\documentclass[reqno]{amsart}
\usepackage{hyperref}
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\AtBeginDocument{{\noindent\small
\emph{Electronic Journal of Differential Equations},
Vol. 2017 (2017), No. 110, pp. 1--12.\newline
ISSN: 1072-6691. URL: http://ejde.math.txstate.edu or http://ejde.math.unt.edu}
\thanks{\copyright 2017 Texas State University.}
\vspace{8mm}}

\begin{document}
\title[\hfilneg EJDE-2017/110\hfil Solution to random differential equations]
{Solution to random differential equations with boundary conditions}

\author[F. Tchier, C. Vetro, F. Vetro  \hfil EJDE-2017/110\hfilneg]
{Fairouz Tchier, Calogero Vetro, Francesca Vetro}

\address{Fairouz Tchier \newline
Mathematics Department College of Science (Malaz),
King Saud University, PO Box 22452,
Riyadh, Saudi Arabia}
\email{ftchier@ksu.edu.sa}

\address{Calogero Vetro (corresponding author) \newline
Department of Mathematics and Computer Science,
University of Palermo,
Via Archirafi 34, 90123 Palermo, Italy}
\email{calogero.vetro@unipa.it}

\address{Francesca Vetro \newline
Department of Energy, Information Engineering and Mathematical Models (DEIM),
University of Palermo,
Viale delle Scienze, 90128 Palermo, Italy}
\email{francesca.vetro@unipa.it}

\dedicatory{Communicated by Mokhtar Kirane}

\thanks{Submitted February 15, 2017. Published April 25, 2017.}
\subjclass[2010]{35R60, 47H10}
\keywords{Measurable space; random differential equation;
\hfill\break\indent random fixed point; vector-valued metric}

\begin{abstract}
 We study a family of random differential equations with boundary conditions.
 Using a random fixed point theorem, we prove an existence theorem that yields
 a unique random solution.
\end{abstract}

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

\section{Introduction}

Let $C([0, 1],\mathbb{R})$ be the set of all continuous real-valued functions
on $[0, 1]$ endowed with the partial order relation:
$x, y \in C([0, 1],\mathbb{R})$, $x \precsim y$ if and only if
 $x(t) \leq y(t)$ for every $t \in [0, 1]$. This is a relation which we can
extend in $C([0, 1],\mathbb{R}) \times C([0, 1],\mathbb{R})$ as follows:
$$
(x, y), (u, v) \in C([0, 1],\mathbb{R}) \times C([0, 1],\mathbb{R}),
\quad (x, y) \precsim (u, v) \Longleftrightarrow x \precsim u, \ y \precsim v.
$$
In this article, we study the following nonlinear boundary value problem
for system of random differential equations:
\begin{equation}\label{EP}
\begin{gathered}
 x''(\omega,t)= f_1(\omega,t,x(\omega,t),y(\omega,t)), \quad
 \text{$0<t <1$, $\omega \in \Omega$},
 \\
 y''(\omega,t)= f_2(\omega,t,x(\omega,t,),y(\omega,t)), \quad
 \text{$0<t <1$, $\omega \in \Omega$},
 \\
 x(\omega,0) =0, \quad x(\omega,1)=\psi_1\Big(\int_0^1 x(\omega,t)dt\Big), \quad 
\omega \in \Omega, \quad \psi_1 \in C(\mathbb{R},\mathbb{R}), \\
 y(\omega,0) =0, \quad y(\omega,1)=\psi_2\Big(\int_0^1 y(\omega,t)dt\Big), \quad 
\omega \in \Omega,\quad \psi_2 \in C(\mathbb{R},\mathbb{R}),
 \end{gathered}
\end{equation}
where $f_1 , f_2 :\Omega \times [0,1] \times \mathbb{R} \times \mathbb{R} 
\to \mathbb{R}$ are two functions with some regularity properties.
By a random solution of system \eqref{EP}, we mean a couple of measurable 
functions $(x,y) : \Omega \to C([0, 1],\mathbb{R}) \times C([0, 1],\mathbb{R})$ 
satisfying \eqref{EP}. The interest for such a kind of equations is motivated 
as follows (see also the books of Bharucha-Reid \cite{BR} and Skorohod \cite{Sk}): 
the mathematical model representation
of natural phenomena arising in biology, physics, engineering processes deal 
with specific parameters which may assume unknown values. If we want to take 
into account this uncertainty, a way to model it is based on the parameter 
$\omega \in \Omega$. From \eqref{EP}, in absence of $\omega$, we retrieve the system
\begin{equation}\label{EL1}
\begin{gathered}
 x''(t) = f_1(t,x(t),y(t)), \quad \text{$0<t <1$},
 \\
 y''(t) = f_2(t,x(t),y(t)), \quad \text{$0<t <1$},
 \\
 x(0) =0, \quad x(1)=\psi_1\Big(\int_0^1 x(t)dt\Big), \quad 
\psi_1 \in C(\mathbb{R},\mathbb{R}),\\
 y(0) =0, \quad y(1)=\psi_2\Big(\int_0^1 y(t)dt\Big), \quad 
\psi_2 \in C(\mathbb{R},\mathbb{R}),
 \end{gathered}
\end{equation}
where $f_1 , f_2 \in C([0,1] \times \mathbb{R} \times \mathbb{R} , \mathbb{R})$. 
So by a solution of system \eqref{EL1}, we mean a couple of functions 
$(x, y) \in C([0, 1],\mathbb{R}) \times C([0, 1],\mathbb{R})$ satisfying \eqref{EL1}. 
Precisely, by using Green's function from the literature, the couple of solutions 
is such that
\begin{gather*}
 x(t) = \int_0^1 K(t,s)f_1(s,x(s),y(s))ds + \psi_1\Big(\int_0^1 x(s)ds\Big)t, 
\quad 0<t <1, \\
 y(t) = \int_0^1 K(t,s)f_2(s,x(s),y(s))ds + \psi_2\Big(\int_0^1 y(s)ds\Big)t, \quad
 0<t <1,
\end{gather*}
where
\begin{equation}\label{En}
K(t,s)=\begin{cases}
 -t(1-s), & \text{$0 \leq t \leq s \leq 1$}, \\
-s(1-t), & \text{$0 \leq s \leq t \leq 1$}. 
 \end{cases}
\end{equation}
System \eqref{EL1} and its equation version are largely studied by many authors, 
with different local and nonlocal conditions. Here, we recall some interesting 
contributions from the existing literature. Multi-point boundary
value problems were studied by Moshinsky \cite{Mo} and Palamides \cite{pa}. 
Existence, localization and multiplicity of solutions for systems of local 
and nonlocal boundary value problems were proved by Agarwal-O'Regan-Wong 
\cite{AOW,AOW1,AOW2}, Bolojan-Nica-Infante-Precup
\cite{BIP}, Henderson-Ntouyas-Purnaras \cite{HNP}, Precup \cite{Pr,Pr1,Pr2}. 
An interesting way of studying differential equations makes use of the fixed 
point theory. For instance, Nieto-Rodr\'{i}guez-L\'{o}pez \cite{NR,NR1} 
studied ordinary differential equations via fixed point theorems in partially 
ordered sets. On the other hand, few authors have investigated the case of random 
differential equations. Here we recall the recently published papers of 
Li-Duan \cite{LD}, Nieto-Ouahab-Rodr\'iguez-L\'opez \cite{NOR} (which is the 
main inspiration of this work) and Sinacer-Nieto-Ouahab \cite{SNQ}. 
These authors consider the problem of fixed points for random operators and 
use this problem to study an equivalent problem of solutions for random 
differential equations. In the references of \cite{NOR,SNQ}, the reader 
can find a good list of manuscripts which point out the cornerstones in the 
development of random fixed point theory and applications; for instance, 
we refer to Itoh \cite{It} and Papageorgiou \cite{Pa}.

In this paper, using iterative methods from the fixed point theory, together 
with the theory of measurable spaces and monotone operators, we study 
problem \eqref{EP}. Precisely, first we prove three
abstract results which are general random fixed point theorems, then we work with
suitable integral operators associated to a large family of random differential 
equations, finally we deduce the existence of a unique random solution 
for problem \eqref{EP}.

\section{Preliminaries}

 In this section, we collect some basic notions and notation from the literature. 
By $\mathcal{B}(X)$ we mean the Borel $\sigma$-algebra on a metric space $X$. 
Given a measurable space $(\Omega, \Sigma)$, by $\Sigma \otimes \mathcal{B}(X)$ 
we mean the smallest $\sigma$-algebra on $\Omega \times X$
containing all the sets $M \times B$ (such that $M \in \Sigma$ and 
$B \in \mathcal{B}(X)$).

 \begin{definition} \label{def2.1}\rm
Let $(\Omega, \Sigma)$ be a measurable space, $X$ and $Y$ two metric spaces. 
A mapping $\widehat{h} : \Omega \times X \to Y$ is called Carath\'{e}odory if, 
for all $x \in X$, the mapping $\omega \to \widehat{h}(\omega,x)$ is 
$(\Sigma, \mathcal{B}(Y))$-measurable $(\Sigma$-measurable, for short$)$ and, 
for all $\omega \in \Omega$, the mapping $x \to \widehat{h}(\omega,x)$ is continuous.
\end{definition}

We need the following results from Denkowski-Mig\'{o}rski-Papageorgiou \cite{DMP}.

\begin{theorem}[{\cite[Theorem 2.5.22]{DMP}}]
If $(\Omega, \Sigma)$ is a measurable space, $X$ is a separable metric space, 
$Y$ is a metric space and $\widehat{h} : \Omega \times X \to Y$ is a 
Carath\'{e}odory mapping, then $\widehat{h}$ is 
$\Sigma \otimes \mathcal{B}(X)$-measurable.
\end{theorem}

\begin{corollary}[{\cite[Corollary 2.5.24]{DMP}}]\label{C1}
If $(\Omega, \Sigma)$ is a measurable space, $X$ is a separable metric space, 
$Y$ is a metric space, $\widehat{h} : \Omega \times X \to Y$ is a Carath\'{e}odory
 mapping and $u : \Omega \to X$ is $\Sigma$-measurable, then 
$\omega \to \widehat{h}(\omega, u(\omega))$ is a $\Sigma$-measurable mapping from 
$\Omega$ into $Y$.
\end{corollary}

Let $(\Omega, \Sigma)$ be a measurable space, $X$ a separable metric space and 
$Y$ a metric space. A mapping $\widetilde{h} : \Omega \times X \to Y$ is said to 
be superpositionally measurable (sup-measurable, for short), if for all 
$\Sigma$-measurable mapping $u : \Omega \to X$, the mapping 
$\omega \to \widetilde{h}(\omega, u(\omega))$ is $\Sigma$-measurable from $\Omega$ 
into $Y$. From Corollary \ref{C1} we deduce that a Carath\'{e}odory mapping is
 sup-measurable. Also every $\Sigma \otimes \mathcal{B}(X)$-measurable mapping 
is sup-measurable 
(see  Denkowski-Mig\'{o}rski-Papageorgiou \cite[Remark 2.5.26]{DMP}).
 Moreover, a mapping $f : \Omega \times X \to X$ is called random operator whenever,
for any $x \in X$, $\omega \to f (\omega, x)$ is $\Sigma$-measurable. 
So, a random fixed point of $f$ is a $\Sigma$-measurable mapping $z : \Omega \to X$ 
such that $z(\omega) = f(\omega,z(\omega))$ for all $\omega \in \Omega$.

\begin{lemma}\label{L1}
Let $X, Y$ be two locally compact metric spaces. A mapping 
$f : \Omega \times X \to Y$ is Carath\'{e}odory if and only if the mapping 
$\omega \to r(\omega)(\cdot) = f (\omega, \cdot)$ is
$\Sigma$-measurable from $\Omega$ to $C(X,Y)$ (i.e., the space of all 
continuous functions from $X$ into $Y$ endowed with the compact-open
topology).
\end{lemma}


\section{Fixed point theorems}

In this section we prove three theorems producing the existence and uniqueness 
of a random fixed point for a given mapping $f : \Omega \times X \to X$, 
where $\Omega$ and $X$ are two nonempty sets.

Later on, we use the following notation. 
If $(\Omega, \Sigma)$ is a measurable space and $X$ a metric space, then 
we denote by $X^\Omega$ the family of all mappings from $\Omega$ into $X$ 
and by $\mathcal{M}(\Omega, X)$ the subset of $X^\Omega$ containing all 
$\Sigma$-measurable mappings. If $X$ is endowed with a partial order $\precsim$, 
then the mappings $g,h \in X^\Omega$ are comparable if, for every 
$\omega \in \Omega$, we have $g(\omega) \precsim h(\omega)$ or 
$h(\omega) \precsim g(\omega)$.
Let $h_0 \in X^\Omega$, if $h_n(\omega)= f(\omega, h_{n-1}(\omega))$ for all 
$\omega \in \Omega$ and $n \in \mathbb{N}$, then we say that $\{h_n\}$ 
is a Picard sequence starting at $h_0$ and $\{h_n(\omega)\}$ is a 
Picard sequence (associate to $\omega$) starting at $h_0(\omega)$.

 The hypotheses on the data of the random fixed point problem are the following:
 \begin{itemize}
 \item[(H0)] $(\Omega, \Sigma)$ is a measurable space, $(X, d,\precsim)$ 
is a separable complete
ordered metric space, and $f : \Omega \times X \to X$ is a random mapping 
such that, for
each $\omega \in \Omega$, $x \to f (\omega, x)$ is a monotone operator;

\item[(H1)] for each $\omega \in \Omega$, there exists a nondecreasing 
function $r_\omega : [0, + \infty[ \to [0, + \infty[$ such that 
$\lim_{n \to + \infty} r_\omega^n(t) =0$ for all $t>0$ and
$$
d(f(\omega, x), f(\omega, y)) \leq r_\omega(d(x, y)),\quad \text{for all } x,y \in X,
 \; x \precsim y;
$$

\item[(H2)] there exists a mapping $x_0 \in \mathcal{M}(\Omega, X)$ with
 ``$x_0(\omega) \precsim f(\omega, x_0(\omega))$, for each $\omega \in \Omega$''
or ``$x_0(\omega) \succsim f(\omega, x_0(\omega))$, for each 
$\omega \in \Omega$'';

\item[(H3)] if $\{x_n\}$ is a monotone sequence in $X$ and $x_n \to x$,
then $x_n$ and $x$ are comparable for all $n \in \mathbb{N}$.
\end{itemize}

 \begin{remark} \rm
Hypothesis (H0) characterizes the space setting that we will use here and 
the monotonic behaviour of the mapping $x \to f (\omega, x)$. 
Hypothesis (H1) is a contraction condition of Matkowski type (see \cite{Ma}).
\end{remark}

\begin{remark} \rm
 Hypothesis (H3) is a regularity condition of the partial order relation, 
that needs to be satisfied whenever we do not assume that $f$ is Carath\'{e}odory.
 \end{remark}

First we establish our theorem with complete proof in the case that $f$ is a 
Carath\'{e}odory mapping. Then, we state the analogous result without this 
assumption.

\begin{theorem}\label{T1}
 If {\rm (H0)--(H2)} hold and $f$ is a Carath\'{e}odory mapping, then there 
exists $z \in \mathcal{M}(\Omega, X)$ which is a random fixed point of $f$. 
Further, if for all $x, y \in \mathcal{M}(\Omega, X)$, there exists 
$u \in X^\Omega$ that is comparable to $x$ and $y$, then $z$ is a unique 
random fixed point of $f$.
\end{theorem}

\begin{proof}
Let $x_0$ and $u_0$ be two comparable elements of $X^\Omega$. 
We consider the Picard sequences $\{x_n\}$ and $\{u_n\}$ starting respectively 
at $x_0$ and $u_0$.
 We claim that
\begin{equation}\label{E0}
\lim_{n \to + \infty} d(x_n(\omega), u_n(\omega))=0, \quad \text{for all } 
\omega \in \Omega.
\end{equation}
Let $\omega \in \Omega$ be fixed, since $x \to f (\omega, x)$ is a monotone 
operator, we obtain that
$x_n(\omega)$ and $u_n(\omega)$ are comparable for each $n \in \mathbb{N}$. 
Clearly, \eqref{E0} holds if $x_n(\omega)=u_n(\omega)$ for some $n \in \mathbb{N}$. 
Thus we assume that $x_n(\omega) \neq u_n(\omega)$ for all $n \in \mathbb{N}$. 
Then by  (H1), we have
\begin{equation}\label{Ec}
d(x_n(\omega), u_{n}(\omega))
\leq r_\omega(d(x_{n-1}(\omega), u_{n-1}(\omega))) 
\leq r_\omega^n(d(x_{0}(\omega), u_{0}(\omega)))
\end{equation}
for all $n \in \mathbb{N}$.
From \eqref{Ec}, using the property of the function $ r_\omega$ (see (H1)),
if we pass to the limit as $n \to + \infty$, we obtain
$$
\lim_{n \to + \infty} d(x_n(\omega), u_n(\omega))=0.
$$
Clearly, this holds for all $\omega \in \Omega$. Next, let 
$x_0 \in \mathcal{M}(\Omega, X)$ be a mapping as in (H2).
If, for each $\omega \in \Omega$, $f(\omega, x_0(\omega)) = x_0(\omega)$, 
then $x_0$ is a random fixed point of $f$. Suppose
that, for some $\omega \in \Omega$, $f(\omega, x_0(\omega)) \neq x_0(\omega)$. 
From  (H2), we have that $x_0$ and $x_1$ are two comparable elements of $X^\Omega$. 
Then, from \eqref{E0}, if we choose $u_0 =x_1$, we deduce
\begin{equation}\label{Ez}
\lim_{n \to + \infty} d(x_n(\omega), x_{n+1}(\omega))=0, \quad \text{for all }
 \omega \in \Omega.
\end{equation}

Now, we show that $\{x_n(\omega)\}$ is a Cauchy sequence for each $\omega \in \Omega$.
Let $\omega \in \Omega$ be fixed. First of all, we note that $r_\omega(t) <t$ 
for all $t>0$ and $r_\omega(0) =0$. Given a real number $\varepsilon >0$, 
by \eqref{Ez}, there exists $n(\varepsilon) \in \mathbb{N}$ such that
\begin{equation*}\label{Eep}
d(x_m(\omega),x_{m+1}(\omega)) < \varepsilon - r_\omega(\varepsilon), \quad 
\text{for all } m \in \mathbb{N}, \; m \geq n(\varepsilon).
\end{equation*}
We claim that
\begin{equation}\label{Eca}
d(x_m(\omega),x_{n+1}(\omega)) < \varepsilon
\end{equation}
whenever $m \geq n(\varepsilon)$ and $n \geq m$. Clearly, \eqref{Eca} 
holds if $n=m$. Now, we suppose that \eqref{Eca} holds for some 
$n \geq m$ and prove that \eqref{Eca} holds also for $n+1$. In fact,
\begin{align*}
d(x_m(\omega),x_{n+2}(\omega)) 
&\leq d(x_m(\omega),x_{m+1}(\omega)) + d(x_{m+1}(\omega),x_{n+2}(\omega))\\
&\leq d(x_m(\omega),x_{m+1}(\omega)) + r_\omega( d(x_{m}(\omega),x_{n+1}(\omega))\\
&< \varepsilon - r_\omega(\varepsilon)+ r_\omega(\varepsilon)
= \varepsilon.
\end{align*}
Thus $\{x_n(\omega)\}$ is a Cauchy sequence for all $\omega \in \Omega$. 
Then there exists $z \in X^\Omega$ such that
$$
z(\omega)= \lim_{n \to + \infty} x_n(\omega), \quad \text{for all } \omega \in \Omega.
$$
By Corollary \ref{C1}, we obtain $x_n \in \mathcal{M}(\Omega,X)$ for all
$n \in \mathbb{N}$ and hence
 $z\in \mathcal{M}(\Omega,X)$. We claim that $z(\omega)=f(\omega, z(\omega))$ 
for each $\omega \in \Omega$. The hypothesis that $f$ is a Carath\'{e}odory 
mapping ensures that
$$
d(z(\omega), f(\omega, z(\omega))) = \lim_{n \to + \infty} d(x_n(\omega), 
f(\omega,x_n(\omega))), \quad \text{for all } \omega \in \Omega.
$$ 
From
\begin{align*}
d(x_n(\omega), f(\omega,x_n(\omega)))
& =d(f(\omega,x_{n-1}(\omega)), f(\omega,x_n(\omega)))\\
&\leq r_\omega (d(x_{n-1}(\omega),x_{n}(\omega)))\\
& \leq d(x_{n-1}(\omega),x_{n}(\omega)),
\end{align*}
letting $n \to + \infty$, we obtain $d(z(\omega), f(\omega, z(\omega))) =0$
for all $\omega \in \Omega$. Thus $z(\omega)=f(\omega, z(\omega))$ for each 
$\omega \in \Omega$, that is, $z$ is a random fixed point of $f$.

We have to prove the uniqueness of this fixed point. So, we assume that 
$v \in \mathcal{M}(\Omega, X)$ is another random fixed point of $f$. 
If $z$ and $v$ are comparable, then from (H1), we deduce that $z=v$.
 Assume that $z$ and $v$ are not comparable, that is $z(\omega)$ is not 
comparable with $v(\omega)$ for some $\omega \in \Omega$. In this case, 
let $u \in X^\Omega$ be comparable with $z$ and $v$ and let $\{u_n\}$ be 
the Picard sequence starting at $u_0=u$.
 By \eqref{E0} with $x_0=z$ and $x_0=v$, we obtain
\begin{equation}\label{Eu}
\lim_{n \to + \infty}d(z(\omega),u_n(\omega))
= \lim_{n \to + \infty}d(v(\omega),u_n(\omega))=0.
\end{equation}
From \eqref{Eu}, we obtain $z=v$ and hence $z$ is a unique random fixed point 
of $f$.
\end{proof}

Now, we are ready  the theorem that produces the existence of a random 
fixed point of $f$, by replacing the Carath\'{e}odory assumption with 
hypothesis (H3) and sup-measurability of $f$.

\begin{theorem}
If {\rm (H0)--(H3)} hold and $f$ is a sup-measurable mapping, then there 
exists a mapping $z \in \mathcal{M}(\Omega, X)$ which is a random fixed point 
of $f$.
\end{theorem}

\begin{proof}
Let $\{x_n\}$ and $z \in \mathcal{M}(\Omega, X)$ as in the proof of
 Theorem \ref{T1}. We note that the hypothesis that $f$ is sup-measurable 
ensures that $x_n \in \mathcal{M}(\Omega, X)$ for all $n \in \mathbb{N}$. 
This implies that $z \in \mathcal{M}(\Omega, X)$.
By (H3), $x_n(\omega)$ and $z(\omega)$ are comparable for all $n \in \mathbb{N}$ 
and $\omega \in \Omega$. Using (H1) we obtain
\begin{align*}
d(z(\omega), f(\omega,z(\omega))) 
&\leq d(z(\omega), f(\omega,x_n(\omega))) + d(f(\omega,x_n(\omega), 
f(\omega,z(\omega)))\\
&\leq d(z(\omega), f(\omega,x_n(\omega))) + r_\omega (d(x_n(\omega), z(\omega)))\\
& \leq d(z(\omega), x_{n+1}(\omega)) + d(x_n(\omega), z(\omega)).
\end{align*}
Letting $n \to + \infty$, we obtain $d(z(\omega), f(\omega,z(\omega))=0$ 
for all $\omega \in \Omega$. This means that $z$ is a random fixed point of $f$.
\end{proof}


Next, we adapt the previous hypotheses for solving the above random fixed point
 problem in the setting of generalized metric spaces
 (see Sinacer-Nieto-Ouahab \cite{SNQ}). 
Let $\mathbb{R}^k_+ :=\{x \in \mathbb{R}^k : x_j \geq 0 \text{ for all } 
j=1, \ldots, k\}$, where $\mathbb{R}^k$ is equipped with the partial order relation:
$$
x=(x_1, \ldots,x_k),\;  y=(y_1, \ldots, y_k) \in \mathbb{R}^k, \quad
 x\preceq y \Longleftrightarrow x_j \leq y_j \text{ for all } j=1, \ldots,k.
$$ 
Also, $x \prec y$ denote that $x \preceq y$ and $x \neq y$; $x \ll y$ 
denote that $x_j <y_j$ for all $j=1, \ldots,k$; $\theta$ denote the zero 
vector in $\mathbb{R}^k$. Let $\mathcal{R}_k$ be the family of all nondecreasing 
functions $r =(r_1, \ldots, r_k) : \mathbb{R}^k_+ \to \mathbb{R}^k_+$ such that
\begin{itemize}
\item[(i)] $\lim_{n \to +\infty} r^n(t) =\theta$ for all 
$t \in \mathbb{R}^k_+$ with $\theta \prec t$;
\item[(ii)] $r(\theta)=\theta$ and $\theta \prec r(t) \prec t$ for 
$t \in \mathbb{R}^k_+ \setminus \{\theta\}$;
\item[(iii)] $\theta\ll t$ implies $r(t) \ll t$.
\end{itemize}

\begin{example}\rm
Let $r: \mathbb{R}^k_+ \to \mathbb{R}^k_+$ be defined by
$$
r(t)= \Big(\frac{t_1}{1+t_1}, \ldots, \frac{t_k}{1+t_k}\Big) \quad 
\text{for all } t=(t_1, \ldots, t_k) \in \mathbb{R}^k_+.
$$
Then $r \in \mathcal{R}_k$.
\end{example}

\begin{example}\rm
Let $r: \mathbb{R}^k_+ \to \mathbb{R}^k_+$ be defined by
$$
r(t)= At^T \quad \text{ for all } t=(t_1, \ldots, t_k) \in \mathbb{R}^k_+
$$
where $A= {\rm diag}(a_1, \ldots, a_k)$ is a diagonal matrix such that 
$0 < a_j <1$ for all $j=1, \ldots, k$.
Then $r \in \mathcal{R}_k$.
\end{example}

 We consider the following set of hypotheses:
\begin{itemize}
\item[(H4)] $(\Omega, \Sigma)$ is a measurable space, $(X, d,\precsim)$ 
is a separable complete ordered generalized metric space, and 
$f : \Omega \times X \to X$ is a random operator such that, for
each $\omega \in \Omega$, $x \to f(\omega, x)$ is a monotone
operator;

\item[(H5)] for each $\omega \in \Omega$, there exists a function 
$r_\omega=(r_{\omega,1}, \ldots,r_{\omega,k}) \in \mathcal{R}_k$ such that
$$
d(f(\omega, x), f(\omega, y)) \preceq r_\omega(d(x, y))\quad \text{for all } 
x,y \in X, \ x \precsim y;
$$

\item[(H6)] there exists a mapping $x_0 \in \mathcal{M}(\Omega, X)$ with
``$x_0(\omega) \precsim f(\omega, x_0(\omega))$, for all $\omega \in \Omega$''
or
``$x_0(\omega) \succsim f(\omega, x_0(\omega))$, for all $ \omega \in \Omega$''.
\end{itemize}

\begin{remark} \rm
For the sake of completeness, we point out that hypotheses (H4) and (H6) 
sound formally as the previous hypotheses (H0) and (H2), but with the difference 
that here $(X, d)$ denotes a generalized metric space. Precisely, a generalized 
metric space $(X, d)$ is a pair, where $X$ is a nonempty set and 
$d : X \times X\to \mathbb{R}^k_+$ is a vector-valued metric, in the sense 
of the following definition.
\end{remark}

\begin{definition} \rm
Let $X$ be a nonempty set. By a vector-valued metric on $X$, we mean a mapping
$d : X \times X\to \mathbb{R}^k_+$ with the following properties:
\begin{itemize}
\item[(i)] if $d(u, v) = \theta$ then $u = v$;
\item[(ii)] $d(u, v) = d(v,u)$ for all $u, v \in X$;
\item[(iii)] $d(u, v)\preceq d(u,w) + d(w, v)$ for all $u, v, w \in X$.
\end{itemize}
\end{definition}

Let $(X,d)$ be a generalized metric space. Let $\{x_n\}$ be a sequence in $X$ 
and $x \in X$. The sequence $\{x_n\}$ converges to $x$ if, for every 
$\varepsilon \in \mathbb{R}_+^k$ with $\theta \ll \varepsilon$, 
there is an $n(\varepsilon) \in \mathbb{N}$ such
that for all $n \geq n(\varepsilon)$ we have $d(x_n, x) \ll \varepsilon$. 
If, for every $\varepsilon \in \mathbb{R}_+^k$ with $\theta \ll \varepsilon$, 
there is an $n(\varepsilon) \in \mathbb{N}$ such
that $d(x_n, x_m) \ll \varepsilon$ for all $n, m \geq n(\varepsilon)$, 
then $\{x_n\}$ is a Cauchy sequence. If every Cauchy sequence is convergent in
$X$, then $X$ is called a complete generalized metric space.

On this basis we prove our third theorem producing the existence of a random 
fixed point of $f$. This result is analogous to the existence part of 
Theorem \ref{T1}.

\begin{theorem}\label{TG}
 If {\rm (H4)--(H6)} hold and $f$ is a Carath\'{e}odory mapping, 
then there exists a mapping $z \in \mathcal{M}(\Omega, X)$ which is a 
random fixed point of $f$.
\end{theorem}

\begin{proof}
As in the proof of Theorem \ref{T1}, we consider two comparable elements of 
$X^\Omega$, say $x_0$ and $u_0$, and the corresponding Picard sequences 
$\{x_n\}$ and $\{u_n\}$ starting respectively at $ x_0$ and $u_0$. 
Also in this theorem, the first step of the proof is to claim that
\begin{equation}\label{E0m}
\lim_{n \to + \infty} d(x_n(\omega), u_n(\omega))=\theta \quad \text{for all } 
\omega \in \Omega.
\end{equation}
Then, we have
\begin{align*} 
& x_n(\omega) \text{ and } u_n(\omega) \text{ are comparable for all }
 n \in \mathbb{N} \text{ ($x \to f(\omega, x)$ is monotone)},\\
&\Rightarrow \; d(x_n(\omega), u_{n}(\omega))\preceq r_\omega(d(x_{n-1}(\omega),
 u_{n-1}(\omega))) \quad \text{ (by (H5))}, \label{Ecm}\\ 
&  \preceq r^n_\omega( d(x_{0}(\omega), u_{0}(\omega)) \quad 
\text{for each $\omega \in \Omega$ and all $n \in \mathbb{N}$}.
\end{align*}

Letting $n \to + \infty$ in the previous inequalities and by property (i)
 of the elements of $\mathcal{R}_k$, we deduce that \eqref{E0m} holds. 
Next let $x_0 \in \mathcal{M}(\Omega, X)$ be a mapping as in (H6).
If, for each $\omega \in \Omega$, $f(\omega, x_0(\omega)) = x_0(\omega)$, 
then $x_0$ is a random fixed point of $f$. Suppose
that, for some $\omega \in \Omega$, $f(\omega, x_0(\omega)) \neq x_0(\omega)$.
 We consider the Picard sequence $\{x_n\}$ starting at $ x_0$. 
From  hypothesis (H6), we obtain that $x_0$ and $x_1$ are two comparable elements
 of $X^\Omega$. Then, from \eqref{E0m}, if we choose $u_0 =x_1$, we obtain
\begin{equation}\label{Ezm}
\lim_{n \to + \infty} d(x_n(\omega), x_{n+1}(\omega))=\theta, \quad 
\text{for each } \omega \in \Omega.
\end{equation}

Now, we show that $\{x_n(\omega)\}$ is a Cauchy sequence for all $\omega \in \Omega$. 
Let $\omega \in \Omega$ be fixed. First of all, we note that 
$\theta \ll t-r_\omega(t)$ whenever $\theta \ll t$ (by property (iii) of the 
elements of $\mathcal{R}_k$). Then given $\varepsilon \in \mathbb{R}^k_+$ with 
$\theta \ll \varepsilon$, by \eqref{Ezm}, there exists 
$n(\varepsilon) \in \mathbb{N}$ such that
\begin{equation*}\label{Eep-b}
d(x_m(\omega),x_{m+1}(\omega)) \ll \varepsilon - r_\omega(\varepsilon), \quad 
\text{for all } m \in \mathbb{N}, \ m \geq n(\varepsilon).
\end{equation*}
We claim that
\begin{equation}\label{Eca-b}
d(x_m(\omega),x_{n+1}(\omega)) \ll \varepsilon
\end{equation}
whenever $m \geq n(\varepsilon)$ and $n \geq m$. Note that $\{x_n(\omega)\}$ 
is a monotone sequence in virtue of (H4) and (H6). Clearly, \eqref{Eca-b} 
holds if $n=m$. So, we suppose that \eqref{Eca-b} holds for some $n \geq m$ 
and prove that \eqref{Eca-b} holds also for $n+1$. In fact,
\begin{align*}
d(x_m(\omega),x_{n+2}(\omega)) 
&\preceq d(x_m(\omega),x_{m+1}(\omega)) + d(x_{m+1}(\omega),x_{n+2}(\omega))\\
 &\preceq d(x_m(\omega),x_{m+1}(\omega)) 
+ r_\omega( d(x_{m}(\omega),x_{n+1}(\omega)))\\
&\ll \varepsilon - r_\omega(\varepsilon)+ r_\omega(\varepsilon)
= \varepsilon.
\end{align*}
Thus $\{x_n(\omega)\}$ is a Cauchy sequence for all $\omega \in \Omega$. 
Then, there exists $z \in X^\Omega$ such that
$$
z(\omega)= \lim_{n \to + \infty} x_n(\omega), \quad \text{for all } 
\omega \in \Omega.
$$

By Corollary \ref{C1}, we obtain $x_n \in \mathcal{M}(\Omega,X)$ for all 
$n \in \mathbb{N}$ and hence
 $z\in \mathcal{M}(\Omega,X)$. The hypothesis that $f$ is a Carath\'{e}odory 
mapping ensures that
$$
d(z(\omega), f(\omega, z(\omega))) = \lim_{n \to + \infty} d(x_n(\omega), 
f(\omega,x_n(\omega))), \quad \text{ for all } \omega \in \Omega.
$$ 
From
\begin{align*}
d(x_n(\omega), f(\omega,x_n(\omega)))
& =d(f(\omega,x_{n-1}(\omega)), f(\omega,x_n(\omega)))\\
&\preceq r_\omega (d(x_{n-1}(\omega),x_{n}(\omega)))\\
&\preceq d(x_{n-1}(\omega),x_{n}(\omega)),
\end{align*}
letting $n \to + \infty$, we obtain $d(z(\omega), f(\omega, z(\omega))) =\theta$ 
for all $\omega \in \Omega$. Thus $z(\omega)=f(\omega, z(\omega))$ for each 
$\omega \in \Omega$, that is, $z$ is a random fixed point of $f$.
\end{proof}

\begin{remark} \rm
In respect of Theorem \ref{TG}, one can establish also the uniqueness of 
random fixed point, by using the additional assumptions in the statement 
of Theorem \ref{T1}. Here, to avoid repetition, we omit details.
\end{remark}


\section{Solution of boundary value problem \eqref{EP}}

In this section, we prove a theorem producing the existence of a unique random 
solution of problem \eqref{EP}, see also 
Nieto-Ouahab-Rodr\'{i}guez-L\'{o}pez \cite{NOR}. 
Let $(\Omega, \Sigma)$ be a measurable space.
Let $f_1 , f_2 : \Omega \times [0,1] \times \mathbb{R} \times \mathbb{R} 
\to \mathbb{R}$ be Carath\'{e}odory functions, which means that 
$\omega \to f_i (\omega,t,u,v)$ is measurable
for all $(t,u,v) \in [0,1] \times \mathbb{R} \times \mathbb{R}$ and 
$(t,u,v) \to f_i (\omega,t,u,v)$ is continuous for all 
$\omega \in \Omega$, $i=1,2$. Denote with $\mathcal{G}$ the family of the 
functions $g: \Omega \times [0,1] \times \mathbb{R} \to \mathbb{R}$ such that 
$g_u: \Omega \times [0,1] \to \mathbb{R}$ is a Carath\'{e}odory function for 
every $u \in C([0, 1],\mathbb{R})$, where $g_u(\omega,t)=g(\omega,t,u(t))$ 
for all $(\omega,t) \in \Omega \times [0,1]$. Then, consider the integral 
operator $F : \Omega \times C([0, 1],\mathbb{R}) \times C([0, 1],\mathbb{R}) 
\to C([0, 1],\mathbb{R}) \times C([0, 1],\mathbb{R})$ defined by
 $$
F(\omega,x,y)(t)=(F_1(\omega,x,y)(t),F_2(\omega,x,y)(t)),\quad x,y \in C([0,1], 
\mathbb{R}), \ t \in [0,1],
$$ 
with
 \begin{gather}\label{Ef1}
 F_1(\omega,x,y)(t)=\int_0^1 K(t,s)f_1(\omega,s,x(s),y(s))ds + g_{1,x}(\omega,t),\\
\label{Ef2}
F_2(\omega,x,y)(t)=\int_0^1 K(t,s)f_2(\omega,s,x(s),y(s))ds + g_{2,y}(\omega,t)
\end{gather}
where $K: \mathbb{R} \times \mathbb{R} \to \mathbb{R}$ is a continuous function 
such that $|K(t,s)| \leq 1$ for all $t,s \in \mathbb{R}$ and 
$g_1,g_2 \in \mathcal{G}$.

\begin{remark} \rm
$F$ is a random operator from 
$ \Omega \times C([0, 1],\mathbb{R}) \times C([0, 1],\mathbb{R})$ into 
$C([0, 1],\mathbb{R}) \times C([0, 1],\mathbb{R})$. In fact, given 
$(x, y) \in C([0, 1],\mathbb{R}) \times C([0, 1],\mathbb{R})$, since 
$f_i$ and $g_i$ ($i=1,2$) are Carath\'{e}odory functions, for $s \in [0,1]$ 
fixed, the function $h:\Omega \times [0,1] \to \mathbb{R}$ defined by 
$h(\omega,t)=K(t,s)f_i(\omega,s,x(s),y(s))$ is Carath\'{e}odory.
By Lemma \ref{L1}, the integrals in \eqref{Ef1} and \eqref{Ef2} are limit 
of a finite sum of measurable functions. So, the mappings
 $\omega \to F_1(\omega, x, y)$ and $\omega \to F_2(\omega, x, y)$
 are measurable and hence $F$ is a random operator.
 \end{remark}

The hypotheses are the following:
\begin{itemize}
\item[(H7)] for each $\omega \in \Omega$ there exists a nondecreasing function 
$\psi_\omega =(\psi_{\omega,1}, \psi_{\omega,2}) : \mathbb{R}^2_+ \to \mathbb{R}_+^2$ 
such that
$$
|f_i (\omega, t, x, y) - f_i (\omega,t,u,v)| \leq \psi_{\omega,i}((|x-u|,|y-v|)), 
\quad i=1,2, 
$$
for each $t \in [0,1]$ and all $x, y,u,v \in \mathbb{R}$ with $(x, y) \preceq (u,v)$;

\item[(H8)] for each $\omega \in \Omega$ there exists a function 
$r_\omega=(r_{\omega,1}, r_{\omega,2}) \in \mathcal{R}_2$ such that
\begin{gather*}
\begin{aligned}
&\psi_{\omega,1}((\|x-u\|_\infty,\|y-v\|_\infty))+|g_{1,x}(\omega,t) 
- g_{1,u}(\omega,t)| \\
&\leq r_{\omega,1}((\|x - u\|_\infty,\|y-v\|_\infty)),
\end{aligned}
 \\
\begin{aligned}
&\psi_{\omega,2}((\|x-u\|_\infty,\|y-v\|_\infty))+|g_{2,y}(\omega,t) 
- g_{2,v}(\omega,t)|\\
& \leq r_{\omega,2}((\|x - u\|_\infty,\|y - v\|_\infty)),
\end{aligned}
\end{gather*}
for each $t \in [0,1]$ and all $x, y,u,v \in C([0,1],\mathbb{R})$;

\item[(H9)] for each $\omega \in \Omega$ fixed, $(x,y) \to f_i (\omega,t, x, y)$ 
and $x \to g_i(\omega,t, x)$, $i=1,2$, (for every $t \in [0, 1]$), 
are all nondecreasing or all nonincreasing operators;

\item[(H10)] one of the following conditions holds:
$$
0 \leq f_i (\omega,t, 0, 0), \quad 0 \leq g_i(\omega,t, 0), \quad \text{for all } 
t \in [0, 1], \; \omega \in \Omega, \; i=1,2,
$$
or
$$
0 \geq f_i (\omega,t, 0, 0), \quad 0 \geq g_i(\omega,t, 0), \quad \text{for all } 
t \in [0, 1], \; \omega \in \Omega, \; i=1,2.
$$
\end{itemize}

Later on, we consider $C([0, 1],\mathbb{R}) \times C([0, 1],\mathbb{R})$ 
equipped with the generalized metric $d$ given by
$$
d((x,y),(u,v))= (\|x-u\|_\infty,\|y-v\|_\infty), 
$$ 
for all $(x,y),(u,v) \in C([0, 1],\mathbb{R}) \times C([0, 1],\mathbb{R})$.
Now, we  have the theorem producing a unique random fixed point.

\begin{theorem}\label{TS}
If the hypotheses {\rm (H7)--(H10)} hold, then the random integral operator $F$
 has a unique random fixed point.
\end{theorem}

\begin{proof}
 For $\omega \in \Omega$ fixed, we show that $(x,y) \to F(\omega,x,y)$ 
is a continuous operator.
Indeed, consider a sequence $\{(x_n, y_n)\}$ in
 $C([0, 1],\mathbb{R}) \times C([0, 1],\mathbb{R})$ with
$(x_n, y_n) \to (x, y) \in C([0, 1],\mathbb{R}) \times C([0, 1],\mathbb{R})$, 
as $n \to + \infty$. For $t \in [0,1]$, we have
\begin{align*}
& |F_1(\omega, x_n, y_n)(t) - F_1(\omega, x, y)(t)|\\
& \leq \int_0^1|f_1(\omega,s,x_n(s),y_n(s))-f_1(\omega,s,x(s),y(s))|ds
  + |g_{1,x_n}(\omega,t) -g_{1,x}(\omega, t)|\\
& \leq \int_0^1\psi_{\omega,1}((|x_n(s)-x(s)|,|y_n(s)-y(s)|))ds
 + |g_{1,x_n}(\omega,t) -g_{1,x}(\omega, t)|\\
& \leq \psi_{\omega,1}((\|x_n(s)-x(s)\|_\infty,\|y_n(s)-y(s)\|_\infty))
 + |g_{1,x_n}(\omega,t) -g_{1,x}(\omega, t)|
\end{align*}
implies 
\[
\|F_1(\omega, x_n, y_n) - F_1(\omega, x, y)\|_\infty
\leq r_{\omega,1}((\|x_n-x\|_\infty,\|y_n-y\|_\infty))
\]
by (H8).

By an analogous reasoning one has
\begin{align*}
\|F_2(\omega, x_n, y_n) - F_2(\omega, x, y)\|_\infty
\leq r_{\omega,2}((\|x_n-x\|_\infty,\|y_n-y\|_\infty)).
\end{align*}
So
$d(F(\omega, x_n, y_n), F(\omega, x, y)) \to (0,0)$, as $n \to + \infty$,
implies that $(x,y) \to F(\omega,x,y)$ is a continuous operator, for each fixed 
$\omega \in \Omega$. 

In addition, for each $\omega \in \Omega$, $(x,y) \to F(\omega,x,y)$ is a
 monotone operator.
Indeed, consider $(x, y), (u, v) \in C([0, 1],\mathbb{R}) \times C([0, 1],\mathbb{R})$
such that $(x, y) \precsim (u,v)$, that is, $x(t) \leq u(t)$, $y(t) \leq v(t)$, 
for all $t \in [0,1]$.
For every $t \in [0,1]$, if $(x,y) \to f_i (\omega,t, x, y)$ and 
$x \to g_i(\omega,t, x)$, $i=1,2$, are nondecreasing operators, then
\begin{gather*}
f_i (\omega,t, x(t), y(t)) \leq f_i (\omega,t, u(t), v(t)), \quad
 \text{for all } t \in [0,1],\; i=1,2,\\
g_1 (\omega,t, x(t)) \leq g_1 (\omega,t, u(t)),\quad  \text{for all } t \in [0,1],\\
g_2 (\omega,t, y(t)) \leq g_2 (\omega,t, v(t)),\quad  \text{for all } t \in [0,1],
\end{gather*}
implies
\[
 F_i(\omega, x, y)(t) \leq F_i(\omega, u, v)(t), \quad \text{for all } t \in [0,1],
 \; i=1,2,
\]
which implies 
$F(\omega,x, y) \precsim F(\omega,u, v)$.

In a similar way, for every $t \in [0,1]$, whenever 
$(x,y) \to f_i (\omega,t, x, y)$ and $x \to g_i(\omega,t, x)$, 
$i=1,2$, are nonincreasing operators, then we deduce that 
$F(\omega,u, v) \precsim F(\omega,x, y)$.

A crucial step of the proof is to show that $F$ satisfies a contraction
 condition (see hypothesis (H5)). Precisely, for every $\omega \in \Omega$ and all
$(x, y), (u,v) \in C([0, 1],\mathbb{R}) \times C([0, 1],\mathbb{R})$ 
such that $(x, y) \precsim (u,v)$, we have to show that
$$
d(F(\omega,x, y),F(\omega,u,v)) \preceq r_\omega( d((x, y), (u,v))).
$$
Again, consider $\omega \in \Omega$ fixed. Let 
$(x, y), (u,v) \in C([0, 1],\mathbb{R}) \times C([0, 1],\mathbb{R})$ 
be such that $(x, y) \precsim (u,v)$, then
\begin{align*}
& |F_1(\omega, x, y)(t) - F_1(\omega, u, v)(t)|\\
& \leq \int_0^1|f_1(\omega,s,x(s),y(s))-f_1(\omega,s,u(s),v(s))|ds 
 + |g_{1,x}(\omega,t)-g_{1,u}(\omega, t)|\\
& \leq \int_0^1\psi_{\omega,1}((|x(s)-u(s)|,|y(s)-v(s)|))ds
 + |g_{1,x}(\omega,t)-g_{1,u}(\omega, t)|\\
& \leq \psi_{\omega,1}((\|x-u\|_\infty,\|y-v\|_\infty))
 + |g_{1,x}(\omega,t)-g_{1,u}(\omega, t)|
\end{align*}
implies
\[
\|F_1(\omega, x, y) - F_1(\omega, x, y)\|_\infty
 \leq r_{\omega,1}((\|x-u\|_\infty,\|y-v\|_\infty)).
\]
By an analogous reasoning 
\begin{gather*}
\|F_2(\omega, x, y) - F_2(\omega, u, v)\|_\infty
\leq r_{\omega,2}((\|x-u\|_\infty,\|y-v\|_\infty)),\\
\Rightarrow \; d(F(\omega,x, y),F(\omega,u,v))
 \preceq r_\omega (d((x, y), (u,v))).
\end{gather*}
Now we prove that condition (H6) holds. Precisely, by (H10),
 we can easily show that
$$
0 \leq F_1(\omega,\cdot, 0, 0)\quad \text{and} \quad 
0 \leq F_2(\omega,\cdot, 0, 0), \quad \text{for all }\omega \in \Omega,
$$
or
$$
0 \geq F_1(\omega,\cdot, 0, 0)\quad \text{and} \quad 
0 \geq F_2(\omega,\cdot, 0, 0), \quad \text{for all }\omega \in \Omega;
$$
that is, $F(\omega,0,0) \succeq (0,0)$ for all $\omega \in \Omega$ or 
$F(\omega,0,0) \preceq (0,0)$ for all $\omega \in \Omega$. This means that, 
for the couple of null random variables defined as 
$(0, 0) : \Omega \to C([0, 1],\mathbb{R}) \times C([0, 1],\mathbb{R})$, 
by $(0,0)(\omega) = (0,0)$, for all $\omega \in \Omega$, one of the following 
two conditions holds:
$$
F(\omega,(0, 0)(\omega)) \succeq (0, 0)(\omega),\quad \text{for all } 
\omega \in \Omega
$$ 
or
$$
F(\omega,(0, 0)(\omega)) \preceq (0, 0)(\omega),\quad \text{for all } 
\omega \in \Omega.
$$ 
Note that the uniqueness condition also holds. Thus all the hypotheses of 
Theorem \ref{TG} are satisfied and so the existence and uniqueness of a 
fixed point of $F$ is a direct consequence of above Theorem \ref{TG}.
\end{proof}

By particularizing the choice of Carath\'{e}odory functions 
$g_1 , g_2 : \Omega \times [0,1] \times \mathbb{R} \to \mathbb{R}$, 
we can have the theorem producing a unique random solution of problem \eqref{EP}. 
Let $g_i(\omega,t,u(t))=\psi_i\big(\int_0^1 u(\omega)(s)ds\big)t$, where 
$\psi_i \in C(\mathbb{R},\mathbb{R})$, for $i=1,2$ and consider the random 
integral operator
 $$
\widetilde{F}(\omega,x,y)(t)=(\widetilde{F}_1(\omega,x,y)(t),
\widetilde{F}_2(\omega,x,y)(t)),\quad x,y \in C([0,1], \mathbb{R}), \; t \in [0,1],
$$ 
with
\begin{gather*}
\widetilde{F}_1(\omega,x,y)(t)=\int_0^1K(t,s)f_1(\omega,s,x(s),y(s))ds 
+ \psi_1\Big(\int_0^1 x(\omega)(s)ds\Big)t, \\
\widetilde{F}_2(\omega,x,y)(t)=\int_0^1 K(t,s)f_2(\omega,s,x(s),y(s))ds 
+ \psi_2\Big(\int_0^1 y(\omega)(s)ds\Big)t
\end{gather*}
where $K: \mathbb{R} \times \mathbb{R} \to \mathbb{R}$ is given by \eqref{En}.

\begin{theorem}
If {\rm (H7)--(H10)} hold, then  problem \eqref{EP}
 has a unique random solution.
\end{theorem}

\begin{proof}
Note that the random fixed points of $\widetilde{F}$ are solutions to \eqref{EP} 
and conversely. Indeed, given a couple of random variables 
$(x, y) : \Omega \to C([0, 1],\mathbb{R}) \times C([0, 1],\mathbb{R})$, 
we obtain that
$$
\widetilde{F}(\omega,x(\omega), y(\omega)) = (x(\omega), y(\omega)),\quad 
\text{for all } \omega \in \Omega,
$$
is equivalent to
\begin{gather*}
x(\omega)(t)=\int_0^1 K(t,s)f_1(\omega,s,x(\omega)(s),y(\omega)(s))ds 
+ \psi_1\Big(\int_0^1 x(\omega)(s)ds\Big)t, \quad \text{$0<t <1$}, \\
y(\omega)(t)=\int_0^1 K(t,s)f_2(\omega,s,x(\omega)(s),y(\omega)(s))ds
 + \psi_2\Big(\int_0^1 y(\omega)(s)ds\Big)t, \quad \text{$0<t <1$},
\end{gather*}
so that the corresponding solution of \eqref{EP} is defined as 
$x(\omega,t) = x(\omega)(t)$, $y(\omega,t) = y(\omega)(t)$, for 
$t \in [0,1]$ and $\omega \in \Omega$. Then, from Theorem \ref{TS}, 
there exists a unique random solution to Problem \eqref{EP}.
\end{proof}

\subsection*{Acknowledgements}
The authors extend their appreciation to the International Scientific 
Partnership Program ISPP at King Saud University for funding this 
research work through ISPP\#0068.

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