(\ref{e13}). For this purpose, we consider the
space
$PC([-r,T],X)$ consisting of functions $x: [-r,T]\to X$
such that $x(t)$ is continuous almost everywhere except
 for some $t_{k}$ at which $x(t^{-}_{k})$
 and  $ x(t^{+}_{k})$, $k=1,\dots,m $ exist and
$x(t^{-}_{k})=x(t_{k})$.

Obviously, for any $t\in [0,T]$   we have $x_t\in \mathcal{D}$ and
$PC([-r,T],X)$ is a Banach space with the norm
$$
\|x\|=\sup\{|x(t)|: t\in [-r,T]\}.
$$
In the following we set for convenience
$$
\Omega=PC([-r,T],X).
$$
Also we denote by $AC(J,X)$ the space of all absolutely continuous
functions $x: J\to X$.

A function $x\in \Omega\cap AC((t_{k},t_{k+1}),X)$,
$k=1, \dots, m$, is
said to be a solution of (\ref{e11})--(\ref{e13}) if $x(t)-f(t,x_t)$ is
absolutely continuous on $J\setminus\{t_1,\dots,t_m\}$ and
(\ref{e11})--(\ref{e13}) are satisfied.


A multi-valued map  $G: J\to \mathcal{P}_{cp, cv}(\mathbb{R}^n)$ is said
to be measurable if for every $y\in \mathbb{R}^n$, the function $t\to
d(y,G(t)) =\inf \{\|y-x\|: x\in G(t)\}$ is measurable.


A multi-valued map $G: J\times\/\mathcal{D} \to \mathcal{P}_{cl}(X)$
is said to  be $L^{1}$-Carath\'e\-odory if
\begin{itemize}
\item[(i)] $t\mapsto G(t,x)$ is measurable for each $x\in \mathcal{D}$,
\item[(ii)] $x\mapsto G(t,x)$ is upper semi-continuous for almost all
$t\in J$, and
\item[(iii)] for each real number $\rho > 0$, there
exists a function $h_{\rho}\in L^{1}(J,\mathbb{R}^{+})$ such that
$$
\|G(t,u)\|:=\sup\{\|v\|: v\in G(t,u)\}\le h_{\rho}(t), \quad a.e.
\quad t\in J
$$
for all $u\in \mathcal{D}$ with $\|u\|_\mathcal{D}\le \rho$.
\end{itemize}


Then we have the following lemmas due to Lasota and Opial \cite{LO}.

\begin{lemma}\label{21}
If $\dim (X)<\infty$ and $F: J\times X\to \mathcal{P}(X)$ is
$L^1$-Carath\'eodory, then $S_{G}^{1}(x)\ne \emptyset$ for each
$x\in X$.
\end{lemma}

\begin{lemma}\label{l22}
Let $X$ be a Banach space,  $G$ an $L^{1}$-Carath\'eodory multi-valued
map with $S_{G}^{1}\ne \emptyset$ where
$$
S_{G}^{1}(x):=\{v\in L^{1}(I,\mathbb{R}^n): v(t)\in G(t,x_t)\;
a.e.\; t\in J\},
$$
and
$\mathcal{K}:L^{1}(J,X)\to C(J,X)$
be a linear continuous mapping. Then the operator
$$
\mathcal{K} \circ S_{G}^{1}:C(J,X)\to \mathcal{P}_{cp,cv}(C(J,X))
$$
is a closed graph operator in $C(J,X)\times C(J,X)$.
\end{lemma}

We need also the following  result from \cite{He}.

\begin{lemma}\label{hen}
Let $v(\cdot), w(\cdot): [0,T]\to [0,\infty)$ be continuous functions. If
$w(\cdot)$ is nondecreasing and there are constants $\theta>0, \,
0<\alpha<1$ such
that
$$
v(t)\le w(t)+\theta\int_0^t\frac{v(s)}{(t-s)^{1-\alpha}}\, ds, \quad t\in
[0,T],
$$
then
$$
v(t)\le e^{\theta^{n}\Gamma(\alpha)^{n}t^{n\alpha}/\Gamma(n\alpha)}
\sum_{J=0}^{n-1} \big(\frac{\theta T^{\alpha}}{\alpha}\big)^{j} w(t),
$$
for every $t\in [0,T]$ and every $n\in \mathbb{N}$ such that $n\alpha>1$, and
$\Gamma(\cdot)$ is the Gamma function.
\end{lemma}

We consider the following set of assumptions in the sequel.

\begin{itemize}
\item [(H1)] There exist constants $0<\beta<1, c_1, c_2, L_f$ such that
$f$ is $X_{\beta}$-valued, $(-A)^{\beta}f$ is continuous, and
\begin{itemize}
\item[(i)] $\|(-A)^{\beta}f(t,x)\|\le c_1\|x\|_\mathcal{D}+c_2$,
$(t,x)\in J\times \mathcal{D}$
\item[(ii)] $\|(-A)^{\beta}f(t,x_1)-(-A)^{\beta}f(t,x_2)\|
\le L_f\|x_1-x_2\|_\mathcal{D}$, $(t,x_i)\in J\times \mathcal{D}$, $i=1,2$,
with
$$
L_f\big\{ \|(-A)^{-\beta}\|
+\frac{C_{1-\beta}T^{\beta}}{\beta}\big\}<1.
$$
\end{itemize}
 \item[(H2)] The multivalued map $G(t,x)$ has compact and convex values
for each $(t,x)\in J\times \mathcal{D}$.
\item[(H3)] The semigroup $S(t)$ is compact for $t>0$, and there exists
$M\ge 1$ such that
$$
\|S(t)\|\le M, \quad \mbox{for all } t\ge 0.
$$
\item[(H4)] $G$ is $L^{1}$-Carath\'eodory.

\item [(H5)] There exists a function $q\in L^1(I,\mathbb{R})$ with $q(t)>0$
for a.e. $t\in J$ and a nondecreasing function
$\psi:\mathbb{R}^+\to (0,\infty)$ such that
$$
\|G(t,x)\|:=\sup\{\|v\|: v\in G(t,x)\}\le q(t)\psi(\|x\|_\mathcal{D})
\,\,\,\mbox{a.e.}\,\,\, t\in J,
$$
for all $x\in \mathcal{D}$.
\item [(H6)] The impulsive functions $I_k $ are continuous and there exist
constants $c_k$ such that $\|I_k(x)\|\le  c_k$,
$k=1,\dots,m$ for each $x\in X$.
\end{itemize}



\begin{theorem}\label{t31}
Assume that (H1)--(H6) hold. Suppose that
$$
bK_2\int_0^T q (s)\, ds< \int_{K_{0}}^{\infty}\frac{ds}{s+ \psi(s)},
$$
where
\begin{gather*}
K_{0}= \frac{F}{1-c_1\|(-A)^{-\beta}\|},\quad
K_2=\frac{M}{1-c_1\|(-A)^{-\beta}\|}, \\
b=e^{K_1^n(\Gamma(\beta))^{n}T^{n\beta}/\Gamma(n\beta)}\sum_{j=0}^{n-1}
\big(\frac{K_1T^{\beta}}{\beta}\big)^{j},
\end{gather*}
and
$$
F= M \|\phi\|_\mathcal{D}\{1+c_1\|(-A)^{-\beta}\|\}
+c_2\|(-A)^{-\beta}\|\{M+1\}+M\sum_{k=1}^{m}c_{k}
+\frac{C_{1-\beta}c_2T^{\beta}}{\beta}.
$$
Then the initial-value problem (\ref{e11})--(\ref{e13}) has at least one
solution on $[-r,T]$.
\end{theorem}

\begin{proof}  Transform the problem (\ref{e11})--(\ref{e13}) into a
fixed point problem. Consider the operator
$N:\Omega\to \mathcal{P}(\Omega)$ defined by
\begin{align*}
Nx(t)=\Big\{&h\in \Omega : h(t)= \phi(t) \mbox{ for $t\in [-r,0]$, and }
 h(t)= S(t)[\phi(0)-f(0,\phi(0))]\\
&+f(t,x_{t})+\int_0^t AS(t-s)f(s,x_s)ds
 +\int_{0}^{t}S(t-s)v(s)ds \\
&+\sum_{0 < t_k < t}S(t-t_k)I_{k}(x(t_{k}^{-}))
\mbox{ for $t\in J$} \Big\},
\end{align*}
where $v\in S_{G}^{1}(x)$.

Now, we define two  operators as follows.
$A: \Omega\to \Omega$ by
 \begin{equation}\label{e32}
 Ax(t) = \begin{cases}
0,  &\mbox{if } t\in [-r,0],\\
 \bigl\{-S(t)f(0, \phi) +f(t, x_t)
+\int_0^t AS(t-s)f(s,x_s)ds\bigr\},
&\mbox{if } t\in J,
\end{cases}
\end{equation}
 and the multi-valued operator $B: \Omega\to \mathcal{P}(\Omega)$
by
\begin{equation}\label{e33}
\begin{aligned}
Bx(t)=\Big\{&h\in \Omega: h(t)=\phi(t)\mbox{ for $t\in [-r,0]$, and }
h(t)=S(t)\phi(0) \\
&+\int_{0}^{t}S(t-s)v(s)\,ds
 +\sum_{0<t_{k}<t}S(t-t_k)I_{k}(x(t_{k}^{-}))
\mbox{ for $t\in J$}\Big\}.
\end{aligned}
\end{equation}

 Then $N=A+B$. We shall show that the operators $A$ and  $B$ satisfy
all the conditions of Theorem \ref{t21} on $[-r,T]$. For better readability,
 we break the proof into a sequence of steps. \smallskip

\noindent {\bf Step I.}
First we remark that   $A$  for each $x\in \Omega$, has closed,
convex values on $\Omega$.
Next we show that $A$ has bounded values for bounded sets
in $X$. To show this, let $S$ be a bounded subset of $\Omega$, with  bound
$\rho$.
Then, for any $x\in S$ one has
\begin{align*}
\|Ax(t)\|\le& M\|f(0,\phi)\|+\|(-A)^{-\beta}\|[c_1\|x_t\|_\mathcal{
 D}+c_2]\\
&+\int_0^t\|(-A)^{1-\beta}S(t-s)\|\|(-A)^{\beta}f(s,x_s)\|ds\\
\le & M\|f(0,\phi)\|+\|(-A)^{-\beta}\|[c_1\|x_t\|_\mathcal{D}+c_2]\\
&+\int_0^t\frac{C_{1-\beta}c_1}{(t-s)^{1-\beta}}\|x_s\|_\mathcal{D}ds
+\frac{C_{1-\beta}c_2T^{\beta}}{\beta}\\
\le& M\|f(0,\phi)\|+\|(-A)^{-\beta}\|[c_1\rho+c_2]+
\frac{C_{1-\beta}T^{\beta}}{\beta}[\rho c_1+c_2],
\end{align*}
and consequently
$$
\|Ax\|\le M\|f(0,\phi)\|+\|(-A)^{-\beta}\|[c_1\rho+c_2]+
\frac{C_{1-\beta}T^{\beta}}{\beta}[\rho c_1+c_2].
$$
Hence $A$ is bounded on bounded subsets of $\Omega$. \smallskip

\noindent {\bf Step II.}
 Next we prove that $Bx$ is a convex subset of $\Omega$ for each $x\in
\Omega$. Let $u_1, u_2\in Bx$. Then there exists $v_1$ and
$v_2$ in $S_{G}^{1}(x)$ such that
$$
u_{j}(t)=S(t)\phi(0)+\sum_{0<t_{k}<t}^{}S(t-s_k)I_{k}(x(t_{k}^{-}))+\int_{0}^{t}
S(t-s)v_{j}(s)\,ds, \quad j=1,2.
$$
Since $G(t,x)$ has convex values, one has for $0\le \mu\le 1$,
$$
[\mu v_1+(1-\mu)v_2](t)\in S_{G}^{1}(x)(t), \quad \forall t\in J.
$$
As a result we have
\begin{align*}
&[\mu u_1+(1-\mu)u_2](t)\\
&=S(t)\phi(0)+\sum_{0<t_{k}<t}^{}S(t-t_k)I_{k}(x(t_{k}^{-}))
+\int_{0}^{t}S(t-s)[\mu v_1(s)+(1-\mu)v_2(s)]\,ds.
\end{align*}
Therefore, $[\mu u_1+(1-\mu)u_2]\in Bx$ and
consequently $Bx$ has convex values in $\Omega$. Thus we have
$B:\Omega\to \mathcal{P}_{cv}(\Omega)$. \smallskip

\noindent{\bf Step III.}  We show that  $A$ is a
contraction on $\Omega$. Let $x,y\in X$. By hypothesis (H1)
\begin{align*}
\|Ax(t) - Ay(t)\|
&\le \|f(t, x_t) - f(t, y_t)\|
+\big\|\int_0^tAS(t-s)[f(s,x_s)-f(s,y_s)]\, ds\big\|\\
&\le \|(-A)^{-\beta}\|L_f\|x_t-y_t\|_\mathcal{D}
+\int_0^t\frac{C_{1-\beta}}{(t-s)^{1-\beta}}\, ds \,
L_f\|x_t-y_t\|_\mathcal{D}\\
&\le  L_f\big\{ \|(-A)^{-\beta}\|
+\frac{C_{1-\beta}T^{\beta}}{\beta}\big\}\|x_t-y_t\|_\mathcal{D}.
\end{align*}
Taking supremum over $t$,
$$
\|Ax - Ay\|\le L_0 \|x-y\|_\mathcal{D}, \quad
L_0:=L_f\big\{\|(-A)^{-\beta}\|
+\frac{C_{1-\beta}T^{\beta}}{\beta}\big\}.
$$
This shows that $A$ is a multi-valued contraction, since $L_0<1$. \smallskip

\noindent{\bf Step IV.}
Now we show that   the multi-valued operator $B$ is
completely continuous on $\Omega$. First we show
that $B$ maps bounded sets into bounded sets in $\Omega$. To
see this, let $Q$ be a bounded set in $\Omega$. Then there
exists a real number $\rho>0$ such that $\|x\|\le \rho, \forall x\in
Q$.

Now  for each $u\in Bx$, there exists a $v\in S_{G}^{1}(x)$ such that
$$
u(t)=S(t)\phi(0)+\sum_{0<t_{k}<t}^{}S(t-t_k)I_{k}(x(t_{k}^{-}))
+\int_{0}^{t} S(t-s)v(s)\,ds,\quad t\in J.
$$
Then for each $t\in J$,
\begin{align*}
\|u(t)\|&\le
M\|\phi(0)\|+M\sum_{k=1}^{m}c_{k}+M\int_{0}^{t}|v(s)|\,ds\\
&\le M\|\phi\|_\mathcal{D}+M\sum_{k=1}^{m}c_{k}+M\int_{0}^{t}
h_{\rho}(s)\,ds\\
&\leq  M\|\phi\|_\mathcal{
D}+M\sum_{k=1}^{m}c_{k}+M\|h_{\rho}\|_{L^{1}}.
\end{align*}
This  implies
$$
\|u\|\le M\|\phi\|_\mathcal{D}+M\sum_{k=1}^{m}c_{k}+M\|h_{\rho}\|_{L^{1}}
$$
for all $u\in  Bx\subset B(Q)=\bigcup_{x\in Q}B(x)$. Hence
$B(Q)$ is bounded.

 Next we show that $B$ maps bounded sets into equi-continuous sets.
Let $Q$ be, as above, a bounded set  and $h\in Bx$ for some $x\in Q$.
Then there exists $v\in S_{G}^{1}(x)$ such that
$$
h(t)=S(t)\phi(0)+\sum_{0<t_{k}<t}^{}S(t-t_k)I_{k}(x(t_{k}^{-}))
+\int_{0}^{t} S(t-s)v(s)\,ds,\quad t\in J.
$$
Let $\tau_{1}, \tau_{2}\in
J\backslash\{t_{1},\dots,t_{m}\}, \ \tau_{1}<\tau_{2}$. Then    we have
\begin{align*}
&\|h(\tau_{2})-h(\tau_{1})\|\\
&\leq \|[S(\tau_2)-S(\tau_1)]\phi(0)\|
+\int^{\tau_1-\epsilon}_{0}\|S(\tau_{2}-s)-S(\tau_{1}-s)\|\varphi_{q}(s)ds\\
&\quad +\int^{\tau_1}_{\tau_{1}-\epsilon}\|S(\tau_{2}-s)
-S(\tau_{1}-s)\|\varphi_{q}(s)ds
 +\int_{\tau_1}^{\tau_2}\|S(\tau_2-s)\|\varphi_{q}(s)ds\\
&\quad +\sum_{0<t_{k}<\tau_2-\tau_1}Mc_k
+ \sum_{0<t_{k}<\tau_2}\|S(\tau_2-t_k)-S(\tau_1-t_k)\|c_k.
\end{align*}
As $\tau_{2}\to \tau_{1}$ and $\epsilon$ becomes sufficiently
small the right-hand
side of the above inequality tends to zero, since $S(t)$ is a
strongly continuous operator and the
compactness of $S(t)$ for $t>0$ implies the continuity in the uniform
operator topology.

This proves the equicontinuity for the case where $t\neq t_{i}$,
$i=1,\dots, m+1$. It remains to examine the equicontinuity at
$t=t_{i}$.
Set
$$
h_{1}(t)=S(t)\phi(0)+\sum_{0<t_k<t}S(t-t_k)I_k(y(t_k^{-}))
$$
and
$$
h_{2}(t)=\int^{t}_{0}S(t-s)v(s)ds.
$$
First we prove equicontinuity at $t=t_{i}^{-}$. Fix $\delta_{1}>0$ such that
$\{t_{k} : \ k\neq i\}\cap [t_{i}-\delta_{1},t_{i}+\delta_{1}]=\emptyset$,
\begin{align*}
h_{1}(t_{i})&=S(t_{i})\phi(0)+\sum_{0<t_k<t_{i}}S(t-t_k)I_k(y(t_k^{-}))\\
&=S(t_{i})\phi(0) +\sum_{k=1}^{i-1}T(t_{i}-t_k)I_k(y(t_k^{-})).
\end{align*}
 For $0<h<\delta_{1}$ we have
\begin{align*}
&\|h_{1}(t_{i}-h)-h_{1}(t_{i})\|\\
&\leq \|(S(t_{i}-h)-S(t_{i}))\phi(0)
+\sum_{k=1}^{i-1}|[S(t_{i}-h-t_k)-S(t_{i}-t_k)]I(y(t_{k}^{-}))\|.
\end{align*}
The right-hand side tends to zero as $h\to 0$.
Moreover
\[
\|h_{2}(t_{i}-h)-h_{2}(t_{i})\|
\leq \int^{t_{i}-h}_{0}\|[S(t_{i}-h-s)-S(t_{i}-s)]v(s)\|ds
+\int_{t_{i}-h}^{t_{i}}M \phi_{q}(s)ds,
\]
which tends to zero as $h\to 0$.
Define
$$
\hat{h}_{0}(t)=h(t), \quad t\in [0,t_{1}]$$
and
$$
\hat{h}_{i}(t)=\begin{cases} h(t), &\mbox{if } t\in(t_{i},t_{i+1}],\\
h(t_{i}^{+}), &\mbox{if } t=t_{i}
\end{cases}
$$
Next we prove equicontinuity at $t=t_{i}^{+}$. Fix $\delta_{2}>0$
such that $\{t_{k} :  k\neq i\}\cap
[t_{i}-\delta_{2},t_{i}+\delta_{2}]=\emptyset $.
 Then
\[
\hat{h}(t_{i})=S(t_{i})\phi(0)+\int^{t_{i}}_{0}S(t_{i}-s)v(s)
+\sum_{k=1}^{i}S(t_{i}-t_k)I_k(y(t_k)).
\]
 For $0<h<\delta_{2}$ we have
\begin{align*}
&\|\hat{h}(t_{i}+h)-\hat{h}(t_{i})\|\\
&\leq \|(S(t_{i}+h)-S(t_{i}))\phi(0)\|
+\int^{t_{i}}_{0}\|[S(t_{i}+h-s)-S(t_{i}-s)]v(s)\|ds\\
&\quad +\int^{t_{i}+h}_{t_{i}}M\varphi_{q}(s)ds
+\sum_{k=1}^{i}\|[S(t_{i}+h-t_k)-S(t_{i}-t_k)]I(y(t_{k}^{-}))\|.
\end{align*}
The right-hand side tends to zero as $h\to 0$.

The equicontinuity for the cases $\tau_{1}<\tau_{2}\leq 0$ and
$\tau_{1}\leq 0\leq \tau_{2}$
follows from the uniform continuity of $\phi$ on the interval $[-r,0]$.
As a consequence of Steps 1 to 3, together
with the Arzel\'a-Ascoli
theorem it suffices to show that $B$  maps $Q$ into a precompact set in
$X$.

Let $0<t\leq b$ be fixed and
let $\epsilon$ be a real number satisfying $0<\epsilon<t$.
For $x\in Q$ we define
\begin{align*}
&h_{\epsilon}(t)\\
&=S(t)\phi(0) +S(\epsilon)\int^{t-\epsilon}_{0}S(t-s-\epsilon)v_1(s)ds
+S(\epsilon)\sum_{0<t_k<t-\epsilon}S(t-t_k-\epsilon)I_{k}(y(t_{k}^{-})),
\end{align*}
where $v_1\in S_{F}^{1}$. Since $S(t)$ is a compact operator, the set
$H_{\epsilon}(t)=\{h_{\epsilon}(t) : h_{\epsilon}\in N(y)\}$
is  precompact in $X$ for every $\epsilon$, $0<\epsilon<t$.
Moreover, for every $h\in N(y)$ we have
$$
|h(t)-h_{\epsilon}(t)|\leq
\int^{t}_{t-\epsilon}\|S(t-s)\|\varphi_{q}(s)ds
+ \sum_{t-\epsilon<t_k<t}\|S(t-t_k)\| c_k.
$$
Therefore, there are precompact sets arbitrarily close to the set
$H(t)=\{h_{\epsilon}(t) : h\in N(y)\}$. Hence the set
 $H(t)=\{h(t) : h\in B(Q)\}$ is precompact in $X$. Hence, the operator\
$B:\Omega\to \mathcal{P}(\Omega)$ is completely continuous.
\smallskip

\noindent{\bf Step V.}
Next we prove that {\em $B$ has a closed graph.}  Let
$\{x_n\}\subset \Omega$ be a sequence such that $x_n\to x_*$
and let $\{y_n\}$ be a sequence defined by $y_n\in Bx_n$ for each
$n\in \mathbb{N}$ such that $y_n\to y_*$. We will show that
$y_*\in Bx_*$. Since $y_n\in Bx_n$, there exists a $v_n\in
S_{G}^{1}(x_n)$ such that
$$
y_n(t)=\phi(0)+\sum_{0<t_{k}<t}^{}S(t-t_{k})I_{k}(y_{n}(t_{k}^{-}))
+\int_{0}^{t}v_n(s)\,ds.
$$
Consider the linear and continuous operator
$\mathcal{K}:L^{1}(J,\mathbb{R}^n)\to C(J,\mathbb{R}^n)$
defined by
$$
\mathcal{K}v(t)=\int_{0}^{t}v_n(s)\,ds.$$
Now
\begin{align*}
&\Bigl\|y_n(t)-\phi(0)-\sum_{0<t_{k}<t}S(t-t_{k})I_{k}(y_{n}(t_{k}^{-}))\\
&-\Bigl(y_*(t)-\phi(0)-\sum_{0<t_{k}<t}^{}S(t-t_{k})I_{k}(y_{*}
(t_{k}^{-}))\Bigr)\Bigr\|
\to 0,
\end{align*}
as $n\to \infty$.
 From Lemma \ref{l22} it follows that $(\mathcal{K}\circ
S_{G}^{1})$ is a closed graph operator and from the definition of
$\mathcal{K}$ one has
$$
y_n(t)-\phi(0)-\sum_{0<t_{k}<t}^{}S(t-t_{k})I_{k}(y_{n}(t_{k}^{-}))\in
(\mathcal{K}\circ S_{F}^{1}(y_n)).
$$
As $x_n\to x_*$ and $y_n\to y_*$, there is a
$v\in S_{G}^{1}(x_*)$ such that
$$
y_*(t)=\phi(0)+\sum_{0<t_{k}<t}^{}S(t-t_{k})I_{k}(y_{*}(t_{k}^{-}))
+\int_{0}^{t}v_*(s)\,ds.
$$
Hence  the multi-valued operator  $B$ is an upper semi-continuous operator on
$\Omega$.\smallskip

\noindent{\bf Step VI.}
Finally we show that  the set
$$
\mathcal{E}=\{u\in \Omega: \lambda u\in Au+Bu
\mbox{ for some }\lambda>1\}
$$
is bounded.
Let $u\in  \mathcal{E}$ be any element. Then there exists $v\in
S_{G}^{1}(u)$  such that
\begin{align*}
u(t)=&\lambda^{-1}S(t)[\phi(0)-f(0,\phi(0))]
+\lambda^{-1}f(t,x_{t})\\[0.3cm]
&+\lambda^{-1}\int_0^t AS(t-s)f(s,x_s)ds
+\lambda^{-1}\int_{0}^{t}S(t-s)v(s)ds\\[0.3cm]
&+\lambda^{-1}\sum_{0 < t_k < t}S(t-t_k)I_{k}(x(t_{k}^{-})).
\end{align*}
Then
\begin{align*}
\|u(t)\|&\le M\|\phi\|_\mathcal{D}+M\|(-A)^{-\beta}\|[c_1\|\phi\|_\mathcal{
D}+c_2] +\|(-A)^{-\beta}\|[c_1\|u_t\|_\mathcal{D}+c_2]\\
&\quad+\int_0^t \|(-A)^{1-\beta}S(t-s)\|\|(-A)^{\beta}f(s,x_s)\|\, ds\\
&\quad+ M\int_{0}^{t}\,
q(s)\psi(\|u_s\|_\mathcal{D}) ds+M\sum_{k=1}^{m}c_{k} \\
&\le M\|\phi\|_\mathcal{D}+M\|(-A)^{-\beta}\|[c_1\|\phi\|_\mathcal{
D}+c_2] +\|(-A)^{-\beta}\|[c_1\|u_t\|_\mathcal{D}+c_2]\\[0.3cm]
&\quad+\int_0^t \frac{C_{1-\beta}c_1}{(t-s)^{1-\beta}}\,
\|u_s\|_\mathcal{D}\, ds
+\frac{C_{1-\beta}c_2T^{\beta}}{\beta}\\[0.3cm]
&\quad+ M\int_{0}^{t}\,
q(s)\psi(\|u_s\|_\mathcal{D}) ds+M\sum_{k=1}^{m}c_{k} \\
&\le F+c_1\|(-A)^{-\beta}\|\|u_t\|_\mathcal{D}\\[0.3cm]
&\quad+\int_0^t
\frac{C_{1-\beta}c_1}{(t-s)^{1-\beta}}\, \|u_s\|_\mathcal{D}\, ds+
M\int_{0}^{t}\, q(s)\psi(\|u_s\|_\mathcal{D}) ds,\quad t\in J,
\end{align*}
where
$$
F= M \|\phi\|_\mathcal{D}\{1+c_1\|(-A)^{-\beta}\|\}
+c_2\|(-A)^{-\beta}\|\{M+1\}+M\sum_{k=1}^{m}c_{k}
+\frac{C_{1-\beta}c_2T^{\beta}}{\beta}.
$$
Put $w(t)= \max\{\|u(s)\|: -r\le s\le t\}$,  $t\in J$. Then
$\|u_t\|_\mathcal{D}\le w(t)$ for all $t\in J $ and there is a point
$t^*\in [-r,t]$ such that $w(t)=\|u(t^*)\|$. Hence we have
\begin{align*}
w(t)&=  \|u(t^*)\|\\
&\leq F+c_1\|(-A)^{-\beta}\|\|u_{t^{*}}\|_\mathcal{D}+C_{1-\beta}
 c_1\int_0^{t^{*}}\frac{\|u_s\|_\mathcal{D}}{(t-s)^{1-\beta}}\,ds\\
&\quad + M\int_{0}^{t^{*}}\, q(s)\psi(\|u_s\|_\mathcal{D}) ds\\
&\le  F+  c_1\|(-A)^{-\beta}\|
 w(t)+C_{1-\beta}c_1\int_{0}^{t}\frac{w(s)}{(t-s)^{1-\beta}}\, ds
 + M\int_{0}^{t}q(s)\psi(w(s))\,ds,
\end{align*}
or
\begin{align*}
w(t)&\le \frac{F}{1-c_1\|(-A)^{-\beta}\|}\\
&\quad
+\frac{1}{1-c_1\|(-A)^{-\beta}\|}\Bigl\{C_{1-\beta}c_1\int_{0}^{t}\frac{w(s)}{
(
t-s)^{1-\beta}}\,
ds+ M\int_{0}^{t}q(s)\psi(w(s))\,ds\Bigr\}\\
&\le K_0+K_1 \int_{0}^{t}\frac{w(s)}{(t-s)^{1-\beta}}\, ds+K_2
\int_{0}^{t}q(s)\psi(w(s))\,ds,\quad t\in I,
\end{align*}
where
$$K_{0}= \frac{F}{1-c_1\|(-A)^{-\beta}\|},\quad
K_1=\frac{C_{1-\beta}c_1}{1-c_1\|(-A)^{-\beta}\|}\quad\mbox{ and}\quad
K_{2}=\frac{M}{1-c_1\|(-A)^{-\beta}\|}.
$$
 From Lemma \ref{hen} we have
$$
w(t)\le b\big(K_0+K_2\int_{0}^{t}q(s)\psi(w(s))\,ds\big),
$$
where
$$
b=e^{K_1^n(\Gamma(\beta))^{n}T^{n\beta}/\Gamma(n\beta)}\sum_{j=0}^{n-1}
\Big(\frac{K_1T^{\beta}}{\beta}\Big)^{j}.
$$
Let
$$
m(t)=b\Big( K_{0}+ K_{2} \int_0^t q(s) \psi(w(s) )\,ds\Big),\quad t\in J.
$$
Then we have $w(t)\le m(t)$ for all $t\in J$. Differentiating
with respect to $t$, we obtain
$$
m'(t)=bK_{2} q(t)  \psi(w(t)),\quad \mbox{a.e. } t\in J,\,\,m(0)=K_{0}.
$$
This implies
$m'(t)\le  bK_{2}q(t) \psi(m(t))$ a.e. $t\in J$; that is,
$$
\frac{m'(t)}{\psi(m(t))}\le bK_{2}q(t),\quad \mbox{a.e. } t\in J.
$$
Integrating from $0$ to $t$, we obtain
$$
\int_0^t \frac{m'(s)}{\psi(m(s))}\,ds\le  b K_{2}\int_0^tq(s)\, ds.
$$
By the change of variable,
$$
\int_{K_{0}}^{m(t)}\frac{ds}{\psi(s)}\le b K_{2}\int_0^T q(t)\, ds<
\int_{K_{0}}^{\infty}\frac{ds}{\psi(s)}.
$$
Hence there exists a constant
$M$ such that
$  m(t)\le M$ for all $t\in J$,
and therefore
$$ w(t)\le  m(t)\le M \quad \mbox{for all } t\in J.
$$
Now from the definition of $w$ it follows that
$$
\|u\|=\sup_{t\in[-r,T]}\|u(t)\|=w(T)\le m(T)\le M,
$$
for all $u\in \mathcal{E}$.
This shows that the set $\mathcal{E}$ is bounded in
$\Omega$. As a result the conclusion (ii) of Theorem \ref{t21} does not
hold.  Hence the conclusion (i) holds and consequently
 the initial value problem  (\ref{e11})--(\ref{e13})  has a solution
$x$ on $[-r,T]$. This completes the proof.
\end{proof}

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\end{document}