Abstract
The measurability of order continuous random mappings in ordered Polish spaces is studied. Using order continuity, some random fixed point theorems and random periodic point theorems for increasing, decreasing, and mixed monotone random mappings are presented.
Highlights
Introduction and PreliminariesThe study of random fixed points forms a central topic in probabilistic functional analysis
Some useful fixed point theorems for monotone mappings were proved by Zhang, Guo and Lakshmikantham, and Bhaskar and Lakshmikantham under some weak assumptions
In this paper, motivated by ideas in 18–21, we study random version of fixed point theorems for increasing, decreasing, and mixed monotone random mappings in ordered Polish spaces
Summary
The study of random fixed points forms a central topic in probabilistic functional analysis. In this paper, motivated by ideas in 18–21 , we study random version of fixed point theorems for increasing, decreasing, and mixed monotone random mappings in ordered Polish spaces. A measurable mapping y : Ω → X is said to be a random fixed point of the random mapping T : Ω × X → X, if T ω, y ω y ω , for ω ∈ Ω a.e. Let 2X be the family of all nonempty subsets of X and F : Ω → 2X a set-valued mapping. Let X, d, φ be an ordered Polish space and Ω, A, P a measure space. If S : Ω × X × X → X is mixed monotone, S ·, ·, x : Ω × X → X is increasing and S ·, x, · : Ω × X → X is decreasing, for every fixed x ∈ X
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