Some random approximations and random fixed point theorems for 1-set-contractive random operators
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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.References
- A. T. Bharucha-Reid, Fixed point theorems in probabilistic analysis, Bull. Amer. Math. Soc. 82 (1976), no. 5, 641–657. MR 413273, DOI 10.1090/S0002-9904-1976-14091-8
- A. T. Bharucha-Reid, Random integral equations, Mathematics in Science and Engineering, Vol. 96, Academic Press, New York-London, 1972. MR 0443086
- Felix E. Browder, Semicontractive and semiaccretive nonlinear mappings in Banach spaces, Bull. Amer. Math. Soc. 74 (1968), 660–665. MR 230179, DOI 10.1090/S0002-9904-1968-11983-4
- Ky Fan, Extensions of two fixed point theorems of F. E. Browder, Math. Z. 112 (1969), 234–240. MR 251603, DOI 10.1007/BF01110225
- 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. MR 528687, DOI 10.1016/0022-247X(79)90023-4
- 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). MR 188994
- Tzu Chu Lin, A note on a theorem of Ky Fan, Canad. Math. Bull. 22 (1979), no. 4, 513–515. MR 563767, DOI 10.4153/CMB-1979-067-x
- Tzu-Chu Lin, Random approximations and random fixed point theorems for non-self-maps, Proc. Amer. Math. Soc. 103 (1988), no. 4, 1129–1135. MR 954994, DOI 10.1090/S0002-9939-1988-0954994-0
- 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. MR 1227521, DOI 10.1090/S0002-9939-1995-1227521-4
- Roger D. Nussbaum, The fixed point index for local condensing maps, Ann. Mat. Pura Appl. (4) 89 (1971), 217–258. MR 312341, DOI 10.1007/BF02414948
- R. D. Nussbaum, The fixed point index and fixed point theorems for $k$-set-contractions, Ph. D. Thesis, Univ. of Chicago (1969).
- Zdzisław Opial, Weak convergence of the sequence of successive approximations for nonexpansive mappings, Bull. Amer. Math. Soc. 73 (1967), 591–597. MR 211301, DOI 10.1090/S0002-9904-1967-11761-0
- Nikolaos S. Papageorgiou, Random fixed point theorems for measurable multifunctions in Banach spaces, Proc. Amer. Math. Soc. 97 (1986), no. 3, 507–514. MR 840638, DOI 10.1090/S0002-9939-1986-0840638-3
- 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. MR 328688, DOI 10.1090/S0002-9947-1973-0328688-2
- 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. MR 796453, DOI 10.1090/S0002-9939-1985-0796453-1
- V. M. Sehgal and Charles Waters, Some random fixed point theorems for condensing operators, Proc. Amer. Math. Soc. 90 (1984), no. 3, 425–429. MR 728362, DOI 10.1090/S0002-9939-1984-0728362-7
Additional Information
- Liu Li-Shan
- Affiliation: Department of Mathematics, Qufu Normal University, Qufu, Shandong, 273165, People’s Republic of China
- Received by editor(s): August 30, 1994
- Received by editor(s) in revised form: August 22, 1995
- Communicated by: Palle E. T. Jorgensen
- © Copyright 1997 American Mathematical Society
- Journal: Proc. Amer. Math. Soc. 125 (1997), 515-521
- MSC (1991): Primary 47H10, 60H25; Secondary 41A50
- DOI: https://doi.org/10.1090/S0002-9939-97-03589-2
- MathSciNet review: 1350953