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

• Main Authors: Pankaj Kumar Jhade, A. S. Saluja
• Format: Article
• Language: English
• Published: Etamaths Publishing 2013-10-01
• Series: International Journal of Analysis and Applications
• Online Access: http://www.etamaths.com/index.php/ijaa/article/view/111

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.