\documentclass[reqno]{amsart}
\usepackage{hyperref}

\AtBeginDocument{{\noindent\small
\emph{Electronic Journal of Differential Equations},
Vol. 2016 (2016), No. 25, pp. 1--12.\newline
ISSN: 1072-6691. URL: http://ejde.math.txstate.edu or http://ejde.math.unt.edu
\newline ftp ejde.math.txstate.edu}
\thanks{\copyright 2016 Texas State University.}
\vspace{9mm}}

\begin{document}
\title[\hfilneg EJDE-2016/25\hfil 
 Random  Hadamard fractional integral equations]
{Existence and Ulam stabilities for  Hadamard fractional integral
equations with \\ random effects}

\author[S. Abbas, W. Albarakati, M. Benchohra, J. Henderson 
\hfil EJDE-2016/25\hfilneg]
{Sa\"id Abbas, Wafaa A. Albarakati, Mouffak Benchohra, Johnny Henderson}

\address{Sa\"id Abbas \newline
Laboratory of Mathematics, University of Sa\"ida,
P.O. Box 138, 20000 Sa\"ida, Algeria}
\email{abbasmsaid@yahoo.fr}

\address{Wafaa A. Albarakati \newline
Department of Mathematics, Faculty of Science,
King Abdulaziz University,
P.O. Box 80203, Jeddah 21589, Saudi Arabia}
\email{wbarakati@kau.edu.sa}

\address{Mouffak Benchohra \newline
Laboratory of Mathematics, University of Sidi Bel-Abb\`es,
 P.O. Box 89, Sidi Bel-Abb\`es 22000, Algeria. \newline
Department of Mathematics, Faculty of Science,
King Abdulaziz University,
P.O. Box 80203, Jeddah 21589, Saudi Arabia}
\email{benchohra@univ-sba.dz}

\address{Johnny Henderson \newline
Department of Mathematics, Baylor University,
Waco, Texas 76798-7328, USA}
\email{Johnny\_Henderson@baylor.edu}

\thanks{Submitted July 16, 2015. Published January 15, 2016.}
\subjclass[2010]{34A08, 34K05}
\keywords{Random integral equation; Hadamard  fractional integral;
\hfill\break\indent  fixed point; Ulam-Hyers-Rassias stability}

\begin{abstract}
 This article concerns the existence  and Ulam stabilities for a
 class of random  integral equations via Hadamard's fractional integral.
 Our main tools is a random fixed point theorem with stochastic domain.
\end{abstract}

\maketitle
\numberwithin{equation}{section}
\newtheorem{theorem}{Theorem}[section]
\newtheorem{lemma}[theorem]{Lemma}
\newtheorem{remark}[theorem]{Remark}
\newtheorem{definition}[theorem]{Definition}
\newtheorem{example}[theorem]{Example}
\allowdisplaybreaks

\section{Introduction}

Fractional calculus is a powerful tool in applied mathematics
to study many problems from fields of science and engineering.
It has produced many break-through results in mathematical physics, finance,
hydrology, biophysics, thermodynamics, control theory, statistical mechanics,
astrophysics, cosmology and bioengineering \cite{Hi,Tar}. 
There has been a significant development in ordinary and partial fractional 
differential and integral equations in recent years; see the monographs
 of Abbas  et al \cite{ABN,ABN1},
Kilbas  et al \cite{KST}, Miller and Ross \cite{MiRo}, and
the papers of Abbas et al \cite{AbBe1,AbBe2,AbBeVi}, Benchohra
 et al \cite{BeHeNtOu},  Vityuk  et al \cite{Vi,ViGo}, and the references therein.

Abbas  et al \cite{ABH1} obtained some existence and uniqueness results 
for determinist integral equations involving the Hadamard fractional 
integral of two independent variables. 
Butzer  et al \cite{BKT1} 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
Pooseh  et al \cite{PAT} 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.

The nature of a dynamic system in engineering or natural sciences depends 
on the accuracy of the information we have concerning the parameters that 
describe that system. If the knowledge about a dynamic system is precise 
then a deterministic dynamical system arises. Unfortunately in most cases 
the available data for the description and evaluation of parameters of a 
dynamic system are inaccurate, imprecise or confusing. 
In other words, evaluation of parameters of a dynamical system is not without 
uncertainties. When our knowledge about the parameters of a dynamic 
system are of statistical nature, that is, the information is probabilistic, 
the common approach in mathematical modeling of such systems is the use of 
random differential equations or stochastic differential equations. 
Random differential equations, as natural extensions of deterministic ones, 
arise in many applications and have been investigated by many mathematicians; 
see \cite{Li, LiS, Liu, Sta, Ya} and references therein. Between them differential 
equations with random coefficients (see, \cite{Sta, Ch}) offer a natural and 
rational approach (see \cite{c7}, Chapter 1), since sometimes we can obtain the 
random distributions of some main disturbances by historical experiences and
 data rather than take all random disturbances into account and assume the 
noise to be white noise.


The stability of functional equations was originally raised by Ulam
in 1940 in a talk given at Wisconsin University. The problem posed
by Ulam was the following: ``Under what conditions does there exist an
additive mapping near an approximately additive mapping?'' (for more
details see \cite{Ula}). The first answer to Ulam's
question was given by Hyers in 1941 in the case of Banach
spaces in \cite{Hy}. 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 concept of stability for a functional
equation arises when we replace the functional equation by an
inequality which acts as a perturbation of the equation. Thus, the
stability question of functional equations becomes, ``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 for all kinds of
functional equations; one can see the monographs of \cite{HIR,Jun}.
Bota-Boriceanu and Petrusel \cite{BP}, Petru et al \cite{PPY},
and Rus \cite{Rus,Rus1} who 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.
More details from a historical point of view, and
recent developments of such stabilities are reported
in \cite{Jun1,Rus} and  \cite{Wang1}-\cite{Wang5}.

 This article concerns the existence and  Ulam stability of solutions to
the  Hada\-mard fractional integral equation
\begin{equation}
\begin{aligned}
u(x,y,w)
&=\mu(x,y,w) +\int_1^x\int_1^y\left(\log\frac{x}{s}\right)^{r_1-1}
\left(\log\frac{y}{t}\right)^{r_{2}-1}\\
&\quad \times \frac{f(s,t,u(s,t,w),w)}{st}\,dt\,ds, \quad
 (x,y)\in J,\; w\in \Omega,
\end{aligned} \label{e1}
\end{equation}
where $J:=[1,a]\times [1,b]$, $a,b>1$, $r_1,r_2>0$, $(\Omega,\mathcal{A})$ 
is a measurable space,
$\mu:J\times\Omega\to \mathbb{R}$ and 
$f:J\times \mathbb{R} \times\Omega\to \mathbb{R}$
are given continuous functions.
In this article we obtain the existence and Ulam stabilities of random solutions
via fixed point techniques.

\section{Preliminaries}

 In this section, we introduce notation, definitions, and preliminary facts which
 are used throughout this article.
Denote by $L^{1}(J,\mathbb{R})$ the Banach space of functions
$u:J\to \mathbb{R}$ that are Lebesgue integrable, with norm
$$
\|u\|_{L^1}=\int_1^{a}\int_1^{b}|u(x,y)|dydx.
$$
Let $L^{\infty}(J)$ be the Banach space of functions $u:J\to \mathbb{R}$
which are measurable and essentially bounded. 
As usual, by $C:=C(J,\mathbb{R})$ we denote the Banach space of all continuous
functions $u:J\to \mathbb{R}$ with the norm
$$
\|u\|_{C}=\sup_{(x,y)\in J}|u(x,y)|.
$$

 Let $\beta_E $ be the $\sigma$-algebra of Borel subsets of $E$.
A mapping $v:\Omega\to E$ is said to be measurable if for any 
$B\in \beta_E$, one has
$$
v^{-1}(B)=\{w\in\Omega:v(w)\in B\}\subset \mathcal{A}.
$$
To define integrals of sample paths of random process, it is necessary
to define a jointly measurable map.

\begin{definition} \label{D0} \rm
	A mapping $T:\Omega\times E\to E$ is called jointly measurable
	if for any $B\in \beta_E$, one has
	$$
	T^{-1}(B)=\{(w,v)\in\Omega\times E:T(w,v)\in B\}
\subset \mathcal{A}\times\beta_E,
	$$
	where  $\mathcal{A}\times\beta_E $ is the direct product of the 
$\sigma$-algebras	$\mathcal{A}$ and $\beta_E$ those defined in $\Omega$
	and $E$ respectively.
\end{definition}

\begin{lemma}[\cite{DhBaNt}] \label{L00} 
Let $T:\Omega\times E\to E$ be a mapping such that $T(.,v)$
is measurable for all $v\in E$, and $T(w,\cdot)$ is continuous for all 
$w\in\Omega$. 
	Then the map $(w,v)\mapsto T(w,v) $ is jointly measurable.
\end{lemma}

\begin{definition}[\cite{HaMaDa}] \rm
	A function $f:J\times E\times\Omega\to E$ is called random Carath\'{e}odory
	if the following conditions are satisfied:
	\begin{itemize}
		\item[(i)] The map $(x,y,w)\to f(x,y,u,w)$ is jointly measurable for 
all $u\in E$, and

		\item[(ii)] The map $u\to f(x,y,u,w)$ is continuous for almost all
		$(x,y)\in J$ and $w\in\Omega$.
	\end{itemize}
\end{definition}

 Let $T:\Omega\times E\to E$ be a mapping. Then $T$ is
called a random operator if $T(w,u)$  is measurable in $w$ for all
$u\in E$ and it is expressed as $T(w)u=T(w,u)$. In this case we also
say that $T(w)$ is a random operator on $E$. A random operator
$T(w)$ on $E$ is called continuous (resp. compact, totally bounded
and completely continuous) if $T(w,u)$ is continuous (resp. compact,
totally bounded and completely continuous) in $u$ for all
$w\in\Omega$. The details of completely continuous random operators
in Banach spaces and their properties appear in Itoh \cite{It}.

\begin{definition}[\cite{En}] \rm
 Let $\mathcal{P}(Y)$ be the family of all
	nonempty subsets of $Y$ and $C$ be a mapping from $\Omega$ into
	$\mathcal{P}(Y)$. 
A mapping $T:\{(w,y):w\in\Omega,\; y\in C(w)\}\to Y$
	is called random operator with stochastic domain $C$ if $C$ is
measurable (i.e., for all closed $A\subset Y,\ \{w\in\Omega,
	C(w)\cap A\neq \emptyset\}$ is measurable) and for all open 
$D	\subset Y$ and all $y\in Y$, $\{w\in\Omega: y\in C(w),  T(w,y)\in
	D\}$ is measurable. $T$ will be called continuous if every $T(w)$ is
	continuous. For a random operator $T$, a mapping $y:\Omega\to Y$ is
	called random (stochastic) fixed point of $T$ if for $P$-almost all
	$w\in\Omega$, $y(w)\in C(w)$ and $T(w)y(w)=y(w)$ and for all open
	$D\subset Y$, $\{w\in\Omega: y(w)\in D\}$ is measurable.
\end{definition}

 Let $\mathcal{M}_X$ denote the class of all bounded subsets of a metric space $X$.

\begin{definition} \rm
	Let $X$ be a complete metric space. A map 
$\alpha:\mathcal{M}_X\to[0,\infty)$ is called a measure
	of noncompactness on $X$ if it satisfies the following properties for all 
$B,B_1,B_2 \in\mathcal{M}_X$.
	\begin{itemize}
		\item[(1)] $\alpha(B)=0$ if and only if $B$ is precompact (Regularity),
		
		\item[(2)] $\alpha(B)=\alpha(\overline{B})$ (Invariance under closure),
		
		\item[(3)] $\alpha(B_1\cup B_2)=\alpha(B_1)+\alpha(B_2)$   (Semi-additivity).
	\end{itemize}
\end{definition}

For more details on  measure of noncompactness and its properties see \cite{App}.

\begin{example} \label{examp2.6} \rm
	In every metric space $X$, the map $\phi:\mathcal{M}_X\to[0,\infty)$ with 
$\phi(B)=0$ if $B$ is relatively compact and $\phi(B)=1$ otherwise is a 
measure of noncompactness, the so-called discrete measure
	of noncompactness \cite[Example 1, p. 19]{AyDoLo}, .
\end{example}

\begin{definition}[\cite{Ha,KST}] \rm
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} \rm
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
$$
(^{H}I_{\sigma}^{r}w)(x,y)=\frac{1}{\Gamma (r_1)\Gamma (r_{2})}
\int_1^x\int_1^y\left(\log\frac{x}{s}\right)^{r_1-1}
\left(\log\frac{y}{t}\right)^{r_{2}-1}
\frac{w(s,t)}{st}\,dt\,ds.
$$
\end{definition}

 Now, we consider the Ulam stability for the Hadamard random integral 
equation \eqref{e1}.
Consider the operator $N:\Omega\times C\to C$ defined by:
\begin{equation}
\begin{aligned}
&(N(w)u)(x,y)\\
&=\mu(x,y,w) +\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,u(s,t,w),w)}{st}\,dt\,ds.
\end{aligned} \label{e8}
\end{equation}
Let $\epsilon$ be a positive real number and $\Phi:J\times\Omega\to [0,\infty)$ 
be a measurable and  bounded function. We consider the following inequalities:
\begin{gather}\label{e2}
|u(x,y,w)-(N(w)u)(x,y)|\leq\epsilon;\quad \text{ for a.a. }(x,y)\in J,\; w\in\Omega,\\
\label{e3}
|u(x,y,w)-(N(w)u)(x,y)|\leq\Phi(x,y,w);\quad \text{ for a.a. }(x,y)\in J,\;
 w\in\Omega, \\
\label{e4}
|u(x,y,w)-(N(w)u)(x,y)|\leq\epsilon\Phi(x,y,w);\quad \text{ for a.a. }(x,y)\in J,\;
 w\in\Omega. 
\end{gather}

\begin{definition}[\cite{ABN,Rus}] \label{D1} \rm
Equation  \eqref{e1} is Ulam-Hyers stable if there exists a real number 
$c_{N}>0$ such that	for each $\epsilon>0$ and for each random solution 
$u:\Omega\to C$ of the inequality \eqref{e2} there exists a random solution 
$v:\Omega\to C$  of the equation  \eqref{e1} with
	$$
|u(x,y,w)-v(x,y,w)|\leq \epsilon c_{N};\quad  (x,y)\in J,\; w\in\Omega.
$$
\end{definition}

\begin{definition}[\cite{ABN,Rus}] \label{D2} \rm
Equation  \eqref{e1} is generalized Ulam-Hyers stable if there exists
	$c_{N}\in C([0,\infty),[0,\infty))$ with $c_{N}(0)=0$ such that for 
each $\epsilon>0$ and for each	random solution $u:\Omega\to C$ 
of  inequality  \eqref{e2} there exists a random solution 
$v:\Omega\to C$ of the equation  \eqref{e1} with
	$$
|u(x,y,w)-v(x,y,w)|\leq c_{N}(\epsilon);\quad (x,y)\in J,\; w\in\Omega.
$$
\end{definition}

\begin{definition}[\cite{ABN,Rus}] \label{D3} \rm
Equation \eqref{e1} is 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 random solution 	$u:\Omega\to C$ of  inequality \emph{\eqref{e4}}
there exists a random solution $v:\Omega\to C$ of  equation \eqref{e1} with
	$$
|u(x,y,w)-v(x,y,w)|\leq \epsilon c_{N,\Phi}\Phi(x,y,w);\quad
 (x,y)\in J,\ w\in\Omega.
$$
\end{definition}

\begin{definition}[\cite{ABN,Rus}] \label{D4} \rm
Equation \eqref{e1} is generalized Ulam-Hyers-Rassias stable with respect 
to $\Phi$ if there exists 	a real number $c_{N,\Phi}>0$ such that for each 
random solution $u:\Omega\to C$  of  inequality \eqref{e3}
	there exists a random solution $v:\Omega\to C$ of  equation \eqref{e1} with
	$$
|u(x,y,w)-v(x,y,w)|\leq c_{N,\Phi}\Phi(x,y,w);\quad (x,y)\in J,\; w\in\Omega.
$$
\end{definition}

\begin{remark}\label{R1} \rm
	It is clear that: 
(i) Definition \ref{D1} implies Definition \ref{D2};
(ii) Definition \ref{D3}  implies Definition \ref{D4};
(iii) Definition \ref{D3} for 	$\Phi(.,.,.)=1$ implies 
 Definition \ref{D1}.
\end{remark}

One can have similar remarks for the inequalities \eqref{e2} and \eqref{e4}.
So, the Ulam stabilities of the fractional random
differential equations are some special types of data dependence of
the solutions of fractional differential equations.

\begin{lemma}[\cite{Bot}] \label{L4}
	If $Y$ is a bounded subset of a Banach space $X$, then for each 
$\epsilon>0$, there is a	sequence $\{y_k\}^{\infty}_{k=1}\subset Y$ such that
	$$
\alpha(Y)\leq2\alpha(\{y_k\}^{\infty}_{k=1})+\epsilon.
$$
\end{lemma}

\begin{lemma}[\cite{Mon,ZhSu}] \label{L5}
	If $\{u_k\}^{\infty}_{k=1}\subset L^{1}(J)$ is uniformly integrable, 
then the function $\alpha(\{u_k\}^{\infty}_{k=1})$ is measurable and for each $(x,y)\in J$,
	$$
\alpha\Big(\Big\{\int_1^x\int_1^yu_k(s,t)\,dt\,ds\Big\}^{\infty}_{k=1}\Big)
\leq2\int_1^x\int_1^y\alpha(\{u_k(s,t)\}^{\infty}_{k=1})\,dt\,ds.
	$$
\end{lemma}

\begin{lemma}[\cite{LGWW}] \label{L0}
	Let $F$ be a closed and convex subset of a real Banach space, let $G: F\to F$
	be a continuous operator and $G(F)$ be bounded.
	If there exists a constant $k\in [0,1)$ such that for each bounded subset
	$B\subset F$,
	$$
	\alpha(G(B))\leq k\alpha(B),
	$$
	then $G$ has a fixed point in $F$.
\end{lemma}

\section{Existence and Ulam stability results}

In this section, we discuss the existence of solutions and we present
conditions for the Ulam stability for the Hadamard integral equation \eqref{e1}.
The following hypotheses will be used in the sequel.
\begin{itemize}
\item[(H1)] The function $w\mapsto\mu(x,y,w)$ is measurable and bounded
for a.e. $(x,y)\in J$,
	
\item[(H2)] The function $f$ is random Carath\'{e}odory
on $J\times\mathbb{R}\times\Omega$,
	
\item [(H3)] There exist functions $p_1,p_2:J\times\Omega\to[0,\infty)$
with $p_i(w)\in C(J,\mathbb{R}_+);\ i=1,2$ such that for
each $w\in\Omega$,
\[
|f(x,y,u,w)|\leq p_1(x,y,w)+\dfrac{p_2(x,y,w)}{1+|u(x,y)|}|u(x,y,w)|,
\]
for all $u\in \mathbb{R}$ and a.e. $(x,y)\in J$,

\item[(H4)] There exists a function $q:J\times\Omega\to[0,\infty)$ with 
$q(w)\in L^{\infty}(J,[0,\infty))$
for each $w\in \Omega$ such that for any bounded $B\subset \mathbb{R}$,
$$
\alpha(f(x,y,B,w))\leq q(x,y,w)\alpha(B), \ for \ a.e. \ (x,y)\in J,
$$

\item[(H5)] There exists a random function $R:\Omega\to(0,\infty)$ such that
$$
R(w)\geq\mu^{*}(w)+\frac{(p_1^{*}(w)+p_2^{*}(w))(\log a)^{r_1}
(\log b)^{r_2}}{\Gamma (1+r_1)\Gamma (1+r_2)},
$$
where
$$
\mu^{*}(w)=\sup_{(x,y)\in J}|\mu(x,y,w)|,\ p_i^{*}(w)
=\operatorname{ess\,sup}_{(x,y)\in J}p_i(x,y,w);\ i=1,2,
$$

\item[(H6)] There exist $q_1,q_2:J\times\Omega\to[0,\infty)$
with $q_i(.,w)\in L^{\infty}(J,[0,\infty));\ i=1,2$ such that for
each $w\in\Omega$, and a.e. $(x,y)\in J$, we have
\[
p_i(x,y,w)\leq q_i(x,y,w,w)\Phi(x,y,w),
\]
	
\item [(H7)] $\Phi(w)\in L^{1}(J,[0,\infty))$ for all $w\in\Omega$, and there
exists $\lambda_{\Phi}>0$ such that for each
$(x,y)\in J$, we have
$$
(^{H}I_{\sigma}^{r}\Phi)(x,y,w)\leq\lambda_\Phi\Phi(x,y,w).
$$
\end{itemize}
 Set
$$q^{*}=\operatorname{ess\,sup}_{(x,y,w)\in J\times\Omega}q(x,y,w).
$$

\begin{theorem}\label{T1}
 Assume that hypotheses {\rm (H1)--(H5)} hold. If
$$
\ell:=\frac{4q^{*}(\log a)^{r_1}(\log b)^{r_2}}{\Gamma(1+r_1)\Gamma(1+r_2)}<1,
$$
then the integral equation  \eqref{e1} has a random solution defined on $J$.
Furthermore, if the hypotheses {\rm (H6)} and {\rm (H7)} hold,
then the random equation  \eqref{e1} is generalized
Ulam-Hyers-Rassias stable.
\end{theorem}

\begin{proof}
 Let $N$ be the operator defined in \eqref{e8}.
From the hypotheses (H2) and (H3), for each $w\in\Omega$ and almost all 
$(x,y)\in J$, we have that $f(x,y,u(x,y,w),w) $ is in $L^{1}$.
Since the function $f$ is continuous, then the indefinite integral is continuous
for all $w\in\Omega$  and almost all $(x,y)\in J$.
Again, as the map $\mu$ is continuous for all $w\in\Omega$ and the
indefinite integral is continuous on $J$, then $N(w)$ defines a mapping
$N:\Omega\times C\to C$. Hence $u$ is a solution for the
integral equation  \eqref{e1} if and only if $u=(N(w))u$. 
We shall show that the operator $N$  satisfies all conditions of Lemma \ref{L0}.
The proof will be given in several steps.
\smallskip

\noindent\textbf{Step 1:}  $N(w)$ is a random operator with stochastic domain 
on $C$.  Since $f(x,y,u,w)$ is random Carath\'{e}odory, the map 
$w\to f(x,y,u,w)$ is measurable in view of Definition \ref{D0}. Similarly,
the product 
$\left(\log\frac{x}{s}\right)^{r_1-1}\left(\log\frac{y}{t}\right)^{r_{2}-1}
\frac{f(s,t,u(s,t,w),w)}{st}$ of a
continuous and a measurable function is again measurable. Further,
the integral is a limit of a finite sum of measurable functions,
therefore, the map
$$
w\mapsto\mu(x,y,w)+\int_1^x\int_1^y
\Big(\log\frac{x}{s}\Big)^{r_1-1}\Big(\log\frac{y}{t}\Big)^{r_{2}-1}
\frac{f(s,t,u(s,t,w),w)}{st\Gamma(r_1)\Gamma(r_2)}\,dt\,ds
$$
is measurable. As a result, $N$ is a random operator on
$\Omega\times C$ into $C$.

 Let $W:\Omega\to\mathcal{P}(C)$ be defined by
$$
W(w)=\{u\in C:\|u\|_C\leq R(w)\},
$$
with $W(w)$ bounded, closed, convex and solid for all $w\in\Omega$.
Then $W$ is measurable by  \cite[Lemma 17]{En}. Let
$w\in\Omega$ be fixed, then from (H4), for any $u\in w(w)$, we obtain
\begin{align*}
	&|(N(w)u)(x,y)|\\
	&\leq|\mu(x,y,w)|+\int_1^x\int_1^y
\big|\log\frac{x}{s}\big|^{r_1-1}\big|\log\frac{y}{t}\big|^{r_{2}-1}
\frac{|f(s,t,u(s,t,w),w)|}{st\Gamma(r_1)\Gamma(r_2)}\,dt\,ds\\
&\leq|\mu(x,y,w)|+\int_1^x\int_1^y
	\big|\log\frac{x}{s}\big|^{r_1-1}\big|\log\frac{y}{t}\big|^{r_{2}-1}
	\frac{|p_1(s,t,w)+p_2(s,t,w)|}{\Gamma(r_1)\Gamma(r_2)}\,dt\,ds\\
&\leq\mu^{*}(w)+\frac{(p_1^{*}(w)+p_2^{*}(w))(\log a)^{r_1}
 (\log b)^{r_2}}{\Gamma (1+r_1)\Gamma (1+r_2)}\\
&\leq R(w).
\end{align*}
Therefore, $N$ is a random operator with stochastic domain $W$
and $N(w):W(w)\to N(w)$. Furthermore, $N(w)$ maps bounded sets into bounded 
sets in $C$.
\smallskip

\noindent\textbf{Step 2:}  $N(w)$ is continuous.
Let $\{u_n\}$ be a sequence such that $u_n\to u$ in
$\mathcal{C}$. Then, for each $(x,y)\in J$ and $w\in \Omega$, we have
\begin{align*}
&|(N(w)u_n)(x,y)-(N(w)u)(x,y)|\\
&\leq\int_1^x\int_1^y
\big|\log\frac{x}{s}\big|^{r_1-1}\big|\log\frac{y}{t}\big|^{r_{2}-1}\\
& \times \frac{|f(s,t,u_n(s,t,w),w)-f(s,t,u(s,t,w),w)|}
 {\Gamma(r_1)\Gamma(r_2)}\,dt\,ds.
\end{align*}
Using the Lebesgue Dominated Convergence Theorem, we obtain
$$
\|N(w)u_n-N(w)u\|_{C} \to 0\quad \text{as }n\to \infty.
$$
As a consequence of Steps 1 and 2, we can conclude that $N(w):W(w)\to N(w)$
is a continuous random operator with stochastic domain $W$,
and $N(w)(W(w))$ is bounded.
\smallskip

\noindent\textbf{Step 3:}  For each bounded subset $B$ of $W(w)$ we have
 $\alpha(N(w)B)\leq \ell\alpha(B)$.
Let $w\in\Omega$ be fixed. From
Lemmas \ref{L4} and \ref{L5}, for any $B\subset W$ and any
$\epsilon>0$, there exists a sequence $\{u_n\}^{\infty}_{n=0}\subset
B$, such that for all $(x,y)\in J$, we have
\begin{align*}
&\alpha((N(w)B)(x,y))\\
&=\alpha\Big(\Big\{\mu(x,y)+
	\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,u(s,t,w),w)}{st\Gamma (r_1)\Gamma (r_{2})}\,dt\,ds;\\
& u\in B  \Big\}\Big)\\
&\leq 2\alpha\Big(\Big\{\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,u_n(s,t,w),w)}{st\Gamma (r_1)\Gamma (r_{2})}\,dt\,ds
\Big\}^{\infty}_{n=1}\Big)+\epsilon\\
&\leq 4\int_1^x\int_1^y
	\alpha\Big(\Big\{\left(\log\frac{x}{s}\right)^{r_1-1}
\left(\log\frac{y}{t}\right)^{r_{2}-1} \\
&\quad\times	\frac{f(s,t,u(s,t,w),w)}{st\Gamma (r_1)\Gamma (r_{2})}\,dt\,ds
\Big\}^{\infty}_{n=1}\Big)\,dt\,ds +\epsilon\\
&\leq 4\int_1^x\int_1^y\left(\log\frac{x}{s}\right)^{r_1-1}
 \left(\log\frac{y}{t}\right)^{r_{2}-1} \\
&\quad\times	\frac{1}{\Gamma (r_1)\Gamma (r_{2})}
	\alpha\left(\{f(s,t,u_n(s,t,w),w)\}^{\infty}_{n=1}\right)\,dt\,ds+\epsilon\\
&\leq 4\int_1^x\int_1^y
	\left(\log\frac{x}{s}\right)^{r_1-1}\left(\log\frac{y}{t}\right)^{r_{2}-1}\\
&\quad\times 	\frac{1}{\Gamma (r_1)\Gamma (r_{2})}
	q(s,t,w)\alpha\left(\{u_n(s,t,w)\}^{\infty}_{n=1}\right)\,dt\,ds+\epsilon\\
&\leq \Big(4\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{1}{\Gamma (r_1)\Gamma (r_{2})}
	q(s,t,w)dsdt\Big)\alpha\left(\{u_n\}^{\infty}_{n=1}\right)\\
&\quad +\epsilon\\
&\leq \Big(4\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{1}{\Gamma (r_1)\Gamma (r_{2})}q(s,t,w)\,dt\,ds\Big)\alpha(B)+\epsilon\\
&\leq \frac{4q^{*}(\log a)^{r_1}(\log b)^{r_2}}{\Gamma(1+r_1)\Gamma(1+r_2)}\alpha(B)+\epsilon\\
&=\ell\alpha(B)+\epsilon.
\end{align*}
Since $\epsilon>0$ is arbitrary, 
$\alpha(N(B))\leq \ell\alpha(B)$.
 Hence, from Lemma \ref{L0} it follows that for each $w\in\Omega$, $N$
has at least one fixed point in $W$.
Since $\bigcap_{w\in\Omega}int W(w)\neq\emptyset$ the hypothesis that a 
measurable selector
of $int W$ exists holds. By Lemma \ref{L0}, $N$ has a stochastic fixed point, i.e.,
the integral equation \eqref{e1} has at least one random solution on $C$.
\smallskip

\noindent\textbf{Step 4:}  The generalized Ulam-Hyers-Rassias stability. Set
$$
q_i^{*}=\operatorname{ess\,sup}_{(x,y,w)\in J\times\Omega}q_i(x,y,w);\quad i=1,2.
$$
Let $u:\Omega\to C$ be a solution of  inequality
\eqref{e3}. By Theorem \ref{T1}, there exists $v$ which is a
solution of the random equation \eqref{e1}. Hence
\begin{align*}
v(x,y,w)
&=\mu(x,y,w) +\int_1^x\int_1^y\left(\log\frac{x}{s}\right)^{r_1-1}
\left(\log\frac{y}{t}\right)^{r_{2}-1} \\
&\quad\times \frac{f(s,t,v(s,t,w),w)}{st\Gamma (r_1)\Gamma (r_{2})}\,dt\,ds;\quad
 (x,y)\in J,\; w\in\Omega.
\end{align*}
From inequality \eqref{e3} and  hypotheses (H6), (H7),
 for each $(x,y)\in J$ and $w\in\Omega$, it follows that
\begin{align*}
&|u(x,y,w)-v(x,y,w)|\leq|u(x,y,w)-N(w)(u)|+|N(w)(u)-N(w)(v)|\\
&\leq \Phi(x,y,w)+\int_1^x\int_1^y\big|\log\frac{x}{s}\big|^{r_1-1}
 \big|\log\frac{y}{t}\big|^{r_{2}-1} \\
&\quad\times \frac{|f(s,t,u(s,t,w))-f(s,t,v(s,t,w))|}{\Gamma (r_1)\Gamma (r_{2})}\,dt\,ds\\
&\leq\Phi(x,y,w)
+\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}\\
&\quad \times\left(2q_1^*+\frac{q_2^*|u(s,t,w)|}{1+|u|}
+\frac{q_2^*|v(s,t,w)|}{1+|v|}\right)\dfrac{\Phi(s,t,w)}{st}\,dt\,ds\\
&\leq\Phi(x,y,w)+2(q_1^*+q_2^*)(^{H}I_{\sigma}^{r}\Phi)(x,y,w)\\
&\leq[1+2(q_1^*+q_2^*)\lambda_\phi]\Phi(x,y,w)\\
&:=c_{N,\Phi}\Phi(x,y,w).
\end{align*}
Hence, the random equation \eqref{e1} is generalized
Ulam-Hyers-Rassias stable.
\end{proof}

\section{An example}

Let $E=\mathbb{R}$ and $\Omega=(-\infty,0)$ be equipped with the usual
$\sigma$-algebra consisting of Lebesgue measurable subsets of
$(-\infty,0)$. Given a measurable function $u:\Omega\to
C([1,e]\times[1,e])$, consider the following  partial random
Hadamard integral equation of the form
\begin{equation}\label{ex}
\begin{aligned}
u(x,y,w)&=\mu(x,y,w)
+\int_1^x\int_1^y\left(\log\frac{x}{s}\right)^{r_1-1}
\left(\log\frac{y}{t}\right)^{r_{2}-1} \\
&\quad\times \frac{f(s,t,u(s,t,w),w)}{st\Gamma (r_1)\Gamma (r_{2})}\,dt\,ds,
\end{aligned}
\end{equation}
for $(x,y)\in [1,e]\times[1,e],\ w\in \Omega$,
where $r_1,r_2>0$, $\mu(x,y,w)=x\sin w+y^{2}\cos w$; 
$(x,y)\in[1,e]\times[1,e]$, and
$$
f(x,y,u(x,y))=\frac{w^{2}xy^{2}}{(1+w^{2}+u(x,y,w)|)e^{x+y+3}}, \quad
(x,y)\in [1,e]\times [1,e],\; w\in \Omega.
$$
The function
$w\mapsto\mu(x,y,w)=x\sin w +y^{2}\cos w$ is measurable
and bounded with
$$
|\mu(x,y,w)|\leq e+e^{2},
$$
hence,  condition (H1) is satisfied.

The mapping $(x,y,w)\mapsto f(x,y,u,w)$ is jointly continuous
for all $u\in\mathbb{R}$ and hence jointly measurable for all
$u\in\mathbb{R}$. Also the map $u\mapsto f(x,y,u,w)$ is continuous for
all $(x,y)\in [1,e]\times[1,e]$ and $w\in\Omega$. So the function $f$ is
Carath\'{e}odory on $[1,e]\times[1,e]\times\mathbb{R}\times\Omega$.
For each $u\in\mathbb{R},\ (x,y)\in [1,e]\times[1,e]$ and $w\in \Omega$, we have
$$
|f(x,y,u,w)|\leq w^{2}xy^{2}(1+\frac{1}{e^{3}}|u|).
$$
Hence the condition (H3) is satisfied with $p_1^{*}=e^{3}$
and $p_1(x,y,w)=p_2^{*}=1$.

 We shall show that the condition $\ell<1$ holds with $a=b=e$ and
$q^{*}=\frac{1}{e^3}$. Indeed,
for each $r_1,r_2>0$ we obtain
\[
\ell=\frac{4q^*(\log a)^{r_1}(\log b)^{r_2}}{\Gamma (1+r_1)\Gamma (1+r_2)}
\leq\frac{4}{e^{3}\Gamma(1+r_1)\Gamma(1+r_2)} <1.
\]
 Also, the hypothesis (H6) is satisfied with
$$
\Phi(x,y,w)=w^{2}w^{2}xy^{2},\quad\text{and}\quad
\lambda_\Phi=\frac{1}{\Gamma(1+r_1)\Gamma(1+r_2)}.
$$
Indeed, for each $(x,y)\in [1,e]\times[1,e]$  we obtain
\[
(^{H}I_{\sigma}^{r}\Phi)(x,y,w)
\leq \frac{w^{2}e^{3}}{\Gamma(1+r_1)\Gamma(1+r_2)} \\
 = \lambda_\Phi\Phi(x,y,w).
\]
Finally, we can see that the hypothesis (H7) is satisfied with
$$
q_1(x,y,w)=1 \quad \text{and} \quad
 q_2(x,y,w)=\frac{1}{e^{3}}.
$$
 Consequently Theorem \ref{T1} implies that the Hadamard integral equation
\eqref{ex} has a solution defined on $[1,e]\times[1,e]$,
and \eqref{ex} is generalized Ulam-Hyers-Rassias stable.

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\end{document}