Fixed Point Theorems for Random Lower Semi-Continuous Mappings
نویسندگان
چکیده
Let X, d be a metric space and S a closed and nonempty subset of X. Denote by 2 resp., C X the family of all nonempty resp., nonempty and closed subsets of X. A mapping T : S → 2 is said to satisfy condition P if, for every closed ball B of S with radius r ≥ 0 and any sequence {xn} in S for which d xn, B → 0 and d xn, T xn → 0 as n → ∞, there exists x0 ∈ B such that x0 ∈ T x0 where d x, B inf{d x, y : y ∈ B}. If Ω is any nonempty set, we say that the operator T : Ω × S → 2 satisfies condition P if for each ω ∈ Ω, the mapping T ω, · : S → 2 satisfies condition P . We should observe that this latter condition is related to a condition that was originally introduced by Petryshyn 1 for single-valued operators, in order to prove existence of fixed points. However, in our case, the condition is used to prove the measurability of a certain operator. On the other hand, in the year 2001, Shahzad cf. 2 using an idea of Itoh cf. 3 , see also 4 , proved that under a somewhat more restrictive condition, named condition A , the following result.
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