\documentclass[reqno]{amsart}
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\AtBeginDocument{{\noindent\small
\emph{Electronic Journal of Differential Equations},
Vol. 2009(2009), No. 136, 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}
\thanks{\copyright 2009 Texas State University - San Marcos.}
\vspace{9mm}}
\begin{document}
\title[\hfilneg EJDE-2009/136\hfil Existence of solutions]
{Existence of local and global solutions to some impulsive
fractional differential equations}
\author[R. Atmania, S. Mazouzi\hfil EJDE-2009/136\hfilneg]
{Rahima Atmania, Said Mazouzi} % in alphabetical order
\address{Rahima Atmania \newline
Laboratory of Applied Mathematics (LMA) \\
Department of Mathematics, University of Badji Mokhtar Annaba \\
P.O. Box 12, Annaba 23000, Algeria}
\email{atmanira@yahoo.fr}
\address{Said Mazouzi \newline
Laboratory of Applied Mathematics (LMA) \\
Department of Mathematics, University of Badji Mokhtar Annaba \\
P.O. Box 12, Annaba 23000, Algeria}
\email{mazouzi\_sa@yahoo.fr}
\thanks{Submitted July 3, 2009. Published October 21, 2009.}
\thanks{Supported by the LMA, University of Badji Mokhtar
Annaba, Algeria}
\subjclass[2000]{26A33, 34A12, 34A37}
\keywords{Fractional derivative; impulsive conditions; fixed point;
\hfill\break\indent
local solution; global solution}
\begin{abstract}
First, by using Schauder's fixed-point theorem we establish the
existence uniqueness of locals for some fractional differential
equation with a finite number of impulses. On the other hand,
by using Brouwer's fixed-point theorem, we establish existence of
the global solutions under suitable assumptions.
\end{abstract}
\maketitle
\numberwithin{equation}{section}
\newtheorem{theorem}{Theorem}[section]
\newtheorem{lemma}[theorem]{Lemma}
\newtheorem{example}[theorem]{Example}
\section{Introduction}
The concept of fractional calculus can be considered as a
generalization of ordinary differentiation and integration to
arbitrary (non-integer) order. However, great efforts must be done
before the ordinary derivatives could be truly interpreted as a
special case of the fractional derivatives. For more details, we
refer to the books by Oldham and Spanier \cite{o1} and by
Miller and Ross \cite{m1}.
Actually, fractional derivatives have been extensively applied in
many fields, for example in Probability, Viscoelasticity,
Electronics, Economics, Mechanics as well as Biology.
Some results on quantitative and qualitative theory of some
fractional differential equations are obtained, we may cite the
references \cite{d1,l1,m1,o1,y1}. On the other hand, the theory of
impulsive differential equations is also an important area of
research which has been investigated in the last few years by
great number of mathematicians. We recall that the impulsive
differential equations may better model phenomena and dynamical
processes subject to a great changes in short times issued, for
instance, in Physics, Biotechnology, Automatics and Robotics. To
learn more about the most recent used techniques for this kind of
problems we refer to the book of Benchohra et al \cite{b1}.
So, we propose to study fractional differential equation subject
to a finite number of impulses. As we know there just few authors
have investigated this subject \cite{m2}. We have obtained some results
regarding local existence and uniqueness for some fractional
integrodifferential problem with a finite number of impulses. For
the existence and uniqueness of local solutions we use the
Schauder's fixed-point theorem, while we use Brouwer's fixed-point
theorem for the global solutions.
\section{Preliminaries}
Among the definitions of fractional derivatives we recall the
Riemann-Liouville definiton as follows.
\[
D^{\alpha }u(t) =\frac{1}{\Gamma (n-\alpha ) }\frac{
d^{n}}{dt^{n}}\int_{t_0}^t(t-s) ^{-\alpha +n-1}u(s) \,ds
\]
where $\Gamma (\cdot) $ is the well known gamma function and
$\alpha \in (n-1,n) $, with $n$ being an integer. One may observe
that the derivative of a constant is not at all equal to zero
which can cause serious problems in both views, theoretical and
practical. For this reason we prefer to use Caputo's definition
which gives better results than those of Riemann-Liouville. So we
define Caputo's derivative of order $\alpha \in (n-1,n) $ of a
function $u(t)$ by
\[
D^{\alpha }u(t) =\frac{1}{\Gamma (n-\alpha ) } \int_{t_0}^t(t-s)
^{-\alpha +n-1}\frac{ d^{n}}{ds^{n}}u(s) \,ds.
\]
Also, we use the fractional integral operator of order $\alpha >0$
given by
\[
D^{-\alpha }u(t) =\frac{1}{\Gamma (\alpha ) }
\int_{t_0}^ t (t-s) ^{\alpha -1}u( s) \,ds.
\]
We shall consider the fractional differential equation
\begin{equation}
D^{\alpha }u(t) =f(t,u(t) ) ;\quad t\in [ t_{0},t_{0}+\tau ],\;
t\neq t_{k},\; k=1,\dots ,m; \label{1}
\end{equation}
with the initial condition
\begin{equation}
D^{\alpha -1}u(t_{0}) =u_{0};\quad
( t-t_{0}) ^{1-\alpha }u(t) \big|_{t=t_{0}}
=\frac{ u_{0}}{\Gamma (\alpha ) };
\label{2}
\end{equation}
subject to the impulsive conditions
\begin{equation} \label{3}
\begin{gathered}
D^{\alpha -1}(u(t_{k}^{+}) -u(t_{k}^{-}) ) = I_{k}(t) ;\quad
t=t_{k},\; k=1,\dots ,m; \\
(t-t_{k}) ^{1-\alpha }u(t) \big|_{t=t_{k}}
=\frac{I_{k}(t_{k}) }{\Gamma (\alpha ) } ,\quad k=1,\dots ,m.
\end{gathered}
\end{equation}
We set the following assumptions
\begin{itemize}
\item[(A1)] $t>t_{0}\geq 0$, $\alpha $ is a real number
such that $0<\alpha \leq 1$, $u_{0}$ is a real constant vector of
$\mathbb{R}^{n}$ (the usual real $n$-dimensional Euclidean
space equipped with its Euclidean norm $\|.\|$);
\item[(A2)] $f(t,u) :I\times \mathbb{R}^{n}\to
\mathbb{R}^{n}$; $I_{k}(t) :I\to \mathbb{R}^{n}$,
$k=1,\dots ,m$, with $I=[t_{0},t_{0}+\tau ]$;
\item[(A3)] $t_{k}\in I$, $k=1,\dots ,m$ and
$t_{0}0\}$.
This is a Banach space with respect to the norm
\[
\|u\| _{\alpha }=\sup_{t\in I'}( t-t_{0})
^{\alpha +1} \prod_{i=1}^m (t-t_{i}) ^{\alpha +1}\|u(t) \| ,
\]
where $I'=(t_{0},t_{0}+\tau ]\backslash \{ t_{k}\}_{k=1,2,\dots }$.
We begin with the following Lemma.
\begin{lemma}\label{lem1}
If $f$ and $I_{k}$, $k=1,\dots m$ are continuous functions, then
$u(t) $ is a solution to problem \eqref{1}-\eqref{3}
in $\mathcal{PC}_{1-\alpha }([t_{0},t_{0}+\tau
] ,\mathbb{R}^{n}) $ if and only if $u(t) $ satisfies the
integrodifferential equation
\begin{equation} \label{4}
\begin{aligned}
u(t) &=\frac{u_{0}}{\Gamma (\alpha ) }( t-t_{0}) ^{\alpha
-1}+\frac{1}{\Gamma (\alpha ) }
\int_{t_0}^t (t-s) ^{\alpha -1}f( s,u(s) )\,ds \\
&\quad +\frac{1}{\Gamma (\alpha ) }\sum_{t_{0}0$ and the properties of
derivative of order $\alpha >0$, and then applying $D^{-1}$ to
\eqref{1} we obtain
\begin{align*}
D^{-1}(D^{\alpha }u(t) )
&=\int_{t_0}^t f(s,u(s) ) \,ds
=\int_{t_0}^t \frac{d}{dt}D^{-(1-\alpha )}u(s)\,ds \\
&= D^{\alpha -1}u(t) -u_{0}-\sum_{t_{0}