Some random fixed point theorems for condensing and nonexpansive operators. (English) Zbl 0716.47029

Xu, Hongkun

Proc. Am. Math. Soc. 110, No. 2, 395-400 (1990).

Let C be a nonempty closed bounded convex subset of a reflexive Banach space X. C is said to have the random fixed point property (RFPP) for nonexpansive random operators if, for any measurable space (\(\Omega\),\(\Sigma\)) with \(\Sigma\) a sigma algebra of subsets of \(\Omega\), every nonexpansive random operator T: \(\Omega\times C\to C\) has a random fixed point (i.e., there exists a measurable operator x: \(\Omega\to C\) such that \(f(\omega,x(\omega))=x(\omega)\) for each \(\omega\in \Omega)\). The author first takes up the question of whether C has the RFPP from random operators when C has the FPP for nonexpansive mappings. He shows that such is the case if either
(i) \(\Sigma\) is closed under the Suslin operation; or
(ii) X is strictly convex and I-T is demiclosed at zero.
He then derives some random fixed point theorems for condensing and nonexpansive operators which are non self-mappings.

Reviewer: W.A.Kirk

MSC:

47H10	Fixed-point theorems
60H25	Random operators and equations (aspects of stochastic analysis)
47H09	Contraction-type mappings, nonexpansive mappings, \(A\)-proper mappings, etc.

Keywords:

closed bounded convex subset of a reflexive Banach space; random fixed point property; nonexpansive random operators; Suslin operation; strictly convex; demiclosed at zero; random fixed point theorems for condensing and nonexpansive operators which are non self-mappings

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