Random Fixed Point Theorem for Weakly Compatible Mappings under Implicit Relation in Cone Random Metric Spaces

(1)

Random Fixed Point Theorem for Weakly Compatible

Mappings under Implicit Relation in Cone Random

Metric Spaces

R. A. Rashwan

1,∗

,

H. A. Hammad

2

1_{Department of Mathematics, Faculty of Science, Assuit University, Assuit 71516, Egypt}

2_{Department of Mathematics, Faculty of Science, Sohag University, Sohag 82524, Egypt}

Abstract

In thispaper, we establish a uniquecommon randomfixedp ointt heoremi nc oner andomm etricspaces for fourweakly compatible mappings byusing an implicit relation. Somecorollariesofthistheoremfortwoandthree random weakly compatible mappings are obtained. Some examples are given to support our generalization. Our results presented in this paper extend and improve several recentresultsinthesettingofconerandommetricspaces.

Keywords

Random Fixed Point, Cone Random Metric Spaces, Random Weakly Compatible Mappings, Implicit Relation

1 Introduction

Random nonlinear analysis is an important mathematical discipline which is mainly concerned with the study of ran-dom nonlinear operators and their properties and is much needed for the study of various classes of random equa-tions. Developments in the investigation on fixed points of non-expansive mappings, contractive mappings in differ-ent spaces like metric spaces, Banach spaces, Fuzzy metric spaces and cone metric spaces have almost been saturated. The study of random fixed point theorems was initiated by the Prague school of probabilistic in 1950’s [9, 10, 31]. The introduction of randomness leads to several new questions of measurability of solutions, probabilistic and statistical as-pects of random solutions. Common random fixed point theorems are stochastic generalization of classical common fixed point theorems. Random methods have revolutionized the financial markets. The survey article by Bharucha-Reid [8] in 1976 attracted the attention of several mathematicians and gave wings to the theory. The results of ˇSpaˇcek and Hanˇs in multi-valued contractive mappings was extended by

Itoh [14]. Now this theory has become the full fledged re-search area and various ideas associated with random fixed point theory are used to give the solution of nonlinear system see [5-7, 11, 25] and nonlinear integral equation see [19-21]. Common random fixed points and random coincidence points of a pair of compatible random operators and fixed point the-orems for contractive random operators in Polish spaces are obtained by Papageorgiou [17, 18] and Beg [3, 4].

Huang and Zhang [12] improved the concept of metric spaces by replacing the set of real numbers with an or-dered Banach space, hence they have defined the cone metric spaces. They also described the convergence of sequences and introduced the notion of completeness in cone metric spaces. They have proved some fixed point theorems of con-tractive mappings on complete cone metric space with the as-sumption of normality of a cone. According to this concept, several other authors [1, 13, 24, 30] studied the existence of fixed points and common fixed points of mappings satisfying contractive type condition on a normal cone metric space. In 2008, the assumption of normality in cone normal spaces is deleted by Rezapour and Hamlbarani [24], which is an im-portant event in developing fixed point theory in cone metric spaces.

Random fixed point results in cone random metric spaces are stochastic generalization of deterministic fixed point re-sults in cone metric space. Several other authors see [15, 22, 26-29] studied the existence of random fixed points and com-mon random fixed points of mappings satisfying contractive type conditions in setting cone random metric spaces.

(2)

2 Preliminaries

2.1

Definition [16]

Let(E, τ)be a topological vector space. A subsetpofE

is called a cone if the following conditions satisfied: (c1)pis closed, nonempty andp6={0};

(c2)a, b∈R,a, b≥0andx, y∈p⇒ax+by∈p;

(c3) Ifx∈pand−x∈p⇒x= 0.

For a given conep⊂E,we define a partial ordering≤with respect topbyx≤y iffy−x∈ p.We shall writex < y

to indicate thatx≤ybutx6=y, whilexywill stand for

y−x∈p◦, wherep◦indicate to the interior ofp.

2.2

Definition [12, 32]

LetX be a nonempty set and the mapping the mapping

d:X×X →Esatisfies:

(d1) 0 ≤ d(x, y)for all x, y ∈ X andd(x, y) = 0 ⇔ x=y;

(d2)d(x, y) =d(y, x)for allx, y∈X;

(d3)d(x, y)≤d(x, z) +d(z, y)x, y, z∈X.

Thendis called a cone metric [12] orK−metric [32] on

Xand(X, d)is called a cone metric space [12].

The concept of a cone metric space is more general than that of a metric space, because each metric space is a cone metric space whereE=_Randp= [0,+∞).

2.3

Example [12]

Let E = _R2_{, p} ₌ _{₍_{x, y}₎ _∈

R2 : x ≥ 0, y ≥ 0}, X = _R and d : X ×X → E defined by d(x, y) = (|x−y|, µ(|x−y|))whereµ≥0is a constant. Then(X, d)

is a cone metric space with normal conepwhereK= 1.

2.4

Example [23] Let E = l2_{, p} ₌ _{{_x

n}n≥1 ∈ E2 : xn ≥ 0, for all

n},(X, ρ)a metric space andd : X ×X → Edefined by

d(x, y) ={ρ(₂x,yn)}n≥1. Then(X, d)is a cone metric space. Clearly, the above examples present that the class of cone metric spaces contains the class of metric spaces.

2.5

Definition [12]

Let(X, d)be a cone metric space. We say that{xn}is:

(i) a Cauchy sequence if for everyεinEwith0ε, then there is anNsuch that for alln, m >N,d(xn, xm)ε;

(ii) a convergent sequence if for everyεinEwith0ε, then there is anNsuch that for alln >N,d(xn, x)εfor

some fixedxinX.

A cone metric space X is said to be complete if every Cauchy sequence inX is convergent inX.

The following definitions are given in [16].

2.6

Definition (Measurable function)

Let(Ω,Σ)be a measurable space withΣ−a sigma algebra of subsets ofΩandMbe a nonempty subset of a metric space

X = (X, d). Let2M _{be the family of nonempty subsets of}

M andC(M)the family of all nonempty closed subsets of

M. A mappingG: Ω→2M _{is called measurable if for each}

open subsetUofM,G−1₍_U₎_∈_Σ_,_where_G−1₍_U_{) =}_{_ω_∈

Ω :G(ω)∩U6=_∅}.

2.7

Definition (Measurable selector)

A mappingξ: Ω→M is called measurable selector of a measurable mappingsG : Ω → 2M _if_ξ_{is measurable and}

ξ(ω)∈G(ω)for eachω∈Ω.

2.8

Definition (Random operator)

The mappingT : Ω×M →Xis called a random operator iff for each fixedx∈ M,the mappingT(., x) : Ω→ X is measurable.

2.9

Definition (Continuous random mapping)

A random operatorT : Ω×M →Xis called continuous random operator if for each fixed x ∈ M andω ∈ Ω,the mappingT(ω, .) : Ω→Xis continuous.

2.10

Definition (Random fixed point)

A measurable mappingsξ : Ω → M is a random fixed point of a random operatorT: Ω×M →XiffT(ω, ξ(ω)) =

ξ(ω)for eachω∈Ω.

2.11

Definition (Cone random metric space)

LetM be a nonempty set and the mappingd: Ω×M →p,

wherepis a cone,ω ∈Ωbe a selector, satisfy the following conditions:

(i)d(x(ω), y(ω))≥0andd(x(ω), y(ω)) = 0⇔x(ω) =

y(ω)for allx(ω), y(ω)∈Ω×M,

(ii) d(x(ω), y(ω)) = d(y(ω), x(ω)) for all x, y ∈ M, ω ∈ Ω andx(ω), y(ω) ∈ Ω×M, (iii) d(x(ω), y(ω)) ≤ d(x(ω), z(ω))+d(z(ω), y(ω))for allx, y, z∈M andω∈Ω

be a selector,

(iv) for any x, y ∈ M, ω ∈ Ω, d(x(ω), y(ω)) is non-increasing and left continuous.

Then dis called cone random metric on M and(M, d)is called a cone random metric space.

2.12

Definition (Weakly compatible [2])

Random operatorsT, S: Ω×X→Xare weakly compat-ible ifT(S(ξ(ω))) =S(T(ξ(ω)))provided thatT(ξ(ω)) =

(3)

2.13

Definition (Implicit relation)

We say an implicit relation betweenxandyis given for allx, y≥0,if there exists a real number0< λ < 1such that

x≤λy,whenevereφ: (_R+₎4 _→

R+be a family of real valued continuous functions, nondecreasing in the first argument

and satisfy the following axioms:

x≤φ(y,x+y

2 ,0, x+y)orx≤φ(y,

x+y

2 , y, x)

orx≤φ(y,x

2, y, x)orx≤φ(x, y, x, x).

3 Main Results

In this section we shall prove a common random fixed point theorem under implicit relation for four weakly compatible mappings satisfying some conditions in the setting of cone random metric spaces.

3.1

Theorem

Let (X, d) be a complete cone random metric space with respect to a cone p and let M be a nonempty separable closed subset ofX.Assume thatS, T, P andQbe four continuous random operators defined onM such that forω ∈ Ω, S(ω, .), T(ω, .), P(ω, .),Q(ω, .) : Ω×M →M satisfying the following conditions:

(i)S(ω, X)⊆Q(ω, X)andT(ω, X)⊆P(ω, X),

(ii) the pairs{S, P}and{T, Q}are random weakly compatible mappings, (iii)

d(S(x(ω), T(y(ω)))≤φ

d(P(x(ω)), Q(y(ω))),1₂{d(P(x(ω)), S(x(ω))) +d(Q(y(ω)), T(y(ω)))}, d(P(x(ω)), T(y(ω))), d(Q(y(ω)), S(x(ω)))

, (3.1)

for allx(ω), y(ω)∈Ω×X. Then the four random mappings have a unique common random fixed point inX.

Proof. For each x◦(ω), x1(ω) ∈ Ω×X andn = 0,1,2, ..we choose y1(ω), y2(ω) ∈ Ω×X such that y1(ω) =

S(x◦(ω)) =Q(x1(ω))andy2(ω) =T(x1(ω)) =P(x2(ω)).In general we construct a sequence of measurable mappings

yn(ω), xn(ω) : Ω→Xdefined by

y2n+1(ω) =S(x2n(ω)) =Q(x2n+1(ω))

y2n+2(ω) =T(x2n+1(ω)) =P(x2n+2(ω))

, (3.2)

Then from (3.1) and (3.2), we get

d(y2n+1(ω), y2n+2(ω)) = d(S(x2n(ω)), T(x2n+1(ω)))

≤ φ





d(P(x2n(ω)), Q(x2n+1(ω))), 1

2{d(P(x2n(ω)), S(x2n(ω))) +d(Q(x2n+1(ω)), T(x2n+1(ω)))},

d(P(x2n(ω)), T(x2n+1(ω))), d(Q(x2n+1(ω)), S(x2n(ω)))





= φ

d(y2n(ω), y2n+1(ω)),1₂{d(y2n(ω), y2n+1(ω)) +d(y2n+1(ω), y2n+2(ω))},

d(y2n(ω), y2n+2(ω)), d(y2n+1(ω), y2n+1(ω))

≤ φ

d(y2n(ω), y2n+1(ω)),12{d(y2n(ω), y2n+1(ω)) +d(y2n+1(ω), y2n+2(ω))},0,

d(y2n(ω), y2n+1(ω)) +d(y2n+1(ω), y2n+2(ω))

.

So, from Definition 2.13, we can write

d(y2n+1(ω), y2n+2(ω))≤λd(y2n(ω), y2n+1(ω)).

Similarly, we have

d(y2n(ω), y2n+1(ω))≤λd(y2n−1(ω), y2n(ω)),

hence

d(y2n+1(ω), y2n+2(ω))≤λ2d(y2n−1(ω), y2n(ω)).

On continuing this process, we have

(4)

Also, forn > m,we get

d(yn(ω), ym(ω)) ≤ d(yn(ω), yn−1(ω)) +d(yn−1(ω), yn−2(ω)) +....+d(ym+1(ω), ym(ω))

≤ (λn−1+λn−2+...+λm)d(y1(ω), y◦(ω))

≤

_λm

1−λ

d(y1(ω), y◦(ω)).

Let0εis given. Choose a natural numberN such that₁λ₋m_λd(y1(ω), y◦(ω))εfor everym≥N,hence

d(yn(ω), ym(ω))≤

λm

1−λ

d(y1(ω), y◦(ω))ε,

this implies that{yn(ω)}is a Cauchy sequence inΩ×X.

Since(X, d)is complete, then there existsz(ω)∈Ω×X such thatyn(ω)→z(ω)asn→ ∞.Then from (3.2), we get

lim

n→∞S(x2n(ω)) = nlim→∞Q(x2n+1(ω)) =z(ω)

and lim

n→∞T(x2n+1(ω)) = nlim→∞P(x2n+2(ω)) =z(ω),

therefore

lim

n→∞S(x2n(ω)) = limn→∞Q(x2n+1(ω)) = limn→∞T(x2n+1(ω)) = limn→∞P(x2n+2(ω)) =z(ω). (3.3)

SinceT(ω, X)⊆P(ω, X),then there existsu(ω)∈Ω×Xsuch that

z(ω) =P(u(ω)). (3.4)

From (3.1), we obtain

d(S(u(ω)), z(ω))

≤ d(S(u(ω)), T(x2n+1(ω))) +d(T(x2n+1(ω)), z(ω))

≤ φ





d(P(u(ω)), Q(x2n+1(ω))), 1

2{d(P(u(ω)), S(u(ω))) +d(Q(x2n+1(ω)), T(x2n+1(ω)))},

d(P(u(ω)), T(x2n+1(ω))), d(Q(x2n+1(ω)), S(u(ω)))



+d(T(x2n+1(ω)), z(ω)).

Taking the limit asn→ ∞in above inequality and using (3.3) and (3.4), we get

d(z(ω), S(u(ω))) ≤ φ

0,d(z(ω), S(u(ω)))

2 ,0, d(z(ω), S(u(ω)))

≤0.

Thus−d(z(ω), S(u(ω))) ∈p.Butd(z(ω), S(u(ω))) ∈p,therefore by Definition 2.1 (c3), we haved(z(ω), S(u(ω))) = 0

and soz(ω) =S(u(ω)).

From (3.4) we have

z(ω) =P(u(ω)) =S(u(ω)). (3.5)

Henceu(ω)is a random coincidence point ofPandS.

Since the pair(P, S)is random weakly compatible, i.e.P(S(u(ω))) =S(P(u(ω)))this gives

P(z(ω)) =S(z(ω)). (3.6)

Again sinceS(ω, X)⊆Q(ω, X),then there existsv(ω)∈Ω×Xsuch that

z(ω) =Q(v(ω)). (3.7)

From (3.1), (3.5) and (3.7), we have

d(z(ω), T(v(ω))) = d(S(u(ω)), T(v(ω)))

≤ φ





d(P(u(ω)), Q(v(ω))),

1

2{d(P(u(ω)), S(u(ω))) +d(Q(v(ω)), T(v(ω)))},

d(P(u(ω)), T(v(ω))), d(Q(v(ω)), S(u(ω)))





= φ

0,d(z(ω), T(v(ω)))

2 , d(z(ω), T(v(ω))),0

(5)

this implies thatd(z(ω), T(v(ω)))≤0,thus−d(z(ω), T(v(ω)))∈p.Butd(z(ω), T(v(ω)))∈p,therefore by Definition 2.1 (c3), we haved(z(ω), T(v(ω))) = 0and soz(ω) =T(v(ω)).

From (3.7) we get

z(ω) =Q(v(ω)) =T(v(ω)). (3.8)

Hencev(ω)is a random coincidence point ofTandQ.

Since the pairTandQare random weakly compatible, i.e.T(Q(v(ω))) =Q(T(v(ω)))this implies that

T(z(ω)) =Q(z(ω)). (3.9)

Now we show thatz(ω)is a random fixed point ofS, we have from (3.1) that

d(S(z(ω)), z(ω)) = d(S(z(ω)), T(v(ω)))

≤ φ





d(P(z(ω)), Q(v(ω))),

1

2{d(P(z(ω)), S(z(ω))) +d(Q(v(ω)), T(v(ω)))},

d(P(z(ω)), T(v(ω))), d(Q(v(ω)), S(z(ω)))



.

Using (3.6) and (3.8), we get

d(S(z(ω)), z(ω))≤φ(d(S(z(ω)), z(ω)),0, d(S(z(ω)), z(ω)), d(z(ω), S(z(ω))))≤0,

it follows thatd(S(z(ω)), z(ω)) = 0,i.e.S(z(ω)) =z(ω).

According to (3.6), we obtain that

P(z(ω)) =S(z(ω)) =z(ω). (3.10)

By a similar way and using (3.10), we can prove that for allω∈Ω,

T(z(ω)) =Q(z(ω)) =z(ω). (3.11)

The equations (3.10) and (3.11) shows thatz(ω)is a common random fixed point ofT, S, PandQ.

For uniqueness. Letq(ω)6=z(ω)be another common random fixed point of the four mappings, then from (3.1), one can write

d(z(ω), q(ω)) = d(S(z(ω)), T(q(ω)))

≤ φ

d(P(z(ω)), Q(q(ω))),1

2{d(P(z(ω)), S(z(ω))) +d(Q(q(ω)), T(q(ω)))},

d(P(z(ω)), T(q(ω))), d(Q(q(ω)), S(z(ω)))

= (d(z(ω), q(ω)),0, d(z(ω), q(ω)), d(q(ω), z(ω)))≤0,

a contradiction. Henceq(ω) = z(ω)and soz(ω)is a unique common random fixed point ofT, S, P andQ.The proof is completed.

If we take,P =Qin above theorem we obtain the following corollary.

3.2

Corollary

Let (X, d) be a complete cone random metric space with respect to a cone p and let M be a nonempty separable closed subset ofX.Assume thatS, T andP are three continuous random operators defined onM such that forω ∈ Ω, S(ω, .), T(ω, .), P(ω, .) : Ω×M →M satisfying the following conditions:

(i)S(ω, X)⊆P(ω, X)andT(ω, X)⊆P(ω, X),

(ii) the pairs{S, P}and{T, P}are random weakly compatible mappings, (iii)

d(S(x(ω)), T(y(ω)))≤φ 



d(P(x(ω)), P(y(ω))),

1

2{d(P(x(ω)), S(x(ω))) +d(P(y(ω)), T(y(ω)))},

d(P(x(ω)), T(y(ω))), d(P(y(ω)), S(x(ω)))



,

for allx(ω), y(ω) ∈ Ω×X.Then the three random mappings have a unique common random fixed point inX.Putting

(6)

3.3

Corollary

Let(X, d)be a complete cone random metric space with respect to a coneCand letM be a nonempty separable closed subset ofX.Assume thatSandPare two continuous random operators defined onM such that forω∈Ω, S(ω, .), P(ω, .) : Ω×M →M satisfying the following conditions:

(i)S(ω, X)⊆P(ω, X),

(ii) the pair{S, P}is random weakly compatible mappings, (iii)

d(S(x(ω)), S(y(ω)))≤φ

d(P(x(ω)), P(y(ω))),1₂{d(P(x(ω)), S(x(ω))) +d(P(y(ω)), S(y(ω)))}, d(P(x(ω)), S(y(ω))), d(P(y(ω)), S(x(ω)))

,

for allx(ω), y(ω)∈Ω×X. Then the two random mappings have a unique common random fixed point inX.

3.4

Remark

LettingP = Q = I(where I is the identity mapping defined byI(ω, x) = x(ω)for allω ∈ Ω)and eliminating the conditions (i) ,(ii) in our theorem, we obtain the results of Saluja and Tripathi [28].

Finally, we present some examples to verify the requirements of our theorem as follows.

3.5

Example

Let(Ω,Σ)denotes a measurable space andM ={1,2,3,4,5} ⊂Rwith the usual metricd.ConsiderΩ ={1,2,3,4,5}

and letP

be the sigma algebra of Lebesgue’s measurable subset ofΩ.DefineT, Q, F, P : Ω×M →M for allω∈Ωby

S(ω, x) =

 



3ifx= 1 4otherwise

and Q(ω, x) =

 



5ifx= 1 4otherwise

T(ω, x) =

 



1ifx= 1 4otherwise

and P(ω, x) =

 



2ifx= 1 4otherwise

.

Taking measurable sequencexn(ω) =x(ω) = 3,it’s clearly thatS(ω, x) ⊆ Q(ω, x)andT(ω, x) ⊆ P(ω, x)and for all

ω ∈ Ω, S(xn(ω)) = P(xn(ω)) = 4,S(P(xn(ω))) = P(S(xn(ω))) = 4,this implies thatP andSare random weakly

compatible mappings, similarlyT andQtoo. To satisfy the condition (1.3), by takingx(ω) = 1andy(ω) =M − {1},we can write

1 = d(S(x(ω), T(y(ω)))≤φ

d(P(x(ω)), Q(y(ω))),1

2{d(P(x(ω)), S(x(ω))) +d(Q(y(ω)), T(y(ω)))},

d(P(x(ω)), T(y(ω))), d(Q(y(ω)), S(x(ω)))

= φ

2,1

2,2,1

.

From Definition 2.13, we can write1≤2λ, henceλ= 1

2 ∈(0,1),therefore all conditions of Theorem 3.1 are satisfied and

4is a unique random fixed point ofS, T, P andQ.

3.6

Example

Let(Ω,Σ)be a measurable space andM ={2,3,4,5,7} ⊂Rwith the usual metricd.ConsiderΩ ={2,3,4,5,7}and letP

be the sigma algebra of Lebesgue’s measurable subset ofΩ.DefineT, Q, S, P: Ω×M →M for allω∈Ω,by

S(ω, x) =

 



2ifx= 2 4otherwise

and Q(ω, x) =

 



5ifx= 2 4otherwise

T(ω, x) =

 



3ifx= 2 4otherwise

and P(ω, x) =

 



7ifx= 2 4otherwise

.

By taking measurable sequencexn(ω) =x(ω) = 1,then it’s obvious thatS(ω, x)⊆Q(ω, x), T(ω, x)⊆P(ω, x)and for

(7)

compatible mappings, similarlyT andQtoo. To justify the condition (1.3), by takingx(ω) = 2andy(ω) =M − {2},it follows that

2 = d(S(x(ω)), T(y(ω)))≤φ 



d(P(x(ω)), Q(y(ω))),

1

2{d(P(x(ω)), S(x(ω))) +d(Q(y(ω)), T(y(ω)))},

d(P(x(ω)), T(y(ω))), d(Q(y(ω)), S(x(ω)))





= φ(3,5

2,3,2) =φ(3, 2 + 3

2 ,3,2).

By Definition 2.13, we get2≤3λ, henceλ= ₃2 ∈(0,1),therefore all requirements of Theorem 3.1 are satisfied and4is a unique random fixed point ofS, T, PandQ.

3.7

Example

Let(Ω,Σ)be a measurable space andM = Ω = [0,1]⊂Rwith the usual metricd,and letP

be the sigma algebra of Lebesgue’s measurable subset ofΩ.DefineT, Q, S, P : Ω×M →Mfor allω∈Ω,by

S(ω, x) =

 



2

3x(ω)ifx(ω)∈Ω− {ω}

x(ω)ifx(ω) =ω

and Q(ω, x) =

 



1ifx(ω)∈Ω− {ω} x(ω)ifx(ω) =ω

T(ω, x) =

 



0ifx(ω)∈Ω− {ω} x(ω)ifx(ω) =ω

and P(ω, x) =

 



1

3x(ω)ifx(ω)∈Ω− {ω}

x(ω)ifx(ω) =ω

.

Letx(ω) =ωbe a measurable mapping,then it’s obvious thatS(ω, x)⊆Q(ω, x), T(ω, x)⊆P(ω, x)and for allω ∈Ω, S(x(ω)) = P(x(ω)) = ω,S(P(x(ω))) = P(S(x(ω))) = ω,this implies thatP andS are random weakly compatible mappings, similarlyT andQtoo. To justify the condition (1.3), by takingx(ω)∈Ω− {ω}andy(ω) =ω,it follows that

x(ω)

3 = d(S(x(ω)), T(y(ω)))≤φ





d(P(x(ω)), Q(y(ω))),

1

2{d(P(x(ω)), S(x(ω))) +d(Q(y(ω)), T(y(ω)))},

d(P(x(ω)), T(y(ω))), d(Q(y(ω)), S(x(ω)))





= φ

₂_x₍_ω₎

3 ,

1 3x(ω)

2 , 2x(ω)

3 ,

x(ω) 3

.

Using Definition 2.13, we obtain 1₃x(ω) ≤ 2

3x(ω)λ, henceλ = 1

(8)

Acknowledgements

The authors would like to thanks the anonymous referees for their efforts and valuable suggestions to improve the qual-ity of the paper.

REFERENCES

[1] M. Abbas, G. Jungck. Common fixed point results for non commuting mappings without continuity in cone metric spaces, J. Math. Anal. Appl., Vol.341, 416-420, 2008.

[2] M. A. Al-Thagafi, N. Shahzad. Generalized I-nonexpansive maps and invariant approximation, Acta Math. Sinica, Vol.25, No.5, 867-876, 2008.

[3] I. Beg. Random fixed points of random operators satisfying semicontractivity conditions, Math. Japonica, Vil.46, 151-155, 1997.

[4] I. Beg. Approximation of random fixed points in normed spaces, Nonlinear Anal., Vol.51, 1363-1372. 2002.

[5] I. Beg, M. Abbas. Equivalence and stability of random fixed point iterative procedures, J. Appl. Math. Stoch. Anal., Vol.2006, 1-19, 2006.

[6] I. Beg, M. Abbas. Iterative procedures for solutions of random operator equations in Banach spaces, J. Math. Anal. Appl., Vol.315, 181-201, 2006.

[7] A. T. Bharucha-Reid. Random Integral equations, Mathemat-ics in Science and Engineering, Vol. 96, Academic Press, New York, 1972.

[8] A. T. Bharucha-Reid. Fixed point theorems in probabilistic analysis, Bull. Amer. Math. Soc., Vol.82, No.5, 641-657, 1976.

[9] O. Hanˇs. Reduzierende zuf˘allige transformationen,

Czechoslov. Math. J., Vol.7, No.82, 154-158, 1957.

[10] O. Hanˇs. Random operator equations, Proceeding of the 4th Berkeiey Symposium on Mathematical Statistics and Probability, Vol.II, University of California Press, California, 185-202, 1961.

[11] C. J. Himmelberg. Measurable relations, Fundamenta Mathe-maticae, Vol.87, 53-72, 1975.

[12] L. G. Huang, X. Zhang. Cone metric spaces and fixed point theorems of contractive mappings, J. Math. Anal. Appl., Vol.332, 1468-1476, 2007.

[13] D. Ilic, V. Rakocevic. Common fixed points for maps on cone metric space, J. Math. Anal. Appl., Vol.341, 876-882, 2008.

[14] S. Itoh. Random fixed point theorems with an application to random differential equations in Banach spaces, J. Math. Anal. Appl., Vol.67, No.2, 261-273, 1979.

[15] S. Mehta, A. D. Singh. Some fixed point theorems in cone random metric spaces, South Asian J. Math., Vol.2, No.3, 248-254, 2012.

[16] S. Mehta, A. D. Singh, Vanita Ben Dhagat. Fixed point theorems for weak contraction in cone random metric spaces, Bull. Math. Soc., Vol.103, No.4, 303-310, 2011.

[17] N. S. Papageorgiou. Random fixed point theorems for measurable multifunctions in Banach spaces, Proc. Amer. Math. Soc., Vol.97, 507-514, 1986.

[18] N. S. Papageorgiou, On measurable multifunctions with stochastic domain, J. Australian Math. Soc., Vol.45, 204-216, 1988.

[19] R. A. Rashwan, D. M. Albaqeri. A common random ?xed point theorem and application to random integral equations, Int. J. Appl. Math. Reser., Vol.3, No.1, 71-80, 2014.

[20] R. A. Rashwan, H. A. Hammad. Random fixed point theorems with an application to a random nonlinear integral equation, Journal of Linear and Topological Algebra, Vol.5, No.2, 119-133, 2016.

[21] R. A. Rashwan, H. A. Hammad. Random common fixed point theorem for random weakly subsequentially continuous generalized contractions with application, Int. J. Pure Appl. Math., Vol.109, No.4, 813-826, 2016.

[22] R. A. Rashwan, H. A. Hammad. A common random fixed point theorem for weakly compatible mappings in cone ran-dom metric spaces, Universal Journal of Computational Math-ematics, Vol.4, No.4, 67-74, 2016.

[23] Sh. Rezapour. A review on topological properties of cone metric spaces, in Procceedingd of the International Confer-ence on Analysis, Topology and Appl. (ATA 08), Vrinjacka Banja, Serbia, May-june 2008.

[24] Sh. Rezapour, R. Hamlbarani. Some notes on the paper, One metric spaces and fixed point theorems of contractive mappings, J. Math. Anal. Appl., Vol.345, 719-724, 2008.

(9)

[26] G. S. Saluja. Random fixed point theorems for Ciric quasi contraction in cone random metric spaces, AAnalele Uni-versit˘at¸ii de Vast, Timis¸oara Seria Matematic˘a-Informatic˘a, Vol.LIII, No.(1), 163-175, 2015.

[27] G. S. Saluja, On common random fixed point theorems under contractive type condition in cone random metric spaces, Gulf J. Math., Vol.3, No.(4), 111-122, 2015.

[28] G. S. Saluja, M. P. Tripathi, On common random fixed point results in cone random metric spaces, J. Fixed Point Theory, Vol.2014:4, 1-11, 2014.

[29] G. S. Saluja, M. P. Tripathi, Some common random fixed point theorems for contractive type conditions in cone

random metric spaces, Acta. Univ. Sapientiae Math., Vol.8, No.1, 174-190, 2016.

[30] P. Vetro, Common fixed points in cone metric spaces, Rend. Circ. Mat. Palermo, Vol.56, 464-468, 2007.

[31] D. H. Wagner, Survey of measurable selection theorems, SIAM J. Control Optim., Vol.15, 859-903, 1977.