Some random approximations and random fixed point theorems for 1-set-contractive random operators. (English) Zbl 0869.47031

Liu, Li-Shan

Proc. Am. Math. Soc. 125, No. 2, 515-521 (1997).

Summary: We prove that the random version of Ky Fan’s Theorem [Math. Z. 112, 234-240 (1969; Zbl 0185.39503)] is true for 1-set-contractive random operators \(f: \Omega \times B_R\rightarrow X \), 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 operators 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.

MSC:

47H10	Fixed-point theorems
60H25	Random operators and equations (aspects of stochastic analysis)
41A50	Best approximation, Chebyshev systems
47H40	Random nonlinear operators

Keywords:

Random approximations and random fixed point theorems; 1-set-contractive random operator; semicontractive random operator; LANE random operators

Citations:

Zbl 0185.39503

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]	Felix E. Browder, Semicontractive and semiaccretive nonlinear mappings in Banach spaces, Bull. Amer. Math. Soc. 74 (1968), 660 – 665. · Zbl 0164.44801
[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]	K. Kuratowski and C. Ryll-Nardzewski, A general theorem on selectors, Bull. Acad. Polon. Sci. Sér. Sci. Math. Astronom. Phys. 13 (1965), 397 – 403 (English, with Russian summary). · Zbl 0152.21403
[7]	Tzu Chu Lin, A note on a theorem of Ky Fan, Canad. Math. Bull. 22 (1979), no. 4, 513 – 515. · Zbl 0429.47019 · doi:10.4153/CMB-1979-067-x
[8]	Tzu-Chu Lin, Random approximations and random fixed point theorems for non-self-maps, Proc. Amer. Math. Soc. 103 (1988), no. 4, 1129 – 1135. · Zbl 0676.47041
[9]	Tzu-Chu Lin, Random approximations and random fixed point theorems for continuous 1-set-contractive random maps, Proc. Amer. Math. Soc. 123 (1995), no. 4, 1167 – 1176. · Zbl 0834.47049
[10]	Roger D. Nussbaum, The fixed point index for local condensing maps, Ann. Mat. Pura Appl. (4) 89 (1971), 217 – 258. · Zbl 0226.47031 · doi:10.1007/BF02414948
[11]	R. D. Nussbaum, The fixed point index and fixed point theorems for \(k\)-set-contractions, Ph. D. Thesis, Univ. of Chicago (1969). · Zbl 0174.45402
[12]	Zdzisław Opial, Weak convergence of the sequence of successive approximations for nonexpansive mappings, Bull. Amer. Math. Soc. 73 (1967), 591 – 597. · Zbl 0179.19902
[13]	Nikolaos S. Papageorgiou, Random fixed point theorems for measurable multifunctions in Banach spaces, Proc. Amer. Math. Soc. 97 (1986), no. 3, 507 – 514. · Zbl 0606.60058
[14]	W. V. Petryshyn, Fixed point theorems for various classes of 1-set-contractive and 1-ball-contractive mappings in Banach spaces, Trans. Amer. Math. Soc. 182 (1973), 323 – 352. · Zbl 0277.47033
[15]	V. M. Sehgal and S. P. Singh, On random approximations and a random fixed point theorem for set valued mappings, Proc. Amer. Math. Soc. 95 (1985), no. 1, 91 – 94. · Zbl 0607.47057
[16]	V. M. Sehgal and Charles Waters, Some random fixed point theorems for condensing operators, Proc. Amer. Math. Soc. 90 (1984), no. 3, 425 – 429. · Zbl 0561.47050