\documentclass[reqno]{amsart} \usepackage{hyperref} \AtBeginDocument{{\noindent\small \emph{Electronic Journal of Differential Equations}, Vol. 2011 (2011), No. 34, 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 2011 Texas State University - San Marcos.} \vspace{9mm}} \begin{document} \title[\hfilneg EJDE-2011/34\hfil Fractional differential equation] {Fractional differential equation with the fuzzy initial condition} \author[S. Arshad, V. Lupulescu\hfil EJDE-2011/34\hfilneg] {Sadia Arshad, Vasile Lupulescu} % in alphabetical order \address{Sadia Arshad \newline Government College University, Abdus Salam School of Mathematical Sciences, Lahore, Pakistan} \email{sadia\_735@yahoo.com} \address{Vasile Lupulescu \newline Constantin Brancusi'' University of Targu Jiu, Romania} \email{lupulescu\_v@yahoo.com} \thanks{Submitted September 23, 2010. Published February 23, 2011.} \subjclass[2000]{34A07, 34A12} \keywords{Fuzzy differential equation; fractional calculus; initial value problem} \begin{abstract} In this paper we study the existence and uniqueness of the solution for a class of fractional differential equation with fuzzy initial value. The fractional derivatives are considered in the Riemann-Liouville sense. \end{abstract} \maketitle \numberwithin{equation}{section} \newtheorem{theorem}{Theorem}[section] \newtheorem{lemma}[theorem]{Lemma} \section{Introduction and preliminaries} Fractional calculus is a generalization of differentiation and integration to an arbitrary order. First works, devoted exclusively to the subject of fractional calculus, are the books \cite{old, smol}. Many recently developed models in areas like rheology, viscoelasticity, electrochemistry, diffusion processes, etc. are formulated in terms of fractional derivatives or fractional integrals. The books \cite{kst,lvd, miros} and \cite{pod} presents the theory of the fractional differential equations and their applications. Some theoretical aspects on the existence and uniqueness results for fractional differential equations have been considered recently by many authors \cite{niet,ar,bel,bon,del,die,ch,lak,lak2,lak3,lak4,nieto,sh}. A differential and integral calculus for fuzzy-valued mappings was developed in papers of Hukuhara \cite{huk}, Dubois and Prade \cite{du1,du2,du3} and Puri and Ralescu \cite{pur1,pur2}. For significant results from the theory of fuzzy differential equations and their applications we refer to the books \cite{dia,lak5} and the papers \cite{al,bede,cal,hul,kal,kh,lak5,miz,seik,xu}. The concept of fuzzy fractional differential equation was introduced by Agarwal, Lakshmikantham and Nieto \cite{fuzzyfac} and \cite{su}. The aim of this paper is to study the existence and uniqueness solution of fuzzy fractional differential equation with fuzzy initial value. Let $E$ be the set of all upper semicontinuous normal convex fuzzy numbers with bounded $\alpha$-level intervals. This means that if $u\in E$ then the $\alpha$-level set, $[u]^{\alpha }=\{x\in \mathbb{R}|u(x)\geq \alpha \}$, $0<\alpha \leq 1$, is a closed bounded interval denoted by $[u]^{\alpha }=[u_{1}^{\alpha },\,\,u_{2}^{\alpha }]$ and there exist a $x_0\in \mathbb{ R}$ such that $u(x_0)=1$. Two fuzzy numbers $u$ and $v$ are called equal, $u=v$, if $u(x)=v(x)$ for all $x\in \mathbb{R}$. It follows that $u=v\$if and only if $[u]^{\alpha }=[v]^{\alpha }$ for all $\alpha \in (0,1]$. The following arithmetic operations on fuzzy numbers are well known and frequently used below. If $u,v\in E$ then \begin{gather*} [ u+v]^{\alpha }=[u_{1}^{\alpha }+v_{1}^{\alpha },u_{2}^{\alpha}+v_{2}^{\alpha }], \\ [ u-v]^{\alpha }=[u_{1}^{\alpha }-v_{2}^{\alpha },u_{2}^{\alpha }-v_{1}^{\alpha }], \\ [ \lambda u]^{\alpha }=\lambda [ u]^{\alpha } =\begin{cases} [ \lambda u_{1}^{\alpha },\lambda u_{2}^{\alpha }] &\text{if }\lambda \geq 0 \\ [ \lambda u_{2}^{\alpha },\lambda u_{1}^{\alpha }] &\text{if }\lambda <0, \end{cases} \quad \lambda \in \mathbb{R}, \end{gather*} \begin{lemma}[\cite{nr}] \label{lem1} If $u\in E$ then the following properties hold: \begin{itemize} \item[(i)] $[u]^{\beta }\subset [ u]^{\alpha }$ if $0<\alpha \leq \beta \leq 1$; \item[(ii)] If $\{\alpha _{n}\}\subset (0,1]$ is a nondecreasing sequence which converges to $\alpha$ then $[u]^{\alpha }=\bigcap_{n\geq 1}[u]^{\alpha _{n}}$ (i.e., $u_{1}^{\alpha}$ and $u_{2}^{\alpha }$ are left-continuous with respect to $\alpha$. \end{itemize} Conversely, if $A_{\alpha }=\{[u_{1}^{\alpha },u_{2}^{\alpha }];\alpha \in (0,1]\}$ is a family of closed real intervals verifying (i) and (ii), then $\{A_{\alpha }\}$ defined a fuzzy number $u\in E$ such that $[u]^{\alpha }=A_{\alpha }$. \end{lemma} For a real inteval $I=[0,a]$, a mapping $u:I\to E$ is called a fuzzy function. We denote $[u(t)]^{\alpha }=[u_1^{\alpha}(t),u_2^{\alpha }(t) ]$, for $t\in I$ and $0<\alpha \leq 1$. the derivative $u'(t)$ of a fuzzy function $u$ is defined by (see \cite{seik}) $$[ u'(t)]^{\alpha }=[(u_1^{\alpha })'(t),(u_2^{\alpha })'(t)],\quad \alpha \in (0,1], \label{fderiv}$$ provided that is equation defines a fuzzy number $u'(t)\in E$. The fuzzy integral $\int_{a}^{b}u(t)dt$, $a,b\in T$, is defined by (see \cite{du1}) $$\Big[ \int_{a}^{b}u(t)dt\Big] ^{\alpha } =\Big[ \int_{a}^{b}u_1^{\alpha }(t)dt,\int_{a}^{b}u_2^{\alpha }(t)dt\Big] \label{fint}$$ provided that the Lebesgue integrals on the right exist. Suppose that $u_1^{\alpha },u_2^{\alpha }\in C((0,a],\mathbb{R})\cap L^{1}((0,a), \mathbb{R})$ for all $\alpha \in [ 0,1]$. Then for $q>0$, we put $$A_{\alpha }=:\frac{1}{\Gamma (q)}\Big[ \int_0^{t}(t-s)^{q-1}u_1^{\alpha }(s)ds,\int_0^{t}(t-s)^{q-1}u_2^{\alpha }(s)ds\Big] . \label{levelint}$$ \begin{lemma} \label{lem2} The family $\{A_{\alpha };\alpha \in [ 0,1]\}$, given by \eqref{levelint}, defined a fuzzy number $x\in E$ such that $[u]^{\alpha }=A_{\alpha }$. \end{lemma} \begin{proof} Since $u\in E$ then, for $\alpha \leq \beta$, we have that $u_{1}^{\alpha}(s)\leq u_{1}^{\beta }(s)$ and $u_{2}^{\alpha }(t)\geq u_{2}^{\beta }(t)$. It follows that $A_{\alpha }\supseteq A_{\beta }$. Since $u_{1}^{0}(t)\leq u_{1}^{\alpha _{n}}(t)\leq u_{1}^{1}(t)$, we have \begin{equation*} | (t-s)^{q-1}u_{i}^{\alpha _{n}}(s)| \leq \max \{a^{q-1}| u_{i}^{0}(s)| ,a^{q-1}| u_{i}^{1}(s)| \}=:g_{i}(s) \end{equation*} for $\alpha _{n}\in (0,1]$ and $i=1,2$. Obviously, $g_{i}$ is Lebesgue integrable on $[0,a]$. Therefore, if $\alpha _{n}\uparrow \alpha$ then by the Lebesgue's Dominated Convergence Theorem, we have \begin{equation*} \lim_{n\to \infty }\int_0^{t}(t-s)^{q-1}u_{i}^{\alpha _{n}}(s)ds =\int_0^{t}(t-s)^{q-1}u_{i}^{\alpha }(s)ds\text{, }i=1,2. \end{equation*} From Lemma \ref{lem1}, the proof is complete. \end{proof} Let $u\in C((0,a],E)\cap L^{1}((0,a),E)$. Define the \textit{fuzzy fractional primitive of order} $q>0$ of $u$, \begin{equation*} I^{q}u(t)=\frac{1}{\Gamma (q)}\int_0^{t}(t-s)^{q-1}u(s)ds, \end{equation*} by \begin{equation*} [ I^{q}u(t)]_{\alpha }=\left[ \int_0^{t}(t-s)^{q-1}u_{1}^{ \alpha }(t)dt,\int_0^{t}(t-s)^{q-1}u_{2}^{\alpha }(t)dt\right] . \end{equation*} For $q=1$ we obtain $I^{1}u(t)=\int_0^{t}u(s)ds$; that is, the integral operator. Let $u\in C((0,a],E)\cap L^{1}((0,a),E)$ be a given function such that $[u(t)]^{\alpha }=[u_{1}^{\alpha }(t),u_{2}^{\alpha }(t)]$ for all $t\in (0,a]$ and $\alpha \in (0,1]$. We define the fuzzy fractional derivative of order \$0