Random approximations and random fixed point theorems for continuous 1- set-contractive random maps. (English) Zbl 0834.47049

Lin, Tzu-Chu

Proc. Am. Math. Soc. 123, No. 4, 1167-1176 (1995).

Summary: Recently, the author [Proc. Am. Math. Soc. 103, No. 4, 1129-1135 (1988; Zbl 0676.47041)] proved random versions of an interesting theorem of Ky Fan [Theorem 2, Math. Z. 112, 234-240 (1969; Zbl 0185.39503)] for continuous condensing random maps and nonexpansive random maps defined on a closed convex bounded subset in a separable Hilbert space.
In this paper, we prove that it is still true for (more general) continuous 1-set-contractive random maps, which include condensing, nonexpansive, locally almost nonexpansive (LANE), semicontractive maps, etc. Then we use these theorems to obtain random fixed points theorems for the above-mentioned maps satisfying weakly inward conditions.
In order to obtain these results, we first need to prove a random fixed point theorem for 1-set-constractive self-maps in a separable Banach space. This leads to the discovery of some new random fixed point theorems in a separable uniform convex Banach space.

MSC:

47H10	Fixed-point theorems
47S50	Operator theory in probabilistic metric linear spaces
41A50	Best approximation, Chebyshev systems
54H25	Fixed-point and coincidence theorems (topological aspects)

Keywords:

continuous condensing random maps; continuous 1-set-contractive random maps; locally almost nonexpansive; semicontractive maps; random fixed point theorem for 1-set-constractive self-maps; random fixed point theorems in a separable uniform convex Banach space

Citations:

Zbl 0676.47041; Zbl 0185.39503