\documentclass[reqno]{amsart} \usepackage{hyperref} \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 $$D^{\alpha }u(t) =f(t,u(t) ) ;\quad t\in [ t_{0},t_{0}+\tau ],\; t\neq t_{k},\; k=1,\dots ,m; \label{1}$$ with the initial condition $$D^{\alpha -1}u(t_{0}) =u_{0};\quad ( t-t_{0}) ^{1-\alpha }u(t) \big|_{t=t_{0}} =\frac{ u_{0}}{\Gamma (\alpha ) }; \label{2}$$ subject to the impulsive conditions $$\label{3} \begin{gathered} D^{\alpha -1}(u(t_{k}^{+}) -u(t_{k}^{-}) ) = I_{k}(t) ;\quad t=t_{k},\; k=1,\dots ,m; \\ (t-t_{k}) ^{1-\alpha }u(t) \big|_{t=t_{k}} =\frac{I_{k}(t_{k}) }{\Gamma (\alpha ) } ,\quad k=1,\dots ,m. \end{gathered}$$ 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 \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}