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

Li-Shan, Liu

doi:10.1090/S0002-9939-97-03589-2

Some random approximations and random fixed point theorems for 1-set-contractive random operators
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Proc. Amer. Math. Soc. 125 (1997), 515-521 Request permission

Abstract:

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 $f : \Omega \times B_R \to X$, where $B_R$ is a weakly compact separable closed ball in a Banach space $X$ and $\Omega$ 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.

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