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Some random fixed point theorems for condensing operators. (English) Zbl 0561.47050

Let (\(\Omega\),\(\Sigma)\) be a measurable space (\(\Sigma\) a sigma algebra) and X be a non empty subset of a Banach space E. In this paper, the authors consider random operators \(T: \Omega \times X\to E\) and give sufficient conditions for the existence of a measurable map \(\phi\) : \(\Omega\) \(\to X\) satisfying a Browder-Fan type result [Ky Fan, Math. Z. 112, 234-240 (1969; Zbl 0185.395)]. As a consequence, a stochastic generalization of the well-known Rothe fixed point theorem is obtained. A random analogue of the Krasnoselski fixed point theorem for the sum of two operators is given for a Hilbert space.
The results improve a recent result of Bharucha-Reid and Mukherjea [A. T. Bharucha-Reid, Bull. Am. Math. Soc. 82, 641-657 (1976; Zbl 0339.60061); A. Mukherjea, C. R. Acad. Sci. Paris, Sér. A 263, 393-395 (1966; Zbl 0139.334) and also some similar results of S. Itoh, J. Math. Anal. Appl. 67, 261-273 (1979; Zbl 0407.60069)].
Reviewer: V.Popa

MSC:

47H10 Fixed-point theorems
60H25 Random operators and equations (aspects of stochastic analysis)
Full Text: DOI

References:

[1] A. T. Bharucha-Reid, Fixed point theorems in probabilistic analysis, Bull. Amer. Math. Soc. 82 (1976), no. 5, 641 – 657. · Zbl 0339.60061
[2] A. T. Bharucha-Reid, Random integral equations, Academic Press, New York-London, 1972. Mathematics in Science and Engineering, Vol. 96. · Zbl 0327.60040
[3] Ward Cheney and Allen A. Goldstein, Proximity maps for convex sets, Proc. Amer. Math. Soc. 10 (1959), 448 – 450. · Zbl 0092.11403
[4] Ky Fan, Extensions of two fixed point theorems of F. E. Browder, Math. Z. 112 (1969), 234 – 240. · Zbl 0185.39503 · doi:10.1007/BF01110225
[5] Shigeru Itoh, Random fixed-point theorems with an application to random differential equations in Banach spaces, J. Math. Anal. Appl. 67 (1979), no. 2, 261 – 273. · Zbl 0407.60069 · doi:10.1016/0022-247X(79)90023-4
[6] Arunava Mukherjea, Transformations aléatoires séparables: Théorème du point fixe aléatoire, C. R. Acad. Sci. Paris Sér. A-B 263 (1966), A393 – A395 (French). · Zbl 0139.33404
[7] C. H. Su and V. M. Sehgal, Some fixed point theorems for nonexpansive mappings in locally convex spaces, Boll. Un. Mat. Ital. (4) 10 (1974), 598 – 601 (English, with Italian summary). · Zbl 0308.47043
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