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

\AtBeginDocument{{\noindent\small
\emph{Electronic Journal of Differential Equations},
Vol. 2009(2009), No. 10, 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 2009 Texas State University - San Marcos.}
\vspace{9mm}}

\begin{document}
\title[\hfilneg EJDE-2009/10\hfil Impulsive fractional differential equations]
{Existence and uniqueness of solutions to impulsive
fractional differential equations}

\author[M. Benchohra, B. A. Slimani\hfil EJDE-2009/10\hfilneg]
{Mouffak Benchohra, Boualem Attou Slimani}  % in alphabetical order

\address{Mouffak Benchohra \newline
Laboratoire de Math\'ematiques, Universit\'e de Sidi Bel-Abb\`es,
B.P. 89, 22000, Sidi Bel-Abb\`es, Alg\'erie}
\email{benchohra@univ-sba.dz}

\address{Boualem Attou Slimani \newline
Facult\'e des Sciences de l'Ing\'enieur, Universit\'e de Tlemcen,
B.P. 119, 13000, Tlemcen, Alg\'erie}
\email{ba\_slimani@yahoo.fr}

\thanks{Submitted October 22, 2008. Published January 9, 2009.}
\subjclass[2000]{26A33, 34A37}
\keywords{Fractional derivative; impulses; Initial value problem;
\hfill\break\indent
 Caputo  fractional integral;
nonlocal conditions; existence; uniqueness; fixed point}

\begin{abstract}
 In this article, we establish sufficient conditions for the
 existence of solutions for a class of initial value problem
 for impulsive fractional differential equations involving
 the Caputo fractional derivative.
\end{abstract}

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

\section{Introduction}

This article studies the existence and uniqueness of solutions for
the initial value problems (IVP for short), for fractional order
differential equations
\begin{gather}\label{e1}
^{c}D^{\alpha}y(t)= f(t,y), \quad   t\in J=[0, T], \;t\neq t_{k},  \\
 \label{e2}
\Delta y\big|_{t=t_{k}}= I_{k}(y(t_{k}^{-})), \\
\label{e3}
y(0)= y_{0},
\end{gather}
where $k=1,\dots,m$, $0<\alpha\leq 1$,
$^{c}D^{\alpha}$ is the Caputo fractional derivative,
$f : J\times \mathbb{R}\to \mathbb{R}$ is a given function,
 $I_{k}:\mathbb{R}\to\mathbb{R}$, and
$ y_{0}\in\mathbb{R}$, $0=t_{0}<t_{1}<\dots <t_{m}<t_{m+1}=T$,
$\Delta y|_{t=t_{k}}=y(t_{k}^{+})-y(t_{k}^{-})$,
$y(t_{k}^{+})=\lim_{h\to 0^+}y(t_k+h)$ and
$y(t_{k}^{-})=\lim_{h\to 0^-}y(t_{k}+h)$ represent the right and
left limits of $y(t)$ at $t=t_{k}$.

Differential equations of fractional order have proved to be valuable
tools in the modelling of many phenomena in various fields of science
and engineering. Indeed, we can find numerous applications in
viscoelasticity, electrochemistry, control, porous media,
electromagnetic, etc. (see \cite{DiFr, GaKlKe, GlNo, Hil, Mai,
MeScKiNo, OlPs}). There has been a significant development in
fractional differential and partial differential equations in recent
years; see the monographs of Kilbas \emph{et al} \cite{KST}, Miller
and Ross \cite{MiRo}, Samko \emph{et al} \cite{SaKiMa} and the papers
of Agarwal \emph{et al} \cite{ABH}, Babakhani and Daftardar-Gejji
\cite{BaDa, BaDa1}, Belmekki \emph{et al} \cite{BeBeGo}, Benchohra
\emph{et al} \cite{BeBe, BeGrHa, BeHaNt, BeHeNtOu1}, Daftardar-Gejji
and Jafari \cite{DaJa1}, Delbosco and Rodino \cite{DeRo}, Diethelm
\emph{et al } \cite{DiFr,DiFo,DiWa}, El-Sayed \cite{El,El1,El2},
Furati and Tatar \cite{FuTa, FuTa1}, Kaufmann and Mboumi \cite{KM},
Kilbas and Marzan \cite{KiMa}, Mainardi \cite{Mai}, Momani and Hadid
\cite{MoHa}, Momani \emph{et al} \cite{MoHaAl}, Podlubny \emph{et al}
\cite{PoPeViLeDo}, Yu and Gao \cite{YuGa} and Zhang \cite{Zha} and
the references therein.

Applied problems require definitions of fractional derivatives
allowing the utilization of physically interpretable initial
conditions, which contain $y(0)$, $y'(0)$, etc.,  the same
requirements of boundary conditions. Caputo's fractional derivative
satisfies these demands.  For more details on the geometric and
physical interpretation for fractional derivatives of both the
Riemann-Liouville and Caputo types see \cite{HePo,Pod1}.

Impulsive differential equations (for $\alpha\in\mathbb{N}$) have become
important in recent years as mathematical models of phenomena in
both the physical and social sciences. There has a significant
development in impulsive theory especially in the area of impulsive
differential equations with fixed moments; see for instance the
monographs by Bainov and Simeonov \cite{BaSi}, Benchohra \emph{et al}
\cite{BeHeNt1}, Lakshmikantham \emph{et al} \cite{LBS}, and
Samoilenko and Perestyuk \cite{SaPe} and the references therein. To
the best knowledge of the authors, no papers exist in the literature
devoted to differential equations with fractional order and
impulses.  Thus the results of the present paper initiate this
study. This paper is organized as follows. In Section 2 we present
some preliminary results about fractional derivation and integration
needed in the following sections. Section 3 will be concerned with
existence and uniqueness results for the IVP \eqref{e1}-\eqref{e3}.
We give three results, the first one is based on Banach fixed point
theorem (Theorem \ref{thm1}), the second one is based on Schaefer's
fixed point theorem (Theorem \ref{thm2}) and the third one on the
nonlinear alternative of Leray-Schauder type (Theorem \ref{thm3}). In
Section 4 we indicate some generalizations to nonlocal initial value
problems. The last section is devoted to an example illustrating the
applicability of the imposed conditions. These results can be
considered as a contribution to this emerging field.

\section{Preliminaries}

 In this section, we introduce notation, definitions,
and preliminary facts which are used throughout this paper. By
$C(J,\mathbb{R})$ we denote the Banach space of all continuous
functions from $J$ into $\mathbb{R}$ with the norm
$$
\|y\|_{\infty}:=\sup\{|y(t)|: t\in J\}.
$$

\begin{definition}[\cite{KST,Pod}] \rm
The fractional (arbitrary) order integral of the function $h\in
L^1([a,b],\mathbb{R}_+)$ of order $\alpha\in\mathbb{R}_+$ is defined by
$$
I^{\alpha}_ah(t)=\int_a^t\frac{(t-s)^{\alpha-1}}{\Gamma(\alpha)}h(s)ds,
$$
where $\Gamma$ is the gamma function. When $a=0$, we write
$I^{\alpha}h(t)=[h*\varphi_{\alpha}](t)$, where
$\varphi_{\alpha}(t)=\frac{t^{\alpha-1}}{\Gamma(\alpha)}$ for $t>0$,
and  $\varphi_{\alpha}(t)=0$ for $t\leq 0$, and $\varphi_{\alpha}\to
\delta(t)$\ as $\alpha\to 0$, where $\delta$ is the delta function.
\end{definition}

\begin{definition}[\cite{KST,Pod}] \rm
For a function $h$ given on the interval $[a,b]$, the $\alpha th$
Riemann-Liouville fractional-order derivative of $h$, is defined by
\[
(D^{\alpha}_{a+}h)(t)=\frac{1}{\Gamma(n-\alpha)}\big(\frac{d}{dt}\big)^{n}\int_
a^t(t-s)^{n-\alpha-1}h(s)ds.
\]
Here $n=[\alpha]+1$ and $[\alpha]$ denotes the integer part of
$\alpha$.
\end{definition}

\begin{definition}[\cite{KiMa}] \rm
For a function $h$ given on the interval $[a,b]$, the Caputo
fractional-order derivative of order $\alpha$ of $h$, is defined by
$$
(^{c}D_{a+}^{\alpha}h)(t)=\frac{1}{\Gamma(n-\alpha)}\int_
a^t(t-s)^{n-\alpha-1}h^{(n)}(s)ds,
$$
where $n=[\alpha]+1$.
\end{definition}

\section{Existence of Solutions}

 Consider the  set of functions
\begin{align*}
PC(J,\mathbb{R})&=\{y: J\to \mathbb{R}: y\in C((t_k,t_{k+1}],\mathbb{R}), \; k=0,\dots,m
\text{ and there exist }\\
&\quad y(t^{-}_{k}) \text{ and }  y(t^{+}_{k}), \; k=1,\dots,m  \text{ with }
y(t^{-}_{k})=y(t_{k})\}.
\end{align*}
This set is a Banach space with the norm
$$
\|y\|_{PC}=\sup_{t\in J}|y(t)|.
$$
 Set $J':=[0,T]\backslash\{t_{1},\dots,t_{m}\}$.

\begin{definition} \rm
A function $y\in PC(J,\mathbb{R})$  whose
$\alpha$-derivative exists on $J'$ is said to be a solution of
\eqref{e1}--\eqref{e3} if $y$ satisfies the equation
$^{c}D^{\alpha}y(t)=f(t,y(t))$ on $J'$, and satisfy the conditions
\begin{gather*}
 \Delta y|_{t=t_{k}}= I_{k}(y(t_{k}^{-})), \  k=1,\dots,m,\\
y(0)= y_{0}
\end{gather*}
\end{definition}

To prove the existence of solutions to \eqref{e1}--\eqref{e3},
we need the following auxiliary lemmas.

\begin{lemma}[\cite{Zha}] \label{l1}
Let $\alpha > 0 $, then the differential equation
$$
^{c}D^{\alpha}h(t)=0
$$
has solutions $h(t)=c_{0}+c_{1}t+c_{2}t^{2}+\dots
+c_{n-1}t^{n-1} , c_{i}\in \mathbb{R}$,
$i=0,1,2,\dots,n-1$, $n=[\alpha]+1$.
\end{lemma}

\begin{lemma}[\cite{Zha}] \label{l2}
Let $\alpha > 0 $, then
$$
{I^{\alpha}}^{c}D^{\alpha}h(t)=h(t)+
c_{0}+c_{1}t+c_{2}t^{2}+\dots+c_{n-1}t^{n-1}
$$
for some $ c_{i}\in \mathbb{R}$,
$i=0,1,2,\dots,n-1$, $n=[\alpha]+1$.
\end{lemma}

As a consequence of Lemma \ref{l1} and Lemma \ref{l2} we have the
following result which is useful in what follows.

\begin{lemma}\label{l3}
 Let  $0< \alpha\leq 1$ and let $ h: J \to\mathbb{R}$ be
continuous. A function $y$ is a solution of the fractional integral
equation
\begin{equation}\label{e4}
y(t)=\begin{cases}
y_{0}+\frac{1}{\Gamma(\alpha)}\int_{0}^{t}(t-s)^{\alpha-1}h(s)ds
& \text{if } t\in [0,t_{1}],
\\[3pt]
y_{0}+\frac{1}{\Gamma(\alpha)}\sum_{i=1}^{k}\int_{t_{i-1}}^{t_{i}}
(t_{i}-s)^{\alpha-1}h(s)ds\\
+ \frac{1}{\Gamma(\alpha)}
\int_{t_{k}}^{t}(t-s)^{\alpha-1}h(s)ds
+\sum_{i=1}^{k}I_{i}(y(t_{i}^{-})),
& \text{if  } t\in (t_{k},t_{k+1}],
\end{cases}
\end{equation}
where $k=1,\dots,m$, if and only if  $y$ is a solution of the fractional  IVP
\begin{gather}\label{e5}
^{c}D^{\alpha}y(t)= h(t), \quad   t\in J', \\
\label{e6}
\Delta y|_{t=t_{k}}= I_{k}(y(t_{k}^{-})), \quad  k=1,\dots,m, \\
\label{e7}
y(0)= y_{0}.
\end{gather}
\end{lemma}

\begin{proof}
 Assume $y$ satisfies \eqref{e5}-\eqref{e7}.
If $t\in [0,t_{1}]$ then
$$
^{c}D^{\alpha}y(t)= h(t).
$$
Lemma \ref{l2} implies
$$
y(t)=y_{0}+\frac{1}{\Gamma(\alpha)}
\int_{0}^{t}(t-s)^{\alpha-1}h(s)ds.
$$
If $t\in (t_{1},t_{2}]$ then
Lemma \ref{l2} implies
\begin{align*}
y(t)&=y(t_{1}^{+})+\frac{1}{\Gamma(\alpha)}
\int_{t_{1}}^{t}(t-s)^{\alpha-1}h(s)ds\\
&= \Delta y|_{t=t_{1}}+y(t_{1}^{-})+\frac{1}{\Gamma(\alpha)}
\int_{t_{1}}^{t}(t-s)^{\alpha-1}h(s)ds\\
&= I_{1}(y(t_{1}^{-}))+y_{0}+\frac{1}{\Gamma(\alpha)}
\int_{0}^{t_{1}}(t_{1}-s)^{\alpha-1}h(s)ds
+\frac{1}{\Gamma(\alpha)} \int_{t_{1}}^{t}(t-s)^{\alpha-1}h(s)ds.
\end{align*}
If $t\in (t_{2},t_{3}]$ then  from Lemma \ref{l2} we get
\begin{align*}
y(t)&= y(t_{2}^{+})+\frac{1}{\Gamma(\alpha)}
\int_{t_{2}}^{t}(t-s)^{\alpha-1}h(s)ds\\
&= \Delta y|_{t=t_{2}}+y(t_{2}^{-})+\frac{1}{\Gamma(\alpha)}
\int_{t_{2}}^{t}(t-s)^{\alpha-1}h(s)ds\\
&= I_{2}(y(t_{2}^{-}))+I_{1}(y(t_{1}^{-}))+y_{0}+\frac{1}{\Gamma(\alpha)}
\int_{0}^{t_{1}}(t_{1}-s)^{\alpha-1}h(s)ds\\
&\quad +\frac{1}{\Gamma(\alpha)}
\int_{t_{1}}^{t_{2}}(t_{2}-s)^{\alpha-1}h(s)ds+\frac{1}{\Gamma(\alpha)}
\int_{t_{2}}^{t}(t-s)^{\alpha-1}h(s)ds.
\end{align*}
If $t\in (t_{k},t_{k+1}]$ then again from Lemma \ref{l2} we get
(\ref{e4}).

Conversely, assume that $y$ satisfies the impulsive fractional
integral equation (\ref{e4}). If $t\in [0,t_{1}]$ then $y(0)=y_{0}$
and using the fact that $^{c}D^{\alpha}$ is the left inverse of
$I^{\alpha}$ we get
$$
^{c}D^{\alpha}y(t)= h(t), \quad \text{for each }
t\in [0,t_{1}].
$$
If $t\in [t_{k},t_{k+1})$, $k=1,\dots,m$ and
using the fact that $^{c}D^{\alpha}C=0$, where $C$ is a constant, we
get
$$
^{c}D^{\alpha}y(t)= h(t), \text{for each }
 t\in [t_{k},t_{k+1}).
$$
Also, we can easily show that
$$
\Delta y|_{t=t_{k}}= I_{k}(y(t_{k}^{-})), \quad  k=1,\dots,m.
$$
\end{proof}
Our first result is based on Banach fixed point theorem.

\begin{theorem}\label{thm1} Assume that
\begin{itemize}
\item[(H1)] There exists a constant $l>0$ such that
$|f(t,u)-f(t,\overline u)|\leq l |u-\overline u|$,
for each $t\in J$, and each $ u,  \overline u \in \mathbb{R}$.

\item[(H2)] There exists a constant $l^*>0$ such that
$|I_{k}(u)-I_{k}(\overline u)|\leq l^*|u-\overline u|$,
for each $u,  \overline u \in \mathbb{R}$ and $k=1,\dots,m$.
\end{itemize}
 If
\begin{equation}\label{eq1}
\big[\frac{T^{\alpha}l(m+1)} {\Gamma(\alpha+1)}+ml^*\big] < 1,
\end{equation}
then  \eqref{e1}-\eqref{e3} has a unique solution on $J$.
\end{theorem}

\begin{proof}
 We transform the problem \eqref{e1}--\eqref{e3} into a
fixed point problem.  Consider the operator
$F:PC(J,\mathbb{R})\to PC(J,\mathbb{R})$ defined by
\begin{align*}
 F(y)(t)&=y_{0}+\frac{1}{\Gamma(\alpha)}\sum_{0<t_{k}<t}
\int_{t_{k-1}}^{t_{k}}(t_{k}-s)^{\alpha-1}f(s,y(s))ds\\
&\quad + \frac{1}{\Gamma(\alpha)}
\int_{t_{k}}^{t}(t-s)^{\alpha-1}f(s,y(s))ds
+\sum_{0<t_{k}<t}I_{k}(y(t_{k}^{-})).
\end{align*}
Clearly, the fixed points of the operator $F$ are solution of the
problem \eqref{e1}-\eqref{e3}. We shall use the Banach contraction
principle  to prove that $F$ has a fixed point. We shall show that
\emph{$F$ is a contraction}.  Let $ x,y \in PC(J,\mathbb{R})$.
Then, for each $t\in J$ we have
\begin{align*}
&|F(x)(t)-F(y)(t)|\\
&\leq \frac{1}{\Gamma(\alpha)}\sum_{0<t_{k}<t}
 \int_{t_{k-1}}^{t_{k}}(t_{k}-s)^{\alpha-1}|f(s,x(s))-f(s,y(s))|ds\\
&\quad + \frac{1}{\Gamma(\alpha)}
 \int_{t_{k}}^{t}(t-s)^{\alpha-1}|f(s,x(s))-f(s,y(s))|ds
 +\sum_{0<t_{k}<t}|I_{k}(x(t_{k}^{-}))-I_{k}(y(t_{k}^{-}))|\\
&\leq \frac{l}{\Gamma(\alpha)}\sum_{k=1}^{m}
 \int_{t_{k-1}}^{t_{k}}(t_{k}-s)^{\alpha-1}|x(s)-y(s)|ds\\
&\quad +\frac{l}{\Gamma(\alpha)}
 \int_{t_{k}}^{t}(t-s)^{\alpha-1}|x(s)-y(s)|ds
 +\sum_{k=1}^{m}l^*|x(t_{k}^{-})-y(t_{k}^{-})|\\
&\leq  \frac{mlT^{\alpha}}{\Gamma(\alpha+1)}\|x-y\|_{\infty}
 +\frac{T^{\alpha}l}{\Gamma(\alpha+1)}\|x-y\|_{\infty}
 + ml^* \|x-y\|_{\infty}.
\end{align*}
Therefore,
$$
\|F(x)-F(y)\|_{\infty} \leq
\big[\frac{T^{\alpha}l(m+1)}{\Gamma(\alpha+1)}+ml^*\big]
\|x-y\|_{\infty}.
$$
Consequently by (\ref{eq1}), $F$ is a contraction. As a
consequence of Banach fixed point theorem, we deduce that $F$ has a
fixed point which is a solution of the problem
$\eqref{e1}-\eqref{e3}$.
\end{proof}

Our second result is based on Schaefer's fixed point theorem.

\begin{theorem}\label{thm2}
 Assume that:
\begin{itemize}
\item[(H3)] The function $f:J\times\mathbb{R}\to\mathbb{R}$ is continuous.
\item[(H4)] There exists a constant $ M >0$ such
that $ |f(t,u)|\leq M$ for each $ t\in J$ and each
$u\in \mathbb{R}$.
\item[(H5)] The functions $I_{k}:\mathbb{R}\to\mathbb{R}$ are continuous and there
exists a constant $ M^* >0$ such
that $|I_{k}(u)|\leq M^*$ for each $u\in \mathbb{R}$, $k=1,\dots,m$.

\end{itemize}
Then  \eqref{e1}-\eqref{e3} has at least one solution on $J$.
\end{theorem}

\begin{proof}  We shall use Schaefer's fixed point theorem to prove
that $F$ has a fixed point. The proof will be given in several
steps.

\textbf{Step 1:}  $F$ is continuous.
 Let $\{y_{n}\}$ be a sequence such that $y_{n}\to y$ in
$PC(J,\mathbb{R})$. Then for each $t\in J$
\begin{align*}
|F(y_{n})(t)-F(y)(t)|
&\leq \frac{1}{\Gamma(\alpha)}\sum_{0<t_{k}<t}
 \int_{t_{k-1}}^{t_{k}}(t_{k}-s)^{\alpha-1}|f(s,y_n(s))-f(s,y(s))|ds \\
&\quad + \frac{1}{\Gamma(\alpha)}
\int_{t_{k}}^{t}(t-s)^{\alpha-1}|f(s,y_n(s))-f(s,y(s))|ds\\
&\quad +\sum_{0<t_{k}<t}|I_{k}(y_n(t_{k}^{-}))-I_{k}(y(t_{k}^{-}))|.
\end{align*}
Since  $f$ and $I_{k}$, $k=1,\dots,m$ are continuous functions, we
have
$$
 \|F(y_{n})-F(y)\|_{\infty}  \to 0 \quad\text{as } n\to\infty.
$$

\textbf{Step 2:} $F$ maps bounded sets into bounded sets in
$PC(J,\mathbb{R})$.   Indeed, it is enough to show that
for any $\eta^*>0$, there exists a positive constant $\ell$ such
that for each $y\in B_{\eta^*}=\{y\in PC(J,\mathbb{R}):
\|y\|_{\infty}\leq \eta^* \}$, we have $\|F(y)\|_{\infty}\leq \ell$.
By (H4) and (H5) we have for each $t\in J$,
\begin{align*}
|F(y)(t)|&\leq  |y_{0}|+\frac{1}{\Gamma(\alpha)}\sum_{0<t_{k}<t}
 \int_{t_{k-1}}^{t_{k}}(t_{k}-s)^{\alpha-1}|f(s,y(s))|ds\\
&\quad + \frac{1}{\Gamma(\alpha)}
  \int_{t_{k}}^{t}(t-s)^{\alpha-1}|f(s,y(s))|ds
 +\sum_{0<t_{k}<t}|I_{k}(y(t_{k}^{-}))|\\
&\leq |y_{0}|+\frac{mMT^{\alpha}}{\Gamma(\alpha+1)}
 +\frac{MT^{\alpha}}{\Gamma(\alpha+1)}+mM^*.
\end{align*}
Thus
 $$
\|F(y)\|_{\infty} \leq
|y_{0}|+\frac{MT^{\alpha}(m+1)}{\Gamma(\alpha+1)}+mM^*:=\ell.
 $$

 \textbf{Step 3:} $F$ maps bounded sets into
equicontinuous sets of $PC(J,\mathbb{R})$.
  Let $\tau_{1}, \tau_{2}\in J$, $\tau_{1}<\tau_{2}$,
$B_{\eta^*}$ be a bounded set of $PC(J,\mathbb{R})$ as in Step  2,
 and let $y\in B_{\eta^*}$. Then
\begin{align*}
&|F(y)(\tau_{2})-F(y)(\tau_{1})|\\
&=\frac{1}{\Gamma(\alpha)}\int_0^{\tau_1}|(\tau_2-s)^{
 \alpha-1}-(\tau_1-s)^{\alpha-1}||f(s,y(s))|ds\\
&\quad + \frac{1}{\Gamma(\alpha)}\int_{\tau_1}^{\tau_{2}}|(\tau_2-s)^{
 \alpha-1}||f(s,y(s))|ds
+\sum_{0<t_{k}<\tau_{2}-\tau_{1}}|I_{k}(y(t_{k}^{-}))|\\
&\leq \frac{M}{\Gamma(\alpha+1)}[2(\tau_2-\tau_1)^{\alpha}
 +\tau_2^{\alpha}-\tau_1^{\alpha}]
 + \sum_{0<t_{k}<\tau_{2}-\tau_{1}}|I_{k}(y(t_{k}^{-}))|.
\end{align*}
 As $\tau_{1}\to \tau_{2}$, the right-hand side of the
above inequality tends to zero. As a consequence of Steps 1 to 3
together with the Arzel\'a-Ascoli theorem, we can conclude that
$F:PC(J,\mathbb{R})\to PC(J,\mathbb{R})$ is
  completely continuous.

\textbf{Step 4:}  A priori bounds.  Now it remains to show that the
set
$$
\mathcal{E}=\{y \in PC(J,\mathbb{R}): y=\lambda F(y) \ \
\text{for some} \ \ 0<\lambda<1 \}
$$
is bounded.
Let $y\in \mathcal{E}$, then $ y=\lambda F(y)$ for some
$0<\lambda<1$. Thus, for each $t\in J$ we have
\begin{align*}
 y(t)&= \lambda y_{0}+\frac{\lambda}{\Gamma(\alpha)}\sum_{0<t_{k}<t}\int_{t_{k-1}}^{t_{k}}(t_{k}-s)^{\alpha-1}f(s,y(s))ds\\
&\quad + \frac{\lambda}{\Gamma(\alpha)}
\int_{t_{k}}^{t}(t-s)^{\alpha-1}f(s,y(s))ds
 +\lambda\sum_{0<t_{k}<t}I_{k}(y(t_{k}^{-})).
\end{align*}
 This implies by (H4) and (H5) (as in Step 2) that for each
$t\in J$ we have
$$
 |y(t)|\leq |y_{0}|+\frac{mMT^{\alpha}}{\Gamma(\alpha+1)}
 +\frac{MT^{\alpha}}{\Gamma(\alpha+1)}+mM^*.
$$
 Thus for every $t\in J$, we have
$$
\|y\|_{\infty} \leq
|y_{0}|+\frac{mMT^{\alpha}}{\Gamma(\alpha+1)}
 +\frac{MT^{\alpha}}{\Gamma(\alpha+1)}+mM^*:=R.
$$
 This shows that the set $\mathcal{E}$ is
bounded. As a consequence of Schaefer's fixed point theorem, we
deduce that $F$ has a fixed point which is a solution of the problem
\eqref{e1}-\eqref{e3}.
\end{proof}

In the following theorem we give an existence result for the problem
\eqref{e1}-\eqref{e3} by  applying the nonlinear
alternative of Leray-Schauder type and which the conditions (H4) and
(H5) are weakened.

\begin{theorem}\label{thm3}
Assume that {\rm (H2)} and the following
conditions hold:
\begin{itemize}
\item[(H6)] There exists $\phi_{f}\in C(J,\mathbb{R}^{+})$
and $\psi:[0,\infty)\to (0,\infty)$ continuous and nondecreasing
such that $$
 |f(t,u)|\leq \phi_{f}(t)\psi(|u|) \quad \text{for all }   t\in J, \;
  u\in \mathbb{R}.
$$

\item[(H7)] There exists $\psi^*:[0,\infty)\to (0,\infty)$ continuous
 and nondecreasing such that
$$
 |I_{k}(u)|\leq \psi^*(|u|) \quad \text{for all}   u\in \mathbb{R}.
$$

\item[(H8)] There exists an number $\overline M>0$ such that
$$
 \frac{\overline M}{|y_{0}|+\psi(\overline M)
 \frac{mT^{\alpha}\phi_{f}^{0}}{\Gamma(\alpha+1)}
+ \psi(\overline M)\frac{T^{\alpha}\phi_{f}^{0}}{\Gamma(\alpha+1)}
+ m\psi^*(\overline M)}>1,
 $$
where $\phi_{f}^{0}=\sup\{\phi_{f}(t): \ t\in J\}$.

\end{itemize}
Then  \eqref{e1}-\eqref{e3} has at least one solution on $J$.
\end{theorem}

\begin{proof}
 Consider the operator $F$ defined in Theorems \ref{thm1}
and \ref{thm2}. It can be easily shown that $F$ is continuous and
completely continuous. For $\lambda\in [0,1]$, let $y$ be such that
for each $t\in J$ we have $y(t)=\lambda (Fy)(t)$. Then from
(H6)-(H7) we have for each $t\in J$,
\begin{align*}
|y(t)|&\leq  |y_{0}|+\frac{1}{\Gamma(\alpha)}\sum_{0<t_{k}<t}\int_{t_{k-1}}^{t_{k}}(t_{k}-s)^{\alpha-1}\phi_{f}(s)\psi(|y(s)|)ds\\
&\quad + \frac{1}{\Gamma(\alpha)}
\int_{t_{k}}^{t}(t-s)^{\alpha-1}\phi_{f}(s)\psi(|y(s)|)ds
 + \sum_{0<t_{k}<t}\psi^*(|y(s)|)\\
&\leq  |y_{0}|+\psi(\|y\|_{\infty})\frac{mT^{\alpha}\phi_{f}^{0}}{\Gamma(\alpha+1)}
+ \psi(\|y\|_{\infty})\frac{T^{\alpha}\phi_{f}^{0}}{\Gamma(\alpha+1)}
 +m\psi^*(\|y\|_{\infty}).
  \end{align*}
 Thus
 $$
\frac{\|y\|_{\infty}}{|y_{0}|+\psi(\|y\|_{\infty})\frac{mT^{\alpha}\phi_{f}^{0}}{\Gamma(\alpha+1)}
+ \psi(\|y\|_{\infty})\frac{T^{\alpha}\phi_{f}^{0}}{\Gamma(\alpha+1)}
+m\psi^*(\|y\|_{\infty})}\leq 1.
$$
 Then by condition (H8), there exists  $\overline M$ such that
 $\|y\|_{\infty} \neq \overline M$.
Let
$$ U=\{y\in PC(J,\mathbb{R}): \|y\|_{\infty}<\overline M\}.
$$
The operator $F: \overline U\to PC(J,\mathbb{R})$ is continuous and completely
continuous. From the choice of $U$, there is no $y\in \partial U$
such that $y=\lambda F(y)$ for some $\lambda\in(0,1)$. As a
consequence of the nonlinear alternative of Leray-Schauder type
\cite{GrDu}, we deduce that $F$
 has a fixed point $y$ in $\overline U$ which is a solution
of the problem \eqref{e1}--\eqref{e3}.
This completes the proof.
\end{proof}

\section{Nonlocal impulsive differential equations}

This section is concerned with a generalization of the results
presented in the previous section to nonlocal impulsive fractional
differential equations. More precisely we shall present some
existence and uniqueness results for the following nonlocal problem
\begin{gather}\label{ne1}
^{c}D^{\alpha}y(t)= f(t,y), \quad \text{for each } t\in J=[0, T], \;
t\neq t_{k},  \\
\label{ne2}
\Delta y\big|_{t=t_{k}}= I_{k}(y(t_{k}^{-})), \\
\label{ne3}
y(0)+g(y)= y_{0},
\end{gather}
where $k=1,\dots,m$, $0<\alpha\leq 1$, $f, I_{k}$,  are as in Section 3 and
$g:PC(J,\mathbb{R})\to\mathbb{R}$ is a continuous function. Nonlocal conditions were
initiated by Byszewski \cite{ByLa} when he proved the existence and
uniqueness of mild and classical solutions of nonlocal Cauchy
problems. As remarked by Byszewski \cite{By1,By2}, the nonlocal
condition can be more useful than the standard initial condition to
describe some physical phenomena. For example, $g(y)$ may be given
by
$$
g(y)=\sum_{i=1}^{p}c_{i}y(\tau_{i})
$$
where $c_{i}$, $i=1,\dots,p$, are given constants and
$0<\tau_{1}<\dots <\tau_{p}\leq T$. Let us introduce the
following set of conditions.
\begin{itemize}

\item[(H9)] There exists a constant $ M^{**} >0$ such
that $ |g(u)|\leq M^{**}$ for each  $u\in PC(J,\mathbb{R})$.

\item[(H10)] There exists a constant $k>0$ such that
$|g(u)-g(\overline u)|\leq l^{**} |u-\overline u|$ for
each $u, \overline u \in PC(J,\mathbb{R})$.

\item[(H11)] There exists $\psi^{**}:[0,\infty)\to (0,\infty)$
continuous and nondecreasing
such that $ |g(u)|\leq \psi^{**}(|u|)$  for each
$u\in PC(J,\mathbb{R})$.

\item[(H12)] There exists an number $\overline M^{*}>0$ such that
$$
\frac{\overline M^*}{|y_{0}|+\psi^{**}(\overline
M^{*})+\psi(\overline
M^{*})\frac{mT^{\alpha}\phi_{f}^{0}}{\Gamma(\alpha+1)} +
\psi(\overline M^{*})\frac{T^{\alpha}\phi_{f}^{0}}{\Gamma(\alpha+1)}
+m\psi^*(\overline M^{*})}>1,
$$
\end{itemize}

\begin{theorem}\label{nt1}
Assume that {\rm (H1), (H2), (H10)} hold.
 If
\begin{equation}\label{neq1}
\big[\frac{T^{\alpha}l(m+1)} {\Gamma(\alpha+1)}+ml^*+l^{**}\big]
< 1,
\end{equation}
 then the nonlocal problem \eqref{ne1}-\eqref{ne3} has a unique solution
on $J$.
\end{theorem}

\begin{proof}
We transform the problem (\ref{ne1})--(\ref{ne3}) into a
fixed point problem.
 Consider the operator
$\tilde F:PC(J,\mathbb{R})\to PC(J,\mathbb{R})$
defined by
\begin{align*}
 \tilde F(y)(t)
&= y_{0}-g(y)+\frac{1}{\Gamma(\alpha)}\sum_{0<t_{k}<t}\int_{t_{k-1}}^{t_{k}}(t_{k}-s)^{\alpha-1}f(s,y(s))ds\\
&\quad + \frac{1}{\Gamma(\alpha)}
\int_{t_{k}}^{t}(t-s)^{\alpha-1}f(s,y(s))ds
 +\sum_{0<t_{k}<t}I_{k}(y(t_{k}^{-})).
\end{align*}
 Clearly, the fixed points of the operator $\tilde F$ are solution of
the problem \eqref{ne1}-\eqref{ne3}. We can easily show the
$\tilde F$ is a contraction.
\end{proof}

\begin{theorem}\label{nt2}
Assume that {\rm (H3)-(H5), (H9)} hold. Then the nonlocal problem
 \eqref{ne1}-\eqref{ne3} has at least one solution on $J$.
\end{theorem}

\begin{theorem}\label{nt3}
Assume that {\rm (H6)-(H7), (H11)-(H12)} hold. Then the nonlocal
problem \eqref{ne1}-\eqref{ne3} has at least one solution on $J$.
\end{theorem}

\section{An Example}

In this section we give an example to illustrate the usefulness of
our main results. Let us consider the  impulsive fractional
initial-value problem,
\begin{gather}\label{ex1}
^{c}D^{\alpha}y(t)=\frac{e^{-t}|y(t)|}{(9+e^{t})(1+|y(t)|)}, \quad
t\in J:=[0,1], \; t\neq \frac{1}{2},  \; 0<\alpha\leq 1, \\
 \label{ex2}
\Delta y|_{t=\frac{1}{2}}=
\frac{|y(\frac{1}{2}^{-})|}{3+|y(\frac{1}{2}^{-})|}, \\
 \label{ex3}
y(0)=0.
\end{gather}
 Set
$$
f(t,x)=\frac{e^{-t}x}{(9+e^{t})(1+x)}, \quad (t,x)\in J\times [0,\infty),
$$
and
$$
I_{k}(x)=\frac{x}{3+x}, \ \ x\in [0,\infty).
$$
 Let $x, y\in [0,\infty)$ and $t\in J$. Then we
have
\begin{align*}
|f(t,x)-f(t,y)|
&=\frac{e^{-t}}{(9+e^{t})}\Bigl|\frac{x}{1+x}-\frac{y}{1+y}\Bigr|\\
&=\frac{e^{-t}|x-y|}{(9+e^{t})(1+x)(1+y)}\\
&\leq \frac{e^{-t}}{(9+e^{t})}|x-y|\\
&\leq \frac{1}{10}|x-y|.
\end{align*}
Hence the condition $(H1)$ holds with $l=1/10$.
Let $x, y\in [0,\infty)$. Then we have
\[
|I_{k}(x)-I_{k}(y)|=\bigl|\frac{x}{3+x}-\frac{y}{3+y}\bigr|
=\frac{3|x-y|}{(3+x)(3+y)}
\leq \frac{1}{3}|x-y|.
\]
Hence the condition $(H2)$ holds with $l^*=1/3$.
  We shall check that condition
(\ref{eq1}) is satisfied with $T=1$ and $m=1$. Indeed
\begin{equation}\label{ex4}
\big[\frac{T^{\alpha}l(m+1)} {\Gamma(\alpha+1)}+ml^*\big] < 1
\Longleftrightarrow \Gamma(\alpha+1)>\frac{3}{10},
\end{equation}
 which is satisfied for some $\alpha\in (0,1]$.
 Then by Theorem \ref{thm1} the problem \eqref{ex1}-\eqref{ex3} has
a unique solution on $[0,1]$ for values of $\alpha$ satisfying
(\ref{ex4}).

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\end{document}