Approximation of fixed points of asymptotically nonexpansive mappings

Schu, Jürgen

doi:10.1090/S0002-9939-1991-1039264-7

Approximation of fixed points of asymptotically nonexpansive mappings
HTML articles powered by AMS MathViewer

by Jürgen Schu PDF

Proc. Amer. Math. Soc. 112 (1991), 143-151 Request permission

Abstract:

Let $T$ be an asymptotically nonexpansive self-mapping of a non-empty closed, bounded, and starshaped (with respect to zero) subset of a smooth reflexive Banach space possessing a duality mapping that is weakly sequentially continuous at zero. Then, if id-$T$ is demiclosed and $T$ satisfies a strengthened regularity condition, the iteration process ${z_{n + 1}}: = {\mu _{n + 1}}{T^n}({z_n})$ converges strongly to some fixed point of $T$, provided $({\mu _n})$ has certain properties.

References

Bernard Beauzamy, Introduction to Banach spaces and their geometry, Notas de Matemática [Mathematical Notes], vol. 86, North-Holland Publishing Co., Amsterdam-New York, 1982. MR 670943
Joseph Diestel, Geometry of Banach spaces—selected topics, Lecture Notes in Mathematics, Vol. 485, Springer-Verlag, Berlin-New York, 1975. MR 0461094
K. Goebel and W. A. Kirk, A fixed point theorem for asymptotically nonexpansive mappings, Proc. Amer. Math. Soc. 35 (1972), 171–174. MR 298500, DOI 10.1090/S0002-9939-1972-0298500-3
Benjamin Halpern, Fixed points of nonexpanding maps, Bull. Amer. Math. Soc. 73 (1967), 957–961. MR 218938, DOI 10.1090/S0002-9904-1967-11864-0
S. K. Samanta, Fixed point theorems in a Banach space satisfying Opial’s condition, J. Indian Math. Soc. (N.S.) 45 (1981), no. 1-4, 251–258 (1984). MR 828875
Jürgen Schu, Iterative approximation of fixed points of nonexpansive mappings with starshaped domain, Comment. Math. Univ. Carolin. 31 (1990), no. 2, 277–282. MR 1077898
P. Vijayaraju, Fixed point theorems for asymptotically nonexpansive mappings, Bull. Calcutta Math. Soc. 80 (1988), no. 2, 133–136. MR 956803