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Random fixed point theorems for various classes of 1-set-contractive maps in Banach spaces. (English) Zbl 0893.47037

The author proves a random fixed point theorem for 1-set contractive random operators (of course satisfying certain conditions). Most random fixed point theorems proved earlier deal with condensing or nonexpansive random operators. The class of 1-set contractive operator includes condensing, nonexpansive, semicontractive type and locally almost nonexpansive random operators. By using the main theorem, some random fixed point theorems have been deduced for various special classes of random operators mentioned. To prove the main theorem, results of H. Xu [Proc. Am. Math. Soc. 110, No. 2, 395-400 (1990; Zbl 0716.47029)] and of K.-K. Tan and X.-Z. Yuan [J. Math. Anal. Appl. 185, No. 2, 378-390 (1994; Zbl 0856.47036)] have been used.
The author has generalized or extended results obtained by S. Itoh [J. Math. Anal. Appl. 67, 261-273 (1979; Zbl 0407.60069)], T.-C. Lin [Proc. Am. Math. Soc. 103, No. 4, 1129-1135 (1988; Zbl 0676.47041); ibid. 123, No. 4, 1167-1176 (1993; Zbl 0834.47049)] and Xu mentioned above.

MSC:

47H10 Fixed-point theorems
47H40 Random nonlinear operators
60H25 Random operators and equations (aspects of stochastic analysis)
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