Tuyển tập báo cáo các nghiên cứu khoa học quốc tế ngành hóa học dành cho các bạn yêu hóa học tham khảo đề tài: ON RANDOM COINCIDENCE AND FIXED POINTS FOR A PAIR OF MULTIVALUED AND SINGLE-VALUED MAPPINGS | ON RANDOM COINCIDENCE AND FIXED POINTS FOR A PAIR OF MULTIVALUED AND SINGLE-VALUED MAPPINGS LJUBOMIR B. CIRIC JEONG S. UME AND SINISA N. JESIC Received 2 February 2006 Revised 21 June 2006 Accepted 22 July 2006 Let X d be a Polish space CB X the family of all nonempty closed and bounded subsets of X and o 2 a measurable space. A pair of a hybrid measurable mappings f o X X X and T o X X CB X satisfying the inequality are introduced and investigated. It is proved that if X is complete T w f w are continuous for all w e o T x f x are measurable for all x e X and f w X X X for each w e o then there is a measurable mapping J o X such that f w J w e T w J w for all w e o. This result generalizes and extends the fixed point theorem of Papageorgiou 1984 and many classical fixed point theorems. Copyright 2006 Ljubomir B. Ciric 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. 1. Introduction and preliminaries Random fixed point theorems are stochastic generalizations of classical fixed point theorems. Random fixed point theorems for contraction mappings on separable complete metric spaces have been proved by several authors Zhang and Huang 25 Hans 6 7 Itoh 8 Lin 12 Papageorgiou 13 14 Shahzad and Hussian 19 20 Spacek 22 and Tan and Yuan 23 . The stochastic version of the well known Schauder s fixed point theorem was proved by Sehgal and Singh 18 . Let X d be a metric space and T X X a mapping. The class of mappings T satisfying the following contractive condition d Tx Ty a max id x y d x Tx d y Ty - - -----------X-1 L z 2 J p max ịd x Tx d y Ty y d x Ty d y Tx for all x y e X where a p Y are nonnegative real numbers such that p 0 Y 0 and a p 2y 1 was introduced and investigated by Ciric 1 . Ciric proved that in a complete Hindawi Publishing Corporation Journal of Inequalities and Applications Volume .
