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\thispagestyle{empty} \pagestyle{myheadings} \markboth{S. Abbas, W.A. Albarakati,
M. Benchohra and A. Petru\c sel} {Ulam Stability for Hadamard Partial Fractional Integral
Inclusions}
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\begin{document}
\begin{center}
{\Large\bf Existence and Ulam Stability Results for Hadamard Partial Fractional Integral Inclusions
via Picard Operators}
\vskip 1cm {\bf Sa\"{\i}d Abbas$^{a}$,  Wafaa A. Albarakati$^{c}$, Mouffak Benchohra$^{b,c}$ and Adrian Petru\c sel$^{d}$}\\[0.3cm]
$^{a}$Laboratory of Mathematics, University of Sa\"{\i}da,\\
PO Box 138, 20000 Sa\"{\i}da, Algeria \\
e-mail: abbasmsaid@yahoo.fr\\[0.3cm]
$^{b}$ Laboratory of Mathematics, University of Sidi
Bel-Abb\`es,\\ P.O. Box 89, Sidi Bel-Abb\`es 22000, Algeria\\
e-mail: benchohra@univ-sba.dz\\[0.3cm]
\ $^{c}$ Department of Mathematics, Faculty of Science, King
Abdulaziz University \\ P.O. Box 80203, Jeddah 21589, Saudi Arabia  \\
e-mail: wbarakati@kau.edu.sa\\[0.3cm]
$^{d}$ Department of Mathematics, Babe\c{s}-Bolyai University,\\
400084 Cluj-Napoca, Romania\\
e-mail: petrusel@math.ubbcluj.ro
\end{center}
\begin{abstract}
In this paper, by using the weakly Picard operators theory, we
investigate some existence  and Ulam type stability results  for a class of  Hadamard
partial fractional integral inclusions.
\end{abstract}
{\bf Key words and phrases:} Hadamard fractional  integral inclusion; multivalued weekly
Picard operator; fixed point inclusion; Ulam-Hyers stability.\\

\noindent {\bf AMS (MOS) Subject Classifications}: 26A33, 34G20,
34A40, 45N05, 47H10.

\bigskip

{\bf Dedicated to Professor Ioan A. Rus on the occasion of his 80th birthday}

\section{Introduction}
The fractional calculus represents a powerful tool in applied mathematics
to study a myriad of problems from different
fields of science and engineering, with many break-through results
found in mathematical physics, finance, hydrology, biophysics,
thermodynamics, control theory, statistical mechanics, astrophysics,
cosmology and bioengineering \cite{Hi,Pod,Tar}. There has been a
significant development in ordinary and partial fractional
differential equations in recent years; see the monographs of Abbas
{\em et al.} \cite{AB,ABN,ABN1}, Kilbas {\em et al.} \cite{KST}, Miller and
Ross \cite{MiRo}, the papers of Abbas {\em et al.}
\cite{ABH,ABP,AbBeSi,ABT}, Vityuk and Golushkov
\cite{ViGo}, and the references therein.
In \cite{BKT1}, Butzer {\em et al.} investigate properties of the Hadamard
fractional integral and the derivative. In \cite{BKT2}, they obtained the Mellin
transforms of the Hadamard fractional integral and differential operators and
in \cite{PAT}, Pooseh {\em et al.} obtained expansion formulas of the Hadamard operators
in terms of integer order derivatives. Many other interesting properties of those
operators and others are summarized in \cite{SaKiMa} and the references therein.

\vskip0.3cm The stability of functional equations was originally
raised by Ulam \cite{Ula} in 1940 and Hyers \cite{Hy} in 1941.
Thereafter, this type of stability is called the Ulam-Hyers stability.
In 1978, Rassias \cite{Ras} provided a remarkable generalization of
the Ulam-Hyers stability of mappings by considering variables.
The stability question of functional equations is how do the solutions
of the inequality differ from those of the given functional
equation? Considerable attention has been given to the study of the
Ulam-Hyers and Ulam-Hyers-Rassias stability of all kinds of
functional equations; one can see the monographs of \cite{HIR,Jun}.
Bota-Boriceanu and Petru\c sel \cite{BP}, Petru {\em et al.}
\cite{PB,PPY}, and Rus \cite{Rus,Rus1} discussed the Ulam-Hyers
stability for operatorial equations and inclusions. Castro and Ramos
\cite{CaRa}, and Jung \cite{Jun2} considered the Hyers-Ulam-Rassias
stability for a class of Volterra integral equations. Ulam stability
for fractional differential equations with Caputo derivative are
proposed by Wang {\em et al.} \cite{WLZ,WLZ1}. Some stability
results for fractional integral equation are obtained by Wei {\em et
al.} \cite{WLL}.  More details from historical point of view, and
recent developments of such stabilities are reported in
\cite{Jun1,Rus,WLL}.

\vskip0.3cm The theory of Picard operators was introduced by Ioan A.
Rus (see \cite{Rus2,Rus3,Rus4} and their references) to study
problems related to fixed point theory. This abstract approach was
used later on by many mathematicians  as a very
powerful method in the study of integral equations and
inequalities, ordinary and partial differential equations
(existence, uniqueness, differentiability of the solutions, ...), see \cite{Rus4} 
and the references therein. The
theory of Picard operators is also a very powerful tool in the study of
Ulam-Hyers stability of functional equations. We only have to define
a fixed point equation from the functional equation we want to
study, then if the defined operator is c-weakly Picard we also have
immediately the Ulam-Hyers stability of the desired equation. Of
course it is not always possible to transform a functional equation
or a differential equation into a fixed point problem and actually
this point shows a weakness of this theory. The uniform approach
with Picard operators to the discuss of the stability problems of
Ulam-Hyers type is due to Rus \cite{Rus1}.

\vskip0.3cm In \cite{ABH,ABP,AbBeSi}, Abbas {\em et al.}
 studied some Ulam stabilities for functional fractional partial differential and integral
inclusions via Picard operators. In this paper, we discuss the Ulam-Hyers and the
Ulam-Hyers-Rassias stability for the following new class of fractional partial integral
inclusions of the form
\begin{equation}\label{e1}
u(x,y)-\mu(x,y)\in(^{H}I_{\sigma}^{r}F)(x,y,u(x,y));\ (x,y)\in
J:=[1,a]\times[1,b],
\end{equation}
where $a,b>1,\ \sigma=(1,1),\ F:J\times E \to {\cal P}(E)$ is a
set-valued function with nonempty values in a (real or complex)
separable Banach space $E,\ {\cal P}(E)$ is the family of all
nonempty subsets of $E,\ ^{H}I_{\sigma}^{r}F$ is the
definite Hadamard integral for the set-valued function $F$ of order
$r=(r_1,r_2)\in(0,\infty)\times(0,\infty),$ and $\mu:J\to E$ is a
given continuous function.

This paper initiates the  existence of the solution and the Ulam stability  
via Picard operators for such new class of fractional integral inclusions.

\section{Basic concepts and auxiliary results}

Let $L^{1}(J)$ be the space of Bochner-integrable functions
$u:J\rightarrow E$ with the norm
$$
\|u\|_{L^{1}}=\int_{1}^{a}\int_{1}^{b}\|u(x,y)\|_Edydx,
$$
where $\|\cdot\|_E$ denotes a complete norm on $E.$ By $L^{\infty}(J)$
we denote the Banach space of measurable functions $u: J \to E$
which are essentially bounded, equipped with the norm
$$
\|u\|_{L^{\infty}}=\inf\{c>0:\|u(x,y)\|_E\leq c,\ a.e. \ (x,y)\in
J\}.
$$
As usual, by ${\cal C}:=C(J)$  we denote the Banach space of all
continuous functions from $J$ into $E$ with the norm
$\|.\|_{\infty}$ defined by
$$
\|u\|_{\infty}=\Sup_{(x,y)\in J}\|u(x,y)\|_E.
$$
\vskip0.3cm Let $(X,d)$ be a metric space induced from the normed space
$(X,\|.\|).$ Denote ${\cal P}_{cl}(X)=\{Y\in {\cal P}(X):
Y\,\hbox{closed}\},$ ${\cal P}_{bd}(X)=\{Y\in {\cal P}(X): Y\,\hbox
{bounded}\},$ ${\cal P}_{cp}(E)=\{Y\in {\cal P}(E): Y\,\hbox{
compact}\}$ and ${\cal P}_{cp,cv}(E)=\{Y\in {\cal P}(E):
Y\,\hbox{compact and convex}\}.$
\begin{definition} A multivalued map $T:X\to {\cal P}(X)$ is convex
(closed) valued if $T(x)$ is convex (closed) for all $x\in X,\ T$ is
called upper semi-continuous (u.s.c.) on $X$ if for each $x_{0}\in
X$, the set $T(x_{0})$ is a nonempty closed subset of $X,$ and if
for each open set $N$ of $X$ containing $T(x_{0}),$ there exists an
open neighborhood $N_0$ of $x_{0}$ such that $T(N_0)\subseteq N.\ T$
is lower semi-continuous (l.s.c.) if the set $ \{t\in X: T(t)\cap
B\not=\emptyset\}$ is open for any open set \ $B$ in \ $X.\ T$ is
said to be completely continuous if $T(B)$ is relatively compact for
every $B\in {\cal P}_{bd}(X).\ T$ has a fixed point if there is
$x\in X$ such that $x\in T(x).$ The fixed point set of the
multivalued operator $T$ will be denoted by $Fix(T).$ The graph of
$T$ will be denoted by $Graph(F):=\{(u,v)\in X\times{\cal P}(X):v\in
T(u)\}.$
\end{definition}

Consider $H_{d}:{\cal P}(X)\times {\cal
P}(X)\to[0,\infty)\cup\{\infty\}$ given by $$
H_{d}(A,B)=\Max\left\{\Sup_{a\in A}d(a,B),\Sup_{b\in
B}d(A,b)\right\},$$ where $d(A,b)=\Inf_{a\in A}d(a,b), \
d(a,B)=\Inf_{b\in B}d(a,b).$ Then $({\cal P}_{bd,cl}(X),H_{d})$ is a
Hausdorff metric space.

\vskip 0.3cm Notice that $A:X\to X$ is a selection for $T:X\to{\cal
P}(X)$ if $A(u)\in T(u);$ for each $u\in X.$ For each $u\in{\cal
C},$ define the set of selections of the multivalued $F:J\times{\cal
C}\to{\cal P}({\cal C})$ by
$$
S_{F,u}=\{v:\in L^{1}(J): v(x,y)\in F(x,y,u(x,y)); \ (x,y)\in J\}.
$$
\begin{definition}
A multivalued map $G:J\to{\cal P}_{cl}(E),$ is said to be measurable
if for every $v\in E$ the function $(x,y)\to
d(v,G(x,y))=\Inf\{d(v,z):z\in G(x,y)\}$ is measurable.
\end{definition}

\vskip0.3cm In what follows we will give some basic definitions and
results on Picard operator theory \cite{Rus4}. Let $(X,d)$ be a
metric space and $A:X\to X$ be an operator. We denote by $F_A$ the
set of the fixed points of $A.$ We also denote $A^{0}:=1_X,\
A^{1}:=A,\ldots,A^{n+1}:=A^{n}\circ A;\ n\in\mathbb{N}$ the iterate operators
of the operator $A.$
\begin{definition} The operator $A:X\to X$ is a Picard operator $(PO)$ if
there exists $x^{*}\in X$ such that:
\begin{itemize}
\item[(i)] $F_A=\{x^{*}\};$
\item[(ii)] The sequence $(A^{n}(x_0))_{n\in\mathbb{N}}$ converges to $x^{*}$ for all
$x_0\in X.$
\end{itemize}
\end{definition}
\begin{definition} The operator $A:X\to X$ is a weakly Picard operator (WPO)
if the sequence $(A^{n}(x))_{n\in\mathbb{N}}$ converges for all $x\in X,$
and its limit (which may depend on $x$) is a fixed point of $A.$
\end{definition}
\begin{definition} If $A$ is weakly Picard operator then we consider the
operator $A^{\infty}$ defined by
$$
A^{\infty}:X\to X;\ A^{\infty}(x)=\Lim_{n\rightarrow\infty}A^{n}(x).
$$
\end{definition}
\begin{remark} It is clear that
$A^{\infty}(X)=F_A.$
\end{remark}
\begin{definition} Let $A$ be a weakly Picard operator and $c>0.$ The
operator $A$ is c-weakly Picard operator if
$$
d(x,A^{\infty}(x))\leq c\ d(x,A(x));\ x\in X.
$$
\end{definition}
In the multivalued case we have the following concepts (see
\cite{RuPeSi}).
\begin{definition} Let $(X,d)$ be a metric space, and $F:X \to{\cal P}_{cl}(X)$ be a
multivalued operator. By definition, $F$ is a multivalued weakly
Picard operator (MWPO), if for each $u\in X$ and each $v\in F(x),$
there exists a sequence $(u_n)_{n\in\N}$ such that
\begin{itemize}
\item[(i)] $u_0=u,\ u_1=v;$
\item[(ii)] $u_{n+1}\in F(u_n),$ for each $n\in\mathbb{N};$
\item[(iii)] the sequence $(u_n)_{n\in\N}$ is convergent and its limit is a fixed point of
$F.$
\end{itemize}
\end{definition}
\begin{remark}
A sequence $(u_n)_{n\in\mathbb{N}}$ satisfying condition $(i)$ and $(ii)$ in
the above Definition is called a sequence of successive
approximations of $F$ starting from $(x,y)\in Graph(F).$
\end{remark}
If $F:X\to {\cal P}_{cl}(X)$ is a (MWPO) then we define $F_1:
Graph(F)\to{\cal P}(Fix(F))$ by the formula $F_1(x,y):=\{u\in
Fix(F):$ there exists a sequence of successive approximations of $F$
starting from $(x,y)$ that converges to $u\}.$
\begin{definition}
Let $(X,d)$ be a metric space and let $\Psi:[0,\infty)\to[0,\infty)$
be an increasing function which is continuous at $0$ and
$\Psi(0)=0.$ Then $F:X\to{\cal P}_{cl}(X)$ is said to be a
multivalued $\Psi-$weakly Picard operator ($\Psi-$MWPO) if it is a
multivalued weakly Picard operator and there exists a selection
$A^{\infty}: Graph(F)\to Fix(F)$ of $F^{\infty}$ such that
$$
d(u,A^{\infty}(u,v))\leq \Psi(d(u,v));\ for\ all\ (u,v)\in Graph(F).
$$
If there exists $c>0$ such that $\Psi(t)=ct,$ for each
$t\in[0.\infty),$ then $F$ is called a multivalued c-weakly Picard
operator $(c-MWPO).$
\end{definition}

\vskip0.3cm Let us recall the notion of comparison function.
\begin{definition} A function $\varphi:[0,\infty)\to[0,\infty)$ is said to be a
comparison function (see \cite{Rus4}) if it is increasing and
$\varphi^{n}(t)\to 0$ as $n\to\infty$, for all $t>0$.
\end{definition}
As a consequence, we have $\varphi(t)<t,$ for each $t>0,\
\varphi(0)=0$ and $\varphi$ is continuous at $0.$
\begin{definition} A function $\varphi:[0,\infty)\to[0,\infty)$ is
said to be a strict comparison function (see \cite{Rus4}) if it is
strictly increasing and $\Sum_{n=1}^{\infty}\varphi^{n}(t)<\infty,$
for each $t>0.$
\end{definition}
\begin{example} The mappings $\varphi_1,\varphi_2:[0,\infty)\to[0,\infty)$
given by $\varphi_1(t)=ct;\ c\in[0,1),$ and
$\varphi_2(t)=\frac{t}{1+t};\ t\in[0,\infty),$ are strict comparison
functions.
\end{example}
\begin{definition} A multivalued operator $N:X\to{\cal P}_{cl}(X)$ is called
\begin{itemize}
\item[a)] $\gamma$-Lipschitz if and only if there
exists $\gamma\geq0$ such that
$$
H_d(N(u),N(v))\leq \gamma d(u,v); \hbox{ for each } \ u,\ v\in X,
$$
\item[b)] a multivalued $\gamma-$contraction if and only if it is
$\gamma$-Lipschitz with $\gamma\in[0,1),$

\item[c)] a multivalued $\varphi-$contraction if and only if there
exists a strict comparison function
$\varphi:[0,\infty)\to[0,\infty)$ such that
$$
H_d(N(u),N(v))\leq \varphi(d(u,v)); \hbox{ for each } \ u,\ v\in X.
$$
\end{itemize}
\end{definition}

\vskip0.3cm Now, we introduce notations and definitions concerning
to partial Hadamard integral of fractional order.

\begin{definition} \cite{Ha,KST}
The Hadamard fractional integral of order $q>0$ for a function $g\in L^{1}([1,a],{\mathbb R}),$ is
defined as
$$
(^{H}I_{1}^{r}g)(x)=\frac{1}{\Gamma (q)}
\int_{1}^{x}\left(\log\frac{x}{s}\right)^{q-1}\frac{g(s)}{s}ds,
$$
where $\Gamma (\cdot)$ is the Euler gamma function.
\end{definition}
\begin{definition}
Let $r_{1},$ $r_{2}\geq0, \ \sigma=(1,1)$ and $r=(r_{1},r_{2}).$
For $w\in L^{1}(J,{\mathbb R}),$ define the Hadamard partial fractional integral of order $r$
by the expression
$$
(^{H}I_{\sigma}^{r}w)(x,y)=\frac{1}{\Gamma (r_{1})\Gamma (r_{2})}
\int_{1}^{x}\int_{1}^{y}\left(\log\frac{x}{s}\right)^{r_{1}-1}\left(\log\frac{y}{t}\right)^{r_{2}-1}
\frac{w(s,t)}{st}dtds.
$$
\end{definition}
\begin{definition} Let $F:J\times E \to {\cal P}(E)$ be a set-valued function with nonempty values in
$E.\ (^{H}I_{\sigma}^{r}F)(x,y,u(x,y))$ is the definite Hadamard integral for the
set-valued functions $F$ of order
$r=(r_1,r_2)\in(0,\infty)\times(0,\infty)$ which is defined as
$$
^{H}I_{\sigma}^{r}F(x,y,u(x,y))=\left\{
\int_{1}^{x}\int_{1}^{y}\left(\log\frac{x}{s}\right)^{r_{1}-1}\left(\log\frac{y}{t}\right)^{r_{2}-1}
\frac{f(s,t)}{st\Gamma (r_{1})\Gamma (r_{2})}dtds:f\in S_{F,u}\right\}.
$$
\end{definition}
\begin{remark}\label{R0}
Solutions of the inclusion (\ref{e1}) are solutions of the fixed
point inclusion $u\in N(u)$ where $N:{\cal C}\to {\cal P}({\cal C})$
is the multivalued operator defined by
$$
(Nu)(x,y)=\left\{\mu(x,y)+(^{H}I_{\sigma}^{r}f)(x,y):\ f\in
S_{F,u}\right\};\ (x,y)\in J.
$$
\end{remark}

\vskip0.3cm Let us give the definition of Ulam-Hyers stability of the
fixed point inclusion $u\in N(u)$,  see for instance \cite{ABH}.
Let $\epsilon$ be a positive real number and $\Phi: J\to [0,\infty)$ be
a continuous function.
\begin{definition}\label{D1}
The fixed point inclusion $u\in N(u)$ is said to be Ulam-Hyers
stable if there exists a real number $c_{N}>0$ such that for each
$\epsilon>0$ and for each solution $u\in {\cal C}$ of the inequality
$H_d(u(x,y),(Nu)(x,y))\leq\epsilon;\ (x,y)\in J,$ there exists a
solution $v\in {\cal C}$ of the inclusion $u\in N(u)$ with
$$\|u(x,y)-v(x,y)\|_E\leq \epsilon c_{N};\ (x,y)\in J.$$
\end{definition}
\begin{definition}\label{D2}
The fixed point inclusion $u\in N(u)$ is said to be generalized
Ulam-Hyers stable if there exists an increasing function
$\theta_{N}\in C([0,\infty),[0,\infty)),\ \theta_{N}(0)=0$ such that
for each $\epsilon>0$ and for each solution $u\in {\cal C}$ of the
inequality $H_d(u(x,y),(Nu)(x,y))\leq\epsilon;\ (x,y)\in J,$ there
exists a solution $v\in {\cal C}$ of the inclusion $u\in N(u)$ with
$$\|u(x,y)-v(x,y)\|_E\leq \theta_{N}(\epsilon);\ (x,y)\in J.$$
\end{definition}
\begin{definition}\label{D3}
The fixed point inclusion $u\in N(u)$ is said to be
Ulam-Hyers-Rassias stable with respect to $\Phi$ if there exists a
real number $c_{N,\Phi}>0$ such that for each $\epsilon>0$ and for
each solution $u\in {\cal C}$ of the inequality
$H_d(u(x,y),(Nu)(x,y))\leq\epsilon\Phi(x,y);\ (x,y)\in J,$ there
exists a solution $v\in {\cal C}$ of the inclusion $u\in N(u)$ with
$$\|u(x,y)-v(x,y)\|_E\leq \epsilon c_{N,\Phi}\Phi(x,y);\ (x,y)\in J.$$
\end{definition}
\begin{definition}\label{D4}
The fixed point inclusion $u\in N(u)$ is said to be generalized
Ulam-Hyers-Rassias stable with respect to $\Phi$ if there exists a
real number $c_{N,\Phi}>0$ such that for each solution $u\in {\cal
C}$ of the inequality $H_d(u(x,y),(Nu)(x,y))\leq\Phi(x,y);\ (x,y)\in
J,$ there exists a solution $v\in {\cal C}$ of the inclusion $u\in
N(u)$ with
$$\|u(x,y)-v(x,y)\|_E\leq c_{N,\Phi}\Phi(x,y);\ (x,y)\in J.$$
\end{definition}
\begin{remark}\label{R1}
It is clear that
\begin{itemize}
\item[(i)] Definition \ref{D1} $\Rightarrow$ Definition \ref{D2},
\item[(ii)] Definition \ref{D3} $\Rightarrow$ Definition \ref{D4},
\item[(iii)] Definition \ref{D3} for $\Phi(x,y)=1 \ \Rightarrow$ Definition \ref{D1}.
\end{itemize}
\end{remark}

\vskip0.3cm The following result, a generalization of Covitz-Nadler
fixed point principle (see \cite{CN}), is known in the literature
as W\c{e}grzyk's fixed point theorem.

\begin{lemma}\label{L0} \cite{We} Let $(X,d)$ be a complete metric space. If $A:X\to{\cal P}_{cl}(X)$
is a $\varphi-$contraction, then $Fix(A)$ is nonempty and for any
$u_0\in X,$ there exists a sequence of successive approximations of
$A$ starting from $u_0$ which converges to a fixed point of $A.$
\end{lemma}
Also, the following result is known in the literature as
W\c{e}grzyk's theorem.
\begin{lemma} \cite{We} \label{L01} Let $(X,d)$ be a Banach space. If an operator $A:X\to {\cal P}_{cl}(X)$ is a
$\varphi-$contraction, then $A$ is a $(MWPO).$
\end{lemma}

\vskip0.3cm Now we present an important characterization Lemma from
the point of view of Ulam-Hyers stability.
\begin{lemma} \cite{PPY} \label{L00} Let $(X,d)$ be a metric space. If $A:X\to{\cal P}_{cp}(X)$ is
a $(\Psi-MWPO),$ then the fixed point inclusion $u\in A(u)$ is
generalized Ulam-Hyers stable. In particular, if $A$ is $(c-MWPO),$
then the fixed point inclusion $u\in A(u)$ is Ulam-Hyers stable.
\end{lemma}

\vskip0.3cm Another Ulam-Hyers stability result, more efficient for
applications, was proved in \cite{La}.
\begin{theorem} \cite{La} \label{T0} Let $(X,d)$ be a complete metric space and $A:X\to {\cal P}_{cp}(X)$
be a multivalued $\varphi-$contraction. Then:
\begin{itemize}
\item[$(i)$] {\em Existence of the fixed point:} $A$ is a $(MWPO);$

\item[$(ii)$] {\em Ulam-Hyers stability
for the fixed point inclusion:} If additionally $\varphi(ct)\leq
c\varphi(t)$ for every $t\in[0,\infty)$ (where $c>1$), then $A$ is a ($\Psi-$MWPO),
with $\Psi(t):=t+\Sum_{n=1}^{\infty}\varphi^{n}(t),$ for each
$t\in[0,\infty);$

\item[$(iii)$] {\em Data dependence of the fixed point set:} Let $S:X\to {\cal P}_{cl}(X)$ be
a multivalued $\varphi-$contraction and $\eta>0$ be such that
$H_d(S(x),A(x))\leq\eta,$ for each $x\in X.$ Suppose that
$\varphi(ct)\leq c\varphi(t)$ for every $t\in[0,\infty)$ (where
$c>1$). Then,
$$H_d(Fix(S),Fix(F))\leq\Psi(\eta).$$
\end{itemize}
\end{theorem}

\section{Existence and Ulam-Hyers stability results}

In this section, we present conditions for the existence and the
Ulam stability of the Hadamard integral inclusion (\ref{e1}).
\begin{theorem}\label{T1} Assume that the multifunction $F:J\times E \to{\cal P}_{cp}(E)$
satisfies the following hypotheses:
\begin{itemize}
\item[$(H_1)$] $(x,y)\longmapsto F(x,y,u)$ is jointly measurable for each $u\in E;$

\item[$(H_2)$] $u\longmapsto F(x,y,u)$ is lower semicontinuous for almost
all $(x,y)\in J;$

\item[$(H_3)$] There exists $p\in
L^{\infty}(J,[0,\infty))$ and a strict comparison function
$\varphi:[0,\infty)\to[0,\infty)$ such that for each $(x,y)\in J$
and each $u,v\in E,$ we have
\begin{equation}\label{e5}
H_d(F(x,y,u(x,y),F(x,y,\overline u)\|\leq
p(x,y)\varphi(\|u-\overline u\|_E),
\end{equation}
and
\begin{equation}\label{e05}
\frac{(\log a)^{r_1}(\log b)^{r_2}\|p\|_{L^{\infty}}}{\Gamma(1+r_1)\Gamma(1+r_2)}\leq1;
\end{equation}

\item[$(H_4)$] There exists an integrable function $q:[1,b]\to[0,\infty)$ such that for
each $x\in [1,a]$ and $u\in E,$ we have $F(x,y,u)\subset
q(y)B(0,1)$, a.e. $y\in[1,b],$ where $B(0,1)=\{u\in E:\|u\|_E<1\}.$
\end{itemize}
Then the following conclusions hold:
\begin{itemize}
\item[$(a)$] The integral inclusion (\ref{e1}) has least one solution and $N$ is a $(MWPO).$

\item[$(b)$] If additionally $\varphi(ct)\leq c\varphi(t)$ for every $t\in[0,\infty)$
(where $c>1$), then the integral inclusion
(\ref{e1}) is generalized Ulam-Hyers stable, and  $N$ is a
($\Psi-$MWPO), with the function $\Psi$ defined by
$\Psi(t):=t+\Sum_{n=1}^{\infty}\varphi^{n}(t),$ for each
$t\in[0,\infty).$ Moreover, in this case the continuous data
dependence of the solution set of the integral inclusion (\ref{e5})
holds.
\end{itemize}
\end{theorem}
\begin{remark} For each $u\in{\cal C},$ the
set $S_{F,u}$ is nonempty since by $(H_1),\ F$ has a measurable
selection (see \cite{CaVa}, Theorem III.6).
\end{remark}

\vskip0.3cm {\bf Proof.} We shall show that $N$ defined in Remark
\ref{R0} satisfies the assumptions of Theorem \ref{T0}. The proof
will be given in two steps.

\vskip 0.3cm {\bf Step 1.} {\em $N(u)\in  P_{cp}({\cal C})$ for each
$u\in {\cal C}.$}\\
From the continuity of $\mu$ and Theorem 2 in Rybi\'{n}ski \cite{Ry}
we have that for each $u\in {\cal C}$ there exists $f\in S_{F,u},$
for all $(x,y)\in J,$ such that $f(x,y)$ is integrable with respect
to $y$ and continuous with respect to $x.$ Then the function
$v(x,y)=\mu(x,y)+^{H}I_{\sigma}^{r}f(x,y)$ has the property $v\in N(u).$
Moreover, from $(H_1)$ and $(H_4),$ via Theorem 8.6.3. in Aubin and
Frankowska \cite{AF}, we get that $N(u)$ is a compact set, for each
$u\in {\cal C}.$

\vskip 0.3cm {\bf Step 2.} {\em $H_d(N(u),N(\overline u))\leq
\varphi(\|u-\overline u\|_{\infty})\ \hbox{for each} \ u, \overline u\in {\cal C}.$}\\
Let $u, \overline u \in{\cal C}$ and $h\in N(u).$ Then, there exists
$f(x,y)\in F(x,y,u(x,y))$ such that for each $(x,y)\in J,$ we have
$$
h(x,y)=\mu(x,y)+^{H}I_{\sigma}^{r}f(x,y).
$$
From $(H_3)$ it follows that
$$
H_d(F(x,y,u(x,y)),F(x,y,\overline u(x,y)))\leq
p(x,y)\varphi(\|u(x,y)-\overline u(x,y)\|_E).
$$
Hence, there exists $w(x,y)\in F(x,y,\overline u(x,y)$ such that
$$
\|f(x,y)-w(x,y)\|_E\leq p(x,y)\varphi(\|u(x,y)-\overline u(x,y)\|_E);\ (x,y)\in J.
$$
Consider $U:J\to {\cal P}(E)$ given by
$$
U(x,y)=\{w\in E: \|f(x,y)-w(x,y)\|_E\leq
p(x,y)\varphi(\|u(x,y)-\overline u(x,y)\|_E)\}.
$$
Since the multivalued operator $u(x,y)=U(x,y)\cap F(x,y,\overline
u(x,y))$ is measurable (see Proposition III.4 in \cite{CaVa}), there
exists a function $\overline f(x,y)$ which is a measurable selection
for $u$. So, ${\overline f}(x,y)\in F(x,y,\overline u(x,y)),$ and
for each $(x,y)\in J,$
$$
\|f(x,y)-{\overline f}(x,y)\|_E\leq p(x,y)\varphi(\|u(x,y)-\overline
u(x,y)\|_E).
$$
Let us define  for each $(x,y)\in J,$
$$
{\overline h}(x,y)=\mu(x,y)+^{H}I_{\sigma}^{r}\overline f(x,y).
$$
Then for each $(x,y)\in J,$ we have
\begin{eqnarray*}
\|h(x,y)-{\overline
h}(x,y)\|_E&\leq&^{H}I_{\sigma}^{r}\|f(x,y)-\overline f(x,y)\|_E\\
&\leq&^{H}I_{\sigma}^{r}(p(x,y)\varphi(\|u(x,y)-\overline u(x,y)\|_E))\\
&\leq&\|p\|_{L^{\infty}}\varphi(\|u-\overline u\|_\infty)
\left(\int_{1}^{x}\int_{1}^{y}\frac{\left|\log\frac{x}{s}\right|^{r_{1}-1}\left|\log\frac{y}{t}\right|^{r_{2}-1}}
{st\Gamma(r_1)\Gamma(r_2)}dtds\right)\\
&\leq&\frac{(\log a)^{r_1}(\log b)^{r_2}\|p\|_{L^{\infty}}
}{\Gamma(1+r_1)\Gamma(1+r_2)}\varphi(\|u-\overline u\|_\infty).
\end{eqnarray*}
Thus, by (\ref{e05}), we get
$$
\|h-{\overline h}\|_{\infty}\leq\varphi(\|u-\overline u\|_\infty).
$$
By an analogous relation, obtained by interchanging the roles of $u$
and $\overline u,$ it follows that
$$
H_d(N(u),N(\overline u))\leq\varphi(\|u-\overline u\|_\infty).
$$
Hence, $N$ is a $\varphi-$contraction.\\

{\bf (a)} By Lemma \ref{L0}, $N$ has a fixed point witch is a
solution of the inclusion (\ref{e1}) on $J,$ and by [Theorem
\ref{T0},(i)], $N$ is a $(MWPO).$\\

{\bf (b)} We will prove that the fixed point inclusion problem
(\ref{e1}) is generalized Ulam-Hyers stable. Indeed, let $\epsilon>
0$ and $v\in{\cal C}$ for which there exists $u\in{\cal C}$ such
that
$$
u(x,y)\in \mu(x,y)+(^{H}I_{\sigma}^{r}F)(x,y,v(x,y));\ \text{ if }(x,y)\in
J,
$$ and
$$
\|u-v\|_\infty\leq\epsilon.
$$
Then $H_d(v,N(v))\leq\epsilon.$ Moreover, by the above proof we have
that $N$ is a multivalued $\varphi-$contraction and using [Theorem
\ref{T0},(i)-(ii)], we obtain that $N$ is a is a $(\Psi-MWPO).$
Then, by Lemma \ref{L00} we obtain that the fixed point problem
$u\in N(u)$ is generalized Ulam-Hyers stable. Thus, the integral
inclusion (\ref{e1}) is generalized Ulam-Hyers stable.\\
Concerning the conclusion of the theorem, we apply [Theorem
\ref{T0},(iii)].

\section{An example}
Let
$$E=l^{1}=\left\{w=(w_{1},w_{2},\ldots,w_{n},\ldots):\sum_{n=1}^{\infty}|w_{n}|<\infty\right\},$$
be the Banach space with norm
$$\|w\|_{E}=\sum_{n=1}^{\infty}|w_{n}|,$$ and consider the following
partial functional fractional order integral inclusion of the form
\begin{equation}\label{ex1}
u(x,y)\in \mu(x,y)+(^{H}I_{\sigma}^{r}F)(x,y,u(x,y));\  \hbox{a.e.} \
(x,y)\in[1,e]\times[1,e],
\end{equation}
where $r=(r_1,r_2),\ r_1,r_2\in (0,\infty),$
$$
u=(u_{1},u_{2},\ldots,u_{n},\ldots),\ \mu(x,y)=(x+e^{-y},0,\ldots,0,\ldots),
$$
and
$$
F(x,y,u(x,y))
$$
$$
=\{v\in C([1,e]\times[1,e],{\Bbb R}): \|f_{1}(x,y,u(x,y))\|_E\leq \|v\|_E\leq \|f_{2}(x,y,u(x,y))\|_E\};
$$
$(x,y)\in[1,e]\times[1,e],$ where $f_1,f_2:[1,e]\times[1,e]\times E\to E,$
$$
f_k=(f_{k,1},f_{k,2},\ldots,f_{k,n},\ldots);\ k\in\{1,2\},\ n\in\N,$$
$$
f_{1,n}(x,y,u_n(x,y))=\frac{xy^{2}u_n}{(1+\|u_n\|_E)e^{10+x+y}};\
n\in\N,
$$
and
$$
f_{2,n}(x,y,u_n(x,y))=\frac{xy^{2}u_n}{e^{10+x+y}};\ n\in\N.
$$
We assume that $F$ is closed and convex valued. We can see that the
solutions of the inclusion(\ref{ex1}) are solutions of the fixed
point inclusion $u\in A(u)$ where $A:C([1,e]\times[1,e],{\Bbb
R})\to{\cal P}( C([1,e]\times[1,e],{\Bbb R}))$ is the multifunction
operator defined by
$$
(Au)(x,y)=\left\{\mu(x,y)+(^{H}I_{\sigma}^{r}f)(x,y);\ f\in
S_{F,u}\right\};\ (x,y)\in[1,e]\times[1,e].
$$
For each $(x,y)\in [1,e]\times[1,e]$ and all $z_1,z_2\in E,$ we have
$$
\|f_2(x,y,z_2)-f_1(x,y,z_1)\|_E\leq xy^{2}e^{-10-x-y}\|z_2-z_1\|_E.
$$
Thus, the hypotheses $(H_1)-(H_3)$ are satisfied with
$p(x,y)=xy^{2}e^{-10-x-y}.$ We shall show that condition (\ref{e05})
holds with $a=b=e.$ Indeed, $\|p\|_{L^{\infty}}=e^{-9},\
\Gamma(1+r_i)>\frac{1}{2};\ i=1,2.$ A simple computation shows that
$$
\zeta:=\frac{(\log a)^{r_{1}}(\log b)^{r_{2}}\|p\|_{L^{\infty}}}{\Gamma(1+r_{1})\Gamma(1+r_{2})}
<4e^{-9}<1.
$$
The condition $(H_4)$ is satisfied with
$q(y)=\frac{y^{2}e^{-10-y}}{\|F\|_{\cal P}};\ y\in[1,e],$ where
$$
\|F\|_{\cal P}=\Sup\{\|f\|_{\cal C}: f\in S_{F,u}\};\ \hbox{ for
all} \, u\in{\cal C}.
$$
Consequently, by Theorem \ref{T1} we concluded that:
\begin{itemize}
\item[$(a)$] The integral inclusion (\ref{ex1})
has least one solution and $A$ is a $(MWPO).$

\item[$(b)$] The function $\varphi:[0,\infty)\to[0,\infty)$ defined
by $\varphi(t)=\zeta t$ satisfies $\varphi(\zeta
t)\leq\zeta\varphi(t)$ for every $t\in[0,\infty)$. Then the integral inclusion
(\ref{ex1}) is generalized Ulam-Hyers stable, and $A$ is a
($\Psi-$MWPO), with the function $\Psi$ defined by
$\Psi(t):=t+(1-\zeta t)^{-1},$ for each $t\in[0,\zeta^{-1}).$
Moreover, the continuous data dependence of the solution set of the
integral inclusion (\ref{e5}) holds.
\end{itemize}

\bigskip

{\bf Acknowledgment.} For the last author this work benefits
 of the financial  support of a grant of the Romanian National Authority for Scientific Research, CNCS-UEFISCDI, 
Project number PN-II-ID-PCE-2011-3-0094.

\bigskip

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\end{document}
