Common random fixed points of compatible random operators

The study of random fixed point theory was initiated by the Prague school of probabilists in the 1950s [12, 13, 26]. Random fixed point theorems are stochastic generalization of classical fixed point theorems. The survey article by Bharucha-Reid [10] attracted the attention of several mathematicians and gave wings to this theory. Itoh [16] extended Spacek’s and Hans’s theorem to multivalued contraction mappings. Now this theory has become the full fledged research area and various ideas associated with random fixed point theory are used to obtain the solution of nonlinear random system (see [9, 19, 25, 27]). Papageorgiou [23], Beg [3, 4], and Beg and Shahzad [6, 8] studied the structure of common random fixed points and random coincidence points of a pair of compatible random operators and proved fixed point theorems for contractive random operators in Polish spaces. Recently Beg and Shahzad [7, 8] had used different iteration processes to obtain common random fixed points. The aim of this paper is to study the necessary conditions for the convergence of random iteration scheme to common random fixed points of two pairs of compatible random operators satisfying Meir-Keeler[18] type conditions in Polish spaces. Also, in Section 3, we establish the existence of unique common random fixed points of random operators under generalized contractive conditions. We first review the following concepts which are essential for our study in this paper. Throughout this paper, (Ω,Σ) denotes a measurable space (Σ—sigma algebra). A symmetric on a set X is a nonnegative real-valued function d on X × X such that for all x, y ∈ X we have