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

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.
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### References:

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