\documentclass[reqno]{amsart} \usepackage{hyperref} \AtBeginDocument{{\noindent\small \emph{Electronic Journal of Differential Equations}, Vol. 2009(2009), No. 155, pp. 1--7.\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-2009/155\hfil Existence and uniqueness of solutions] {Existence and uniqueness of solutions to fractional semilinear mixed Volterra-Fredholm integrodifferential equations with nonlocal conditions} \author[M. M. Matar\hfil EJDE-2009/155\hfilneg] {Mohammed M. Matar} \address{Mohammed M. Matar \newline Department of Mathematics, Al-Azhar University of Gaza, P. O. Box 1277, Gaza, Palestine} \email{mohammed\_mattar@hotmail.com} \thanks{Submitted September 12, 2009. Published December 1, 2009.} \subjclass[2000]{45J05, 26A33, 34A12} \keywords{Fractional integrodifferential equations; mild solution; \hfill\break\indent nonlocal condition; Banach fixed point} \begin{abstract} In this article we study the fractional semilinear mixed Volterra-Fredholm integrodifferential equation $$\frac{d^{\alpha }x(t)}{dt^{\alpha }} =Ax(t)+f\Big(t,x(t), \int_{t_0}^tk(t,s,x(s))ds,\int_{t_0}^{T}h(t,s,x(s))ds\Big) ,$$ where $t\in [t_0,T]$, $t_0\geq 0$, $0<\alpha <1$, and $f$ is a given function. We prove the existence and uniqueness of solutions to this equation, with a nonlocal condition. \end{abstract} \maketitle \numberwithin{equation}{section} \newtheorem{theorem}{Theorem}[section] \newtheorem{lemma}[theorem]{Lemma} \newtheorem{definition}[theorem]{Definition} \allowdisplaybreaks \section{Introduction} The problem of existence and uniqueness of solution of fractional differential equations have been considered by many authors; see for example \cite{Del,Yu,Ng,Bal,Fur,Mat,momani,Lak}). In particular, fractional differential equations with nonlocal conditions have been studied by N'Guerekata \cite{Ng}, Balachandran, and Park \cite{Bal}, Furati and Tatar \cite{Fur}, and by many others. In \cite{momani}, the authors investigated the existence for a semilinear fractional differential equation with kernels in the nonlinear function by using the Banach fixed point theorem. The nonlocal Cauchy problem is discussed by authors in \cite{Bal} using the fixed point concepts. Tidke \cite{Tid} studied the non-fractional mixed Volterra-Fredholm integrodifferential equations with nonlocal conditions using Leray-Schauder theorem. Motivated by these works, we study the existence of solutions for nonlocal fractional semilinear integrodifferential equations in Banach spaces by using fractional calculus and a Banach fixed point theorem. Consider the fractional semilinear integrodifferential equation $$\begin{gathered} \frac{d^{\alpha }x(t)}{dt^{\alpha }}=Ax(t)+f(t,x(t), \int_{t_0}^tk(t,s,x(s))ds,\int_{t_0}^{T}h(t,s,x(s))ds), \\ x(t_0)=x_0\in X. \end{gathered} \label{eq1}$$ where $t\in J=[t_0,T]$, $t_0\geq 0,0<\alpha <1$, $x\in Y=C(J,X)$ is a continuous function on $J$ with values in the Banach space $X$ and $\| x\| _{Y}=\max_{t\in J}\| x(t)\| _X$, and the nonlinear functions $f:J\times X\times X\times X\to X$, $k:D\times X\to X$, and $h:D_0\times X\to X$ are continuous. Here $D=\{ (t,s)\in \mathbb{R}^{2}:t_0\leq s\leq t\leq T\}$, and $D_0=J\times J$. The operator $\frac{d^{\alpha }}{ dt^{\alpha }}$ denotes the Caputo fractional derivative of order $\alpha$. For brevity let $Kx(t)=\int_{t_0}^tk(t,s,x(s))ds,\quad Hx(t)= \int_{t_0}^{T}h(t,s,x(s))ds.$ and we use the common norm $\| \cdot \|$. The paper is organized as follows. In section 2, some definitions, lemmas, and assumptions are introduced to be used in the sequel. Section 3 will involve the main results and proofs of existence problem of \eqref{eq1}, together with a nonlocal condition. \section{Preliminaries} In this section, present some definitions and lemmas to be used later. \begin{definition} \label{def1} \rm A real function $f(x)$, $x>0$, is said to be in the space $C_{\mu}$, $\mu \in\mathbb{R}$ if there exists a real number $p (>\mu)$, such that $f(x)=x^{p} f_1(x)$, where $f_1 (x) \in{C[0,\infty)}$, and it is said to be in the space $C_{\mu}^{n}$ if $f^{(n)}\in C_{\mu}$, $n\in\mathbb{N}$. \end{definition} \begin{definition} \label{def2} \rm A function $f\in C_{\mu}$, $\mu\geq-1$ is said to be fractional integrable of order $\alpha>0$ if $I^{\alpha}f(t)=\frac{1}{\Gamma\left( \alpha\right) } \int_{t_0}^t (t-s) ^{\alpha-1}f(s)ds<\infty,$ where $t_0\geq 0$; and if $\alpha=0$, then $I^{0}f(t)=f(t)$. \end{definition} Next, we introduce the Caputo fractional derivative. \begin{definition} \label{def3} \rm The fractional derivative in the Caputo sense is defined as $\frac{d^{\alpha}f(t)}{dt^{\alpha}}=I^{1-\alpha}\Big( \frac{df(t)} {dt}\Big) =\frac{1}{\Gamma(1-\alpha) } \int_{t_0}^t(t-s) ^{-\alpha}f'(s) ds$ for $0<\alpha\leq 1$, $t_0 \geq 0$, $f'\in C_{-1}$. \end{definition} The properties of the above operators can be found in \cite{Miller} and the general theory of fractional differential equations can be found in \cite{Podlubny}. Next we introduce the so-called Mild Solution'' for fractional integrodifferential equation \eqref{eq1} (see \cite[Definition 1.3]{momani}). \begin{definition} \label{def4} \rm A continuous solution $x(t)$ of the integral equation $$x(t)=T(t-t_0)x_0+\frac{1}{\Gamma(\alpha)} \int_{t_0}^t(t-s)^{\alpha-1}T(t-s)f(s,x(s),Kx(s),Hx(s))ds \label{eq3(mild)}$$ is called a mild solution of \eqref{eq1}. \end{definition} To proceed, we need the following assumptions: \begin{itemize} \item[(A1)] $T(\cdot)$ is a $C_0-$semigroup generated by the operator $A$ on $X$ which satisfies $M=\max_{t\in J}\| T(t)\|$. \item[(A2)] $f$ is a continuous function and there exist positive constants $L_1$, $L_2$, and $L$ such that $\| f(t,x_1,y_1,z_1)-f(t,x_2,y_2,z_2)\| \leq L_1(\| x_1-x_2\| +\| y_1-y_2\| +\| z_1-z_2\| )$ for all $x_1,y_1,z_1,x_2,y_2,z_2\in Y$, $L_2=\max_{t\in J}\| f(t,0,0,0)\|$, and\\ $L=\max\{L_1,L_2\}$. \item[(A3)] $k$ is a continuous function and there exist positive constants $N_1$, $N_2$, and $N$ such that $\| k(t,s,x_1)-k(t,s,x_2)\| \leq N_1\| x_1-x_2\|$ for all $x_1,x_2\in Y$, $N_2=\max_{(t,s)\in D}\| k(t,s,0)\|$, and $N=\max\{N_1,N_2\}$. \item[(A4)] $h$ is a continuous function and there exist positive constants $C_1$, $C_2$, and $C$ such that $\| h(t,s,x_1)-h(t,s,x_2)\| \leq C_1\| x_1-x_2\|$ for all $x_1,x_2\in Y$, $C_2=\max_{(t,s)\in D_0}\|h(t,s,0)\|$, and $C=\max\{C_1,C_2\}$. \end{itemize} \section{Existence of solutions} In this section, we prove the main results on the existence of solutions to \eqref{eq1}. Firstly, we obtain the following estimates. \begin{lemma} \label{estimates} If {\rm (A3), (A4)} are satisfied, then the estimates \begin{gather*} \| Kx(t)\| \leq(t-t_0)(N_1\|x\|+N_2)\\ \| Kx_1(t)-Kx_2(t)\| \leq N_1(t-t_0) \| x_1-x_2\| \end{gather*} and \begin{gather*} \| Hx(t)\| \leq(T-t_0)(C_1\|x\|+C_2)\\ \| Hx_1(t)-Hx_2(t)\| \leq C_1(T-t_0) \| x_1-x_2\| \end{gather*} are satisfied for any $t\in J$, and $x,x_1,x_2\in Y$. \end{lemma} \begin{proof} By (A3), we have \begin{align*} \| Kx(t)\| & \leq\int_{t_0}^t\|k(t,s,x(s))\| ds \\ & =\int_{t_0}^t\| k(t,s,x(s))-k(t,s,0)+k(t,s,0)\| ds \\ & \leq\int_{t_0}^t\| k(t,s,x(s))-k(t,s,0)\| ds+\int_{t_0}^t\| k(t,s,0)\| ds \\ & \leq N_1(t-t_0)\| x\| +N_2(t-t_0)\leq (T-t_0)(N_1\| x\| +N_2). \end{align*} On the other hand, \begin{align*} \| Kx_1(t)-Kx_2(t)\| & \leq\int_{t_0}^t\| k(t,s,x_1(s))-k(t,s,x_2(s))\| ds \\ & \leq N_1\int_{t_0}^t\| x_1(s)-x_2(s)\| ds \\ & \leq N_1\left( t-t_0\right) \| x_1-x_2\| . \end{align*} Similarly, for the other estimates, we use assumption (A4), to get $\| Hx(t)\| \leq\int_{t_0}^{T}\| h(t,s,x(s))\| ds \leq(T-t_0)(C_1\| x\| +C_2)$ and $\| Kx_1(t)-Kx_2(t)\| \leq C_1( T-t_0)\| x_1-x_2\| .$ \end{proof} The existence result for \eqref{eq1} and its proof is as follows. \begin{theorem} \label{existance} If {\rm (A1)-(A4)} are satisfied, and \[ q\Gamma(\alpha+1)\geq ML\Big( 1+C(T-t_0)+\frac{N}{\alpha+1}(T-t_0 )\Big) (T-t_0)^{\alpha},\quad 0