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**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.

(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
Full Text:
DOI

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