Random fixed point theorems for Hardy-Rogers self-random operators with applications to random integral equations

TL;DR

This paper establishes new random fixed point theorems for Hardy-Rogers operators in Banach spaces and applies these results to demonstrate solutions for random integral equations, extending deterministic fixed point results to stochastic settings.

Contribution

It introduces novel random fixed point theorems for Hardy-Rogers operators and applies them to solve random nonlinear integral equations in Banach spaces.

Findings

Existence of solutions for certain random nonlinear integral equations.

Extension of deterministic fixed point theorems to stochastic frameworks.

Development of stochastic versions of Hardy-Rogers fixed point theorems.

Abstract

In this paper, we prove some random fixed point theorems for Hardy-Rogers self-random operators in separable Banach spaces and, as some applications, we show the existence of a solution for random nonlinear integral equations in Banach spaces. Some stochastic versions of deterministic fixed point theorems for Hardy-Rogers self mappings and stochastic integral equations are obtained.

Equations

$$d(Tx,Ty)\le \alpha d(x,y)$$

$$d(Tx,Ty)\le \beta [d(x,Tx)+d(y,Ty)]$$

$$d(Tx,Ty)\le \alpha d(x,Tx)+\beta d(y,Ty)+\gamma d(y,Ty)$$

$$d(Tx,Ty)\le \alpha d(x,y)+\beta d(x,Tx)+\gamma d(y,Ty)+\delta [d(x,Ty)+d(y,Tx)]$$

$$d(Tx,Ty)\le \beta [d(x,Ty)+d(y,Tx)]$$

$$d(Tx,Ty)<\max \{d(x,y),[d(x,Tx)+d(y,Ty)]/2,[d(x,Ty)+d(y,Tx)]/2\}.$$

$$d(Tx,Ty)\le \alpha d(x,y)+\beta d(x,Tx)+\gamma d(y,Ty)+\delta d(x,Ty)+\eta d(y,Tx)$$

$$d(Tx,Ty)\le ad(x,y)+b\max \{d(x,Tx),d(y,Ty)\}+c[d(x,Ty)+d(y,Tx)]$$

$$\Vert T(\omega ,x_1)-T(\omega ,x_2)\Vert$$

$$\underset{n\to \infty }{\lim }\Vert x_n(\omega )-x(\omega )\Vert =0$$

$$\Vert T(\omega ,x_1(\omega ))-T(\omega ,x_2(\omega ))\Vert$$

$$A=\{\omega \in \Omega :T(\omega ,x_1)\text{is a continuous function of}x\},$$

$$\displaystyle B=\Big{\{}\omega\in\Omega:\sum^{5}_{i=1}\alpha_{i}(\omega)<1\Big{\}}$$

$$C_{x_1,x_2}$$

$$\Vert T(\omega ,s_1(\omega ))-T(\omega ,s_2(\omega ))\Vert$$

$$\Vert s_1(\omega )-s_2(\omega )\Vert \le \Vert s_1(\omega )-x_1(\omega )\Vert +\Vert x_1(\omega )-x_2(\omega )\Vert +\Vert x_2(\omega )-s_2(\omega )\Vert ,$$

$$\Vert s_1(\omega )-T(\omega ,s_1(\omega ))\Vert$$

$$\Vert s_2(\omega )-T(\omega ,s_2(\omega ))\Vert$$

$$\Vert s_1(\omega )-T(\omega ,s_2(\omega ))\Vert$$

$$\Vert s_2(\omega )-T(\omega ,s_1(\omega ))\Vert$$

$$\Vert T(\omega ,s_1(\omega ))-T(\omega ,s_2(\omega ))\Vert$$

$$\Vert T(\omega ,x_1(\omega ))-T(\omega ,x_2(\omega ))\Vert$$

$$\Vert T(\omega ,x_1(\omega ))-T(\omega ,s_1(\omega ))\Vert <\frac{\epsilon }{8}$$

$$\Vert T(\omega ,x_2(\omega ))-T(\omega ,s_2(\omega ))\Vert <\frac{\epsilon }{8}$$

$$\displaystyle\delta_{1}=\min\Big{\{}\delta_{1}(x_{1}(\omega)),\frac{\varepsilon}{8}\Big{\}}$$

$$\displaystyle\delta_{2}=\min\Big{\{}\delta_{2}(x_{2}(\omega)),\frac{\varepsilon}{8}\Big{\}},$$

$$\Vert T(\omega ,x_1(\omega ))-T(\omega ,x_2(\omega ))\Vert$$

Full text

∗Corresponding author: poom.kum@kmutt.ac.th (P. Kumam)

This project was supported by the Theoretical and Computational Science (TaCS) Center.

1. Introduction

Some well known random fixed point theorems are stochastic generalizations of Banach’s fixed point theorem and Banach’s type fixed point theorems in complete metric spaces. In 1955, Spacek [32] and Hans [9, 10] initiated to prove random fixed point theorems for random contraction mappings in separable complete metric spaces. In 1966, Mukherjee [22] proved a random fixed point theorem in the sense of Schaduer’s fixed point theorem in atomic probability measure spaces. Especially, in 1976, the work of Bharucha-Reid [5] has been developed by various mathematicians. In 1979, Itoh [11] extended some random fixed point theorems of Spacek and Hans to the setting of multi-valued contraction mappings and applied random fixed point theorems to solve some random differential equations in Banach spaces. In 1984, Sehgal and Waters [31] proved some random fixed point theorems including classical results given by Rothe [26].

Recently, Beg and Shahzad [4] showed the existence of random common fixed points and random coincidence points of a pair of compatible random multi-valued mappings in Polish spaces. Especially, Kumam et al. [15, 17, 18, 14, 16] proved many random fixed point theorems for multi-valued nonexpansive nonself-mappings satisfying the inwardness condition in Banach spaces (see [19]). Jung et al. [13] proved random fixed point theorems for a certain class of mappings in banach spaces. Cho et al. [6] proved random Ishikawa iterative sequence with errors for approximating random fixed points. Likewise, Kumam and Plubtieng [20] showed the existence of a random coincidence point for a pair of reciprocally continuous and compatible single-valued and multi-valued mappings and Saha [27], Saha and Debnath [28] established some random fixed point theorems in separable Hilbert spaces and separable Banach spaces, respectively. On the other hand, Padgett [23], Achari [1], Saha and Dey [29] applied some random fixed point theorems to show the existence of solutions of random nonlinear integral equations in Banach spaces.

Recently, Saha and Ganguly [30] proved some random fixed point theorems for a class of contractive mappings in separable Banach spaces equipped with a complete probability measure.

In fact, Banach’s contraction principle ([3]) is very important to show the existence of solutions of some nonlinear equations, differential and integral equations, and other nonlinear problems. Since Banach’s contraction principle, many authors have studied in several ways.

Theorem 1.1**.**

(Banach’s contraction principle)* If $(X,d)$ is a complete metric space and $T:X\rightarrow X$ be a mapping such that, for some $\alpha\in[0,1)$,*

[TABLE]

for each $x,y\in X$, then $T$ has a unique fixed point in $X$.

Note that the mapping $T$ satisfying the Banach contraction condition is continuous, but the mappings $T$ satisfying the following contractions conditions are not continuous.

(1)  In 1968, Kannan’s contraction ([25]): for some $\beta\in[0,\frac{1}{2})$,

[TABLE]

for each $x,y\in X$;

(2)  In 1971, Reich’s contraction ([25]): for some $\alpha,\beta,\gamma\geq 0$ with $\alpha+\beta+\gamma<1$,

[TABLE]

for each $x,y\in X$;

(3)  In 1971, $\acute{\text{C}}$iri$\acute{\text{c}}$’s contraction ([25]): for some $\alpha,\beta,\gamma,\delta\geq 0$ with $\alpha+\beta+\gamma+2\delta<1$,

[TABLE]

for each $x,y\in X$;

(4)  In 1972, Chatterjea’s contraction ([25]): for some $\beta\in[0,\frac{1}{2})$,

[TABLE]

for each $x,y\in X$;

(5)  In 1972, Zamfirescu contractive conditions ([25]): there exist real numbers $\alpha,\beta,\gamma,0\leq\alpha<1,0\leq\beta<1,\gamma<\frac{1}{2}$, such that, for each $x,y\in X$, at least one of the following is true:

1. (i)

$d(Tx,Ty)\leq\alpha d(x,y)$; 2. (ii)

$d(Tx,Ty)\leq\beta[d(x,Tx)+d(y,Ty)]$; 3. (iii)

$d(Tx,Ty)\leq\gamma[d(x,Ty)+d(y,Tx)]$.

For each $x,y\in X,x\neq y$,

[TABLE]

(6)  In 1973, Hardy and Rogers’s contraction ([25]): for some $\alpha,\beta,\gamma,\delta,\eta\geq 0$ with $\alpha+\beta+\gamma+\delta+\eta<1$,

[TABLE]

for each $x,y\in X$:

In 2000, $\acute{\text{C}}$iri$\acute{\text{c}}$ [7] dealt with a class of mappings (not necessarily continuous) satisfying Gregus type contraction in metric spaces ([8]) and proved the following fixed point theorem:

Theorem 1.2**.**

Let $C$ be a closed convex subset of a complete convex metric space $X$ and $T:C\rightarrow C$ be a mapping satisfying

[TABLE]

for all $x,y\in C$, where $0<a<1$, $a+b=1$ and $c\leq\frac{4-a}{8-b}$. Then $T$ has a unique fixed point in $X$.

Moreover, Common fixed points under contractive conditions in cone metric spaces was studied by Radenovi$\acute{\text{c}}$ (see in [24]).

Recently, Saha and Ganguly [30] proved some random fixed point theorems for a certain class of contractive mappings in a separable Banach space equipped with a complete probability measure as follows:

Theorem 1.3**.**

Let $X$ be a separable Banach space and $(\Omega,\beta,\mu)$ be a complete probability measure space. Let $T:\Omega\times X\rightarrow X$ be a continuous random operator such that for all $\omega\in\Omega$, $T$ satisfies

[TABLE]

for all random variables $x_{1},x_{2}\in X$ where $a(\omega),b(\omega),c(\omega)$ are real-valued random variables such that $0<a(\omega)<1$, $a(\omega)+b(\omega)=1$, $c(\omega)\leq\frac{4-a(\omega)}{8-a(\omega)}$ almost surely. Then there exist unique random fixed point of $T$ in $X$.

Note that, if $\beta=0$ or $\gamma=0$ and $\delta=\eta$, then fixed point theorems for Hardy and Roger’s contraction (1.7) reduced to fixed point theorems for Gregus type contraction (1.8).

The purpose of this paper is to prove some random fixed point theorems for random Hardy-Rogers self-mappings in separable Banach spaces and, by using our main results, we show the existence of solutions of random nonlinear integral equations.

2. Preliminaries

Throughout this paper, X will denote a separable Banach over the real. Let $\beta_{X}$ be a $\sigma$-algebra of Borel subsets of $X$. Let $(\Omega,\beta,\mu)$ denote a complete probability measure space with the measure $\mu$ and $\beta$ be a $\sigma$-algebra of subsets of $\Omega$. For more details, see Joshi and Bose [12].

Definition 2.1**.**

(1)  A mapping $x:\Omega\rightarrow X$ is called an $X$-valued random variable if the inverse image under the mapping $x$ of every Borel set $B$ of $X$ belongs to $\beta$, that is, $x^{-1}(X)\in\beta$ for all $B\in\beta_{X}$.

(2) A mapping $x:\Omega\rightarrow X$ is called a finitely-valued random variable if it is constant on each finite number of disjoint sets $A_{i}\in\beta$ and is equal to [math] on $\Omega-(\bigcup_{i=1}^{n}A_{i})$. The mapping $x$ is called a simple random variable if it is finitely valued and $\mu\{\omega:\|x(\omega)\|>0\}<\infty$.

(3)  A mapping $x:\Omega\rightarrow X$ is called a strong random variable if there exists a sequence $\{x_{n}(\omega)\}$ of simple random variables which converges to $x(\omega)$ almost surely, that is, there exists a set $A_{0}\in\beta$ with $\mu(A_{0})=0$ such that $\lim_{n\rightarrow\infty}x_{n}(\omega)=x(\omega)$ for any $\omega\in\Omega-A_{0}$.

(4)  A mapping $x:\Omega\rightarrow X$ is called a weak random variable if the function $x^{*}(x(\omega))$ is a real-valued random variable for each $x^{*}\in X^{*}$, where $X^{*}$ denots the first normed dual space of $X$.

In a separable Banach space $X$, the notions of strong and weak random variables $x:\Omega\rightarrow X$ ([12]) coincide and, in $X$, $x$ is termed as a random variable.

Now, we recall the following:

Theorem 2.2**.**

([12])* Let $x,y:\Omega\rightarrow X$ be strong random variables and $\alpha,\beta$ be constants. Then the following statements hold:*

(1)*  $\alpha x(\omega)+\beta y(\omega)$ is a strong random variable.*

(2)*  If $f(\omega)$ is a real-valued random variable and $x(\omega)$ is a strong random variable, then $f(\omega)x(\omega)$ is a strong random variable.*

(3)*  If $x_{n}(\omega)$ is a sequence of strong random variables converging strongly to $x(\omega)$ almost surely, that is, if there exists a set $A_{0}\in\beta$ with $\mu(A_{0})=0$ such that*

[TABLE]

for any $\omega\notin A_{0}$, then $x(\omega)$ is a strong random variable.

Remark 2.3*.*

If $X$ is a separable Banach space, then every strong and also weak random variable is measurable in the sense of Definition 2.1.

Let $Y$ be an another Banach space. We also need the following definitions (see Joshi and Bose [12]).

Definition 2.4**.**

(1)  A mapping $F:\Omega\times X\rightarrow Y$ is called a random mapping if $F(\omega,x)=Y(\omega)$ is a $Y$-valued random variable for all $x\in X$.

(2)  A mapping $F:\Omega\times X\rightarrow Y$ is called a continuous random mapping if the set of all $\omega\in\Omega$ for which $F(\omega,x)$ is a continuous function of $x$ has measure one.

(3)  A mapping $F:\Omega\times X\rightarrow Y$ is said to be demicontinuous at the $x\in X$ if $\|x_{n}-x\|\rightarrow 0$ implies $F(\omega,x_{n})\rightharpoonup F(\omega,x)$ almost surely.

Theorem 2.5**.**

([12])* Let $F:\Omega\times X\rightarrow Y$ be a demicontinuous random mapping where a Banach space $Y$ is separable. Then, for any $X$-valued random variable $x$, the function $F(\omega,x(\omega))$ is a $Y$-valued random variable.*

Remark 2.6*.*

([12]) Since a continuous random mapping is a demicontinuous random mapping, Theorem 2.2 is also true for a continuous random mapping.

Also, we recall the following definitions (see Joshi and Bose [12]):

Definition 2.7**.**

(1)  An equation of the type $F(\omega,x(\omega))=x(\omega)$, where $F:\Omega\times X\rightarrow X$ is a random mapping, is called a random fixed point equation.

(2)  Any mapping $x:\Omega\rightarrow X$ which satisfies the random fixed point equation $F(\omega,x(\omega))=x(\omega)$ almost surely is called a wide sense solution of the fixed point equation.

(3)  Any $X$-valued random variable $x(\omega)$ which satisfies

$\mu\{\omega:F(\omega,x(\omega))=x(\omega)\}=1$

is called a random solution of the fixed point equation or a random fixed point of $F$.

Remark 2.8*.*

A random solution is a wide sense solution of the fixed point equation. But the converse is not necessarily true. This is evident from an example, under Remark 1, in Joshi and Bose [12].

3. The main results

Motivated and inspired by Theorem 1.3, we proposed the definition as follows:

Definition 3.1**.**

Let $T:\Omega\times X\rightarrow X$ be a continuous random mapping. The random mapping $T$ is called Hardy-Rogers’ contraction if, for any $\omega\in\Omega$,

[TABLE]

for all random variables $x_{1},x_{2}:\Omega\rightarrow X$ and $\alpha_{i}:\Omega\rightarrow\mathbb{R}_{+}\cup\{0\}$ for $i=1,2,3,4,5$ such that $\sum^{5}_{i=1}\alpha_{i}(\omega)<1$.

Theorem 3.2**.**

Let $X$ be a separable Banach space and $(\Omega,\beta,\mu)$ be a complete probability measure space. Let $T:\Omega\times X\rightarrow X$ be a continuous random mapping satisfying Hardy-Rogers’ contraction. Then there exists a unique random fixed point of $T$ in $X$.

Proof.

Let

[TABLE]

[TABLE]

and

[TABLE]

Let $S$ be a countable dense subset of $X$. Now, we prove that

$\bigcap_{x_{1},x_{2}\in X}(C_{x_{1},x_{2}}\cap A\cap B)=\bigcap_{s_{1},s_{2}\in S}(C_{s_{1},s_{2}}\cap A\cap B)$.

Now, for all $s_{1},s_{2}\in S$, we have

[TABLE]

Since $S$ is dense in $X$, for any $\delta_{i}(x_{i})>0$, there exist $s_{1},s_{2}\in S$ such that $\|x_{i}-s_{i}\|<\delta_{i}(x_{i})$ for each $i=1,2$. Note that, for any $x_{1},x_{2}\in X$,

[TABLE]

[TABLE]

[TABLE]

[TABLE]

and

[TABLE]

Suppose that

[TABLE]

Since

[TABLE]

substituting (3) in (3), we have

[TABLE]

Thus, from (3.3), (3), (3), (3), (3), (3), it follows that

[TABLE]

For any $\omega\in\Omega$, sice $T(\omega,x(\omega))$ is a continuous function of $x(\omega)$, for any $\varepsilon>0$, there exists $\delta_{i}(x_{i}(\omega))>0~{}~{}~{}(i=1,2)$ such that

[TABLE]

whenever $\|x_{1}(\omega)-s_{1}(\omega)\|<\delta_{1}(x_{1}(\omega))$ and

[TABLE]

whenever $\|x_{2}(\omega)-s_{2}(\omega)\|<\delta_{1}(x_{2}(\omega))$. Now, choosing

[TABLE]

and

[TABLE]

by (3), we have

[TABLE]

and so

[TABLE]

Since $\varepsilon>0$ is arbitrary, it follows that

[TABLE]

Thus we have $\omega\in\bigcap_{x_{1},x_{2}\in X}(C_{x_{1},x_{2}}\cap A\cap B)$, which implies that

$\bigcap_{s_{1},s_{2}\in S}(C_{s_{1},s_{2}}\cap A\cap B)\subset\bigcap_{x_{1},x_{2}\in X}(C_{x_{1},x_{2}}\cap A\cap B)$.

Also, we have

$\bigcap_{x_{1},x_{2}\in X}(C_{x_{1},x_{2}}\cap A\cap B)\subset\bigcap_{s_{1},s_{2}\in S}(C_{s_{1},s_{2}}\cap A\cap B)$.

Therefore, we have

$\bigcap_{s_{1},s_{2}\in S}(C_{s_{1},s_{2}}\cap A\cap B)=\bigcap_{x_{1},x_{2}\in X}(C_{x_{1},x_{2}}\cap A\cap B)$.

Let $N^{\prime}=\bigcap_{s_{1},s_{2}\in S}(C_{s_{1},s_{2}}\cap A\cap B)$. Then $\mu(N^{\prime})=1$, which implies that $T(\omega,x)$ is a deterministic mapping. Hence $T$ has a unique random fixed point in $X$. This completes the proof.

If $\alpha_{4}(\omega)=\alpha_{5}(\omega)=0$ in Theorem 3.2, then we obtain the following random fixed point theorem for Reich’s contraction:

Corollary 3.3**.**

Let $X$ be a separable Banach space and $(\Omega,\beta,\mu)$ be a complete probability measure space. Let $T:\Omega\times X\rightarrow X$ be a continuous random mapping satisfying the following condition: for any $\omega\in\Omega$,

[TABLE]

for all random variables $x_{1},x_{2}:\Omega\rightarrow X$ and $\alpha_{i}:\Omega\rightarrow\mathbb{R}_{+}\cup\{0\}$ for $i=1,2,3$ such that $\sum^{3}_{i=1}\alpha_{i}(\omega)<1$. Then there exists a unique random fixed point of $T$ in $X$.

If $\alpha_{1}(\omega)=\alpha_{4}(\omega)=\alpha_{5}(\omega)=0$ in Theorem 3.2, then we obtain the following random fixed point theorem for Kannan’s contraction:

Corollary 3.4**.**

Let $X$ be a separable Banach space and $(\Omega,\beta,\mu)$ be a complete probability measure space. Let $T:\Omega\times X\rightarrow X$ be a continuous random mapping satisfying the following condition: for any $\omega\in\Omega$,

[TABLE]

for all random variables $x_{1},x_{2}:\Omega\rightarrow X$ and $\alpha_{i}:\Omega\rightarrow\mathbb{R}_{+}\cup\{0\}$ for $i=2,3$ such that $\alpha_{2}(\omega)+\alpha_{3}(\omega)<1$. Then there exists a unique random fixed point of $T$ in $X$.

If $\alpha_{1}(\omega)=\alpha_{2}(\omega)=\alpha_{3}(\omega)=0$ in Theorem 3.2, then we obtain the following random fixed point theorem for Chatterjea’s contraction:

Corollary 3.5**.**

Let $X$ be a separable Banach space and $(\Omega,\beta,\mu)$ be a complete probability measure space. Let $T:\Omega\times X\rightarrow X$ be a continuous random mapping satisfying the following condition: for all $\omega\in\Omega$,

[TABLE]

for all random variables $x_{1},x_{2}:\Omega\rightarrow X$ and $\alpha_{i}:\Omega\rightarrow\mathbb{R}_{+}\cup\{0\}$ for $i=4,5$ such that $\alpha_{4}(\omega)+\alpha_{5}(\omega)<1$. Then there exists a unique random fixed point of $T$ in $X$.

Remark 3.6*.*

The random fixed point theorems for Hardy-Rogers’s contraction reduced to the random fixed point theorems for $\acute{\text{C}}$iri$\acute{\text{c}}$’s contraction.

4. Applications to random nonlinear integral equations

In this section, we give an application of Theorem 3.2 to show the existence and uniqueness of a solution of a nonlinear stochastic integral equation of the Hammerstein type ([23]):

[TABLE]

where

(a)  $S$ is a locally compact metric space with metric d defined on $S\times S$ and $\mu_{0}$ is a complete $\sigma$-finite measure defined on the collection of Borel subsets of $S;$

(b)  $\omega\in\Omega$ where $\omega$ is the supporting set of the probability measure space $(\Omega,\beta,\mu);$

(c)  $x(t;\omega)$ is the unknown vector-valued random variable for each $t\in S;$

(d)  $h(t;\omega)$ is the stochastic free term defined for $t\in S;$

(e)  $k(t,s;\omega)$ is the stochastic kernel defined for $t$ and $s$ in $S;$

(f)  $f(t,x)$ is a vector-valued function of $t\in S$ and $x.$

Note that the integral in the equation (4.1) is interpreted as a Bochner integral ([33]).

Further, we assume that the union of a countable family $\{C_{n}\}$ of compact sets with $C_{n+1}\subset C_{n}$ is defined as $S$ such that, for each other compact set in $S$, there exists $C_{i}$ which contains it (see [2]).

We define $C=C(S,L_{2}(\Omega,\beta,\mu))$ as a space of all continuous functions from $S$ into the space $L_{2}(\Omega,\beta,\mu)$ with the topology of uniform convergence on compact sets of $S$, that is, $x(t;\omega)$ is a vector-valued random variable for each fixed $t\in S$ such that

$\|x(t;\omega)\|^{2}_{L_{2}(\Omega,\beta,\mu)}=\int_{\Omega}|x(t;\omega)|^{2}d\mu(\omega)<\infty.$

Noted that $C(S,L_{2}(\Omega,\beta,\mu))$ is a space of locally convex ([33]) whose topology is defined by the countable family of semi-norms given by

$\|x(t;\omega)\|_{n}=\sup_{t\in C_{n}}\|x(t;\omega)\|_{L_{2}(\Omega,\beta,\mu)}$

for each $n\geq 1$. Furthermore, since $L_{2}(\Omega,\beta,\mu)$ is complete, $C(S,L_{2}(\Omega,\beta,\mu))$ is complete relative to this topology.

Next, we define $BC=BC(S,L_{2}(\Omega,\beta,\mu))$ as a Banach space of all bounded continuous functions from $S$ into $L_{2}(\Omega,\beta,\mu)$ with the norm

$\|x(t;\omega)\|_{BC}=\sup_{t\in S}\|x(t;\omega)\|_{L_{2}(\Omega,\beta,\mu)}.$

The space $BC\subset C$ is a space of all second order vector-valued stochastic processes defined on $S$ which are bounded and continuous in mean-square.

Now, we consider the functions $h(t;\omega)$ and $f(t,x(t;\omega))$ to be in the $C(S,L_{2}(\Omega,\beta,\mu))$ space with respect to the stochastic kernel and assume that, for each pair $(t,s)$, $k(t,s;\omega)\in L_{\infty}(\Omega,\beta,\mu)$ and the norm denoted by

$\||k(t,s;\omega)|\|=\|k(t,s;\omega)\|_{L_{\infty}(\Omega,\beta,\mu)}=\mu-ess\sup_{\omega\in\Omega}|k(t,s;\omega)|.$

Also, we suppose that $k(t,s;\omega)\in L_{\infty}(\Omega,\beta,\mu)$ is such that

[TABLE]

is $\mu$-integrable with respect to $s$ for each $t\in S$ and $x(s;\omega)\in C(S,L_{2}(\Omega,\beta,\mu))$ and there exists a real-valued function $G$ $\mu$-a.e. on $S$ such that $G(S)\|x(s;\omega)\|_{L_{2}(\Omega,\beta,\mu))}$ is $\mu$-integrable and, for each pair $(t,s)\in S\times S$,

[TABLE]

Forward, assume that, for almost all $s\in S$, $k(t,s;\omega)$ is continuous in $t$ from $S$ into $L_{\infty}(\Omega,\beta,\mu)$.

Now, we define the random integral operator $T$ on $C(S,L_{2}(\Omega,\beta,\mu))$ by

[TABLE]

where the integral is a Bochner integral. From the conditions on $k(t,s;\omega)$, it follows that, for each $t\in S$, $(Tx)(t;\omega)\in L_{2}(\Omega,\beta,\mu)$ and $(Tx)(t;\omega)$ is continuous in mean square by Lebesgue’s dominated convergence theorem, that is, $(Tx)(t;\omega)\in C(S,L_{2}(\Omega,\beta,\mu))$.

Lemma 4.1**.**

([23])* The linear operator $T$ defined by the equation $(4.2)$ is continuous from $C(S,L_{2}(\Omega,\beta,\mu))$ into itself.*

Proof.

See [23].

Definition 4.2**.**

([1], [21]) Let $B$ and $D$ be Banach spaces. The pair $(B,D)$ is said to be admissible with respect to a linear operator $T$ if $T(B)\subset D$.

Lemma 4.3**.**

([23])* If $T$ is a continuous linear operator from $C(S,L_{2}(\Omega,\beta,\mu))$ into itself and $B,D\subset C(S,L_{2}(\Omega,\beta,\mu))$ are Banach spaces stronger than $C(S,L_{2}(\Omega,\beta,\mu))$ such that $(B,D)$ is admissible with respect to $T$, then $T$ is continuous from $B$ into $D$.*

By a random solution of the equation (4.1), we mean a function

[TABLE]

which satisfies the equation (4.1) $\mu-a.e.$

Now, by using Theorem 3.2, we prove the following:

Theorem 4.4**.**

If the stochastic integral equation $(4.1)$ is subject to the following conditions:

(1)*  $B$ and $D$ are Banach spaces stronger than $C(S,L_{2}(\Omega,\beta,\mu))$ such that $(B,D)$ is admissible with respect to the integral operator defined by $(4.2)$;*

(2)*  $x(t;\omega)\mapsto f(t,x(t;\omega))$ is an operator from the set $Q(\rho)=\{x(t;\omega):x(t;\omega)\in D,\|x(t;\omega)\|_{D}\leq\rho\}$ into the space $B$ satisfying*

[TABLE]

for all $x_{1}(t,\omega),x_{2}(t,\omega)\in Q(\rho)$ and $\alpha_{i}:\Omega\rightarrow\mathbb{R}_{+}\cup\{0\}$ for $i=1,2,3,4,5$ such that $\sum^{5}_{i=1}\alpha_{i}(\omega)<1$ almost surely;

(3)*  $h(t;\omega)\in D$,*

then there exists a unique random solution of the equation $(4.1)$ in $Q(\rho)$ provided

[TABLE]

where the norm of $T(\omega)$ denoted by $l(\omega)$.

Proof.

Let a mapping $\mathcal{U}(\omega):Q(\rho)\rightarrow D$ defined by

[TABLE]

Then we have

[TABLE]

Thus it follows from (4.3) that

[TABLE]

and so

[TABLE]

Hence we have

[TABLE]

Therefore, by (4.4), we have

[TABLE]

and so, by (4.5), $(\mathcal{U}x)(t,\omega)\in Q(\rho)$. Thus, for any $x_{1}(t,\omega)$, $x_{2}(t,\omega)\in Q(\rho)$ and, by the condition (2), we have

[TABLE]

Consequently, $\mathcal{U}(\omega)$ is a random contractive mapping on $Q(\rho)$. Hence, by Theorem 3.2, there exists a random fixed point of $\mathcal{U}(\omega)$, which is the random solution of the equation (4.1). This completes the proof.

Open Problem: Can Theorems 1.3 and 3.2 be generalized to non-separable Banach spaces?

Acknowledgements

The authors are gratefully thankful for referee’s valuable comments, which significantly improve materials in this paper. The first author was supported by Rajamangala University of Technology Lanna (RMUTL) for Ph.D. program at King Mongkut’s University of Technology Thonburi (KMUTT). Yeol Je Cho was supported by Basic Science Research Program through the National Research Foundation of Korea (NRF) funded by the Ministry of Science, ICT and future Planning (2014R1A2A2A01002100). This work was carried out while the third author (YJ. Cho) was visiting Theoretical and Computational Science Center (TaCS), Science Laboratory Building, Faculty of Science, King Mongkut’s University of Technology Thonburi (KMUTT), Bangkok, Thailand, during 15 January-2 march, 2016. He thanks Professor Poom Kumam and the University for their hospitality and support.

Moreover, Poom Kumam was supported by the Thailand Research Fund (TRF) and the King Mongkut’s University of Technology Thonburi (KMUTT) under the TRF Research Scholar Award (Grant No. RSA6080047).