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

Copyright © 2006 Ljubomir B. Ćirić et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

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.