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Random fixed point theorems and approximation. (English) Zbl 0892.47060

Summary: We first prove some random fixed point theorems for random nonexpansive operators in Banach spaces. As applications, some random approximation theorems for random 1-set-contractions or random continuous condensing mappings defined on closed balls of a separable Banach space, or on separable closed convex subsets of a Hilbert space, or on spheres of infinite-dimensional separable Banach spaces, are established. Our results are generalizations, improvements or stochastic versions of the corresponding results of Bharucha-Reid (1976), Lin (1988, 1989), Lin and Yen (1988), Massatt (1983), Seghal and Waters (1984) and Xu (1990).

MSC:

47H10 Fixed-point theorems
47H40 Random nonlinear operators
41A50 Best approximation, Chebyshev systems
54H25 Fixed-point and coincidence theorems (topological aspects)
47H04 Set-valued operators
47H09 Contraction-type mappings, nonexpansive mappings, \(A\)-proper mappings, etc.
54C60 Set-valued maps in general topology
28A05 Classes of sets (Borel fields, \(\sigma\)-rings, etc.), measurable sets, Suslin sets, analytic sets
28A20 Measurable and nonmeasurable functions, sequences of measurable functions, modes of convergence
60H25 Random operators and equations (aspects of stochastic analysis)
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References:

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