Journal of Inequalities and Applications 
Volume 2006 (2006), Article ID 81045, 12 pages
doi:10.1155/JIA/2006/81045

On random coincidence and fixed points for a pair of multivalued and single-valued mappings

Ljubomir B. Ćirić,1 Jeong S. Ume,2 and Siniša N. Ješić3

 1Faculty of Mechanical Engineering, University of Belgrade, Aleksinačkih Rudara 12-35, Belgrade 11070, Serbia and Montenegro
2Department of Applied Mathematics, Changwon National University, Changwon 641-773, Korea
3Faculty of Electrical Engineering, University of Belgrade, Bulevar Kralja Aleksandra 73, Belgrade 11000, Serbia and Montenegro
Received 2 February 2006; Revised 21 June 2006; Accepted 22 July 2006

Abstract

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.