RANDOM COMMON FIXED POINT THEOREMS FOR TWO PAIRS OF NONLINEAR CONTRACTIVE MAPS IN POLISH SPACES

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.


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 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.

Preliminaries
Let (Ω, φ) be a measurable space, A a separable metric space and σ n : 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 : 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] respectively.

Main Results
In this section we present our results as follow: Proof: Consequently, two sequences a n (v) and b n (v) in A can be generated such that; For a given k ∈ N and employing (2.1) we get Continuing the process and by induction we have For n > m and using the triangle inequality we have As k → ∞ we have a contradiction, hence we have d(B(v, z(v)) = ω(v). This shows that We claim that D(v, u(v)) = ω(v). On the other hand, we assume that D(v, u(v)) = ω(v).
So using (1) we obtain, As k → ∞ we obtain This is a contradiction according to the condition of Ω. Therefore D(v, u(v)) = ω(v Now we prove that the points of coincidence are unique. This shows that ω 1 (v) = ω 2 (v) by the property of Ω. Therefore 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)).
Suppose ω(v) = D(v, ω(v)) then using (2.1) we have, This contradict our assumption that ω and using (2.1) we get 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. Conflicts of Interest: The author(s) declare that there are no conflicts of interest regarding the publication of this paper.