On Random Coincidence and Fixed Points for a Pair of Multivalued and Single-valued Mappings

Let (X ,d) be a Polish space, CB(X) the family of all nonempty closed and bounded subsets of X , and (Ω,Σ) a measurable space. A pair of a hybrid measurable mappings f : Ω×X → X and T : Ω×X → CB(X), satisfying the inequality (1.2), 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 f (ω×X) = X for each ω ∈Ω, then there is a measurable mapping ξ : Ω → X such that f (ω,ξ(w)) ∈ T(ω,ξ(w)) for all ω ∈Ω. This result generalizes and extends the fixed point theorem of Papageorgiou (1984) and many classical fixed point theorems.