A Common Random Fixed Point Theorem for Weakly Compatible Mappings in Cone Random Metric Spaces

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A Common Random Fixed Point Theorem for Weakly

Compatible Mappings 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, weproveauniquecommon ran-domfixedp ointt heoremi nt hef rameworko fc onerandom metricspacesforfourweaklyrandomcompatiblemappings under strict contractive condition. Some corollaries of thistheorem for three andtwo weakly randomcompatible mappings and for one random mapping are derived. Two examples to justify our theorem are given. Our results extend someprevious work relatedto cone random metric spacesfromthecurrentexistingliterature.

Keywords

Common Random Fixed Point, Cone Random Metric Spaces, Weakly Random Compatible Mappings

1 Introduction

Fixed point theory has the diverse applications in differ-ent branches of mathematics, statistics, engineering, and eco-nomics in dealing with the problems arising in approximation theory, potential theory, game theory, theory of differential equations, theory of integral equations, and others. Develop-ments in the investigation on fixed points of non-expansive mappings, contractive mappings in different spaces like met-ric spaces, Banach spaces, Fuzzy metmet-ric spaces and cone metric spaces have almost been saturated. The study of ran-dom fixed point theorems was initiated by the Prague school of probabilistic in 1950’s [9, 10, 23]. The introduction of ran-domness leads to several new questions of measurability of solutions, probabilistic and statistical aspects of random so-lutions. Common random fixed point theorems are stochas-tic 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 research area and various ideas associated with random fixed point theory are used to give the solution of nonlinear system see [5-7, 11, 20]. Com-mon 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 [15, 16] and Beg [3, 4].

In [12] Huang and Zhang generalized the concept of metric spaces, replacing the set of real numbers by an ordered Ba-nach space, hence they have defined the cone metric spaces. They also described the convergence of sequences and in-troduced the notion of completeness in cone metric spaces. They have proved some fixed point theorems of contractive mappings on complete cone metric space with the assump-tion of normality of a cone. According to this concept, sev-eral other authors [1, 13, 19, 22] studied the existence of fixed points and common fixed points of mappings satisfying con-tractive type condition on a normal cone metric space. In 2008, the assumption of normality in cone normal spaces is deleted by Rezapour and Hamlbarani [19], which is an im-portant event in developing fixed point theory in cone metric spaces.

The aim of this paper is to extends the contractive condi-tion (2.1) for four, three and two random mappings and estab-lish a unique random fixed point results under this condition in random cone metric spaces using the concept of weakly random compatible mappings.

2 Preliminaries

2.1

Definition [21]

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 ≤yiffy−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, 24]

LetX be a nonempty set. Assume that the mappingd : X×X →Esatisfies:

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

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(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 [24] onX

and(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 [18]

LetE = l2, p = {{xn}n≥1 ∈ E2 : xn ≥ 0, for all n},(X, ρ)a metric space andd : X×X → E defined 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 [15]

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 inXis convergent inX.

The following definitions are given in [21].

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 subsetU ofM,G−1(U)∈Σ,whereG−1(U) ={ω ∈

Ω :G(ω)∩U 6=∅}.

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 →X is called continuous random operator if for each fixedx ∈ 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)

LetMbe 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 allx, y∈M, ω∈

Ωandx(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∈Ωbe 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 operators T, S : Ω × X → X (where

X be a separable Banach space) are weakly compatible if T(ω, S(ω, ξ(ω))) = S(ω, T(ω, ξ(ω))) provided that

T(ω, ξ(ω)) =S(ω, ξ(ω))for everyω∈Ω.

Recently, Rashwan and Hammad [17] proved a unique ran-dom fixed point in a separable Hilbert space under the follow-ing general contractive condition: LetT : Ω×X →Xbe a continuous operator such that forω∈Ω,

kT(ω, x)−T(ω, y)k ≤α(ω) max

     

    

kx−yk,

β(ω)

2 [kx−T(ω, x)k+ ky−T(ω, y)k], γ(ω)

2 [kx−T(ω, y)k

+ky−T(ω, x)k]

     

    

,

(2.1) for allx, y∈Xwhereα(ω), β(ω)andγ(ω)are nonnegative real valued random variables such thatα(ω), β(ω), γ(ω) ∈

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3 Main Results

In this section we shall prove a common random fixed point theorem under a generalized contraction condition for four 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 weakly random compatible, (iii)

d(S(x(ω), T(y(ω)))≤α(ω) max

 



d(P(x(ω)), Q(y(ω))), β(ω)

2 [d(P(x(ω)), S(x(ω))) +d(Q(y(ω)), T(y(ω)))],

γ(ω)

2 [d(P(x(ω)), T(y(ω))) +d(Q(y(ω)), S(x(ω)))]

 



, (3.1)

for all x(ω), y(ω) ∈ Ω×X where α(ω), β(ω) and γ(ω) are nonnegative real valued random variables such that

α(ω), β(ω), γ(ω) ∈ (0,1) andα(ω).β(ω) < α(ω), α(ω).γ(ω) < α(ω). Then the four random mappings have a unique

common random fixed point inX.

Proof. For each x◦(ω), x1(ω) ∈ Ω×X and n = 0,1,2, .. we choosey1(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→X defined 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)))

≤ α(ω) max

     

    

d(P(x2n(ω)), Q(x2n+1(ω))),

β(ω)

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

γ(ω)

2 [d(P(x2n(ω)), T(x2n+1(ω)))

+d(Q(x2n+1(ω)), S(x2n(ω)))]

     

    

≤ α(ω) max

 



d(y2n(ω), y2n+1(ω)),

β(ω)

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

γ(ω)

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

 



≤ α(ω) max

 



d(y2n(ω), y2n+1(ω)),

β(ω)

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

γ(ω)

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

 



.

For nonnegative real numbersa,bandc,ifmax{a, b, c}=a,then

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

Ifmax{a, b, c}=b,it follows that

d(y2n+1(ω), y2n+2(ω))≤

α(ω)β(ω)

2−α(ω)β(ω)d(y2n(ω), y2n+1(ω))< α(ω)

2−α(ω)d(y2n(ω), y2n+1(ω)).

Ifmax{a, b, c}=c,we can write

d(y2n+1(ω), y2n+2(ω))≤

α(ω)γ(ω)

2−α(ω)γ(ω)d(y2n(ω), y2n+1(ω))< α(ω)

2−α(ω)d(y2n(ω), y2n+1(ω)).

Putting

λ(ω) = max{α(ω), α(ω)

2−α(ω)}, (3.3)

it’s clearly that0< λ(ω)<1.

According to (3.3), we can easy write

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Similarly, we have

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

hence

d(y2n+1(ω), y2n+2(ω))≤λ2(ω)d(y2n−1(ω), y2n(ω)).

On continuing this process, we have

d(y2n+1(ω), y2n+2(ω))≤λ2n(ω)d(y0(ω), y1(ω)).

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(ω) +λn−2(ω) +...+λm(ω))d(y1(ω), y◦(ω))

≤

λm(ω)

1−λ(ω)

d(y1(ω), y◦(ω)).

Let0εis given. Choose a natural numberNsuch that₁λ₋m_λ(₍ω_ω)₎d(y1(ω), y◦(ω))εfor everym≥N,hence

d(yn(ω), ym(ω))≤

λm(ω) 1−λ(ω)

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×Xsuch 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). (3.4)

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

z(ω) =P(u(ω)). (3.5)

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))

≤ α(ω) max

     

    

d(P(u(ω)), Q(x2n+1(ω))),

β(ω)

2 [d(P(u(ω)), S(u(ω)))+

d(Q(x2n+1(ω)), T(x2n+1(ω)))],

γ(ω)

2 [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 (3.4) and (3.5), we have

d(z(ω), S(u(ω))) ≤ α(ω) max

0,β(ω)

2 [d(z(ω), S(u(ω)))], γ(ω)

2 [d(z(ω), S(u(ω)))]

< α(ω)

2 d(z(ω), S(u(ω))),

or, (1 − α(₂ω))d(z(ω), S(u(ω))) ≤ 0, this implies that d(z(ω), S(u(ω))) ≤ 0, since 0 < 1 − α(₂ω) < 1. Thus −d(z(ω), S(u(ω))) ∈ p. Butd(z(ω), S(u(ω))) ∈ p, therefore by Definition 2.1 (c3), we have d(z(ω), S(u(ω))) = 0

and soz(ω) =S(u(ω)).

From (3.5) we get

z(ω) =P(u(ω)) =S(u(ω)). (3.6)

Henceu(ω)is a random coincidence point ofPandS.

Since the pairPandSare weakly random compatible, i.e.P(S(u(ω))) =S(P(u(ω)))this implies that

P(z(ω)) =S(z(ω)). (3.7)

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

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From (3.1), (3.6) and (3.8), we have

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

≤ α(ω) max

 



d(P(u(ω)), Q(v(ω))), β(ω)

2 [d(P(u(ω)), S(u(ω))) +d(Q(v(ω)), T(v(ω)))],

γ(ω)

2 [d(P(u(ω)), T(v(ω))) +d(Q(v(ω)), S(u(ω)))]

 



≤ α(ω) max

0,β(ω)

2 [0 +d(z(ω), T(v(ω)))], γ(ω)

2 [d(z(ω), T(v(ω)))]

< α(ω)

2 d(z(ω), T(v(ω))),

or, (1 − α(₂ω))d(z(ω), T(v(ω))) ≤ 0, this implies that d(z(ω), T(v(ω))) ≤ 0, since 0 < 1 − λ(ω) < 1. Thus −d(z(ω), T(v(ω))) ∈ p.But d(z(ω), T(v(ω))) ∈ p,therefore by Definition 2.1 (c3), we have d(z(ω), T(v(ω))) = 0

and soz(ω) =T(v(ω)).

From (3.8) we get

z(ω) =Q(v(ω)) =T(v(ω)). (3.9)

Hencev(ω)is a random coincidence point ofT andQ.

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

T(z(ω)) =Q(z(ω)). (3.10)

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)))

≤ α(ω) max

 



d(P(z(ω)), Q(v(ω))), β(ω)

2 [d(P(z(ω)), S(z(ω))) +d(Q(v(ω)), T(v(ω)))],

γ(ω)

2 [d(P(z(ω)), T(v(ω))) +d(Q(v(ω)), S(z(ω)))]

 



.

Using (3.7) and (3.9), we get

d(S(z(ω)), z(ω)) ≤ α(ω) max

_d₍_S₍_z₍_ω₎₎_{, z}₍_ω₎₎_,₀_,

γ(ω)

2 [d(S(z(ω)), z(ω)) +d(z(ω), S(z(ω)))]

< α(ω)

2 d(S(ω, z(ω)), z(ω)),

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

According to (3.7), we obtain that

P(z(ω)) =S(z(ω)) =z(ω). (3.11)

By a similar way and using (3.11), we can prove that for allω∈Ω

T(z(ω)) =Q(z(ω)) =z(ω). (3.12)

The equations (3.11) and (3.12) shows thatz(ω)is a common random fixed point ofT, S, P andQ.

Now, we show the 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)))

≤ α(ω) max

 



d(P(z(ω)), Q(q(ω))), β(ω)

2 [d(P(z(ω)), S(z(ω))) +d(Q(q(ω)), T(q(ω)))],

γ(ω)

2 [d(P(z(ω)), T(q(ω))) +d(Q(q(ω)), S(z(ω)))]

 



≤ α(ω) max

d(z(ω), q(ω)),0,γ(ω)

2 [d(z(ω), q(ω)) +d(q(ω), z(ω))]

< α(ω)

2 d(z(ω), q(ω))< d(z(ω), q(ω)),

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

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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 weakly random compatible, (iii)

d(S(x(ω)), T(y(ω)))≤α(ω) max

 



d(P(x(ω)), P(y(ω))), β(ω)

2 [d(P(x(ω)), S(x(ω))) +d(P(y(ω)), T(y(ω)))],

γ(ω)

2 [d(P(x(ω)), T(y(ω))) +d(P(y(ω)), S(x(ω)))]

 



,

for all x(ω), y(ω) ∈ Ω × X where α(ω), β(ω) and γ(ω) are nonnegative real valued random variables such that

α(ω), β(ω), γ(ω) ∈ (0,1) andα(ω).β(ω) < α(ω), α(ω).γ(ω) < α(ω).Then the three random mappings have a unique

PuttingP=QandS=T in above theorem we get the following corollary.

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 pairs{S, P}is weakly random compatible, (iii)

d(S(x(ω)), S(y(ω)))≤α(ω) max

 



d(P(x(ω)), P(y(ω))), β(ω)

2 [d(P(x(ω)), S(x(ω))) +d(P(y(ω)), S(y(ω)))],

γ(ω)

2 [d(P(x(ω)), S(y(ω))) +d(P(y(ω)), S(x(ω)))]

 



,

α(ω), β(ω), γ(ω) ∈ (0,1)andα(ω).β(ω) < α(ω), α(ω).γ(ω) < α(ω).Then the two random mappings have a unique

LettingP=I(whereIis the identity mapping defined byI(ω, x) =x(ω)for allω∈Ω)in Corollary 3.3, we have

3.4

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 thatSbe a continuous random operators defined onM such that forω ∈Ω, S(ω, .) : Ω×M →M

satisfying the condition

d(S(x(ω)), S(y(ω)))≤α(ω) max

 



d(x(ω), y(ω)), β(ω)

2 [d(x(ω), S(x(ω))) +d(y(ω), S(y(ω)))],

γ(ω)

2 [d(x(ω), S(y(ω))) +d(y(ω), S(x(ω)))]

 



,

α(ω), β(ω), γ(ω)∈(0,1)andα(ω).β(ω)< α(ω), α(ω).γ(ω)< α(ω).ThenShas a unique random fixed point inX.

Finally, we present some examples to verify the requirements of Theorem 3.1 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, S, P: Ω×M →M by

S(ω, x) = 3ifx=1

4otherwise andQ(ω, x) =

5ifx=1 4otherwise,

T(ω, x) = 1₄if_otherwisex=1 andP(ω, x) =2₄if_otherwisex=1 for allω∈Ω.

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 andS are weakly random

compatible mappings, similarlyT andQtoo. To satisfy the condition (1.3), by takingx(ω) = 1andy(ω) = 2,we can write

1 = d(S(x(ω)), T(y(ω)))≤α(ω) max

 



d(P(x(ω)), Q(y(ω))), β(ω)

2 [d(P(x(ω)), S(x(ω))) +d(Q(y(ω)), T(y(ω)))],

γ(ω)

2 [d(P(x(ω)), T(y(ω))) +d(Q(y(ω)), S(x(ω)))]

 



= α(ω) max

2,β(ω) 2 (1),

γ(ω) 2 (3)

= 2α(ω) = 1.

Henceα(ω) = 1₂ ∈ (0,1),therefore all conditions of Theorem 3.1 are satisfied and4 is a unique random fixed point of

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3.6

Example

Let(Ω,Σ)be a measurable space andM ={0,1,2,3,4} ⊂Rwith the usual metricd.ConsiderΩ ={0,1,2,3,4}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) = 02ifotherwisex=2 andQ(ω, x) =

4ifx=2 0otherwise,

T(ω, x) = 1ifx=2

0otherwise andP(ω, x) =

3ifx=2 0otherwise.

By taking measurable sequencexn(ω) =x(ω) = 4,then it’s obvious thatS(ω, x)⊆Q(ω, x), T(ω, x)⊆P(ω, x)and for all

ω∈Ω, S(xn(ω)) =P(xn(ω)) = 0,S(P(xn(ω))) =P(S(xn(ω))) = 0,this givesPandSare weakly random compatible mappings, similarlyTandQtoo. To justify the condition (1.3), by takingx(ω) = 2andy(ω) = 3,it follows that

2 = d(S(x(ω)), T(y(ω)))≤α(ω) max

 



d(P(x(ω)), Q(y(ω))), β(ω)

2 [d(P(x(ω)), S(x(ω))) +d(Q(y(ω)), T(y(ω)))],

γ(ω)

2 [d(P(x(ω)), T(y(ω))) +d(Q(y(ω)), S(x(ω)))]

 



= α(ω) max

3,β(ω) 2 (1),

γ(ω) 2 (5)

= 3α(ω) = 2.

Henceα(ω) = 2₃ ∈ (0,1),therefore all requirements of Theorem 3.1 are satisfied and0is a unique random fixed point of

S, T, P andQ.

Acknowledgements

The authors are very grateful to the anonymous referee for his careful reading and useful suggestions on the manuscript.

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