International Journal of Analysis and Applications

Volume 18, Number 1 (2020), 63-70

URL: https://doi.org/10.28924/2291-8639

DOI: 10.28924/2291-8639-18-2020-63

RANDOM COMMON FIXED POINT THEOREMS FOR TWO PAIRS OF

NONLINEAR CONTRACTIVE MAPS IN POLISH SPACES

KANAYO STELLA EKE∗, HUDSON AKEWE AND JIMEVWO GODWIN OGHONYON

Department of Mathematics, Covenant University, Canaanland, KM 10 Idiroko Road, P. M. B. 1023, Ota,

Ogun State, Nigeria

∗Corresponding author: kanayo.eke@covenantuniversity.edu.ng

Abstract. This research work proves the random common fixed point theorem for two pairs of random

weakly compatible mappings fulfilling certain generalized random nonlinear contractive conditions in Polish

spaces. An example is given to support the validity of our results. Our results generalize and extend some

works in literature.

1. Introduction

The random fixed point theory introduced in 1950 by Prague School of Probabilistic plays very important

role in the theory of random integral, random differential equations and other areas of applied mathematics.

Some classical fixed point theorems in different abstract spaces are proved in the context of random fixed

point theory (see; Akewe et al.[1] , Rashwan and Albaqeri [2], Hans [3] and Nieto et al. [4]). The common

fixed point of two pairs of weakly compatible mappings satisfying certain contractive conditions in G-partial

metric spaces without assuming the continuity of any of the maps involved was proved by Eke and Akinlabi

[8]. The random common fixed point of two pairs of random subsequentially continuous mappings with

compatibility of type (E) satisfying certain generalized contractive conditions in Polish spaces ( separable

metric space) was established by Rashwan and Hammed [9]. In this paper, we prove the random version of

Received 2019-01-18; accepted 2019-02-25; published 2020-01-02.

2010 Mathematics Subject Classification. 47H10.

Key words and phrases. Polish space; random operators; random weakly compatible maps; random coincidence point;

random common fixed point.

c©2020 Authors retain the copyrights

of their papers, and all open access articles are distributed under the terms of the Creative Commons Attribution License.

63

https://doi.org/10.28924/2291-8639
https://doi.org/10.28924/2291-8639-18-2020-63


Int. J. Anal. Appl. 18 (1) (2020) 64

the result of Eke and Akinlabi [8] in the context of Polish spaces by using the contractive maps of Rashwan

and Hammed [9]. Our results are extension and an improvement on some related results in the literature.

2. Preliminaries

Let (Ω,φ) be a measurable space, A a separable metric space and σn : Ω → A a measurable sequence. An

operator f : Ω ×A → A is random operator, if for every x ∈ A, the mapping F(.,x) : Ω → A is measurable.

A measurable mapping ω : Ω → A is a random fixed point of a random operator F : Ω × A → A if

F(v,ω(v)) = ω(v) for each v ∈ ω, (details of these definitions can be found in Beg and Abbas [5], Choudhury

and Ray [6] and Choudhury and Upadhyah [7]).

The following theorems are the results of Eke and Akinlabi [8] and Rashwan and Hammed [9] re-

spectively.

Theorem 1.1 [8]: Let B, D, E and F be self-maps of a G-partial metric space A satisfying B(A) ⊂ F(A),

D(A) ⊂ E(A) and

Gp(Ba,Ba,Db) ≤ hua,a,b(B,D,E,F),

and

Gp(Ba,Db,Db) ≤ hua,b,b(B,D,E,F),

where h ∈ (0, 1) and

ua,a,b(B,D,E,F) ∈ {Gp(Ea,Ea,Fb),Gp(Ba,Ba,Ea),Gp(Db,Db,Fb),

Gp(Ba,Ba,Fb) + Gp(Db,Db,Ea)

2
} (1.1)

and

ua,b,b(B,D,E,F) ∈ {Gp(Ea,Fb,Fb),Gp(Ba,Ea,Ea),Gp(Db,Fb,Fb),

Gp(Ba,Fb,Fb) + Gp(Db,Ea,Ea)

2
} (1.2)

for all a,b ∈ A. If one of B(A), D(A), E(A) or F(A) is a complete subspace of A, then {B,E} and {D,F}

have a unique point of coincidence in X. Moreover if {B,E} and {D,F} are weakly compatible, then B,

D, E and F have a unique common fixed point.

Theorem 1.2 [9]: Let A be a Polish space and B,D,E,F : Ω × A → A are four random map-

pings satisfy

d(B(v,a),F(v,b)) ≤

φ(max{d(E(v,a),F(v,a)),d(E(v,a),F(v,a)),d(F(v,b),D(v,b)), d(E(v,a),D(v,b))+d(F (v,y),B(v,x))
2

}),



Int. J. Anal. Appl. 18 (1) (2020) 65

for all a,b ∈ A, and v ∈ Ω, where φ : [0,∞) → [0,∞) is contractive modulus and non decreasing such that

φ(0) = 0. If the two pairs {B,E} and {D,F} are weakly random subsequentially continuous and random

compatible of type (E). Then B,D,E and F have a unique common random fixed point in A.

3. Main Results

In this section we present our results as follow:

Theorem 2.1 : Let A be a Polish space and B,D,E,F, : φ × A → A are two pairs random operators

fulfilling B(v,A) ⊂ F(v,A), D(v,A) ⊂ E(v,A) and

d(B(v,a),D(v,b)) ≤

Ω(max{d(E(v,a),F(v,b)),d(E(v,a),B(v,a)),d(F(v,b),D(v,b)), d(E(v,a),D(v,b))+d(F (v,b),B(v,a))
2

}), (2.1)

for every a,b ∈ A, and v ∈ φ where Ω : [0,∞) → [0,∞) is a comparison function. If one of B(v, A), D(v,

A), E(v, A) or F(v, A) is a complete subspaces of A, then {B,E} and {D,F} have a unique random point

of coincidence in A. Additionally, if {B,E} and {D,F} are random weakly compatible, then B,D,E and

F have a unique random common fixed point in A.

Proof: Let a0(v) ∈ A be an arbitrary random variable in A. Consider a1(v) ∈ A such that B(v,a0) =

F(v,a1), since B(A) ⊂ F(A). Suppose D(A) ⊂ E(A) then there is a2(v) ∈ A such that D(v,a1) = E(v,a2).

Consequently, two sequences an(v) and bn(v) in A can be generated such that;

B(v,a2k(v)) = F(v,a2k+1(v)) = b2k+1(v)

D(v,a2k+1(v)) = E(v,a2k+2(v)) = b2k+2(v).

For a given k ∈ N and employing (2.1) we get

d(bk(v),bk+1(v)) = d(B(v,ak−1(v)),D(v,ak(v)))

≤ Ω(max{d(E(v,ak−1(v)),F(v,ak(v))),

d(E(v,ak−1(v)),B(v,ak−1(v))),d(F(v,ak(v)),D(v,ak(v))),

d(E(v,ak−1(v)),D(v,ak(v))) + d(F(v,ak(v)),B(v,ak−1(v)))

2
})

= Ω(max{d(bk−1(v),bk(v)),d(bk−1(v),bk(v)),d(bk(v),bk+1(v)),

d(bk−1(v),bk+1(v)) + d(bk(v),bk(v))

2
})

≤ Ω(max{d(bk−1(v),bk(v)),d(bk(v),bk+1(v)),
d(bk−1(v),bk(v)) + d(bk(v),bk+1(v))

2
})

≤ Ω(d(bk−1(v),bk(v)))

Continuing the process and by induction we have

d(bk(v),bk+1(v)) ≤ Ωkd(b0(v),b1(v)). (2.2)



Int. J. Anal. Appl. 18 (1) (2020) 66

For n > m and using the triangle inequality we have

d(bm(v),bn(v)) ≤ d(bm(v),bm+1(v)) + d(bm+1(v),bm+2(v)) + d(bm+2(v),bm+3(v))

+ · · · + d(bn−1(v),bn(v))

≤ Ωmd(b0(v),b1(v)) + Ωm+1d(b0(v),b1(v)) + Ωm+2d(b0(v),b1(v))

+ · · · + Ωm+n−1d(b0(v),b1(v))

< (Ωm + Ωm+1 + Ωm+2 + · · · + Ωm+n−1)d(b0(v),b1(v))

≤
Ωm

1 − Ωm
d(b0(v),b1(v)).

With the condition of Ω we have that {bk(v)} is a Cauchy sequence in A. If E(v, A) is a complete subspace

of A, then there is b0 ∈ A such that E(v,ak(v)) = bk(v) converges to ω.

Likewise, E(v,ak(v)) = D(v,ak+1(v)) = bk → ω

and F(v,ak−1(v)) = B(v,ak−2(v)) = bk−1 → ω

as k →∞. If there is z in A such that E(v,z(v)) = ω(v). Then we prove that B(v,z(v)) = ω(v). Otherwise,

we show that B(v,z(v)) 6= ω(v).

d(B(v,z(v)),ω(v)) ≤ d(B(v,z(v)),D(v,ak(v))) + d(D(v,ak(v),ω(v))

≤ Ω(max{d(E(v,z(v)),F(v,ak(v))),d(E(v,z(v)),B(v,z(v))),

d(F(v,ak(v)),D(v,ak(v))),

d(E(v,z(v)),D(v,ak(v))) + d(F(v,ak(v)),B(v,z(v)))

2
})

+d(D(v,ak(v),ω(v)).

As k →∞ we have

d(B(v,z(v)),ω(v)) ≤ Ω(max{d(ω(v),ω(v)),d(ω(v),B(v,z(v))),d(ω(v),ω(v)),

d(ω(v),ω(v)) + d(ω(v),B(v,z(v)))

2
})

≤ Ω(d(B(v,z(v)),ω(v))) < d(B(v,z(v)),ω(v)),

a contradiction, hence we have d(B(v,z(v)) = ω(v). This shows that

B(v,z(v)) = E(v,z(v)) = ω(v). Since ω(v) ∈ B(v,A) ⊂ F(v,A), then there is a u(v) ∈ A such that

F(v,u(v)) = ω(v). We claim that D(v,u(v)) = ω(v). On the other hand, we assume that D(v,u(v)) 6= ω(v).



Int. J. Anal. Appl. 18 (1) (2020) 67

So using (1) we obtain,

d(ω(v),D(v,u(v))) ≤ d(ω(v),B(v,ak(v))) + (B(v,ak(v)),D(v,u(v)))

≤ Ω(max{d(E(v,ak(v)),F(v,u(v))),d(E(v,u(v)),B(v,ak(v))),

d(D(v,u(v)),F(v,u(v))),

d(E(v,ak(v)),D(v,u(v))) + d(F(v,u(v)),B(v,ak(v)))

2
})

+d(B(v,ak(v),ω(v)).

As k →∞ we obtain

d(ω(v),D(v,u(v))) ≤ Ω(max{d(ω(v),D(v,u(v))),d(D(v,u(v)),ω(v)),

d(D(v,u(v)),ω(v))

2
})

≤ Ω(d(ω(v),D(v,u(v))) < d(ω(v),D(v,u(v)).

This is a contradiction according to the condition of Ω. Therefore D(v,u(v)) = ω(v) . This shows that

{B,E} and {D,F} have a common point of coincidence. Consider {B,E} and {D,F} being random weakly

compatible, then

B(v,ω(v)) = B(v,E(v,z(v))) = E(v,B(v,z(v))) = E(v,ω(v)) = ω1(v) (say) and

D(v,ω(v)) = D(v,F(v,u(v))) = F(v,D(v,u(v))) = F(v,ω(v)) = ω2(v) (say).

Now we prove that the points of coincidence are unique.

d(ω1(v),ω2(v)) = d(B(v,ω1(v)),F(v,ω2(v))

≤ Ω(max{d(E(v,ω1(v)),F(v,ω2(v))),d(E(v,ω1(v)),B(v,ω1(v))),

d(F(v,ω2(v)),D(v,ω2(v))),

d(E(v,ω1(v))),D(v,ω2(v))) + d(F(v,ω2(v)),B(v,ω1(v)))

2
})

≤ Ω(max{d(ω1(v),ω2(v)),d(ω1(v),ω1(v)),d(ω2(v),ω2(v)),

d(ω1(v),ω2(v)) + d(ω2(v),ω1(v))

2
})

≤ Ω(d(ω1(v),ω2(v))) < d(ω1(v),ω2(v)).

This shows that ω1(v) = ω2(v) by the property of Ω. Therefore

B(v,ω(v)) = E(v,ω(v)) = D(v,ω(v)) = F(v,ω(v))

Now, we prove that ω(v) is the common fixed point of B,D,E and F in A. We claim that ω(v) = D(v,ω(v)).



Int. J. Anal. Appl. 18 (1) (2020) 68

Suppose ω(v) 6= D(v,ω(v)) then using (2.1) we have,

d(ω(v),D(v,ω(v))) = d(B(v,ω(v)),D(v,ω(v)))

≤ Ω(max{d(E(v,ω(v)),F(v,ω(v))),d(E(v,ω(v)),B(v,ω(v))),

d(F(v,ω(v)),D(v,ω(v))),

d(E(v,ω(v))),D(v,ω(v))) + d(F(v,ω(v)),B(v,ω(v)))

2
})

≤ Ω(max{d(ω(v),D(v,ω(v))),d(D(v,ω(v)),ω(v)),

d(D(v,ω(v)),ω(v))

2
})

≤ Ω(d(ω(v),D(v,ω(v))) < d(ω(v),D(v,ω(v))

This contradict our assumption that ω(v) 6= D(v,ω(v)). Hence ω(v) = D(v,ω(v)). Thus ω(v) is the random

common fixed point of B,D,E and F.

For the uniqueness of the random common fixed point of B,D,E and F, we assume a different random

common fixed point of B,D,E and F say ω′(v) such that ω(v) = ω′(v) .On the other hand, let ω(v) 6= ω′(v)

and using (2.1) we get

d(ω(v),ω′(v)) = d(B(v,ω(v)),D(v,ω′(v)))

≤ Ω(max{d(E(v,ω(v)),F(v,ω′(v))),d(E(v,ω(v)),B(v,ω(v))),

d(F(v,ω′(v)),D(v,ω′(v))),

d(E(v,ω(v))),D(v,ω′(v))) + d(F(v,ω′(v)),B(v,ω(v)))

2
})

≤ Ω(max{d(ω(v),ω′(v)),d(ω(v),ω(v)),d(ω′(v),ω′(v)),

d(ω(v),ω′(v)) + d(ω′(v),ω(v))

2
})

≤ Ω(d(ω(v),ω′(v)) < d(ω(v),ω′(v)),

a contradiction, hence ω(v) = ω′(v).

Remark : Theorem 2.1 gives an independent version of the result of Rashwan and Hammed [9](Theorem

9) because the result is proved without the assumption of weakly random subsequential continuity and

compatibility of type (E). Theorem 2.1 proves the random version of the result of Eke and Akinlabi[8]

(Theorem 2.1) with general contractive mappings in the context of Polish space.

If Ω(t) = k(t) in Theorem 2.1 then we obtain the following corollary.

Corollary 2.2 : Let A be a Polish space and B,D,E,F, : φ × A → A are two pairs random operators

fulfilling

d(B(v,a),D(v,b)) ≤



Int. J. Anal. Appl. 18 (1) (2020) 69

k(max{d(E(v,a),F(v,b)),d(E(v,a),B(v,a)),d(F(v,b),D(v,b)), d(E(v,a),D(v,b))+d(F (v,b),B(v,a))
2

}),

for every a,b ∈ A, and v ∈ φ where k ∈ [0, 1). If one of B(v, A), D(v, A), E(v, A) or F(v, A) is a complete

subspaces of A, then {B,E} and {D,F} have a unique random point of coincidence in A. Additionally,

if {B,E} and {D,F} are random weakly compatible, then B,D,E and F have a unique random common

fixed point in A.

Example: Let X = R with the usual metric d and φ = [0, 1]. Define Ω : (0,∞) → [0,∞) by Ω(t) = t. Let

the random operator B,D,E,F : φ×A → A be defined by

E(v,a(v)) = F(v,a(v)) =




0, if a(v) ≤ 1

2, otherwise

(3.1)

B(v,a(v)) = D(v,a(v)) =




0, if a(v) ≤ 1

1
2
, otherwise

(3.2)

The pairs {B,E} and {D,F} are weakly compatible, B(v,A) ⊂ F(v,A), and D(v,A) ⊂ E(v,A). The

contractive condition is satisfied and the unique random common fixed point of B,D,E and F is 0.

Conclusion: The random common fixed point of two pairs of generalized random nonlinear contractive

mappings employing the property of random weakly compatible mappings is proved in the context of Polish

space. Further work can be done using different contractive mappings in this space. We provided an example

to support our result.

Acknowledgement: The authors are grateful to Covenant University for supporting this research finan-

cially.

Conflicts of Interest: The author(s) declare that there are no conflicts of interest regarding the publication

of this paper.

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	1. Introduction
	2. Preliminaries
	3. Main Results
	References