An iteration process for nonexpansive mappings with applications to fixed point theory in product spaces

Kirk, W. A.

doi:10.1090/S0002-9939-1989-0941325-6

An iteration process for nonexpansive mappings with applications to fixed point theory in product spaces
HTML articles powered by AMS MathViewer

by W. A. Kirk PDF

Proc. Amer. Math. Soc. 107 (1989), 411-415 Request permission

Abstract:

A uniform transfinite iteration procedure for selecting fixed points of nonexpansive mappings is introduced. This procedure, which applies to arbitrary nonexpansive mappings in Banach spaces having Kadec-Klee norm and to strictly contractive mappings in reflexive Banach spaces, is used to generalize a fixed point theorem of Kirk and Sternfeld for nonexpansive mappings in product spaces.

References

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
D. van Dulst and Brailey Sims, Fixed points of nonexpansive mappings and Chebyshev centers in Banach spaces with norms of type (KK), Banach space theory and its applications (Bucharest, 1981) Lecture Notes in Math., vol. 991, Springer, Berlin-New York, 1983, pp. 35–43. MR 714171
R. Huff, Banach spaces which are nearly uniformly convex, Rocky Mountain J. Math. 10 (1980), no. 4, 743–749. MR 595102, DOI 10.1216/RMJ-1980-10-4-743
Shiro Ishikawa, Fixed points and iteration of a nonexpansive mapping in a Banach space, Proc. Amer. Math. Soc. 59 (1976), no. 1, 65–71. MR 412909, DOI 10.1090/S0002-9939-1976-0412909-X
John L. Kelley, General topology, D. Van Nostrand Co., Inc., Toronto-New York-London, 1955. MR 0070144
W. A. Kirk and Y. Sternfeld, The fixed point property for nonexpansive mappings in certain product spaces, Houston J. Math. 10 (1984), no. 2, 207–214. MR 744905
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

On the points of continuity and the points of sequential continuity