\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})  
 \begin{equation}
[ u'(t)]^{\alpha }=[(u_1^{\alpha })'(t),(u_2^{\alpha })'(t)],\quad 
\alpha \in (0,1], \label{fderiv}
\end{equation}
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})
\begin{equation}
\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}
\end{equation}
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
\begin{equation}
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}
\end{equation}


\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 $0<q<1$ of $u$,
\begin{equation*}
D^{q}u(t)=\frac{1}{\Gamma (1-q)}\frac{d}{dt}\int_0^{t}(t-s)^{-q}u(s)ds,
\end{equation*}
by
\begin{equation*}
[ D^{q}u(t)]^{\alpha }=:\frac{1}{\Gamma (1-q)}
\Big[ \frac{d}{dt}
\int_0^{t}(t-s)^{-q}u_{1}^{\alpha }(s)ds,\frac{d}{dt}
\int_0^{t}(t-s)^{-q}u_{2}^{\alpha }(s)ds\Big] ,
\end{equation*}
provided that equation defines a fuzzy number $D^{q}u(t)\in E$. In fact,
\begin{equation*}
[ D^{q}u(t)]^{\alpha }:=[D^{q}u_{1}^{\alpha }(t),D^{q}u_{2}^{\alpha
}(t)].
\end{equation*}
Obviously, $D^{q}u(t)=\frac{d}{dt}I^{1-q}u(t)$ for $t\in (0,a]$.

\section{Main result}

Let $0<q<1$. We shall consider the initial value problem
\begin{equation}
D^{q}u(t)=f(t,u(t)),\quad \lim_{t\to 0^{+}}t^{1-q}u(t)=v_0  \label{IVP}
\end{equation}
where $f$ is a continuous mapping from $[0,a]\times \mathbb{R}$ into
$\mathbb{R}$ and $v_0$ is a fuzzy number with $\alpha $-level intervals
$[v_0]^{\alpha }=[v_{01}^{\alpha },v_{02}^{\alpha }]$,
$0<\alpha \leq 1$. The extension principle of Zadeh leads to the
following definition of $f(t,u)$ when $u$ is a fuzzy number
\begin{equation*}
f(t,u)(y)=\sup \{u(x):y=f(t,x)\},\quad  x\in \mathbb{R}.
\end{equation*}
It follows that
\begin{equation*}
[ f(t,u)]^{\alpha }=[\min \{f(t,x):x\in [ u_{1}^{\alpha
},u_{2}^{\alpha }]\},\quad
\max \{f(t,x):x\in [ u_{1}^{\alpha},u_{2}^{\alpha }]\}]
\end{equation*}
for $u\in E$ with $\alpha $-level sets $[u]^{\alpha }=[u_{1}^{\alpha
},u_{2}^{\alpha }]$, $0<\alpha \leq 1$. We call $u:(0,a]\to E$ a
fuzzy solution of \eqref{IVP}, if
\begin{equation}
\begin{gathered}
D^{q}u_{1}^{\alpha }(t)= \min \{f(t,x):x\in [ u_{1}^{\alpha
}(t),u_{2}^{\alpha }(t)]\},\quad
\lim_{t\to 0^{+}} t^{1-q}u_{1}^{\alpha }(t)=v_{01}^{\alpha } \\
D^{q}u_{2}^{\alpha }(t)=\max \{f(t,x):x\in [ u_{1}^{\alpha
}(t),u_{2}^{\alpha }(t)]\},\quad\lim_{t\to 0^{+}}t^{1-q}u_{2}^{\alpha }(t)
=v_{02}^{\alpha }
\end{gathered}\label{IVP-S0}
\end{equation}
 for $t\in (0,a]$ and $0<\alpha \leq 1$. Denote
$\widetilde{f}=(f_{1},f_{2})$, $f_{1}(t,u)=\min \{f(t,x):x\in [
u_{1},u_{2}]\}$ and $f_{2}(t,u)=\max \{f(t,x):x\in [
u_{1},u_{2}]\}$ where $u=(u_{1},u_{2})\in \mathbb{R}^{2}$. Thus
for fixed $\alpha $, we have an initial value problem in
$\mathbb{R}^{2}$:
\begin{equation}
\begin{gathered}
D^{q}u_{1}^{\alpha }(t)=\widetilde{f}(t,u_{1}^{\alpha }(t)
,u_{2}^{\alpha }(t) ),\quad
\lim_{t\to 0^{+}} t^{1-q}u_{1}^{\alpha }(t)=v_{01}^{\alpha } \\
D^{q}u_{2}^{\alpha }(t)=\widetilde{f}(t,u_{1}^{\alpha }(t)
,u_{2}^{\alpha }(t) ),\quad
\lim_{t\to 0^{+}} t^{1-q}u_{2}^{\alpha }(t)=v_{02}^{\alpha }
\end{gathered}  \label{IVP-S}
\end{equation}
If we can solve it (uniquely), we have only to verify that the intervals
$[u_{1}^{\alpha }(t),u_{2}^{\alpha }(t)]$, $0<\alpha \leq 1$, define a fuzzy
number $u(t)$ in $E$. Since $f$ is assumed continuous, the initial value
problem \eqref{IVP-S} is equivalent to the following fractional integral
equation
\begin{equation}
u(t)=v_0(t)+\frac{1}{\Gamma (q)}\int_0^{t}(t-s)^{q-1}\widetilde{f}
(s,u(s))ds,\quad 0\leq t\leq a,  \label{FIEq}
\end{equation}
where $v_0(t)=t^{q-1} v_0 /\Gamma (q)$.

\begin{theorem} \label{thm1}
Assume that
\begin{itemize}

\item[(a)] $f\in C([0,a]\times \mathbb{R},\mathbb{R})$ and
 $|f(t,u)| \leq M_0$ on $[0,a]\times [ 0,b]$;

\item[(b)] $g\in C([0,a]\times [ 0,b],\mathbb{R}_{+}),g(t,r)\leq
M_{1}$  on $[0,a]\times [ 0,b],g(t,0)\equiv 0$, $g(t,r)$
is nondecreasing in $r$  for each $t$and $r(t)\equiv 0$  is the only
solution of
\begin{equation}
D^{q}r(t)=g(t,r(t)),\quad t\in (0,a]  \label{kamke}
\end{equation}
with the initial condition $\lim_{t\to 0^{+}}t^{1-q}r(t)=0$;

\item[(c)]
\begin{equation}
| f(t,u)-f(t,\overline{u})| \leq g(t,| u-
\overline{u}| ),\quad t\geq 0,\, u,\overline{u}\in \mathbb{R},
\label{lip}
\end{equation}

\item[(d)] solutions $r(t,r_0)$ of \eqref{kamke} are
continuous with respect to the initial condition
$r_0=\lim_{t\to 0^{+}} t^{1-q}r(t)$.
\end{itemize}
 Then the initial value problem \eqref{IVP} has a unique fuzzy solution.
\end{theorem}

\begin{proof}
It can be shown that \eqref{lip} implies
\begin{equation}
\| \widetilde{f}(t,u)-\widetilde{f}(t,\overline{u})\|
\leq g(t,\| u-\overline{u}\| ),\quad t\geq 0,\;
u,\overline{u }\in \mathbb{R}^{2}  \label{lip1}
\end{equation}
where the $\| \cdot \| $ is defined by
$\|u\| =\max \{| u_{1}| ,|u_{2}| \}$. It is well known that
\eqref{lip1} and the assumptions on $g$
\cite[Theorems 2.1 and 2.2]{lak} guarantee the existence,
uniqueness and continuous dependence on initial value of a solution to
\begin{equation}
D^{q}u(t)=\widetilde{f}(t,u(t)),\quad \lim_{t\to 0^{+}}
t^{1-q}u(t)=v_0\in \mathbb{R}^{2}  \label{CRISP-IVP}
\end{equation}
and that for any continuous function $u_0:R_{+}\to R^{2}$ the
successive approximations
\begin{equation}
u_{n+1}(t)=v_0(t)+\frac{1}{\Gamma (q)}\int_0^{t}(t-s)^{q-1}
\widetilde{f}(s,u_{n}(s))ds,\quad n=0,1,\dots,  \label{approx}
\end{equation}
converge uniformly on closed subintervals of $\mathbb{R}_{+}$ to the
solution of \eqref{CRISP-IVP} \cite[Theorem 2.1]{lak}. By choosing
$ v_0=(v_{01}^{\alpha },v_{02}^{\alpha })$ in \eqref{CRISP-IVP} we get a
unique solution $u^{\alpha }(t)=(u_{1}^{\alpha }(t),u_{2}^{\alpha }(t))$
to \eqref{IVP-S0} for each $\alpha \in (0,1]$. Next we will show that the
intervals $[u_{1}^{\alpha }(t),u_{2}^{\alpha }(t)],0<\alpha \leq 1$, define
a fuzzy number $u(t)\in E$ for each $t\geq 0$; i.e., that $u$ is a fuzzy
solution to \eqref{IVP}. The successive approximations
$u_0(t)=v_0\in E$,
\begin{equation*}
u_{n+1}(t)=v_0(t)+\frac{1}{\Gamma (q)}\int_0^{t}(t-s)^{q-1}
\widetilde{f}(s,u_{n}(s))ds,\quad t\geq 0,\; n=0,1,\dots,
\end{equation*}
where the integral is the fuzzy integral, define a sequence of fuzzy numbers
$u_{n}(t)\in E$ for each $t\geq 0$. Hence
\begin{equation*}
[ u_{n}(t)]^{\alpha }\supset [ u_{n}(t)]^{\beta}\quad\text{if }
0<\alpha \leq \beta \leq 1,
\end{equation*}
which implies that
\begin{equation*}
[ u_{1}^{\alpha }(t),u_{2}^{\alpha }(t)]\supset [ u_{1}^{\beta
}(t),u_{2}^{\beta }(t)],\quad 0<\alpha \leq \beta \leq 1,
\end{equation*}
since by the convergence of sequence \eqref{approx}, the end points of
$ [u_{n}(t)]_{\alpha }$ converge to $u_{1}^{\alpha }(t)$ and
 $u_{2}^{\alpha }(t)$ respectively. Thus the inclusion property (i)
 of Lemma \ref{lem1} holds for the intervals
$[u_{1}^{\alpha }(t),u_{2}^{\alpha }(t)],0<\alpha \leq 1$. For the
proof of continuity property (ii) of Lemma \ref{lem1},
let $(\alpha _{k})$ be a
nondecreasing sequence in $(0,1]$ converging to $\alpha $. Then
$v_{01}^{\alpha _{k}}\to v_{01}^{\alpha }$ and
$v_{02}^{\alpha_{k}}\to v_{02}^{\alpha }$ because $v_0\in E$.
But then by the continuous dependence on the initial value of the
solution of \eqref{CRISP-IVP},
$u_{1}^{\alpha _{k}}(t)\to u_{1}^{\alpha }(t)$ and
$u_{2}^{\alpha _{k}}(t)\to u_{2}^{\alpha }(t)$, i.e.
(ii) holds for the intervals
$[u_{1}^{\alpha }(t),u_{2}^{\alpha }(t)]$, $0<\alpha \leq 1$.
Hence, by Lemma \ref{lem1}, $u(t)\in E$ and so $u$ is a fuzzy solution of
\eqref{IVP}. The uniqueness follows from the uniqueness of the
 solution of \eqref{CRISP-IVP}.
\end{proof}

\section{Example}

Consider the crisp differential equation
\begin{equation}
D^{q}u(t)=-u(t)  \label{crex}
\end{equation}
with the fuzzy initial condition
\begin{equation}
\underset{t\to 0^{+}}{\lim }t^{1-q}u(t)=(1|2|3),  \label{icex}
\end{equation}
where $t\in (0,a]$, $0<q\leq 1$, and $v_0=(1|2|3)\in E$ is a
fuzzy triangular number, that is,
$[v_0]^{\alpha }=[1+\alpha ,3-\alpha ]$ for $\alpha \in (0,1]$.
If we put $[u(t)]^{\alpha }=[u_{1}^{\alpha }(t),u_{2}^{\alpha }(t)]$,
then $[D^{q}u(t)]^{\alpha }=[D^{q}u_{1}^{\alpha }(t),D^{q}u_{2}^{\alpha
}(t)]$. We obtain the  system
\begin{gather*}
D^{q}u_{1}^{\alpha }(t)=-u_{2}^{\alpha }(t), \quad
 \lim_{t\to 0^{+}} t^{1-q}u_{1}^{\alpha }(t)=1+\alpha  \\
D^{q}u_{2}^{\alpha }(t)=-u_{1}^{\alpha }(t), \quad
\lim_{t\to 0^{+}} t^{1-q}u_{2}^{\alpha }(t)=3-\alpha ,
\end{gather*}
or
\begin{equation}
D^{q}y(t)=Ay(t),\quad \lim_{t\to 0^{+}} t^{1-q}y(t)=c,
\label{matrix}
\end{equation}
where
\begin{equation*}
y(t)=\begin{bmatrix}
u_{1}^{\alpha }(t) \\
u_{2}^{\alpha }(t)
\end{bmatrix},\quad
 A= \begin{bmatrix}
0 & -1 \\
-1 & 0
\end{bmatrix},\quad
 c=\begin{bmatrix}
1+\alpha \\
3-\alpha
\end{bmatrix}.
\end{equation*}
Using the same method that in \cite{jjj}, we obtain the
solution of \eqref{matrix}. It is given by
\begin{equation*}
y(t)=t^{q-1}E_{q,q}(At^{q})c=t^{q-1}E_{q,q}(At^{q})
\begin{bmatrix}
1+\alpha \\
3-\alpha
\end{bmatrix},
\end{equation*}
where
\begin{align*}
E_{q,q}(At^{q})
&=\sum_{k=0}^{\infty }\frac{(At^{q})^{k}}{\Gamma (q(k+1))}\\
&=\begin{bmatrix}
\sum_{n=0}^{\infty }\frac{t^{2qn}}{\Gamma (q(2n+1))} & 0 \\
0 & \sum_{n=0}^{\infty }\frac{t^{2qn}}{\Gamma (q(2n+1))}
\end{bmatrix}\\
 &\quad 
+\begin{bmatrix}
0 & -\sum_{n=0}^{\infty }\frac{t^{(2n+1)q}}{\Gamma (q(2n+2))} \\
-\sum_{n=0}^{\infty }\frac{t^{(2n+1)q}}{\Gamma (q(2n+1))} & 0
\end{bmatrix}.
\end{align*}
Then we obtain
\begin{gather*}
u_{1}^{\alpha }(t)=\sum_{n=0}^{\infty }\frac{t^{(2n+1)q-1}}{\Gamma (q(2n+1))
}(1+\alpha )-\sum_{n=0}^{\infty }\frac{t^{(2n+2)q-1}}{\Gamma (q(2n+2))}
(3-\alpha ),
\\
u_{2}^{\alpha }(t)=\sum_{n=0}^{\infty }\frac{t^{(2n+1)q-1}}{\Gamma (q(2n+1))
}(3-\alpha )-\sum_{n=0}^{\infty }\frac{t^{(2n+2)q-1}}{\Gamma (q(2n+2))}
(1+\alpha ).
\end{gather*}
It easy to see that $[u_{1}^{\alpha }(t),u_{2}^{\alpha }(t)]$
define the $\alpha $-level intervals of a fuzzy number.
So $[u(t)]^{\alpha }$ are the $ \alpha $-level intervals of the
fuzzy solution of \eqref{crex}-\eqref{icex}.

\subsection{Conclusion}

Using the Hukuhara derivative, we given a result for the
existence and uniqueness of the solution for a class of fractional
differential equations with fuzzy initial value. This approach
based on Hukuhara derivative has the disadvantage that any
solution of a fuzzy differential equation has increasing length of
its support. Consequently, this approach cannot really reflect any
of the rich behavior of ordinary differential equations
\cite{dia}. Moreover, there exist simple fuzzy functions (e.g.,
$F(t)=cg(t)$, where $c$ is a fuzzy number and
$g:[a,b]\to [ 0,\infty )$ is a function with $g'(t)<0$)
which are not Hukuhara differentiable. Bede and Gal \cite{bede}
(see also \cite{cal,kh,sb,sb2}), solved the above mentioned
shortcoming under strongly generalized differentiability of
fuzzy-number-valued functions. In this case the derivative exists
and the solution of a fuzzy differential equation may have
decreasing length of the support, but the uniqueness is lost.
Another approach consists in interpreting a fuzzy differential
equations as a family of differential inclusions \cite{hul},
\cite{miz}. The main shortcoming of using differential inclusions
is that we do not have a derivative of a fuzzy-number-valued
function. In future, the study of fuzzy fractional equations,
using different approaches mentioned above can help to develop
this theory.

\subsection*{Acknowledgements} The authors are thankful to the
anonymous referees for their very helpful comments and
suggestions.

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