\documentclass[reqno]{amsart} \usepackage{hyperref} \AtBeginDocument{{\noindent\small \emph{Electronic Journal of Differential Equations}, Vol. 2012 (2012), No. 65, pp. 1--8.\newline ISSN: 1072-6691. URL: http://ejde.math.txstate.edu or http://ejde.math.unt.edu \newline ftp ejde.math.txstate.edu} \thanks{\copyright 2012 Texas State University - San Marcos.} \vspace{9mm}} \begin{document} \title[\hfilneg EJDE-2012/65\hfil Integral equations of fractional order] {Integral equations of fractional order with multiple time delays in Banach spaces} \author[M. Benchohra, D. Seba\hfil EJDE-2012/65\hfilneg] {Mouffak Benchohra, Djamila Seba} \address{Mouffak Benchohra \newline Laboratoire de Math\'ematiques, Universit\'e de Sidi Bel-Abb\es\\ BP 89, 22000 Sidi Bel-Abb\es, Alg\'erie} \email{benchohra@univ-sba.dz} \address{Djamila Seba \newline D\'epartement de Math\'ematiques, Universit\'e de Boumerd\es\\ Avenue de l'ind\'ependance, 35000 Boumerd\es, Alg\'erie} \email{djam\_seba@yahoo.fr} \thanks{Submitted March 7, 2012. Published April 27, 2012.} \subjclass[2000]{26A33, 45N05} \keywords{Integral equation; left sided mixed Riemann Liouville integral; \hfill\break\indent measure of noncompactness; fixed point; Banach space} \begin{abstract} In this article, we give sufficient conditions for the existence of solutions for an integral equation of fractional order with multiple time delays in Banach spaces. Our main tool is a fixed point theorem of M\"onch type associated with measures of noncompactness. Our results are illustrated by an example. \end{abstract} \maketitle \numberwithin{equation}{section} \newtheorem{theorem}{Theorem}[section] \newtheorem{lemma}[theorem]{Lemma} \newtheorem{definition}[theorem]{Definition} \allowdisplaybreaks \section{Introduction} Fractional differential and integral equations play an important role in characterizing many chemical, physical, viscoelasticity, control and engineering problems. For more details, see \cite{BDST, Die, Hi, Ma, Tar}, and references therein. In consequence, the subject of fractional differential and integral equations is gaining much importance and attention; see, for instance, the monograph of Abbas \emph{et al.} \cite{ABN}, Kilbas \emph{et al.} \cite{KST}, and the papers of Abbas and Benchohra \cite{AbBe1}, Agarwal \emph{et al.} \cite{ABH1}, Bana\`{s} and Zaj\c{a}c \cite{BaZa}, Benchohra and Seba \cite{BeSe, BeSe1}, Vityuk and Golushkov \cite{ViGo} and the references therein. Ibrahim and Jalab \cite{IbJa} studied the existence of solutions of the fractional integral inclusion $$u(t)-\sum_{i=1}^{m}b_i(t)u(t-\tau_i)\in I^{\alpha}F(t,u(t)),\quad t\in [0,T],$$ where $\tau_i0$, $\theta=(0,0)$, $\xi_i,\mu_i\geq 0$; $i=1,\dots,m$, $\xi:=\max_{i=1,\dots,m}\{\xi_i\}$, $\mu:=\max_{i=1,\dots,m}\{\mu_i\}$, $f:J\times {E} \to {E}$ is a function satisfying some assumptions specified later, $I_{\theta}^{r}$ is the left-sided mixed Riemann-Liouville integral of order $r=(r_1,r_2)\in(0,\infty)\times(0,\infty)$, $g_i:J\to{E}$; $i=1,\dots m$, are continuous functions, $\Phi:\tilde J\to E$ is a continuous function such that \begin{gather*} \Phi(x,0)=\sum_{i=1}^{m}g_i(x,0)\Phi(x-\xi_i,-\mu_i),\quad x\in [0,a], \\ \Phi(0,y)=\sum_{i=1}^{m}g_i(0,y)\Phi(-\xi_i,y-\mu_i), \quad y\in[0,b], \end{gather*} and $E$ is a real Banach space with norm $\|\cdot\|$. Using properties of the Kuratowski measure of noncompactness and a fixed point theorem of M\"onch type, we prove the existence of solutions to \eqref{e1}-\eqref{e1'}. Let us note here that the technique of measures of noncompactness is a very important tool for finding solutions for differential and integral equations; for more details see \cite{AgBeSe, BeSe, BeSe1} and references therein. \section{Preliminaries} In this section, we collect a few auxiliary results which will be needed in the sequel. By $C(J, E)$ we denote the Banach space of continuous functions $u:J\to E$, with the norm $$\|u\|_{\infty}=\sup_{(x,y)\in J}\|u(x,y)\|.$$ Let $L^1(J,E)$ be the space of Lebesgue integrable functions $u: J \to E$ with the norm $$\| u\|_{L^1}=\int_{0}^{a}\int_{0}^{b}\|u(x,y)\|dxdy.$$ Let $C([-\xi,a]\times[-\mu,b],E)$ be a Banach space endowed with the norm $$\|u\|_{C}=\sup_{(x,y)\in [-\xi,a]\times[-\mu,b]}\|u(x,y)\|.$$ \begin{definition}[\cite{ViGo}] \rm Let $r=(r_1,r_2)\in (0,\infty)\times(0,\infty)$, $\theta=(0,0)$ and $u\in L^{1}(J, E).$ The left-sided mixed Riemann-Liouville integral of order $r$ of $u$ is defined by $$(I_{\theta}^{r}u)(x,y)=\frac{1}{\Gamma (r_{1})\Gamma (r_{2})}\int_{0}^{x}% \int_{0}^{y}(x-s)^{r_{1}-1}(y-t)^{r_{2}-1}u(s,t)dtds.$$ \end{definition} In particular, $$( I_{\theta}^{\theta}u)(x,y)=u(x,y), \ ( I_{\theta}^{\sigma}u)(x,y) =\int_{0}^{x}\int_{0}^{y}u(s,t)dtds;$$ for almost all $(x,y)\in J$, where $\sigma=(1,1)$. For instance, $I_{\theta}^{r}u$ exists for all $r_1,r_2\in(0,\infty)$, when $u\in L^{1}(J,E).$ Note also that when $u\in C(J,E)$, then $(I_{\theta}^{r}u)\in C(J,E)$, moreover $$(I_{\theta}^{r}u)(x,0)=(I_{\theta}^{r}u)(0,y)=0, \quad x\in [0,a], \; y\in [0,b].$$ Now we recall some fundamental facts of the notion of Kuratowski measure of noncompactness. \begin{definition}[\cite{AkKaPaRoSa, BaGo}] \rm Let $F$ be a Banach space and let $\Omega_{F}$ be the family of bounded subsets of $F$. The Kuratowski measure of noncompactness is the map $\alpha:\Omega_{F}\to [0,\infty]$ defined by $$\alpha(B)=\inf\{\epsilon>0: B\subseteq\cup_{i=1}^{n}B_{i} \text{ and } \operatorname{diam}(B_{i})\leq\epsilon\}, \quad \text{here } B\in \Omega_{E}.$$ \end{definition} The Kuratowski measure of noncompactness satisfies the following properties (For more details see \cite{AkKaPaRoSa,BaGo}). \begin{itemize} \item[(a)] $\alpha(B)= 0 \Leftrightarrow \overline{B} \;$ is compact ($B$ is relatively compact). \item[(b)]$\alpha(B)=\alpha(\overline{B})$. \item[(c)]$A \subset B \Rightarrow\alpha(A)\leq\alpha(B)$. \item[(d)]$\alpha(A+B)\leq\alpha(A)+\alpha(B)$ \item[(e)]$\alpha(cB)=|c|\alpha(B);\,c\in\mathbb{R}$. \item[(f)]$\alpha(\operatorname{conv} B)=\alpha(B)$. \end{itemize} For our purpose we will need the following auxiliary results. \begin{theorem}[\cite{Mon}] \label{thm1} Let $D$ be a bounded, closed and convex subset of a Banach space such that $0\in{D}$, and let $N$ be a continuous mapping of $D$ into itself. If the implication $$V=\overline{\operatorname{conv}} N(V) \quad \text{or} \quad V=N(V)\cup\{0\}\Rightarrow\alpha(V)=0$$ holds for every subset $V$ of $D$, then $N$ has a fixed point. \end{theorem} \begin{lemma}[\cite{GuLaLi}] \label{lem3} Let $V\subset C(J,E)$ be bounded and equicontinuous on $J$. Then the map $(s,t)\mapsto \alpha(V(s,t))$ is continuous on $J$ and $$\alpha\Big(\int_{J}V(s,t)\,ds\,dt\Big)\leq \int_{J}\alpha(V(s,t))\,ds\,dt,$$ where $V(s,t)=\{u(s,t): u\in V\}$. \end{lemma} \section{Main Results} \begin{definition} \rm A function $u\in C(J, E)$ is said to be a solution of \eqref{e1}-\eqref{e1'} if $u$ satisfies equation \eqref{e1} on $J$ and condition \eqref{e1'}. \end{definition} Set $$B=\max_{i=1,\dots m}\big\{\sup_{(x,y)\in J}\|g_i(x,y)\|\big\}.$$ Let us impose two conditions for convenience. \begin{itemize} \item[(H1)] $f: J\times E\to E$ is a continuous map. \item[(H2)] There exists $p\in C(J, \mathbb{R}_+)$, such that $$\|f(x,y, u)\|\leq p(x,y)\|u\|, \quad \text{for (x,y)\in {J} and each } u\in{E}.$$ \end{itemize} Let $p^* = \|p\|_{\infty}$. The main result in this paper reads as follows. \begin{theorem} \label{thm2} Assume that assumptions {\rm (H1)} and {\rm (H2)} hold. If $$\label{e5} m B+\frac{p^*a^{r_1}b^{r_2}}{\Gamma (r_1+1)\Gamma (r_2+1)}<1$$ then the problem \eqref{e1}-\eqref{e1'} has at least one solution. \end{theorem} \begin{proof} Transform the problem \eqref{e1}-\eqref{e1'} into a fixed point problem. Consider the operator $N:C(J,E)\to C(J, E)$ defined by $$\label{e2} N(u)(x,y)=\sum_{i=1}^{m}g_i(x,y)u(x-\xi_i,y-\mu_i)+I_{\theta}^{r}f(x,y,u(x,y)).$$ Since $f$ is continuous, the operator $N$ is well defined; i.e., $N$ maps $C(J,E)$ into itself. The problem of finding the solutions of equation \eqref{e1}-\eqref{e1'} is reduced to finding the solutions of the operator equation $N(u)=u.$ Let $R>0$ and consider the set $$D_R=\{u\in C(J,E):\|u\|_{\infty}\leq R\}.$$ It is clear that $D_R$ is a closed bounded and convex subset of $C(J, E).$ We shall show that $N$ satisfies the assumptions of Theorem \ref{thm1}. The proof will be given in three steps. \end{proof} \noindent\textbf{Step 1:} $N$ is continuous. Let $\{u_n\}$ be a sequence such that $u_n\to u$ in ${C(J, E)}$, then for each $(x,y)\in J$, \begin{align*} &\|N(u_{n})(x,y)-N(u)(x,y)\|\\ &\leq \frac{1}{\Gamma (r_{1})\Gamma (r_{2})}\int_{0}^{x} \int_{0}^{y}(x-s)^{r_{1}-1}(y-t)^{r_{2}-1}\|f(s,t,u_n)-f(s,t,u)\|\,ds\,dt. \end{align*} Let $\rho>0$ be such that $$\|u_{n}\|_{\infty}\leq \rho, \ \|u\|_{\infty}\leq \rho.$$ By (H2) we have $$(x-s)^{r_{1}-1}(y-t)^{r_{2}-1}\|f(s,t,u_n)-f(s,t,u)\|\leq 2\rho p^{*}(x-s)^{r_{1}-1}(y-t)^{r_{2}-1}$$ which belongs to $L^{1}(J,\mathbb{R}_+)$. Since $f$ is continuous, then by the Lebesgue dominated convergence theorem we have $$\| N(u_{n})-N(u)\|_{\infty }\to{0} \text{ as } n\to\infty.$$ \noindent\textbf{Step 2:} $N$ maps $D_R$ into itself. For each $u\in D_R$, by (H2) and \eqref{e5} we have for each $(x,y)\in J$, \begin{align*} &\|N(u)(x,y)\|\\ &\leq \sum_{i=1}^{m}\|g_i(x,y)\|\|u(x-\xi_i,y-\mu_i)\|\\ &\quad +\frac{1}{\Gamma (r_{1})\Gamma (r_{2})} \int_{0}^{x}\int_{0}^{y}(x-s)^{r_{1}-1}(y-t)^{r_{2}-1}\|f(s,t,u(s,t))\|\,ds\,dt\\ &\leq mB\|u\|_{\infty}+\frac{1}{\Gamma (r_{1})\Gamma (r_{2})} \int_{0}^{x}\int_{0}^{y}(x-s)^{r_{1}-1}(y-t)^{r_{2}-1}p(s,t)\|u\|_{\infty}\,ds\,dt\\ &\leq m B R+\frac{p^*R}{\Gamma (r_{1})\Gamma (r_{2})} \int_{0}^{x}\int_{0}^{y}(x-s)^{r_{1}-1}(y-t)^{r_{2}-1}\,ds\,dt\\ &\leq m B R+\frac{p^*R\, a^{r_{1}}b^{r_{2}}} {\Gamma (r_{1}+1)\Gamma (r_{2}+1)} < R. \end{align*} \noindent\textbf{Step 3:} $N(D_R)$ is bounded and equicontinuous. By Step 2 we have $N(D_R)=\{ N(u): u\in D_R\}\subset D_R$. Thus, for each $u\in D_R$ we have $\|N(u)\|_{\infty}\leq R$ which means that $N(D_R)$ is bounded. For the equicontinuity of $N(D_R)$, let $(x_1,y_1),(x_2,y_2)\in J$, \$x_1