# Some Random Fixed-Point Theorems for Weakly Contractive Random Operators in a Separable Banach Space

Authors: Kenza Benkirane; Abderrahim EL Adraoui; El Miloudi Marhrani
Journal: International Journal of Mathematics and Mathematical Sciences
 (2021)

---

## Abstract

The aim of this paper is to prove a common random fixed-point and some random fixed-point theorems for random weakly contractive operators in separable Banach spaces. A random Mann iterative process is introduced to approximate the fixed point. Finally, the main result is supported by an example and used to prove the existence and the uniqueness of a solution of a nonlinear stochastic integral equation system.


---

## Body

## 1. Introduction

The fixed-point theory has been revealed as a very powerful and important tool in the study of different mathematical models expressed in the forms of differential equations [1], integral equations [2, 3], fractional differential equations [4, 5], matrix equations [6], etc. Also, its applications are very useful and interesting in economics, in game theory, in computer science, and in other domains.Probabilistic functional analysis is one of the essential mathematical disciplines that are applied to solving problems, characterized with uncertainties, known as probabilistic models. The random fixed-point theorems are stochastic generalizations of classical fixed-point theorems which are known as deterministic results and are required for the theory of random equations, random matrices, random partial differential equations, and various classes of random operators.The theory of random fixed point was initiated by the Prague School of Probability in the 1950s. The random fixed-point theory finds its roots in the work of Špaček [7] and Hanš [8, 9]. They established a stochastic generalization of Banach contraction principle (BCP), and they applied their results to study the existence of a solution of random linear Fredholm integral equations. In 1976, Bharucha-Reid published his review article [10] which has attracted the attention of several researchers and which has led to the development of random fixed-point theory. In 1979, Itoh [11] extended Špaček’s and Hanš’s theorems to multivalued contraction random mappings. The result obtained by Itoh in [11] was applied to solve a random differential equation in Banach space. In the recent past, random differential equations and random integral equations have been solved by random fixed-point theorems (see, for example, [12–16]). For some important contributions in the random fixed-point theory, we invite the reader to consult [17–25] and the references therein.It is necessary to mention that the BCP is the first fundamental deterministic fixed-point theorem in a metric space.Theorem 1 (see [26]).
IfX,d is a complete metric space and T:X⟶X is a self-mapping such that(1)dTx,Ty≤kdx,y,for all x,y∈X and k∈0,1, then T has a unique fixed point.Among the generalizations of this principle, we find the following theorem established in 2001 by Rhoades.Theorem 2 (see [27]).
IfX,d is a complete metric space and T:X⟶X is a self-mapping such that(2)dTx,Ty≤dx,y−ϕdx,y,for all x,y∈X, and ϕ:0,∞⟶0,∞ is a continuous and nondecreasing function such that ϕt=0 if and only if t=0, then T has a unique fixed point.Several interesting weak contractions were considered in various frameworks (see, for example, [28–31] and references therein). Among all these weak contractions, we are interested in the one studied by Eslamian and Abkar in [32]. We state the result in the following.Theorem 3 (see [32]).
IfX,d is a complete metric space and T:X⟶X is a ψ1,ψ2,φ-weakly contractive self-mapping, i.e.,(3)ψ1dTx,Ty≤ψ2dx,y−φdx,y,for allx,y∈X,then T has a unique fixed point.Here, the three functions ψ1,ψ2,φ:ℝ+⟶ℝ+, called control functions, satisfy the following conditions:(a)
ψ1 and ψ2 are continuous(b)
φ is lower semicontinuous(c)
ψ1 is increasing(d)
Fori∈1,2, ψit=0 if and only if t=0(e)
For allt∈ℝ+, φt=0 if and only if t=0(f)
For allt>0, ψ1t−ψ2t+φt>0
In this study, we prove a common random fixed-point theorem and some random fixed-point theorems for randomψ1,ψ2,φ-weakly contractive operators in a separable Banach space, where the three control functions ψ1,ψ2,andφ satisfy all the conditions a−f except the condition (d) which is replaced by the weak condition:(4)ψ2t=0if and only ift=0.As an application, we show the existence and the uniqueness of a random solution for a system of nonlinear integral equations. To prove our main results, we need to recall the following concepts and results. For more details, the reader may consult [33, 34].


## 2. Preliminaries

LetX,‖.‖ be a separable Banach space, βX be the σ-algebra of all Borel subsets of X, and Ω,β,μ be a complete probability measure space with the measure μ and β be the σ-algebra of subsets of Ω. Let C be a nonempty subset of X.Definition 1 (see [33, 35]).
(i)
A mappingx:Ω⟶X is said to be a random variable with values in X if the inverse image under the mapping x of every Borel set B of X belongs to β, that is, x−1B∈β for all B∈βX(ii)
A mappingx:Ω⟶C is called a random variable with values in C if x−1B∩C∈β for all B∈βXDefinition 2 (see [33, 34]).
(i)
A mappingx:Ω⟶X is said to be a finitely-valued random variable if it is constant on each of a finite number of disjoint sets Ai∈β and equal to 0 on Ω−∪i=1nAi.(ii)
A mappingx:Ω⟶X is said to be a simple random variable if it is finitely valued.(iii)
A mappingx:Ω⟶X is said to be a strong (or Bochner) random variable if there exists a sequence xnn of simple random variables which converges to x almost surely, that is, there exists a set A0∈β with μA0=0 such that(5)limn⟶∞xnω−xω=0for everyω∈Ω−A0.
Next, we introduce the notion of a weak random variable.Definition 3 (see [33]).
A mappingx:Ω⟶X is said to be a weak (or Pettis) random variable if the functions x∗x are real-valued random variables for each x∗∈X∗, where X∗ denotes the first normed dual space of X.Remark 1 (see [33]).
In this work, we restrict our attention to the case whereX is a separable Banach space. In this setting, the concept of weak and strong random variables is equivalent.
The following definition of the mode of convergence for Banach space-valued random variables, which we use in the sequel, is borrowed from [33].Definition 4.
Letxnn and x be X-valued random variables. The sequence xnn converges to x in Ω strongly almost surely if there exists a set A0∈β with μA0=0 such that(6)limn⟶∞xnω−xω=0for anyω∈Ω−A0.
We recall the following results from the study by Joshi and Bose ([34], Theorem 6.1.2).Theorem 4.
Letx and y be two strong random variables and α and β be two constants. Then, the following statements hold:(i)
αx+βy is a strong random variable(ii)
Iff is a real-valued random variable and x is a strong random variable, then fx is a strong random variable(iii)
Ifxnn is a sequence of strong random variables converging strongly to x almost surely, then x is a strong random variableDefinition 5 (see [35]).
(i)
A mappingT:Ω×C⟶C is said to be random operator if for each x∈C, the mapping T.,x:Ω⟶C is measurable(ii)
A random operatorT:Ω×C⟶C is continuous if the set of all ω∈Ω for which Tω,. is continuous has measure one
Throughout this paper, we denoteℛVX as the set of all X−valued random variables and we adopt the following definition of the random fixed point given by Joshi and Bose in [34].Definition 6.
Letx∈ℛVX. x is said to be a random fixed point of T if(7)μω:Tω,xω=xω=1.


## 3. Main Results

In this section, we prove a common random fixed-point theorem and some random fixed-point theorems forψ1,ψ2,φ−weakly contractive mappings in a separable Banach space.Definition 7.
LetX,. be a separable Banach space and Ω,β,μ be a complete probability measure space. The mapping T:Ω×X⟶X is called a ψ1,ψ2,φ-weakly contractive random operator if T satisfies the following inequality:(8)ψ1Tω,x1ω−Tω,x2ω≤ψ2x1ω−x2ω−φx1ω−x2ω,almost surely, for all x1,x2∈ℛVX.Theorem 5.
LetX,. be a separable Banach space, C be a nonempty closed subset of X, and Ω,β,μ be a complete probability measure space. Let T,S:Ω×C⟶C be two continuous random operators satisfying(9)ψ1Tω,x1ω−Sω,x2ω≤ψ2x1ω−x2ω−φx1ω−x2ω,almost surely, for all x1,x2∈ℛVC. Then, there exists a unique common random fixed point of S and T.Proof.
Letx0∈ℛVC. Let x1ω=Sω,x0ω and x2ω=Tω,x1ω. Since S and T are two continuous random operators in a separable space, it follows that x1 and x2 are in ℛVC. Now, consider the sequence defined, for each n∈ℕ, by x2n+1ω=Sω,x2nω and x2n+2ω=Tω,x2n+1,ω.
By induction,xnn is a sequence in ℛVC. Consider the set A=X∩Y such that(10)X=ω∈Ω:x⟶Tω,xis continuous onC,Y=ω∈Ω:x⟶Sω,xis continuous onC.
For ally,z∈ℛVC, we denote by Ey,z the set of elements ω∈Ω such that(11)ψ1Tω,yω−Sω,zω≤ψ2yω−zω−φyω−zω.
LetM=∩n∈ℕ∗A∩Exn,xn+1. As stated, μA=1 and μExn,xn+1=1, for each n∈ℕ∗. Then, μM=1. Let ω∈M. For all n∈ℕ∗, we have(12)ψ1xn+1ω−xnω≤ψ2xnω−xn−1ω−φxnω−xn−1ω.This implies that(13)xn+1ω−xnω≤xnω−xn−1ω,for alln∈ℕ∗.
Indeed, let us assume that there existsn0∈ℕ∗ such that(14)xn0+1ω−xn0ω>xn0ω−xn0−1ω.
Sinceψ1 is increasing, we get(15)ψ1xn0+1ω−xn0ω>ψ1xn0ω−xn0−1ω.
Then,(16)ψ1xn0+1ω−xn0ω>ψ2xn0ω−xn0−1ω−φxn0ω−xn0−1ω,which is a contradiction. Hence, for all n∈ℕ∗, we have(17)xn+1ω−xnω≤xnω−xn−1ω.
It follows that the sequence‖xn+1ω−xnω‖n is decreasing and consequently there exists l≥0 such that(18)limn⟶∞xn+1ω−xnω=l.
Since(19)limsupn⟶+∞ψ1xn+1ω−xnω≤limsupn⟶+∞ψ2xnω−xn−1ω−φxnω−xn−1ω≤limsupn⟶+∞ψ2xnω−xn−1ω−liminfn⟶+∞φxnω−xn−1ωand by using the continuity of ψ1 and ψ2 and the lower semicontinuity of φ, we obtain that ψ1l≤ψ2l−φl, which is a contradiction unless l=0.
Hence,(20)limn⟶∞xn+1ω−xnω=0,for allω∈M.
Now, fixω in M and let us prove that xnωn is a Cauchy sequence in C. For this, it is sufficient to show that the subsequence x2nωn is a Cauchy sequence. If we assume the contrary, then(21)∃∫>0,∀k∈ℕ,∃mk>nk≥ksuch thatx2mkω−x2nkω≥∫.
Furthermore, corresponding tonk, we can choose mk in such a way that it is the smallest integer with mk>nk satisfying(22)x2mkω−x2nkω≥∫.
Consequently,(23)x2mk−2ω−x2nkω<∫.
By using the triangular inequality, we obtain(24)ϵ≤x2mkω−x2nkω≤x2mkω−x2mk−1ω+x2mk−1ω−x2nkω≤x2mkω−x2mk−1ω+x2mk−1ω−x2mk−2ω+x2mk−2ω−x2nkω<x2mkω−x2mk−1ω+x2mk−1ω−x2mk−2ω+ϵ.Hence,(25)limk⟶∞x2mkω−x2nkω=ε.
Also,(26)x2mkω−x2nkω−x2mk+1ω−x2mkω≤x2mk+1ω−x2nkω,(27)x2mk+1ω−x2nkω≤x2mk+1ω−x2mkω+x2mkω−x2nkω.
Lettingk⟶+∞, we obtain(28)limk⟶∞x2mk+1ω−x2nkω=ϵ.
By the same way, we have(29)x2mkω−x2nk−1ω≤x2nk−1ω−x2nkω+x2mkω−x2nkω,(30)x2mkω−x2nkω−x2nk−1ω−x2nkω≤x2mkω−x2nk−1ω.
Then,(31)limk⟶∞x2mkω−x2nk−1ω=∫.
We have, for allk∈ℕ,(32)ψ1‖x2mk+1ω−x2nkω‖=ψ1Sω,x2mkω−Tω,x2nk−1ω≤ψ2x2mkω−x2nk−1ω−φx2mkω−x2nk−1ω.
By passing to the upper limit, we getψ1ϵ≤ψ2ϵ−φϵ, which is a contradiction since ϵ>0. This shows that x2nωn is a Cauchy sequence in C, for each ω∈M. Using (20), it is easy to check that xnωn is a Cauchy sequence in C, for each ω∈M.
SinceC is a closed subset of the Banach space X, then C is complete, which implies that, for all ω∈M, the sequence xnωn converges by norm in C. Let x:Ω⟶C be the mapping such that xω=limn⟶+∞xnω, for each ω∈M. Since the sequence xnn converges strongly almost surely to x, then, according to ([33], Theorem 1.6), x is a C-valued random variable.
Letω∈M. For all n∈ℕ, we have x2n+1=Sω,x2nω and x2n+2=Tω,x2n+1ω.
Sincexω=limn⟶+∞xnω and by using the continuity of S and T, we get Sω,xω=Tw,xω=xω, for each ω∈M.
Hence,(33)μω∈Ω:Sω,xω=Tw,xω=xω=1.
It means thatx is a common random fixed point of S and T.
To prove the uniqueness of this common fixed point, lety be another common random fixed point of S and T. Consider the two sets(34)Fx=ω∈Ω:Sω,xω=Tw,xω=xω,Fy=ω∈Ω:Sω,yω=Tw,yω=yω.
Then,μFx∩Fy∩M=1. Let N=Fx∩Fy∩M. For each ω∈N, we have(35)ψ1xω−yω=ψ1Tω,xω−Sω,yω≤ψ2xω−yω‖−φ‖xω−yω.This implies that xω−yω=0. Therefore, x=y almost surely. This proves the uniqueness of the common random fixed point of S and T.
IfT=S in Theorem 5, we obtain the following random fixed-point theorem for ψ1,ψ2,φ-weakly contractive random mapping.Corollary 1.
LetX,. be a separable Banach space, C be a nonempty closed subset of X, and Ω,β,μ be a complete probability measure space. Let T:Ω×C⟶C be a continuous ψ1,ψ2,φ-weakly contractive random mapping. Then, there exists a unique random fixed point of T.Example 1.
LetX=ℝ2 with the norm 1 defined, for all x,y∈ℝ2, as follows:(36)x,y1=x+y.
Ω=0,1 and C=0,1×0,1. Let β be a σ-algebra of Lebesgue measurable subsets of ℝ2. Consider the three functions ψ1,ψ2,φ:≤0,+∞⟶0+∞ defined for all t≥0 as follows:(37)ψ1t=t,ψ2t=2t3andφt=t6if0≤t≤21ift>2.
Consider the random operatorT:Ω×C⟶C defined by Tω,xω=Tω,x1ω,x2ω=x1ω+ω/3;x1ω+ω/3, where ω∈Ω and x=x1,x2∈ℛVC.
Letω∈Ω. We have(38)ψ1Tω,x1ω,x2ω−Tω,y1ω,y2ω1=Tω,x1ω,x2ω−Tω,y1ω,y2ω1,=x1ω+ω3;x2ω+ω3−y1ω+ω3;y2ω+ω31,=x1ω+ω3−y1ω+ω3;x2ω+ω3−y2ω+ω31,=x1ω−y1ω3;x2ω−y2ω31,=x1ω−y1ω+x2ω−y2ω3.(39)ψ2x1ω;x2ω−y1ω;y2ω1=ψ2x1ω−y1ω;x2ω−y2ω1,=ψ2x1ω−y1ω+x2ω−y2ω,=2x1ω−y1ω+x2ω−y2ω3.φx1ω;x2ω−y1ω;y2ω1=φx1ω−y1ω;x2ω−y2ω1,=φx1ω−y1ω+x2ω−y2ω,=x1ω−y1ω+x2ω−y2ω6.
Then,(40)ψ2x1ω;x2ω−y1ω;y2ω1−φx1ω;x2ω−y1ω;y2ω1=2x1ω−y1ω+x2ω−y2ω3,−x1ω−y1ω+x2ω−y2ω6,=x1ω−y1ω+x2ω−y2ω2,≥x1ω−y1ω+x2ω−y2ω3.
Consequently, for allx=x1;x2,y=y1;y2∈ℛVC and for each ω∈Ω,(41)ψ1Tω,xω−Tω,yω1≤ψ2xω−yω1−φxω−yω1.
All conditions of Corollary1 are satisfied and T has a random fixed point which is(42)x:ω⟶ω2;ω2.
In Corollary1, if ψ1t=ψ2t=t, for all t≥0, we obtain the following corollary which is an improvement of ([36], Theorem 5.2) in a separable Banach space.Corollary 2.
LetX,. be a separable Banach space, C be a nonempty closed subset of X, and Ω,β,μ be a complete probability measure space. Let T:Ω×C⟶C be a continuous random operator satisfying(43)Tω,x1ω−Tω,x2ω≤x1ω−x2ω−φx1ω−x2ω,almost surely, for all x1,x2∈ℛVC. Then, there exists a unique random fixed point of T.In Corollary1, if ψ1=ψ2, we obtain the following random fixed-point theorem for ψ,φ-weakly contractive random mapping.Corollary 3.
LetX,. be a separable Banach space, C be a nonempty closed subset of X, and Ω,β,μ be a complete probability measure space. Let T:Ω×C⟶C be a continuous random operator satisfying the following condition:(44)ψTω,x1ω−Tω,x2ω≤ψx1ω−x2ω−φx1ω−x2ω,almost surely, for all x1,x2∈ℛVC. Then, there exists a unique random fixed point of T.In Corollary1, if ψ1t=t and ψ2−φt=kt, for all t≥0 and for some k∈0,1, we obtain the following random fixed-point theorem for Banach’s contraction.Corollary 4.
LetX,. be a separable Banach space, C be a nonempty closed subset of X, and Ω,β,μ be a complete probability measure space. Let T:Ω×C⟶C be a continuous random operator satisfying the following condition:(45)Tω,x1ω−Tω,x2ω≤kx1ω−x2ω,almost surely, for all x1,x2∈ℛVC and k∈0,1. Then, there exists a unique random fixed point of T.


## 4. Random Mann Iteration Scheme

In the following, we investigate the convergence of random Mann iteration scheme applied to aψ1,ψ2,φ-weakly contractive random operator.Definition 8.
(random Mann iteration scheme [35]). Let T:Ω×C⟶C be a random operator, where C is a nonempty convex subset of a separable Banach space X. A random Mann iteration scheme is the sequence of C-valued random variables xnn defined, for all ω∈Ω, by(46)xn+1ω=1−cnxnω+cnTω,xnω,for alln∈ℕx0:Ω⟶Cis an arbitrary measurable mapping,where 0≤cn≤cn+1≤1 and 0≤limn⟶∞cn=c≤1, for all n∈ℕ.
In particular, ifcn=1, for all n∈ℕ, the sequence xnn is said to be a random Picard iteration scheme.Theorem 6.
LetX,. be a separable Banach space, C be a nonempty closed convex subset of X, and Ω,β,μ be a complete probability measure space.LetT:Ω×C⟶C be a continuous ψ1,ψ2,φ-weakly contractive random operator. Assume that ψ1 is convex. Then, the following two statements hold:(i)
There exists a unique random fixed pointx:Ω⟶C of T(ii)
The random Mann iteration scheme converges strongly almost surely to the unique random fixed pointx of TProof.
(i)
From Corollary1, T has a unique random fixed point x:Ω⟶C. Consider the set F=ω∈Ω:Tω,xω=xω. Then, μF=1.(ii)
Consider the setA=ω∈Ω:x⟶Tω,xis continuous onC.
LetEx1,x2 be the set of elements ω∈Ω such that(47)ψ1‖Tω,x1ω−Tω,x2ω‖≤ψ2x1ω−x2ω−φx1ω−x2ω.
As stated,μM=1, where M=∩n∈ℕExn,xn+1∩A∩F.
Letω∈M. We claim that, for all n∈ℕ,(48)Tω,xnω−Tω,xω≤xnω−xω.
Indeed, let us assume that there existsn0∈ℕ such that(49)xn0ω−xω<Tω,xn0ω−Tω,xω.
Sinceψ1 is increasing, we get(50)ψ1xn0ω−xω<ψ1Tω,xn0ω−Tω,xω≤ψ2xn0ω−xω−φ‖xn0ω−xω‖.This is a contradiction, since for each t>0, ψ1t−ψ2t+ϕt>0.Hence, for all n∈ℕ,(51)Tω,xnω−Tω,xω≤xnω−xω.
Then, for alln∈ℕ,(52)xn+1ω−xω=Tω,xnω−Tω,xω≤xnω−xω.
It follows that the sequence‖xnω−xω‖n is decreasing and consequently there exists l≥0 such that(53)limn⟶∞xnω−xω=l.Since xnn is a random Mann iteration scheme, we have, for all n∈ℕ,(54)xn+1ω−xω=1−cnxnω+cnTω,xnω−xω=1−cnxnω+cnTω,xnω−1−cnxω−cnxω≤1−cnxnω−xω+cnTω,xnω−xω.
Then,(55)xn+1ω−xω≤1−cnxnω−xω+cnTω,xnω−Tω,xω.Since ψ1 is nondecreasing and convex,(56)ψ1xn+1ω−xω≤1−cnψ1xnω−xω+cnψ1Tω,xnω−Tω,xω.(57)ψ1xn+1ω−xω≤1−cnψ1xnω−xω+cnψ2xnω−xω−cnφ‖xnω−xω‖.
Then,
By passing to the upper limit, we obtain(58)ψ1l≤1−cψ1l+cψ2l−cφl.
Then,(59)ψ1l≤ψ2l−φl.This is a contradiction, since for each t>0, ψ1t−ψ2t+ϕt>0. Then, l=0.
Then, for allω∈M,(60)limn⟶∞xnω−xω=0.
Consider the set(61)L=ω∈Ω:limn⟶∞xnω−xω=0.
SinceM⊆L and μM=1, then μL=1. This shows that the sequence xnn of the C-valued random variable converges strongly almost surely to the unique random fixed point x.


## 5. Applications to Nonlinear Stochastic Integral Equations System

In this section, we give an application of Theorem5 to show the existence and the uniqueness of a solution of a nonlinear stochastic integral equations system (NSIE) presented as follows:(62)xt;ω=ht;ω+∫ℝkt,s;ωfs,ys;ωdλs,N.S.I.Eyt;ω=ht;ω+∫ℝkt,s;ωgs,xs;ωdλs,where we have the following:(a)
ℝ is the locally compact real space with the usual norm of reals and λ is the Lebesgue measure on ℝ(b)
ω∈Ω, where Ω is the supporting set of the probability measure space Ω,β,μ(c)
For allt∈ℝ, xt;. and yt;. are two unknown elements in ℛVℝ(d)
ht;ω is the stochastic free term defined for t∈ℝ(e)
kt,s;ω is the stochastic kernel defined for t and s in ℝ(f)
f and g are two real-valued functionsRemark 2 (see [37]).
The topological spaceℝ is the union of a countable family of compact subsets Kn having the properties that Kn⊂Kn+1 and that for any other compact set in ℝ, there is a Ki which contains it.
LetCℝ,L2Ω,β,μ be the space of all continuous functions from ℝ into the space L2Ω,β,μ with the topology of uniform convergence on compact sets of ℝ. Note that Cℝ,L2Ω,β,μ is a locally convex space (see [38]) and so it can be endowed with a topology induced by a countable family of seminorms .nn∈ℕ defined by xn=supt∈Knxt;.L2Ω,β,μ, for each n∈ℕ and x∈Cℝ,L2Ω,β,μ.
Here,(63)xt;.L2Ω,β,μ2=∫Ωxt;ω2dμω<∞.
Note that furthermore, sinceL2Ω,β,μ is complete, Cℝ,L2Ω,β,μ is complete with respect to this topology.
We assume that, for each pairt,s, kt,s;.∈L∞Ω,β,μ and denote the norm in L∞Ω,β,μ by(64)‖|kt,s;.‖=‖kt,s;.‖L∞Ω,β,μ=μ−esssupω∈Ω|kt,s;ω.
Also, we suppose the following:(i)
For almost alls∈ℝ, the function t⟼kt,s;. is continuous from ℝ into L∞Ω,β,μ(ii)
The functions⟼kt,s;..xs;.L2Ω,β,μ is λ-integrable, for each x in Cℝ,L2Ω,β,μ and almost all t∈ℝ(iii)
There exists a real-valued functionG defined λ-a.e. on ℝ, such that Gsxs;.L2Ω,β,μ is λ-integrable and for each pair t,s,u∈ℝ3,(65)kt,u;.−ks,u;.|.xu;.L2Ω,β,μ≤Guxu;.L2Ω,β,μλ−a.e.
Consider a random operatorT defined on Ω×Cℝ,L2Ω,β,μ such that, for all x∈Cℝ,L2Ω,β,μ and t∈ℝ,(66)Tωxt;ω=∫ℝkt,s;ωxs;ωdλs,a.sinΩ.
However, for eacht∈ℝ, the function s⟼t,s;..xt;.L2Ω,β,μ is λ-integrable; then, Tωxt;.∈L2Ω,β,μ a.s. From (iii) and by using Lebesgue’s dominated convergence theorem, t⟼Tωxt;. is a continuous in mean square, so Tωx∈Cℝ,L2Ω,β,μ a.s.
LetB and D be two Banach spaces. The pair B,D is said to be admissible with respect to the linear operator T if TωB⊂D a.s.Lemma 1 (see [39]).
(1)
The linear operatorTω is continuous from Cℝ,L2Ω,β,μ into itself a.s.(2)
IfB,D⊂Cℝ,L2Ω,β,μ are two Banach spaces stronger than the space Cℝ,L2Ω,β,μ such that B,D is admissible with respect to Tω, then Tω is continuous from B to D a.s.Definition 9.
By a random solution of NSIE, we will mean a pair of functionsx,y in Cℝ,L2Ω,β,μ2 which satisfies the two equations of NSIE μ-a.e.
Letω∈Ω. We consider the two functions F,G:Cℝ,L2Ω,β,μ⟶Cℝ,L2Ω,β,μ defined, for each t∈ℝ and ω∈Ω, by Fxt;ω=ft,xt;ω and Gxt;ω=gt,xt;ω.
Letρ be a positive real number. Consider the ball is centered on 0^ and the neutral element in D is(67)Mρ=x∈D:xD≤ρ.Theorem 7.
The following conditions hold:(1)
B and D are Banach spaces stronger than Cℝ,L2Ω,β,μ so that B,D is admissible by respect to the integral operator imposed by (66).(2)
The functionsF and G maps from Mρ into B and there exists three control functions ψ1, ψ2, and φ such that for any x1,x2∈Mρ,(68)ψ1Fx1−Gx2B≤ψ2x1−x2D−φx1−x2D.(3)
h∈D.(4)
cω∈0,1 a.s. and(69)hD+cωG0^B≤ρ1−cωa.s.,hD+cωF0^B≤ρ1−cωwhere the norm of Tω is denoted by cω.Then, if the above conditions hold, system NSIE admits a unique solution inMρ2.Proof.
Consider the mappingsU,V:Ω×Mρ⟶D such that for all x∈Mρ,(70)Uωxt;ω=ht;ω+∫ℝkt,s;ωfs,xs;ωdλsa.s.(71)Vωxt;ω=ht;ω+∫ℝkt,s;ωgs,xs;ωdλsa.s.
Letx∈Mρ. Since(72)UωxD≤hD+cωFxBa.s.≤hD+cωG0^B+cωFx−G0^Ba.s.≤hD+cωG0^B+cωxDa.s.≤hD+cωG0^B+cωρa.s.≤ρ1−cω+cωρ≤ρa.s.,then Uωx∈Mρ a.s. By the same argument, we prove that Vωx∈Mρa.s.
Letx1,x2∈Mρ. We have(73)Uωx1−Vωx2D≤cωFx1−Gx2B≤Fx1−Gx2Ba.s.
Then,(74)ψ1Uωx1−Vωx2D≤ψ1Fx1−Gx2B≤ψ1x1−x2D−φx1−x2Da.s.,which shows that Uω and Vω are ψ1,ψ2,φ-weakly contractive mappings almost surely on Mρ. Then, the two operators U,V:Ω×Mρ⟶Mρ are ψ1,ψ2,φ-weakly contractive random operators. Therefore, by Theorem 5, there is a unique common random fixed point of random operators U and V, which is the unique stochastic solution of NSIE. This completes the proof.


## 6. Conclusion

In this paper, we have the following:Theorem5 is a random generalization of the main result of [32] in a separable Banach spaceCorollary 1 generalizes and improves ([36], Theorem 2.5) in the setting of Banach spacesCorollary4 is a random version of Banach contraction principle in a separable Banach spaceTheorem6 extends ([36], Theorem 5.3) to ψ1,ψ2,ϕ-weakly contractive random operatorsTheorem7 shows the utility of our main result in solving a system of nonlinear stochastic integral equationsThis work will open the door for other deterministic results that can be randomized, for example, [29].



---

*Source: 9484771-2021-12-07.xml*