Random approximations and random fixed point theorems for continuous 1-set-contractive random maps

Lin, Tzu-Chu

doi:10.1090/S0002-9939-1995-1227521-4

Random approximations and random fixed point theorems for continuous $1$-set-contractive random maps
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Proc. Amer. Math. Soc. 123 (1995), 1167-1176 Request permission

Abstract:

Recently the author [Proc. Amer. Math. Soc. 103 (1988), 1129-1135] proved random versions of an interesting theorem of Ky Fan [Theorem 2, Math. Z. 112 (1969), 234-240] 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-contractive 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.

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