×
Tan, Kok-Keong; Yuan, Xian-Zhi

Random fixed point theorems and approximation. (English) Zbl 0892.47060

Stochastic Anal. Appl. 15, No. 1, 103-123 (1997).
Summary: We first prove some random fixed point theorems for random nonexpansive operators in Banach spaces. As applications, some random approximation theorems for random 1-set-contractions or random continuous condensing mappings defined on closed balls of a separable Banach space, or on separable closed convex subsets of a Hilbert space, or on spheres of infinite-dimensional separable Banach spaces, are established. Our results are generalizations, improvements or stochastic versions of the corresponding results of Bharucha-Reid (1976), Lin (1988, 1989), Lin and Yen (1988), Massatt (1983), Seghal and Waters (1984) and Xu (1990).

Cited in 1 Review
Cited in 14 Documents

MSC:

47H10 Fixed-point theorems
47H40 Random nonlinear operators
41A50 Best approximation, Chebyshev systems
54H25 Fixed-point and coincidence theorems (topological aspects)
47H04 Set-valued operators
47H09 Contraction-type mappings, nonexpansive mappings, \(A\)-proper mappings, etc.
54C60 Set-valued maps in general topology
28A05 Classes of sets (Borel fields, \(\sigma\)-rings, etc.), measurable sets, Suslin sets, analytic sets
28A20 Measurable and nonmeasurable functions, sequences of measurable functions, modes of convergence
60H25 Random operators and equations (aspects of stochastic analysis)

Keywords:

measurable selection; \(k\)-set-contraction; condensing densifying operator; proximity map; random demicompact map; hemi-compact map; demiclosed; random fixed point; random nonexpansive operators; random approximation; random 1-set-contraction; random continuous condensing mappings
Cite Review PDF
Full Text: DOI

References:

[1] Altman M., Bull. Polish Acad. Sci 5 pp 12– (1957)
[2] DOI: 10.1090/S0002-9904-1976-14091-8 · Zbl 0339.60061 · doi:10.1090/S0002-9904-1976-14091-8
[3] Bharucha-Reid A.T., Random integral equations (1972) · Zbl 0327.60040
[4] DOI: 10.1090/S0002-9904-1968-11983-4 · Zbl 0164.44801 · doi:10.1090/S0002-9904-1968-11983-4
[5] DOI: 10.1073/pnas.74.11.4749 · Zbl 0375.47028 · doi:10.1073/pnas.74.11.4749
[6] DOI: 10.1090/S0002-9947-1976-0394329-4 · doi:10.1090/S0002-9947-1976-0394329-4
[7] DOI: 10.1090/S0002-9939-1959-0105008-8 · doi:10.1090/S0002-9939-1959-0105008-8
[8] DOI: 10.1007/BF01110225 · Zbl 0185.39503 · doi:10.1007/BF01110225
[9] DOI: 10.1007/BF01458545 · Zbl 0515.47029 · doi:10.1007/BF01458545
[10] Furi M., Atti Accad. Naz. Lincei Rend. Cl. Sci. Fis. Mat. Natur 47 pp 465– (1969)
[11] DOI: 10.1007/BF01159159 · Zbl 0551.47024 · doi:10.1007/BF01159159
[12] DOI: 10.1007/BF01350295 · Zbl 0191.14701 · doi:10.1007/BF01350295
[13] DOI: 10.1090/S0002-9947-1968-0221345-0 · doi:10.1090/S0002-9947-1968-0221345-0
[14] DOI: 10.1016/0022-247X(79)90023-4 · Zbl 0407.60069 · doi:10.1016/0022-247X(79)90023-4
[15] Kuratowski C., Fund. Math 15 pp 301– (1930)
[16] DOI: 10.1090/S0002-9939-1988-0954994-0 · doi:10.1090/S0002-9939-1988-0954994-0
[17] DOI: 10.1090/S0002-9939-1989-0973838-5 · doi:10.1090/S0002-9939-1989-0973838-5
[18] DOI: 10.1016/0021-9045(88)90053-6 · Zbl 0643.47053 · doi:10.1016/0021-9045(88)90053-6
[19] Massatt P., Proc. Roy. Soc. Edinburgh sect.(A) 94 pp 323– (1983)
[20] Nussbaum R.D., Doctoral Dissertation (1969)
[21] DOI: 10.1007/BF02414948 · Zbl 0226.47031 · doi:10.1007/BF02414948
[22] DOI: 10.1090/S0002-9904-1967-11761-0 · Zbl 0179.19902 · doi:10.1090/S0002-9904-1967-11761-0
[23] DOI: 10.1090/S0002-9939-1986-0840638-3 · doi:10.1090/S0002-9939-1986-0840638-3
[24] Petryshyn W.V., Proc. Amer. Math. Soc 97 pp 189– (1972)
[25] Petryshyn W.V., Trans. Amer. Math. Soc 182 pp 323– (1973)
[26] DOI: 10.1016/0022-247X(76)90232-8 · Zbl 0328.47034 · doi:10.1016/0022-247X(76)90232-8
[27] DOI: 10.1016/0022-247X(78)90222-6 · Zbl 0375.47031 · doi:10.1016/0022-247X(78)90222-6
[28] DOI: 10.1016/0022-247X(89)90163-7 · Zbl 0681.60056 · doi:10.1016/0022-247X(89)90163-7
[29] DOI: 10.1090/S0002-9939-1985-0796453-1 · doi:10.1090/S0002-9939-1985-0796453-1
[30] DOI: 10.1090/S0002-9939-1984-0728362-7 · doi:10.1090/S0002-9939-1984-0728362-7
[31] Sehgal V.M., Contemporary Math 21 pp 215– (1983) · Zbl 0541.47041 · doi:10.1090/conm/021/729519
[32] DOI: 10.1016/0021-9045(83)90069-2 · Zbl 0518.47040 · doi:10.1016/0021-9045(83)90069-2
[33] Tan, K.K. and Yuan, X.Z. 1992.Some random fixed point theorems, Fixed Point Theory and Applications, Edited by: Tan, K.K. 334–345. World Scientific.
[34] DOI: 10.1006/jmaa.1994.1256 · Zbl 0856.47036 · doi:10.1006/jmaa.1994.1256
[35] DOI: 10.1090/S0002-9939-1990-1021908-6 · doi:10.1090/S0002-9939-1990-1021908-6
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.