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In this paper, we will prove that the random version of Fan's Theorem (Math. Z. 112 (1969), 234-240) is true for 1-set-contractive random operator f: Q x BR -+ X, where BR is a weakly compact separable closed ball in a Banach space X and Q is a measurable space. This class of 1-set-contractive random operator includes condensing random operators, semicontractive random operators, LANE random operators, nonexpansive random operators and others. As applications of our theorems, some random fixed point theorems of non-self-maps are proved under various well-known boundary conditions.

Some random approximations and random fixed point theorems for 1-set-contractive random operators