A random fixed point theorem for multivalued nonexpansive operators in uniformly convex Banach spaces

Xu, Hong Kun

doi:10.1090/S0002-9939-1993-1123670-8

A random fixed point theorem for multivalued nonexpansive operators in uniformly convex Banach spaces
HTML articles powered by AMS MathViewer

by Hong Kun Xu PDF

Proc. Amer. Math. Soc. 117 (1993), 1089-1092 Request permission

Abstract:

Let $(\Omega ,\Sigma )$ be a measurable space with $\Sigma$ a sigma-algebra of subsets of $\Omega$, and let $C$ be a nonempty, bounded, closed, convex, and separable subset of a uniformly convex Banach space $X$. It is shown that every multivalued nonexpansive random operator $T:\Omega \times C \to K(C)$ has a random fixed point, where $K(C)$ is the family of all nonempty compact subsets of $C$ endowed with the Hausdorff metric induced by the norm of $X$.

References

Asuman G. Aksoy and Mohamed A. Khamsi, Nonstandard methods in fixed point theory, Universitext, Springer-Verlag, New York, 1990. With an introduction by W. A. Kirk. MR 1066202, DOI 10.1007/978-1-4612-3444-9
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
Michael Edelstein, The construction of an asymptotic center with a fixed-point property, Bull. Amer. Math. Soc. 78 (1972), 206–208. MR 291917, DOI 10.1090/S0002-9904-1972-12918-5
Kazimierz Goebel, On a fixed point theorem for multivalued nonexpansive mappings, Ann. Univ. Mariae Curie-Skłodowska Sect. A 29 (1975), 69–72 (1977) (English, with Russian and Polish summaries). MR 461225
Shigeru Itoh, A random fixed point theorem for a multivalued contraction mapping, Pacific J. Math. 68 (1977), no. 1, 85–90. MR 451228
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
John L. Kelley, General topology, D. Van Nostrand Co., Inc., Toronto-New York-London, 1955. MR 0070144
W. A. Kirk, An iteration process for nonexpansive mappings with applications to fixed point theory in product spaces, Proc. Amer. Math. Soc. 107 (1989), no. 2, 411–415. MR 941325, DOI 10.1090/S0002-9939-1989-0941325-6
W. A. Kirk and Silvio Massa, Remarks on asymptotic and Chebyshev centers, Houston J. Math. 16 (1990), no. 3, 357–364. MR 1089022
W. A. Kirk and Carlos Martínez-Yañez, Nonexpansive and locally nonexpansive mappings in product spaces, Nonlinear Anal. 12 (1988), no. 7, 719–725. MR 947884, DOI 10.1016/0362-546X(88)90024-7
Teck Cheong Lim, A fixed point theorem for multivalued nonexpansive mappings in a uniformly convex Banach space, Bull. Amer. Math. Soc. 80 (1974), 1123–1126. MR 394333, DOI 10.1090/S0002-9904-1974-13640-2
Teck Cheong Lim, Remarks on some fixed point theorems, Proc. Amer. Math. Soc. 60 (1976), 179–182 (1977). MR 423139, DOI 10.1090/S0002-9939-1976-0423139-X
Teck Cheong Lim, Characterizations of normal structure, Proc. Amer. Math. Soc. 43 (1974), 313–319. MR 361728, DOI 10.1090/S0002-9939-1974-0361728-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
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
Hong Kun Xu, Some random fixed point theorems for condensing and nonexpansive operators, Proc. Amer. Math. Soc. 110 (1990), no. 2, 395–400. MR 1021908, DOI 10.1090/S0002-9939-1990-1021908-6