On Random Coincidence & Fixed Points for a Pair of Multi-Valued & Single-Valued Mappings

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Pankaj Kumar Jhade, A. S. Saluja, On Random Coincidence & Fixed Points for a Pair of Multi-Valued & Single-Valued Mappings, Int. J. Anal. Appl., 4 (1) (2014), 26-35.

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Pankaj Kumar Jhade, A. S. Saluja

Abstract

Let (X,d) be a Polish space, CB(X) the family all nonempty closed and bounded subsets of X and (Ω,Σ) be a measurable space. In this paper a pair of hybrid measurable mappings f : Ω×X → X and T : Ω×X →CB(X), satisfying the inequality (2.1) below are introduced and investigated. It is proved that if X is complete, T(ω, ·), f(ω, ·) are continuous for all ω ∈ Ω, T( ·,x), f( ·,x) are measurable for all x ∈ X and T(ω,ξ(ω)) ⊆ f(ω × X) and f(ω ×X) = X for each ω ∈ Ω, then there is a measurable mapping ξ : Ω → X such that f(ω,ξ(ω)) ∈ T(ω,ξ(ω)) for all ω ∈ Ω.

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