An Iteration for Finding a Common Random Fixed Point

The study of random fixed points has been an active area of contemporary research in mathematics. Some of the recent works in this field are noted in [1, 2, 3, 11]. In particular, random iteration schemes leading to random fixed points were introduced in [3]. After that, random iterations for finding solutions of random operator equations and fixed points of random operators have been discussed, as, for example, in [4, 5, 6, 7, 13]. In the present context, we define an iteration scheme for two random operators on a nonempty closed convex subset of a separable Hilbert space and consider its convergence to a common random fixed point of the two random operators. The two random operators satisfy some contractive inequality. Contractive mappings have often been subjects of fixed point studies. For a review of the subject matter, we refer to [12]. The case where the domain is further compact has also been discussed. We first review the following concepts which are essentials for our study in this paper. These concepts are obtainable in [1, 3, 6]. Throughout this paper, (Ω, ∑ ) denotes a measurable space and H stands for a separable Hilbert space. C is a nonempty subset of H . A function f : Ω→ C is said to be measurable if f −1(B∩C) ∈∑ for every Borel subset B of H . A function F : Ω×C→ C is said to be a random operator if F(·,x) : Ω→ C is measurable for every x ∈ C. A measurable function g : Ω→ C is said to be a random fixed point of the random operator F : Ω×C→ C if F(t,g(t)) = g(t) for all t ∈Ω. The following result was established in [10]. We present this result as a lemma and omit its proof.