Random fixed point theorem on a Ćirić-type contractive mapping and its consequence

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R E S E A R C H

Open Access

Random fixed point theorem on a Ćirić-type

contractive mapping and its consequence

M Saha

1

_{and Anamika Ganguly}

2*

*_{Correspondence:} [email protected] 2_{Burdwan Railway Balika Vidyapith} High School, Khalasipara, Burdwan, West Bengal 713101, India Full list of author information is available at the end of the article

Abstract

Purpose:The main purpose of this paper is to prove a random ﬁxed point theorem in a separable Banach space equipped with a complete probability measure for a certain class of contractive mappings.

Results:The main finding of this paper is the identification of some random fixed point theorems and the relevant application with appropriate supporting examples.

Conclusion:A random ﬁxed point theorem is useful to determine the existence of a solution in a Banach space of a random nonlinear integral equation.

MSC: Primary 47H10; secondary 60H25

Keywords: complete probability measure space; random variable; random solution; random ﬁxed point equation; Bochner integral

1 Introduction

The application of fixed point theory in different branches of mathematics, statistics, en-gineering and economics relating to problems associated with approximation theory, the-ory of differential equations, thethe-ory of integral equations,etc.has been recognized in the existing literature [, ] and []. Progress in the study on fixed points of non-expansive mappings, contractive mappings in various spaces like a metric space, a Banach space, a fuzzy metric space, a cone metric spaceetc.has been saturated at large. After the initial im-petus given by the Prague school of Probability in s, considerable attention has been given to the study of random fixed point theorems. This arises because of the significance of fixed point theorems in probabilistic functional analysis and probabilistic models along with several applications. Issues relating to measurability of solutions, probabilistic and statistical aspects of random solutions have arisen due to the introduction of randomness. It is no denying the fact that random fixed point theorems are stochastic generalizations of classical fixed point theorems that have been described as deterministic results.

Špaček [] and Hanš [, ] first proved random fixed point theorems for random con-traction mappings on separable complete metric spaces. The article by Bharucha-Reid [] in  attracted the attention of several mathematicians and led to the development of this theory. Špaček’s and Hanš’s theorems have been extended to multivalued contraction mappings by Itoh []. A random version of Schaduer’s fixed point theorem on an atomic probability measure space has been provided by Mukherjee []. The results of this work have been generalized by Bharucha-Reid [, ] on a general probability measure space. Itoh [] obtained random fixed point theorems with an application to random differential

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equations in Banach spaces. Several random fixed point theorems including random ana-logue of the classical results based on Rothe [] have been obtained by Sehgal and Waters []. Kumam in a series of papers (see [–]) proved some remarkable results on random fixed point theorems. In a couple of papers [] and [], he along with his coauthor proved some random fixed point theorems for multivalued non-expansive non-self operators in the framework of Banach spaces satisfying inwardness conditions. In another paper, Ku-mam and Plubtieng [] proved some random coincidence points and random common fixed point theorems for nonlinear multivalued random operators. They also proved the existence of a random coincidence point for a pair of reciprocally continuous and compat-ible single-valued and multivalued operators. Saha [], Saha and Debnath [] in their works established some random fixed point theorems over a separable Banach space and a separable Hilbert space. On the other hand, Padgett [] applied a random fixed point theorem to prove the existence of a solution in a Banach space of a random nonlinear integral equation. Achari [], Saha and Dey [] developed this new area of application. Banach’s contraction principle [] is one of the pivotal results of nonlinear analysis. It has been the source of metric fixed point theory and its significance rests in its vast applicability in different branches of mathematics. In the general setting of a complete metric space, this theorem runs as follows (see Theorem . [] or Theorem .. []).

Theorem .(Banach’s contraction principle) Let(X,d)be a complete metric space,c∈ [, )and f :X→X be a mapping such that for each x,y∈X,d(fx,fy)≤cd(x,y).

Then f has a unique ﬁxed point a∈X,and for each x∈X,limn→∞fnx=a. On the other hand, Greguš [] proved the following ﬁxed point theorem.

Theorem . Let X be a Banach space,C be a closed convex subset of X and T:C→C be a mapping satisfying

Tx–Ty ≤ax–y+pTx–x+pTy–y

for all x,y∈C,where <a< ,p≥and a+ p= .Then T has a unique ﬁxed point.

During the eighties, many theorems which are closely related to the Greguš theorem have appeared in several literatures (see [–] and []). Also, Ćirić [] dealt with a class of mappings (not necessarily continuous) which are deﬁned on a metric space and proved the following ﬁxed point theorem which is a double generalization of Greguš [].

Theorem . Let C be a closed convex subset of a complete convex metric space X and T:C→C be a mapping satisfying

d(Tx,Ty)≤amaxd(x,y),cd(x,Ty) +d(y,Tx) +bmaxd(x,Tx),d(y,Ty),

where <a< ,a+b= ,c≤–_–a_a for all x,y∈C.Then T has a unique ﬁxed point.

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Theorem . Let C be a closed convex metric space X and T:C→C be a mapping sat-isfying

d(Tx,Ty)≤ad(x,y) +bmaxd(x,Tx),d(y,Ty) +cd(x,Ty) +d(y,Tx)

for all x,y∈C,where≤a< ,b≥,c≥,a+c> and a+b+

c= .Then T has a

unique ﬁxed point.

Ćirić also introduced several contractive operators on metric spaces and proved many fixed point theorems on such operators. Inspired by Ćirić’s contractive operators, many researchers have obtained fixed point theorems on Ćirić’s operators in different settings. In this context, Karapinar [] proved some non-unique fixed point theorems on Ćirić-type contractive operators in cone metric spaces. Also Karapinaret al.[] proved a fixed point theorem on a metric space for a class of maps that satisfy the Ćirić-type contractive condition.

In this paper, our main objective is to prove some random ﬁxed point theorems in a separable Banach space equipped with a complete probability measure for a certain class of contractive mappings. The results are stochastic generalizations of deterministic ﬁxed point theorems of Ćirić []. The result obtained in this paper will also be useful in applica-tion to a random nonlinear integral equaapplica-tion. Also, we have introduced some appropriate supporting examples.

In order to make the paper self-contained, we state some important deﬁnitions and an example that are available in Joshi and Bose [] and Debnath and Mikusinski [].

2 Preliminaries

Let (X,βX) be a separable Banach space, whereβX is aσ-algebra of Borel subsets ofX, and let (,β,μ) denote a complete probability measure space with measureμandβbe a

σ-algebra of subsets of. For more details, one can see Joshi and Bose [].

Deﬁnition . A mappingx:→X is said to be anX-valued random variable if the inverse image under the mappingxof every Borel setBofXbelongs toβ, that is,x–_(B)_∈_β

for allB∈βX.

Definition . A mappingx:→Xis said to be a finitely-valued random variable if it is constant on each finite number of disjoint setsAi∈βand is equal to  on– (

n

i=Ai). xis called a simple random variable if it is ﬁnitely valued andμ{ω:x(ω)> }<∞.

Definition . A mappingx:→Xis said to be a strong random variable if there exists a sequence{xn(ω)}of simple random variables which converges tox(ω) almost surely, that is, there exists a setA∈βwithμ(A) =  such thatlimn→∞xn(ω) =x(ω),ω∈–A. Definition . A mappingx:→Xis said to be a weak random variable if the function x*_(x(_ω_{)) is a real-valued random variable for each}_x*_∈_X*_{, the space}_X*_{denoting the first}

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In a separable Banach spaceX, the notions of strong and weak random variablesx:→

X(see Corollary  of Joshi and Bose []) coincide, and in respect of such a spaceX,xis termed as a random variable.

We recall the following results.

Theorem .(see Theorem ..(a) of Joshi and Bose []) Let x,y:→X be strong ran-dom variables andα,βbe constants.Then the following statements hold:

(a) αx(ω) +βy(ω)is a strong random variable.

(b) Iff(ω)is a real-valued random variable andx(ω)is a strong random variable,then f(ω)x(ω)is a strong random variable.

(c) If{xn(ω)}is a sequence of strong random variables converging strongly tox(ω)almost surely,i.e.,if there exists a setA∈βwithμ(A) = such that

limn→∞xn(ω) –x(ω)= for everyω∈/A,thenx(ω)is a strong random variable. Remark . IfXis a separable Banach space, then every strong and also weak random variable is measurable in the sense of Deﬁnition ..

LetYbe another Banach space. We also need the following deﬁnitions as cited in Joshi and Bose [].

Deﬁnition . A mappingF:×X→Yis said to be a random mapping ifF(ω,x) =Y(ω) is aY-valued random variable for everyx∈X.

Deﬁnition . A mappingF:×X→Yis said to be a continuous random mapping if the set of allω∈for whichF(ω,x) is a continuous function ofxhas measure one.

Deﬁnition . A mappingF:×X→Yis said to be demi-continuous at thex∈Xif xn–x → impliesF(ω,xn)

weakly

–→ F(ω,x) almost surely.

Theorem .(see Theorem .. of Joshi and Bose []) Let F:×X→Y be a demi-continuous random mapping where a Banach space Y is separable.Then,for any X-valued random variable x,the function F(ω,x(ω))is a Y -valued random variable.

Remark .(see []) Since a continuous random mapping is a demi-continuous random mapping, Theorem . is also true for a continuous random mapping.

We shall also recall the following deﬁnitions as seen in Joshi and Bose [].

Deﬁnition . An equation of the typeF(ω,x(ω)) =x(ω), whereF:×X→Xis a random mapping, is called a random ﬁxed point equation.

Definition . Any mappingx:→Xwhich satisfies the random fixed point equa-tionF(ω,x(ω)) =x(ω) almost surely is said to be a wide sense solution of the fixed point equation.

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Remark . A random solution is a wide sense solution of the ﬁxed point equation. But the converse is not necessarily true. This is evident from the following example as found under Remark  in the work of Joshi and Bose [].

Example . LetXbe the set of all real numbers and letEbe a non-measurable subset ofX. LetF:×X→Ybe a random mapping defined asF(ω,x) =x+x–  for allω∈. In this case, the real-valued functionx(ω), defined asx(ω) =  for allω∈, is a random fixed point ofF. However, the real-valued functiony(ω) defined as

y(ω) = ⎧ ⎨ ⎩

–, ω∈/E,

, ω∈E

is a wide sense solution of the ﬁxed point equationF(ω,x(ω)) =x(ω) without being a ran-dom ﬁxed point ofF.

3 Main results

Theorem . Let X be a separable Banach space and(,β,μ)be a complete probability measure space.Let T:×X→X be a continuous random operator such that forω∈, T satisﬁes

T(ω,x) –T(ω,x)≤a(ω)max

x–x,c(ω)x–T(ω,x)+x–T(ω,x)

+b(ω)maxx–T(ω,x),x–T(ω,x)

for all x,x∈X,where a(ω),b(ω),c(ω)are real-valued random variables such that <

a(ω) < ,a(ω) +b(ω) = ,c(ω)≤–_–_aa(ω)_(ω) almost surely. Then there exists a unique random ﬁxed point of T in X.

Proof LetA={ω∈:T(ω,x) is a continuous function ofx}

B=ω∈:  <a(ω) < ∩ω∈:a(ω) +b(ω) = 

∩

ω∈:c(ω)≤ –a(ω)  –a(ω)

Cx,x=

ω∈:T(ω,x) –T(ω,x)

≤a(ω)maxx–x,c(ω)x–T(ω,x)+x–T(ω,x)

+b(ω)maxx–T(ω,x),x–T(ω,x)

.

LetSbe a countable dense subset ofX. We now prove that

x,x∈X

(Cx,x∩A∩B) =

s,s∈S

(Cs,s∩A∩B).

Then for alls,s∈S,

T(ω,s) –T(ω,s)≤a(ω)max

s–s,c(ω)s–T(ω,s)+s–T(ω,s)

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SinceSis dense inX, givenδi(xi) >  (i= , ), there exists,s∈Ssuch thatxi–si<δi(xi); i= , .

Letx,x∈X.

Note that

s–T(ω,s)≤ s–x+x–T(ω,x)+T(ω,x) –T(ω,s), (.)

s–s ≤ s–x+x–x+x–s, (.)

s–T(ω,s)≤ s–x+x–T(ω,x)+T(ω,x) –T(ω,s), (.)

s–T(ω,s)≤ s–x+x–T(ω,x)+T(ω,x) –T(ω,s), (.)

s–T(ω,s)≤ s–x+x–T(ω,x)+T(ω,x) –T(ω,s). (.)

We now examine the following cases. Case I:

Suppose

T(ω,s) –T(ω,s)≤a(ω)s–s+b(ω)s–T(ω,s).

Now

_T(ω,x) –T(ω,x)≤T(ω,x) –T(ω,s)+T(ω,s) –T(ω,s)

+T(ω,s) –T(ω,x)

≤T(ω,x) –T(ω,s)+T(ω,s) –T(ω,x)

+a(ω)s–s+b(ω)s–T(ω,s). (.)

Using (.), (.), (.), we get by routine calculation

T(ω,x) –T(ω,x)≤

 +b(ω)T(ω,x) –T(ω,s)+T(ω,s) –T(ω,x)

+a(ω) +b(ω)s–x+b(ω)x–T(ω,x)

+a(ω)x–x+a(ω)x–s. (.)

Since for a particularω∈,T(ω,x) is a continuous function ofx, so for anyε> , there existsδi(xi) >  (i= , ) such that

T(ω,x) –T(ω,s)<

ε

 wheneverx–s<δ(x) (.)

and

T(ω,x) –T(ω,s)<

ε

 wheneverx–s<δ(x). (.)

Now choose

δ=min

δ(x),

ε



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and

δ=min

δ(x),

ε



. (.)

For such a choice ofδ,δby (.), we get

_T₍ω,x) –T(ω,x)≤

 +b(ω)ε +

ε

+

a(ω) +b(ω)ε 

+a(ω)ε

+b(ω)x–T(ω,x)+a(ω)x–x

= ε

+a(ω)x–x+b(ω)x–T(ω,x).

Asε>  is arbitrary, it follows that

T(ω,x) –T(ω,x)≤a(ω)x–x+b(ω)x–T(ω,x). (.)

Case II: Suppose

T(ω,s) –T(ω,s)≤a(ω)c(ω)s–T(ω,s)+s–T(ω,s)

+b(ω)s–T(ω,s).

Now

T(ω,x) –T(ω,x)≤T(ω,x) –T(ω,s)+T(ω,s) –T(ω,s)

+T(ω,s) –T(ω,x)

≤T(ω,x) –T(ω,s)+T(ω,s) –T(ω,x)

+a(ω)c(ω)s–T(ω,s)+s–T(ω,s)

+b(ω)s–T(ω,s). (.)

Using (.), (.), (.), (.), by routine calculation, we get T(ω,x) –T(ω,x)≤

 +a(ω)c(ω) +b(ω)T(ω,x) –T(ω,s)

+ +a(ω)c(ω)T(ω,x) –T(ω,s)

+a(ω)c(ω) +b(ω)s–x+a(ω)c(ω)s–x

+a(ω)c(ω)x–T(ω,x)+x–T(ω,x)

+b(ω)x–T(ω,x). (.)

Since for a particularω∈,T(ω,x) is a continuous function ofx, by using (.), (.), (.), (.) and for such a choice ofδ,δ, we get by (.)

T(ω,x) –T(ω,x)≤

 +a(ω)c(ω) +b(ω)ε +

 +a(ω)c(ω)ε 

+a(ω)c(ω) +b(ω)ε

+a(ω)c(ω)

ε

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+a(ω)c(ω)x–T(ω,x)+x–T(ω,x)

+b(ω)x–T(ω,x)

<  – a(ω)  –a(ω)

ε

+a(ω)c(ω)x–T(ω,x)+x–T(ω,x)

+b(ω)x–T(ω,x)

<ε+a(ω)c(ω)x–T(ω,x)+x–T(ω,x)

+b(ω)x–T(ω,x).

T(ω,x) –T(ω,x)≤a(ω)c(ω)x–T(ω,x)+x–T(ω,x)

+b(ω)x–T(ω,x). (.)

Case III: Suppose

T(ω,s) –T(ω,s)≤a(ω)s–s+b(ω)s–T(ω,s).

Now

+T(ω,s) –T(ω,x)

≤T(ω,x) –T(ω,s)+T(ω,s) –T(ω,x)+a(ω)s–s

+b(ω)s–T(ω,s). (.)

Using (.), (.), (.), by a routine check-up, we get

T(ω,x) –T(ω,x)≤T(ω,x) –T(ω,s)+

 +b(ω)T(ω,s) –T(ω,x)

+a(ω)s–x+

a(ω) +b(ω)x–s

+a(ω)x–x+b(ω)x–T(ω,x). (.)

Since for a particularω∈,T(ω,x) is a continuous function ofx, by using (.), (.), (.), (.) and for such a choice ofδ,δ, we get from relation (.)

T(ω,x) –T(ω,x)≤

ε



 + a(ω) + b(ω)+a(ω)x–x+b(ω)x–T(ω,x)

= ε

+a(ω)x–x+b(ω)x–T(ω,x).

T(ω,x) –T(ω,x)≤a(ω)x–x+b(ω)x–T(ω,x). (.)

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Suppose

T(ω,s) –T(ω,s)≤a(ω)c(ω)s–T(ω,s)+s–T(ω,s)

+b(ω)s–T(ω,s).

Now

+T(ω,s) –T(ω,x)

≤T(ω,x) –T(ω,s)+T(ω,s) –T(ω,x)

+a(ω)c(ω)s–T(ω,s)+s–T(ω,s)

+b(ω)s–T(ω,s). (.)

Using (.), (.), (.) and (.), we get by a routine check-up

T(ω,x) –T(ω,x)≤

 +a(ω)c(ω)T(ω,x) –T(ω,s)

+ +a(ω)c(ω) +b(ω)T(ω,x) –T(ω,s)

+a(ω)c(ω)s–x+

a(ω)c(ω) +b(ω)s–x

+a(ω)c(ω)x–T(ω,x)+x–T(ω,x)

+b(ω)x–T(ω,x). (.)

Since for a particularω∈,T(ω,x) is a continuous function ofx, by using (.), (.), (.), (.) and for such a choice ofδ,δ, we get by (.)

T(ω,x) –T(ω,x)≤

 +a(ω)c(ω)ε +

 +a(ω)c(ω) +b(ω)ε 

+a(ω)c(ω)ε +

a(ω)c(ω) +b(ω)ε 

+a(ω)c(ω)x–T(ω,x)+x–T(ω,x)

+b(ω)x–T(ω,x)

<  – a(ω)  –a(ω)

ε

+a(ω)c(ω)x–T(ω,x)+x–T(ω,x)

+b(ω)x–T(ω,x)

<ε+a(ω)c(ω)x–T(ω,x)+x–T(ω,x)

+b(ω)x–T(ω,x).

T(ω,x) –T(ω,x)≤a(ω)c(ω)x–T(ω,x)+x–T(ω,x)

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Combining (.), (.), (.) and (.), we get

T(ω,x) –T(ω,x)≤a(ω)max

x–x,c(ω)x–T(ω,x)+x–T(ω,x)

+b(ω)maxx–T(ω,x),x–T(ω,x).

Thus,ω∈_x

,x∈XCx,x∩A∩B, which implies

s,s∈S

Cs,s∩A∩B⊂

x,x∈X

Cx,x∩A∩B.

Also,

x,x∈X

Cx,x∩A∩B⊂

s,s∈S

Cs,s∩A∩B.

Thus,

x,x∈X

Cx,x∩A∩B=

s,s∈S

Cs,s∩A∩B.

LetN=_s__,_s_∈_SCs,s∩A∩B, thenμ(N) = .

So, for eachω∈N,T(ω,x) is a deterministic operator due to Ćirić []. Hence,Thas

a unique ﬁxed point inX.

Theorem . Let X be a separable Banach space and(,β,μ)be a complete probability measure space.Let T:×X→X be a continuous random operator such that forω∈, T satisﬁes

T(ω,x) –T(ω,x)≤a(ω)x–x+b(ω)maxx–T(ω,x),x–T(ω,x)

+c(ω)x–T(ω,x)+x–T(ω,x) (.)

for all x,y∈X,where a(ω),b(ω),c(ω)are real-valued random variables such that

 <a(ω) < , b(ω)≥, c(ω)≥ satisfying a(ω) +c(ω) >  (.)

and

a(ω) +b(ω) +

c(ω) =  almost surely, (.)

then T has a unique random ﬁxed point in X.

Proof Seta(ω) +_c(ω) =a(ω).

Thena(ω) +b(ω) =  and we have

a(ω)x–x+b(ω)maxx–T(ω,x),x–T(ω,x)

+c(ω)· ·



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≤

a(ω) + c(ω)

max

x–x,



x–T(ω,x)+x–T(ω,x)

+b(ω)maxx–T(ω,x),x–T(ω,x).

Since_<–_–a_a_(ω)(ω), the relation (.), (.) and (.) would imply Theorem . witha(ω)+

b(ω) = . Therefore, we can apply Theorem . and consequentlyThas a unique random

ﬁxed point inX.

We now give a couple of examples in support of Theorem . and Theorem ..

Example . LetE= [–, ]⊂Rwith the usual norm of reals.

Consider= [–, ] and letβbe aσ-algebra of Lebesgue measurable sets of [–, ]. DeﬁneT:×E→EbyT(ω,x) =_x, wherex∈Eandω∈.

By a routine check-up, we see that the condition of Theorem . is satisﬁed whenever a(ω) =_,b(ω) = _ and ≤c(ω)≤_. The functionx:→Ewithx(ω) =  is a unique random ﬁxed point ofT.

By consideringEandas above, we takea(ω) =_,b(ω) =_ andc(ω) = . We see that condition (.) of Theorem . is satisﬁed and the functionx:→E withx(ω) =  is the unique random ﬁxed point ofT.

Example . LetX=Rwith the usual norm of reals. Let =R. β be aσ-algebra of Lebesgue measurable sets ofR.

DeﬁneT:×X→XbyT(ω,x) =_.

All the conditions of Theorem . and Theorem . are satisﬁed. In both of the cases, we see that the functionx:→Ewithx(ω) =_ is the unique random ﬁxed point ofT.

4 Application to a random nonlinear integral equation

Here we apply Theorem . to prove the existence of a solution in a Banach space of a random nonlinear integral equation of the following form:

x(t;ω) =h(t;ω) +

S

k(t,s;ω)fs,x(s;ω)dμ(s), (.)

where

(i) Sis a locally compact metric space with a metricdonS×Sequipped with a completeσ-ﬁnite measureμdeﬁned on the collection of Borel subsets ofS; (ii) ω∈, whereωis the supporting element of a set of probability measure space

(,β,μ);

(iii) x(t;ω)is the unknown vector-valued random variable for eacht∈S; (iv) h(t;ω)is the stochastic free term deﬁned fort∈S;

(v) k(t,s;ω)is the stochastic kernel deﬁned fortandsinSand (vi) f(t,x)is a vector-valued function oft∈Sandx.

The integral in equation (.) is interpreted as a Bochner integral [].

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Deﬁnition . We deﬁne the spaceC(S,L(,β,μ)) to be the space of all continuous

func-tions fromSintoL(,β,μ) with the topology of uniform convergence on compact sets

ofS, that is, for each ﬁxedt∈S,x(t;ω) is a vector-valued random variable such that

x(t;ω)_L

(,β,μ)=

x(t;ω)dμ(ω) <∞.

It may be noted thatC(S,L(,β,μ)) is a locally convex space (see []) whose topology

is deﬁned by a countable family of semi-norms given by

x(t;ω)_n=sup

t∈Cn

x(t;ω)_L

(,β,μ), n= , , . . . .

Moreover,C(S,L(,β,μ)) is complete relative to this topology sinceL(,β,μ) is

com-plete.

We further deﬁneBC=BC(S,L(,β,μ)) to be the Banach space of all bounded

con-tinuous functions fromSintoL(,β,μ) with the norm

x(t;ω)_BC=sup

t∈S

x(t;ω)_L (,β,μ).

Here the spaceBC⊂Cis the space of all second-order vector-valued functions deﬁned onSwhich are bounded and continuous in mean square. We will consider the function h(t;ω) andf(t,x(t;ω)) to be in the spaceC(S,L(,β,μ)) with respect to the stochastic

kernel. We assume that for each pair (t,s),k(t,s;ω)∈L∞(,β,μ) and denote the norm by

k(t,s;ω)=k(t,s;ω)_L_∞_(,β,μ)

=μ–ess sup

ω∈

k(t,s;ω).

Let us suppose thatk(t,s;ω) is such that|k(t,s;ω)|·x(s;ω)L(,β,μ)isμ-integrable with

respect tosfor eacht∈Sandx(s;ω) inC(S,L(,β,μ)), and let there exist a real-valued

functionGdeﬁnedμ-a.e. onS, so thatG(S)x(s;ω)L(,β,μ)isμ-integrable so that for each pair (t,s)∈S×S,

k(t,u;ω) –k(s,u;ω)·x(u,ω)_L

(,β,μ)≤G(u)x(u,ω)L(,β,μ),

μ-a.e. Further, for almost all s ∈ S, k(t,s;ω) will be continuous in t from S into

L∞(,β,μ).

We now deﬁne the random integral operatorTonC(S,L(,β,μ)) by

(Tx)(t;ω) =

S

k(t,s;ω)x(s;ω)dμ(s), (.)

where the integral is a Bochner integral. Moreover, we have that for eacht∈S, (Tx)(t;ω)∈ L(,β,μ) and that (Tx)(t;ω) is continuous in mean square by the Lebesgue dominated

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Deﬁnition .(see [] and []) LetBandDbe two Banach spaces. The pair (B,D) is said to be admissible with respect to a random operatorT(ω) ifT(ω)(B)⊂D.

Lemma .(see []) The linear operator T deﬁned by(.)is continuous from C(S,L(,

β,μ))into itself.

Lemma .(see [] and []) If T is a continuous linear operator from C(S,L(,β,μ))

into itself and B,D⊂C(S,L(,β,μ))are Banach spaces stronger than C(S,L(,β,μ))

such that(B,D)is admissible with respect to T,then T is continuous from B into D.

Remark .(see []) The operatorTdeﬁned by (.) is a bounded linear operator from BintoD.

A random solution of equation (.) will mean a function x(t;ω) in C(S,L(,β,μ))

which satisﬁes equation (.)μ-a.e.

We are now in a position to prove the following theorem.

Theorem . We consider the stochastic integral equation(.)subject to the following conditions:

(a)B and D are Banach spaces stronger than C(S,L(,β,μ))such that(B,D)is

admis-sible with respect to the integral operator deﬁned by(.);

(b) x(t;ω)→ f(t,x(t;ω)) is an operator from the set Q(ρ) = {x(t;ω) : x(t;ω) ∈ D, x(t;ω)D≤ρ}into the space B satisfying

ft,x(t;ω)

–ft,x(t;ω)_B

≤a(ω)maxx(t;ω) –x(t;ω)_D,

c(ω)x(t;ω) –f

t,x(t;ω)_D+x(t;ω) –f

t,x(t;ω)_D

+b(ω)maxx(t;ω) –f

t,x(t;ω)_D+x(t;ω) –f

t,x(t;ω)_D

(.)

for x(t;ω),x(t;ω)∈Q(ρ),a(ω),b(ω),c(ω)are real-valued random variables where <

a(ω) < and a(ω) +b(ω) = with c(ω)≤–_–_aa(ω)_(ω) almost surely,and (c)h(t;ω)∈D.

Then there exists a unique random solution of(.)in Q(ρ),provided _–_a_(ω)l_c(ω)_(ω)–_b_(ω) <  and

h(t;ω)_D+l(ω)f(t, )_B

  –b(ω)+

 +b(ω)

+  +a(ω)c(ω)  –a(ω)c(ω) –b(ω)+

 +b(ω) +a(ω)c(ω)  –a(ω)c(ω)

≤ρ

 – l(ω)  –a(ω)c(ω)

, where l(ω)is the norm of T(ω).

Proof Deﬁne the operatorU(ω) fromQ(ρ) intoDby

(Ux)(t;ω) =h(t;ω) +

S

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Now

(Ux)(t;ω)_D≤h(t;ω)_D+l(ω)ft,x(t;ω)_B

≤h(t;ω)_D+l(ω)f(t; )_B+l(ω)ft,x(t;ω)–f(t; )_B.

Then from (.) of this theorem,

ft,x(t;ω)–f(t, )_B

≤a(ω)maxx(t;ω)_D,c(ω)x(t;ω) –f(t; )_D+ft,x(t;ω)_D +b(ω)maxx(t;ω) –ft,x(t;ω)_D,f(t; )_D.

Suppose

ft,x(t;ω)–f(t; )_B≤a(ω)x(t;ω)_D+b(ω)x(t;ω) –ft,x(t;ω)_D ≤a(ω)x(t;ω)_D+b(ω)x(t;ω)_D

+b(ω)ft,x(t;ω)–f(t; )_D+b(ω)f(t; )_D.

Hence,

ft,x(t;ω)–f(t; )_B≤a(ω) +b(ω)  –b(ω) ρ+

b(ω)

 –b(ω)f(t; )B. (.)

So, by (.)

(Ux)(t;x)_D≤h(t;ω)_D+l(ω)f(t; )_B

+a(ω) +b(ω)  –b(ω) l(ω)ρ+

b(ω)l(ω)

 –b(ω)f(t; )B

<h(t;ω)_D+ 

 –b(ω)l(ω)f(t; )B+ 

 –b(ω)l(ω)ρ

<ρ. (.)

Again, suppose

ft,x(t;ω)–f(t; )_B≤a(ω)x(t;ω)_D+b(ω)f(t; )_D.

So,

ft,x(t;ω)–f(t; )_B≤a(ω)ρ+b(ω)f(t; )_B. (.)

Therefore, by (.)

(Ux)(t;ω)_D≤h(t;ω)_D+l(ω)f(t; )_B+l(ω)a(ω)ρ+l(ω)b(ω)f(t; )_B

=h(t;ω)_D+ +b(ω)l(ω)f(t; )_B+l(ω)a(ω)ρ

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Also, suppose

ft,x(t;ω)–f(t; )_B≤a(ω)c(ω)x(t;ω) –f(t; )_D+ft,x(t;ω)_D

+b(ω)x(t;ω) –ft,x(t;ω)_D

≤a(ω)c(ω)x(t;ω)_D+a(ω)c(ω)f(t; )_D

+a(ω)c(ω)ft;x(t;ω)–f(t; )_D

+a(ω)c(ω)f(t; )_D+b(ω)x(t;ω)_D

+b(ω)ft,x(t;ω)–f(t; )_D+b(ω)f(t; )_D.

Then

ft,x(t;ω)–f(t; )_B≤ a(ω)c(ω) +b(ω)  –a(ω)c(ω) –b(ω)ρ

+

a(ω)c(ω) +b(ω)  –a(ω)c(ω) –b(ω)

f(t; )_B. (.)

Therefore, by (.)

(Ux)(t;ω)_D≤h(t;ω)_D+l(ω)f(t; )_B

+l(ω) a(ω)c(ω) +b(ω)  –a(ω)c(ω) –b(ω)ρ

+ a(ω)c(ω) +b(ω)

 –a(ω)c(ω) –b(ω)l(ω)f(t; )B

=h(t;ω)_D+  +a(ω)c(ω)

 –a(ω)c(ω) –b(ω)l(ω)f(t; )B

+l(ω) a(ω)c(ω) +b(ω)  –a(ω)c(ω) –b(ω)ρ

<ρ. (.)

Suppose

ft,x(t;ω)–f(t; )_B≤a(ω)c(ω)x(t;ω) –f(t; )_D+ft,x(t;ω)_D

+b(ω)f(t; )_D

≤a(ω)c(ω)x(t;ω)_D+a(ω)c(ω)f(t; )_D

+a(ω)c(ω)ft,x(t;ω)–f(t; )_D

+a(ω)c(ω)f(t; )_D+b(ω)f(t; )_D.

So,

ft,x(t;ω)–f(t; )_B≤ a(ω)c(ω)  –a(ω)c(ω)ρ+

b(ω) + a(ω)c(ω)

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Therefore, by (.)

(Ux)(t;ω)_D≤h(t;ω)_D+l(ω)f(t; )_B+ a(ω)c(ω)  –a(ω)c(ω)l(ω)ρ

+b(ω) + a(ω)c(ω)

 –a(ω)c(ω) l(ω)f(t; )B

=h(t;ω)_D+ a(ω)c(ω)

 –a(ω)c(ω)l(ω)ρ+

 +b(ω) +a(ω)c(ω)

 –a(ω)c(ω) l(ω)f(t; )B

<ρ. (.)

Then by (.), (.), (.) and (.), we get (Ux)(t;ω)∈Q(ρ). Then forx(t;ω),x(t;ω)∈Q(ρ). We have by condition (b)

(Ux)(t;ω) – (Ux)(t;ω)_D

=

S

k(t,s;ω)fs,x(s;ω)

–fs,x(s;ω)

dμ(s)

D

≤l(ω)ft,x(t;ω)

–ft,x(t;ω)_B

≤a(ω)maxx(t;ω) –x(t;ω),

c(ω)x(t;ω) – (Ux)(t;ω)_D+x(t;ω) – (Ux)(t;ω)_D

+b(ω)maxx(t;ω) – (Ux)(t;ω)_D,x(t;ω) – (Ux)(t;ω)_D

.

Since_–_a_(ω)l_c(ω)_(ω)–_b_(ω)< .

Therefore,U(ω) is a random contractive nonlinear operator onQ(ρ). Hence, by Theo-rem ., there exists a random ﬁxed point ofU(ω), which is the random solution of

equa-tion (.).

Example . Consider the following nonlinear stochastic integral equation:

x(t;ω) =

_∞



e–t–s ( +|x(s;ω)|)ds.

Comparing with (.), we see that

h(t;ω) = , k(t,s;ω) =  e

–t–s_, _f_s,_x(s;_ω₎₌  ( +|x(s;ω)|).

By routine calculation, it is easy to show that (.) is satisﬁed witha(ω) =  ,b(ω) =

  and

≤c(ω)≤_.

Comparing with integral operator equation (.), we see that the norm of T(ω) is l(ω) =_.

Also, we see that_–_a_(ω)l_c(ω)_(ω)–_b_(ω)< . So, all the conditions of Theorem . are satisﬁed and hence there exists a random ﬁxed point of the integral operatorTsatisfying (.).

Competing interests

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Authors’ contributions

Both the authors are jointly responsible for the present research work and have equally contributed to the research work. Both the authors have read and approved the ﬁnal manuscript.

Author details

1_{Department of Mathematics, The University of Burdwan, Burdwan, West Bengal 713104, India.}2_{Burdwan Railway Balika} Vidyapith High School, Khalasipara, Burdwan, West Bengal 713101, India.

Acknowledgements

Authors remain grateful to the honorable reviewers for their kind suggestions for improvement of our paper.

Received: 25 June 2012 Accepted: 6 November 2012 Published: 22 November 2012 References

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