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Let ( Open image in new window ) be a Polish space, Open image in new window  the family of all nonempty closed and bounded subsets of Open image in new window , and ( Open image in new window ) a measurable space. A pair of a hybrid measurable mappings Open image in new window  and Open image in new window , satisfying the inequality (1.2), are introduced and investigated. It is proved that if Open image in new window  is complete, Open image in new window , Open image in new window  are continuous for all Open image in new window , Open image in new window , Open image in new window  are measurable for all Open image in new window , and Open image in new window  for each Open image in new window , then there is a measurable mapping Open image in new window  such that Open image in new window  for all Open image in new window . This result generalizes and extends the fixed point theorem of Papageorgiou (1984) and many classical fixed point theorems.

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