Fixed point iteration processes for asymptotically nonexpansive mappings

Tan, Kok-Keong; Xu, Hong Kun

doi:10.1090/S0002-9939-1994-1203993-5

Fixed point iteration processes for asymptotically nonexpansive mappings
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by Kok-Keong Tan and Hong Kun Xu

Proc. Amer. Math. Soc. 122 (1994), 733-739

DOI: https://doi.org/10.1090/S0002-9939-1994-1203993-5

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Abstract:

Let X be a uniformly convex Banach space which satisfies Opial’s condition or has a Fréchet differentiable norm, C a bounded closed convex subset of X, and $T:C \to C$ an asymptotically nonexpansive mapping. It is then shown that the modified Mann and Ishikawa iteration processes defined by ${x_{n + 1}} = {t_n}{T^n}{x_n} + (1 - {t_n}){x_n}$ and ${x_{n + 1}} = {t_n}{T^n}({s_n}{T^n}{x_n} + (1 - {s_n}){x_n}) + (1 - {t_n}){x_n}$, respectively, converge weakly to a fixed point of T.

References

S. C. Bose, Weak convergence to the fixed point of an asymptotically nonexpansive map, Proc. Amer. Math. Soc. 68 (1978), no. 3, 305–308. MR 493543, DOI 10.1090/S0002-9939-1978-0493543-4
Ronald E. Bruck, A simple proof of the mean ergodic theorem for nonlinear contractions in Banach spaces, Israel J. Math. 32 (1979), no. 2-3, 107–116. MR 531254, DOI 10.1007/BF02764907
D. van Dulst, Equivalent norms and the fixed point property for nonexpansive mappings, J. London Math. Soc. (2) 25 (1982), no. 1, 139–144. MR 645871, DOI 10.1112/jlms/s2-25.1.139
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
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 by a new iteration method, Proc. Amer. Math. Soc. 44 (1974), 147–150. MR 336469, DOI 10.1090/S0002-9939-1974-0336469-5
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
Gregory B. Passty, Construction of fixed points for asymptotically nonexpansive mappings, Proc. Amer. Math. Soc. 84 (1982), no. 2, 212–216. MR 637171, DOI 10.1090/S0002-9939-1982-0637171-7
Simeon Reich, Weak convergence theorems for nonexpansive mappings in Banach spaces, J. Math. Anal. Appl. 67 (1979), no. 2, 274–276. MR 528688, DOI 10.1016/0022-247X(79)90024-6
J. Schu, Weak and strong convergence to fixed points of asymptotically nonexpansive mappings, Bull. Austral. Math. Soc. 43 (1991), no. 1, 153–159. MR 1086729, DOI 10.1017/S0004972700028884
Jürgen Schu, Iterative construction of fixed points of asymptotically nonexpansive mappings, J. Math. Anal. Appl. 158 (1991), no. 2, 407–413. MR 1117571, DOI 10.1016/0022-247X(91)90245-U
Kok-Keong Tan and Hong Kun Xu, The nonlinear ergodic theorem for asymptotically nonexpansive mappings in Banach spaces, Proc. Amer. Math. Soc. 114 (1992), no. 2, 399–404. MR 1068133, DOI 10.1090/S0002-9939-1992-1068133-2
Kok-Keong Tan and Hong Kun Xu, A nonlinear ergodic theorem for asymptotically nonexpansive mappings, Bull. Austral. Math. Soc. 45 (1992), no. 1, 25–36. MR 1147241, DOI 10.1017/S0004972700036972
Hong Kun Xu, Existence and convergence for fixed points of mappings of asymptotically nonexpansive type, Nonlinear Anal. 16 (1991), no. 12, 1139–1146. MR 1111624, DOI 10.1016/0362-546X(91)90201-B