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

\AtBeginDocument{{\noindent\small
\emph{Electronic Journal of Differential Equations},
Vol. 2011 (2011), No. 107, 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-2011/107\hfil Existence of mild solutions]
{Existence of mild solutions for impulsive fractional-order semilinear
evolution equations with nonlocal conditions}

\author[A. Chauhan, J. Dabas \hfil EJDE-2011/107\hfilneg]
{Archana Chauhan, Jaydev Dabas}  % in alphabetical order

\address{Archana Chauhan\newline
Department of Mathematics, Motilal Nehru National Institute
of Technology, Allahabad - 211 004, India}
\email{archanasingh.chauhan@gmail.com}

\address{Jaydev Dabas \newline
Department of Paper Technology, IIT Roorkee,
Saharanpur Campus, Saharanpur - 247001, India}
\email{jay.dabas@gmail.com}

\thanks{Submitted April 29, 2011. Published August 24, 2011.}
\subjclass[2000]{34K05, 34A12, 34A37, 26A33}
\keywords{Fractional order differential equation; nonlocal
conditions; \hfill\break\indent
contraction mapping; mild solution; impulsive conditions}

\begin{abstract}
 In this work we consider a class of impulsive  fractional-order
 semilinear evolution equations with a nonlocal initial condition.
 By means of solution operator and application of fixed point
 theorems we established the existence and uniqueness of a mild
 solution.
\end{abstract}

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



\section{Introduction}

Recently fractional differential equations attracted many authors
(see for instance \cite{da,mb,b1,dd,mm,mg,m,gmn,wj,yz} and
references in these papers). Many phenomena in engineering,
physics,continuum mechanics, signal processing, electromagnetics,
economics and science describes efficiently by fractional order
differential equations. Impulsive differential equations have become
important in recent years as mathematical models
of phenomena in both physical and social sciences
(see for instance \cite{aa,b3,ZF,jl,mg,chen} and references in these papers).
There has been a significant development in impulsive theory especially in the area
of impulsive differential equations with fixed moments.


In this article, we are concerned with the existence and uniqueness
of the solution for the fractional order differential equation in
a complex Banach space $X$,
\begin{gather}\label{ME}
\frac{d^{\alpha}}{dt^{\alpha}}x(t)+Ax(t)
 =f(t,x(t),x(a_1(t)),\dots,x(a_m(t))),\quad t\in J=[0,T],\; t\neq t_i,\\
 x(0)+g(x)=x_0,\label{me}\\
\Delta x(t_i)=I_i(x(t_i^-)), \label{ME1}
\end{gather}
where $\frac{d^{\alpha}}{dt^\alpha}$ is Caputo's fractional
derivative of order $0<\alpha<1$,  $i=1,2,\dots,p$,
$0=t_0<t_1<t_2<\dots <t_p<t_{p+1}=T$. Linear operator $A$, defined
from the domain $D(A)\subset X$ into $X$, is such that $-A$
generates $\alpha-$resolvent family $\{S_\alpha(t):t\geq0\}$ of
bounded linear operators in $X$, the nonlinear map $f$ is defined
from $J\times X^{m+1}$ into $X$, for each of $i$ the map $a_i$ is
defined on $[0,T]$ into $[0,T]$ and $ \Delta x(t_i)=
x(t_{i}^{+})-x(t_{i}^{-}),x(t_{i}^{+}),x(t_{i}^{-})$ denotes the
right and the left limit of $x$ at $t_i$, respectively. In general
the derivatives $x'(t_i)$ do not exist, we assume that
$x'(t_i)=x'(t_i-0)$ at the point of discontinuity $t_i$ of the
solution $t\to x(t)$.

The nonlocal condition $g:X\to X$, defined as
$g(x)=\sum_{k=1}^pc_kx(t_k)$, where $c_k$, $k=1,\dots,p$, are
given constants and $0<t_1<t_2<\dots<t_p<T$. Let us recall that
such nonlocal conditions were first used by Deng \cite{deng}.
In this paper, Deng indicated that using the nonlocal condition
$x(0)+g(x)=x_0$ to describe, for instance, the diffusion
phenomenon of a small amount of gas in a transparent tube can give
better result than using the usual local Cauchy Problem
$x(0)=x_0$.

The study of the impulsive fractional order semilinear
functional differential problem of the type \eqref{ME} is
motivated by the paper of Byszewski and Akca \cite{BL1} and Sui, Lai and Chen \cite{chen}. In
\cite{BL1} the authors have considered the nonlocal Cauchy problem
\begin{equation} \label{ba}
\begin{gathered}
u'(t)+Au(t)=f(t,u(t),u(a_1(t)),\dots,u(a_m(t))),\quad t\in J=[0,T], \\
u(0)+g(u)=u_0,
\end{gathered}
\end{equation}
where $-A$ is the generator of a compact semigroup in $X$,
$g:C(J,X)$ into $X$, $u_0\in X$ and for each $i=1,2,\dots,m$,
$a_i:J\to J$. Further, the results obtained in \cite{BL1} have been
extended by Bahuguna in \cite{db}. For more results on nonlocal
conditions we refer the papers \cite{db,bb1,b2,BL,BL1,deng,ek}
and references therein.

In \cite{chen}, the authors have investigated the existence
of mild solutions of the following system
\begin{gather*}
D^{\alpha}x(t)=Ax(t)+f(t,x(t)), \quad t\in[0,T],\;t\neq t_k,\\
x(0)=x_0\in X,\\
\Delta x|_{t=t_k}=I_k(x(t_k^-)),\quad k=1,\dots,m,
\end{gather*}
and corrected the errors in Mophu paper \cite{mg}, and generalized
some previous results.

The organization of this paper is as follows. In Section
$2$, we present some necessary definitions and preliminary results
that will be used to prove our main results. The proofs of our
main results are given in Section $3$.

\section{Preliminaries}

Throughout, in this paper $X$ will be a complex Banach space provided
with the norm $\|\cdot \|_X$ and $L(X)$ is the Banach space of
bounded linear operators from $X$ into $X$. In addition, $B_r(x,X)$
represents the closed ball in $X$ with the center at $x$ and the
radius $r$.  $-A$ is the infinitesimal generator
of an analytic $\alpha-$resolvent family
$\{{S_\alpha(t)}\}_{t\geq0}$ of operators on $X$ . For the theory
of resolvent operator one can see the monograph by Pazy \cite{ap}. 
The Mittag-Leffler
function is an important function that finds widespread use in the
world of fractional calculus. Just as the exponential naturally
arises out of the solution to integer order differential
equations, the Mittag-Leffler function plays an important role in
the solution of non-integer order differential equations. The
standard definition of the Mittag-Leffler function (see\cite{ip})
is given as
$$
E_\alpha(z)=\sum_{k=0}^\infty\frac{z^k}{\Gamma(\alpha k+1)}.
$$
It is also common to represent the Mittag-Leffler function
 in two arguments, $\alpha$ and $\beta$, such that
$$
E_{\alpha,\beta}(z)=\sum_{k=0}^\infty
\frac{z^k}{\Gamma(\alpha k+\beta)}=
\frac{1}{2\pi i}\int_{H_a}e^\mu
\frac{\mu^{\alpha-\beta}}{\mu^\alpha-z}d\mu,\quad
\alpha,\beta>0,\;z\in \mathbb{C},
$$
where $H_a$ is a Hankel path, that is a contour
which starts and ends at $-\infty$ and encircles the disc
$|\mu|\leq|z|^{1\over \alpha}$ counter clockwise. It is an entire
function which provides a generalization of several usual
functions, for example: Exponent function: $E_{1,1}(z)=e^z$;
cosine functions: $E_{2,1}(z^2)=\cosh(z)$ and
$E_{2,1}(-z^2)=\cos(z);$ Sine functions: $zE_{2,2}(z^2)=\sinh(z)$
and $zE_{2,2}(-z^2)=\sin(z)$. The Laplace transform of the
Mittag-Leffler function is given as:
$$
L(t^{\beta-1}E_{\alpha,\beta}(-\rho^\alpha t^\alpha))
=\frac{\lambda^{\alpha-\beta}}
{\lambda^\alpha+\rho^\alpha},\quad
\operatorname{Re}\lambda>\rho^{1/\alpha},\;\rho>0.
$$
 To begin with the analysis we need some basic definitions and
properties from the fractional calculus theory (see \cite{ip}).

\begin{definition} \label{def2.1} \rm
 Caputo's derivative of order $\alpha$ for a function $f:
[0,\infty)\to\mathbb{R}$ is defined as
\[
{d^{\alpha}f(t)\over dt^{\alpha}}=\frac{1}{\Gamma(m-\alpha)}
\int_0^t(t-s)^{n-\alpha-1}f^{(n)}(s)ds,
\]
for $n-1<\alpha<n,\;n\in N$. If $0<\alpha\le 1$, then
\[
{d^{\alpha}f(t)\over dt^{\alpha}}=\frac{1}{\Gamma(1-\alpha)}
\int_0^t(t-s)^{-\alpha}f^{(1)}(s)ds.
\]
The Laplace transform of the Caputo derivative of order $\alpha>0$
is given as
$$
L\{D_t^\alpha f(t);\lambda\}=\lambda^\alpha\widehat{f}(\lambda)
-\sum_{k=0}^{n-1}\lambda^{\alpha-k-1}f^{(k)}(0);\quad
n-1<\alpha\leq n.
$$
\end{definition}

\begin{definition}[{\cite[Definition 2.3]{da}}]\label{def2.2} \rm
Let $A$ be a closed and linear
operator with domain $D(A)$ defined on a Banach space $X$ and
$\alpha>0$. Let $\rho(A)$ be the resolvent set of $A$. We call $A$
the generator of an $\alpha-$resolvent family if there exists
$\omega\geq 0$ and a strongly continuous function $S_\alpha:R_+\to
L(X)$ such that
$\{\lambda^\alpha:Re\lambda>\omega\}\subset\rho(A)$ and
$$
(\lambda^\alpha I-A)^{-1}x=\int_0^\infty e^{-\lambda t}S_\alpha(t)x
dt,\quad\operatorname{Re}\lambda>\omega,\; x\in X.
$$
In this case, $S_\alpha(t)$ is
called the $\alpha-$resolvent family generated by $A$.
\end{definition}

\begin{definition}[{\cite[Definition 2.1]{rp}}] \label{def2.3}\rm
Let $A$ be a closed and linear
operator with domain $D(A)$ defined on a Banach space $X$ and
$\alpha>0$. Let $\rho(A)$ be the resolvent set of $A$, then we say
that $A$ is the generator of a solution operator if there exists
$\omega\geq 0$ and a strongly continuous function $S_\alpha:R_+\to
L(X)$ such that
$\{\lambda^\alpha:Re\lambda>\omega\}\subset\rho(A)$ and
$$
\lambda^{\alpha-1}(\lambda^\alpha I-A)^{-1}x
=\int_0^\infty e^{-\lambda t}S_\alpha(t)x
dt,\quad\operatorname{Re}\lambda>\omega,\; x\in X.
$$
In this case, $S_\alpha(t)$ is
called the solution operator generated by $A$.
\end{definition}

The concept of solution operator is closely related to the concept
of a resolvent family (see \cite[Chapter 1]{jp}). For more
details on $\alpha$-resolvent family and solution operators, we
refer to \cite{cl,jp} and the references therein.


\section{Main results}

In \cite{chen}, if $\alpha\in(0,1)$ and $A\in
A^\alpha(\theta_0,\omega_0)$, then for any $x\in X$ and $t>0$, we
have
$$
\|T_\alpha(t)\|_{L(X)}\leq Me^{\omega t},\quad
\|S_\alpha(t)\|_{L(X)}\leq Ce^{\omega t}(1+t^{\alpha-1}),\quad
t>0,\; \omega>\omega_0.
$$
Let
$$
\widetilde{M}_T=\sup_{0\leq t\leq T}\|T_\alpha(t)\|_{L(X)},\quad
\widetilde{M}_S=\sup_{0\leq t\leq
T}Ce^{\omega t}(1+t^{1-\alpha}),
$$
where $L(X)$ is the Banach space
of bounded linear operators from $X$ into $X$ equipped with its
natural topology. So we have
$$
\|T_\alpha(t)\|_{L(X)}\leq \widetilde{M}_T,\quad
 \|S_\alpha(t)\|_{L(X)}\leq t^{\alpha-1}\widetilde{M}_S.
$$
Let us consider the set of functions
\begin{align*}
 PC(J,X)
&=  \{x:J\to X: x \in C((t_k,t_{k+1}],X),k=0,1,\dots p\text{ and
there exist}\\
&\quad x(t_k^-)\text{ and } x(t_k^+),\;
k=1,\dots,p\text{ with }x(t_k^-)=x(t_k)\}.
\end{align*}
Endowed with the norm
$$
\|x\|_{PC}=\sup_{t\in J} \|x(t)\|_X,
$$
the space $(PC(J,X),\|\cdot\|_{PC})$ is a Banach space.

\begin{lemma}[\cite{chen}] \label{lem3.1}
Consider the  Cauchy problem
\begin{gather*}
D_t^\alpha x(t)+Ax(t)=f(t,x(t),x(a_1(t)),\dots,x(a_m(t))),\quad
t>t_0,\;t_0\ge 0,\; 0<\alpha<1,\\
x(t_0)=x_0\in X,
\end{gather*}
if $f$ satisfies the uniform Holder condition with exponent
$\beta\in(0,1]$ and $A$ is a sectorial operator, then the unique
solution of this Cauchy problem  is
$$
x(t)=T_\alpha(t-t_0)x(t_0^+)+\int_{t_0}^tS_\alpha(t-\theta)
f(\theta,x(\theta),x(a_1(\theta)),\dots,x(a_m(\theta)))d\theta,
$$
where
\begin{gather*}
T_\alpha(t)= E_{\alpha,1}(-At^\alpha)
=\frac{1}{2\pi i}\int_{\widehat{B}_r}e^{\lambda t}
\frac{\lambda^{\alpha-1}}{\lambda^\alpha+A}d\lambda, \\
S_{\alpha}(t)= t^{\alpha-1}E_{\alpha,\alpha}(-At^\alpha)
=\frac{1}{2\pi i}\int_{\widehat{B}_r}e^{\lambda t}
\frac{1}{\lambda^\alpha+A}d\lambda,
\end{gather*}
where $\widehat{B}_r$ denotes the Bromwich path.
$S_\alpha(t)$ is called the $\alpha-$resolvent
family and $T_\alpha(t)$ is the solution operator, generated by
$-A$.
\end{lemma}

\begin{proof}
Let $t-t_0=s$, then
$$
D_s^\alpha x(s+t_0)+Ax(s+t_0)=f(s+t_0,x(s+t_0),x(a_1(s+t_0)),
\dots,x(a_m(s+t_0))),
$$
for $s>0$.
Now, applying the Laplace transform, we have
\begin{equation} \label{d1}
\begin{aligned}
&\lambda^\alpha L\{x(s+t_0)\}-\lambda^{\alpha-1}x(t_0^+)
+A L\{x(s+t_0)\}\\
&=L\{f(s+t_0,x(s+t_0),x(a_1(s+t_0)),\dots,x(a_m(s+t_0)))\}.
\end{aligned}
\end{equation}
Since $(\lambda^\alpha I+A)^{-1}$ exists, that is
$\lambda^\alpha\in\rho(A)$,  from  \eqref{d1}, we
obtain
\begin{align*}
L\{x(s+t_0)\}
&=\lambda^{\alpha-1}(\lambda^\alpha I+A)^{-1}x(t_0^+)
+(\lambda^\alpha I+A)^{-1}\\
&\quad\times L\{f(s+t_0,x(s+t_0),x(a_1(s+t_0)),
\dots,x(a_m(s+t_0)))\}.
\end{align*}
Therefore, by the inverse Laplace transform, we have
\begin{align*}
x(s+t_0)&=E_{\alpha,1}(-A
s^\alpha)x(t_0^+)+\int_0^s(s-\tau)^{\alpha-1}
E_{\alpha,\alpha}(-A(s-\tau)^\alpha)\\
&\quad\times f(\tau+t_0,x(\tau+t_0),x(a_1(\tau+t_0)),
\dots,x(a_m(\tau+t_0)))d\tau.
\end{align*}
Let $s+t_0=t$, we obtain
\begin{align*}
x(t)&=E_{\alpha,1}(-A(t-t_0)^\alpha)x(t_0^+)
 +\int_0^{t-t_0}(t-t_0-\tau)^{\alpha-1}E_{\alpha,
 \alpha}(-A(t-t_0-\tau)^\alpha)\\
&\quad\times f(\tau+t_0,x(\tau+t_0),x(a_1(\tau+t_0)),
 \dots,x(a_m(\tau+t_0)))d\tau.
\end{align*}
This is the same as
\begin{align*}
x(t)&=E_{\alpha,1}(-A(t-t_0)^\alpha)x(t_0^+)
 +\int_{t_0}^t(t-\theta)^{\alpha-1}E_{\alpha,\alpha}
(-A(t-\theta)^\alpha) \\
&\quad\times f(\theta,x(\theta),x(a_1(\theta)),
\dots,x(a_m(\theta)))d\theta.
\end{align*}
Let $T_\alpha(t)=E_{\alpha,1}(-At^\alpha)$ and
$S_\alpha(t)=t^{\alpha-1}E_{\alpha,\alpha}(-At^\alpha)$, then we
have
$$
x(t)=T_\alpha(t-t_0)x(t_0^+)+\int_{t_0}^tS_\alpha(t-\theta)
f(\theta,x(\theta),x(a_1(\theta)),\dots,x(a_m(\theta)))d\theta.
$$
This completes the proof of the Lemma.
\end{proof}

 Now, we define the definition of mild solution of \eqref{ME}.

\begin{definition}\label{mild} \rm
A function  $x\in$ $PC(J,X)$ solution of the fractional integral
equation
\begin{align*}
x(t)=\begin{cases}
T_\alpha(t)(x_0-g(x))\\
+\int_0^tS_\alpha(t-s)f(s,x(s),x(a_1(s)),
\dots,x(a_m(s)))ds,& t\in[0,t_1]; \\[3pt]
T_\alpha(t-t_1)[x(t_1^-)+I_1(x(t_1^-))]\\
+\int_{t_1}^t  S_\alpha(t-s)f(s,x(s),x(a_1(s)),\dots,x(a_m(s)))ds,
&t\in(t_1,t_2];\\
\dots \\
T_\alpha(t-t_p)[x(t_p^-)+I_p(x(t_p^-))]\\
+\int_{t_p}^tS_\alpha(t-s)f(s,x(s),x(a_1(s)),\dots,x(a_m(s)))ds,
&t\in(t_p,T].
\end{cases}
\end{align*}
will be called a mild solution of problem \eqref{ME}. From Lemma
\ref{lem3.1} we can verify this definition.
\end{definition}

 Now we introduce the following assumptions:
\begin{itemize}
\item[(H1)] There exists a constant $L_g>0$ such that
$\|g(x)-g(y)\|_X\leq L_g\|x-y\|_X$.

\item[(H2)] The nonlinear map
$f:[0,T]\times X^{m+1}\to X$ is continuous and there exist a
constant $L_f$ such that
\begin{align*}
&\|f(t,x_1,x_2,\dots,x_{m+1})- f(s,y_1,y_2,\dots,y_{m+1})\|_X\\
&\leq L_f\big[|t-s|+\sum_{i=1}^{m+1}\|x_i-y_i\|_X\big]
\end{align*}
for all $(x_1,\dots,x_{m+1})$ and $(y_1,\dots,y_{m+1})$ in
$X^{m+1}$ and $t\in[0,T]$.

\item[(H3)] The function
$I_k:X\to X$ are continuous and there exists $L_k>0$ such
that  $\|I_k(x)-I_k(y)\|_X\leq L_k\|x-y\|_X,\quad x,y \in
X,k=1,2,\dots,p,\;L=\max{\{L_k\}}>L_g$.
\end{itemize}

\begin{theorem} \label{thm3.3}
Assume {\rm (H1)--(H3)} are satisfied and
$$
\big[\widetilde{M}_T(1+L)+\widetilde{M}_SL_f(m+1)
\frac{T^\alpha}{\alpha}\big]<1.
$$
Then impulsive problem \eqref{ME} has a unique mild
solution $x\in PC(J,X)$.
\end{theorem}

\begin{proof}
Define a mapping $N$ from $PC(J,X)$ into itself by
\[
(N x)(t)=\begin{cases}
T_\alpha(t)(x_0-g(x))\\
+\int_0^tS_\alpha(t-s)f(s,x(s),x(a_1(s)),
\dots,x(a_m(s)))ds,& t\in[0,t_1]; \\[3pt]
T_\alpha(t-t_1)[x(t_1^-)+I_1(x(t_1^-))]\\
+\int_{t_1}^tS_\alpha(t-s)f(s,x(s),x(a_1(s)),\dots,x(a_m(s)))ds,
&t\in(t_1,t_2];\\
\dots \\
T_\alpha(t-t_p)[x(t_p^-)+I_p(x(t_p^-))]\\
+\int_{t_p}^tS_\alpha(t-s)f(s,x(s),x(a_1(s)),\dots,x(a_m(s)))ds,
&t\in(t_p,T].
\end{cases}
\]
Now we show that $N$ is a contraction on $PC(J,X)$. We
have
\begin{align*}
&\|Nx(t)-Ny(t)\|_X\\
&\leq\begin{cases}
\|T_\alpha(t)\|_{L(X)}(\|g(x)-g(y)\|_X)
 +\int_0^t\|S_\alpha(t-s)\|_{L(X)} \\
\times \|f(s,x(s),x(a_1(s)),\dots,x(a_m(s)))\\
-f(s,y(s),y(a_1(s)),\dots,y(a_m(s)))\|_Xds,
& t\in[0,t_1]; \\[3pt]
\|T_\alpha(t-t_1)\|_{L(X)}(\|x(t_1^-)-y(t_1^-)\|_X
+\|I_1(x(t_1^-))-I_1(y(t_1^-))\|_X)\\
+\int_{t_1}^t\|S_\alpha(t-s)\|_{L(X)}\|f(s,x(s),x(a_1(s)),
 \dots,x(a_m(s)))\\
-f(s,y(s),y(a_1(s)),\dots,y(a_m(s)))\|_Xds,
& t\in(t_1,t_2];\\
\dots \\
\|T_\alpha(t-t_p)\|_{L(X)}(\|x(t_p^-)-y(t_p^-)\|_X+\|I_p(x(t_p^-))
-I_p(y(t_p^-))\|_X)\\
+\int_{t_p}^t\|S_\alpha(t-s)\|_{L(X)}\|f(s,x(s),x(a_1(s)),
\dots,x(a_m(s)))\\
-f(s,y(s),y(a_1(s)),\dots,y(a_m(s)))\|_X ds,
& t\in(t_p,T];
\end{cases}
\end{align*}
Applying Assumptions (H1)--(H3), we obtain
\[
\|Nx(t)-Ny(t)\|_X\leq \begin{cases}
[\widetilde{M}_T[L_g+\widetilde{M}_SL_f(m+1)
 \frac{T^\alpha}{\alpha}] \|x-y\|_{PC}, &t\in[0,t_1];\\
[\widetilde{M}_T(1+L_1)+\widetilde{M}_SL_f(m+1)
\frac{T^\alpha}{\alpha}]\|x-y\|_{PC}, &t\in(t_1,t_2]; \\
\dots  \\
[\widetilde{M}_T(1+L_p)+\widetilde{M}_SL_f(m+1)
\frac{T^\alpha}{\alpha}]\|x-y\|_{PC}, &t\in(t_p,T].
\end{cases}
\]
Which implies that for $t\in[0,T]$,
$$
\|Nx-Ny\|_{PC}\leq
[\widetilde{M}_T(1+L)+\widetilde{M}_SL_f(m+1)
\frac{T^\alpha}{\alpha}]\|x-y\|_{PC}.
$$
Since $[\widetilde{M}_T(1+L)+\widetilde{M}_SL_f(m+1)
\frac{T^\alpha}{\alpha}]<1$, $N$ is a contraction. Therefore,
$N$ has a unique fixed point by Banach contraction principle.
This completes the proof of the theorem.
\end{proof}

Our second result is based on the following Krasnoselkii's fixed
point theorem.

\begin{theorem} \label{thm3.4}
Let $B$ be a closed convex and nonempty subset of a Banach space
$X$. Let $P$ and $Q$ be two operators such that:
\begin{itemize}
\item[(1)] $Px+Qy\in B$ whenever $x,y\in B$;

\item[(2)] $P$ is compact and continuous;

\item[(3)] $Q$ is a contraction mapping;
\end{itemize}
 Then there exists $z\in B$ such that $z=Pz+Qz$.
\end{theorem}

Now, we make the following assumptions:
\begin{itemize}
\item[(H4)] $f\in C(J\times X^{m+1},X),\;g\in C(X,X)$, and $b_i\in
C(J,J)$ $(i=1,\dots,m)$. Moreover, there are $C_i>0\;(i=1,2)$ such that
$\|f(s,z_0,z_1,\dots,z_m)\|\leq C_1$ for $s\in J$,
$z_i\in B_r\;(i=0,1,\dots,m)$ and $\|g(w)\|\leq C_2$ for $w\in X$.

\item[(H5)] The function $I_k:X\to X$ are continuous and
there exists $\rho>C_2$ such that
$$
\rho=\max_{1\leq k\leq m,x\in B_r}\{\|I_k(x)\|_X\}.
$$
\end{itemize}

\begin{theorem}\label{thm3.5}
Assume {\rm (H2), (H4), (H5)} are satisfied and
$$
[\widetilde{M}_SL_f(m+1) {T^\alpha\over\alpha}]<1.
$$
Then the impulsive problem \eqref{ME} has at least one
mild solution on $J$.
\end{theorem}

\begin{proof}
Choose $r\geq [\widetilde{M}_T(r+\rho)
 +\widetilde{M}_SC_1\frac{T^\alpha}{\alpha}]$
and consider $B_r=\{x\in PC(J,X):\|x\|_{PC}\leq r,\}$ then $B_r$
is a bounded, closed convex subset in $PC(J,X)$. Define on $B_r$
the operators $P$ and $Q$ by:
\begin{gather*}
(Px)(t)= \begin{cases} T_\alpha(t)(x_0-g(x)), & t\in[0,t_1];\\
T_\alpha(t-t_1)[x(t_1^-)+I_1(x(t_1^-))], & t\in(t_1,t_2];\\
\dots  \\
T_\alpha(t-t_p)[x(t_p^-)+I_p(x(t_p^-))], & t\in(t_p,T],
\end{cases}
\\
(Qx)(t)= \begin{cases}
\int_0^tS_\alpha(t-s)f(s,x(s),x(a_1(s)),\dots,x(a_m(s)))ds,
&t\in[0,t_1];\\
\int_{t_1}^tS_\alpha(t-s)f(s,x(s),x(a_1(s)),\dots,x(a_m(s)))ds,
&t\in(t_1,t_2]; \\
\dots \\
\int_{t_p}^tS_\alpha(t-s)f(s,x(s),x(a_1(s)),\dots,x(a_m(s)))ds,
&t\in(t_p,T].
\end{cases}
\end{gather*}
Now we present the proof in five steps:


\textbf{Step 1.} We show that $Px+Qy\in B_r$ whenever $x,y\in B_r$.
Let $x,y\in B_r$, then
\begin{align*}
&\|Px+Qy\|_{PC}\\
&\leq \begin{cases}
\|T_\alpha(t)\|_{L(X)}(\|x_0\|_X+\|g(x)\|_X)\\
+\int_0^t\|S_\alpha(t-s)\|_{L(X)}\|f(s,y(s),y(a_1(s)),
\dots,y(a_m(s)))\|_Xds, & t\in[0,t_1];
\\[3pt]
\|T_\alpha(t-t_1)\|_{L(X)}[\|x(t_1^-)\|_X+\|I_1(x(t_1^-))\|_X]\\
+\int_{t_1}^t\|S_\alpha(t-s)\|_{L(X)}\|f(s,y(s),y(a_1(s)),
\dots,y(a_m(s)))\|_Xds,
& t\in(t_1,t_2]; \\
\dots  \\
\|T_\alpha(t-t_p)\|_{L(X)}[\|x(t_p^-)\|_X+\|I_p(x(t_p^-))\|_X]\\
+\int_{t_p}^t\|S_\alpha(t-s)\|_{L(X)}\|f(s,y(s),y(a_1(s)),
\dots,y(a_m(s)))\|_Xds, & t\in(t_p,T].
\end{cases}
\\
&\leq \begin{cases}
\widetilde{M}_T(r+C_2)+\widetilde{M}_SC_1\frac{T^\alpha}{\alpha},
& t\in[0,t_1];\\
\widetilde{M}_T(r+\rho)+\widetilde{M}_SC_1\frac{T^\alpha}{\alpha},
&t\in(t_1,t_2];\\
\dots  \\
\widetilde{M}_T(r+\rho)+\widetilde{M}_SC_1\frac{T^\alpha}{\alpha},
& t\in(t_p,T].
\end{cases}
\end{align*}
Which implies
\[
\|Px+Qy\|_{PC}\leq[\widetilde{M}_T(r+\rho)
+\widetilde{M}_SC_1\frac{T^\alpha}{\alpha}] \leq r.
\]

\textbf{Step 2.} Continuity of $P$. For this purpose, let
$\{x^n\}_{n=0}^\infty$ be a sequence in $B_r$ with $\lim x^n\to x$
in $B_r$. Then for every $t\in J$, we have
\[
\|(Px^n)(t)-(Px)(t)\|_X
\leq \begin{cases}
\|T_\alpha(t)\|_{L(X)}\|g(x^n)-g(x)\|_X, & t\in[0,t_1];\\
\|T_\alpha(t-t_1)\|_{L(X)}[\|x^n(t_1^-)-x(t_1^-)\|_X\\
+\|I_1(x^n(t_1^-))-I_1x(t_1^-)\|_X],
&t\in(t_1,t_2];\\
\dots \\
\|T_\alpha(t-t_p)\|_{L(X)}[\|x^n(t_p^-)-x(t_p^-)\|_X\\
+\|I_p(x^n(t_p^-))-I_px(t_p^-)\|_X],
&t\in(t_p,T].
\end{cases}
\]
Since the functions $g$ and $I_k$, $k=1,\dots,p$ are continuous,
$\lim_{n\to\infty}\|Px^n-Px\|_{PC}=0$ in $B_r$.
This implies that the mapping $P$ is continuous on $B_r$.


\textbf{Step 3.} $P$ maps bounded sets into bounded sets in $PC(J,X)$.
So, let us prove that for any $r>0$ there exists a $\gamma>0$ such
that for each $x\in B_r=\{x\in PC(J,X):\|x\|_{PC}\leq r\}$, we
have $\|Px\|_{PC}\leq \gamma$.
 Indeed, we have for any $x\in B_r$,
\begin{align*}
\|Px(t)\|_X
&\leq \begin{cases}
\|T_\alpha(t)\|_{L(X)}(\|x_0\|_X+\|g(x)\|_X),
& t\in[0,t_1]; \\
\|T_\alpha(t-t_1)\|_{L(X)}[\|x(t_1^-)\|_X+\|I_1(x(t_1^-))\|_X],
& t\in(t_1,t_2]; \\
\dots \\
\|T_\alpha(t-t_p)\|_{L(X)}[\|x(t_p^-)\|_X+\|I_p(x(t_p^-))\|_X],
&t\in(t_p,T].
\end{cases}\\
&\leq \begin{cases}
\widetilde{M}_T(r+C_2), & t\in[0,t_1];\\
\widetilde{M}_T(r+\rho), & t\in(t_1,t_2];\\
\dots  \\
\widetilde{M}_T(r+\rho), & t\in(t_p,T].
\end{cases}
\end{align*}
Which implies that $\|Px\|_{PC}\leq
\widetilde{M}_T(r+\rho)=\gamma$.

\textbf{Step 4.} We prove that $P(B_r)$ is equicontinuous with $B_r$.
For $0\leq u<v\leq T$, we have
\begin{align*}
&\|(Px)(v)-(Px)(u)\|_X \\
&\leq \begin{cases}
\|T_\alpha(v)-T_\alpha(u)\|_{L(X)}[\|x_0\|_X+\|g(x)\|_X],
& 0\leq u<v\leq t_1;\\[3pt]
\|T_\alpha(v-t_1)-T_\alpha(u-t_1)\|_{L(X)}\\
\times[\|x(t_1^-)\|_X+\|I_1(x(t_1^-))\|_X],
&t_1<u<v\leq t_2;\\
\dots \\
\|T_\alpha(v-t_p)-T_\alpha(u-t_p)\|_{L(X)}\\
\times[\|x(t_p^-)\|_X+\|I_p(x(t_p^-))\|_X, & t_p<u<v\leq T.
\end{cases}\\
&\leq \begin{cases}
(r+C_2)\|T_\alpha(v)-T_\alpha(u)\|_{L(X)}, &0\leq u<v\leq t_1;\\
(r+\rho)\|T_\alpha(v-t_1)-T_\alpha(u-t_1)\|_{L(X)}, &
 t_1<u<v\leq t_2\\
\dots \\
(r+\rho)\|T_\alpha(v-t_p)-T_\alpha(u-t_p)\|_{L(X)}, &
 t_p<u<v\leq T.
\end{cases}
\end{align*}
Therefore, the continuity of the function $t\mapsto\|T(t)\|$ allows
us to conclude that $\lim_{u\to
v}\|T_\alpha(v-t_i)-T_\alpha(u-t_i)\|_{L(X)}=0$, $i=1,\dots,p$ and
$\lim_{u\to v}\|T_\alpha(v)-T_\alpha(u)\|_{L(X)}=0$.
Finally, combining Step 2 to Step 4 with the Ascoli's Theorem,
we deduce that the operator $P$ is a compact.


\textbf{Step 5.} We show that $Q$ is a contraction mapping.
 Let $x,y\in B_r$ and we have
\begin{align*}
&\|(Qx)(t)-(Qy(t))(t)\|_X \\
&\leq \begin{cases}
\int_0^t\|S_\alpha(t-s)\|_{L(X)}\|f(s,x(s),x(a_1(s)),\dots,x(a_m(s)))\\
-f(s,y(s),y(a_1(s)),\dots,y(a_m(s)))\|_X ds,
& t\in[0,t_1];\\[3pt]
\int_{t_1}^t\|S_\alpha(t-s)\|_{L(X)}\|f(s,x(s),x(a_1(s)),
\dots,x(a_m(s)))\\
-f(s,y(s),y(a_1(s)),\dots,y(a_m(s)))\|_X ds,
 &t\in(t_1,t_2]; \\[3pt]
\dots  \\
\int_{t_p}^t\|S_\alpha(t-s)\|_{L(X)}\|f(s,x(s),x(a_1(s)),\dots,
x(a_m(s)))\\
-f(s,y(s),y(a_1(s)),\dots,y(a_m(s)))\|_X ds, & t\in(t_p,T].
\end{cases} \\
&\leq \begin{cases}
\widetilde{M}_SL_f(m+1)\frac{T^\alpha}{\alpha}\|x-y\|_{PC},
& t\in[0,t_1];\\
\widetilde{M}_SL_f(m+1)\frac{T^\alpha}{\alpha}\|x-y\|_{PC},
& t\in(t_1,t_2];\\
\dots  \\
\widetilde{M}_SL_f(m+1)\frac{T^\alpha}{\alpha}\|x-y\|_{PC},
& t\in(t_p,T].
\end{cases}
\end{align*}
Since $(\widetilde{M}_SL_f(m+1)\frac{T^\alpha}{\alpha})<1$ then
$Q$ is a contraction mapping. Hence, by the Krasnoselkii theorem,
we can conclude that \eqref{ME} has at least one solution on
 $[0,T]$. This completes the proof of the theorem.
\end{proof}

\subsection*{Acknowledgements}
The authors wish to express their deep gratitude to the
anonymous referees for their valuable suggestions and comments for
improving the original manuscript.


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\end{document}