Remote Access Proceedings of the American Mathematical Society
Green Open Access

Proceedings of the American Mathematical Society

ISSN 1088-6826(online) ISSN 0002-9939(print)

 
 

 

Fixed point iteration processes for asymptotically nonexpansive mappings


Authors: Kok-Keong Tan and Hong Kun Xu
Journal: Proc. Amer. Math. Soc. 122 (1994), 733-739
MSC: Primary 47H17; Secondary 47H09, 47H10
DOI: https://doi.org/10.1090/S0002-9939-1994-1203993-5
MathSciNet review: 1203993
Full-text PDF Free Access

Abstract | References | Similar Articles | Additional Information

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 [Enhancements On Off] (What's this?)

  • [1] S. C. Bose, Weak convergence to the fixed point of an asymptotically nonexpansive map, Proc. Amer. Math. Soc. 68 (1978), 305-308. MR 0493543 (58:12538)
  • [2] R. E. Bruck, A simple proof of the mean ergodic theorem for nonlinear contractions in Banach spaces, Isreal J. Math. 32 (1979), 107-116. MR 531254 (80j:47066)
  • [3] D. van Dulst, Equivalent norms and the fixed point property for nonexpansive mappings, J. London Math. Soc. (2) 25 (1982), 139-144. MR 645871 (83e:47040)
  • [4] K. Goebel and W. A. Kirk, A fixed point theorem for asymptotically nonexpansive mappings, Proc. Amer. Math. Soc. 35 (1972), 171-174. MR 0298500 (45:7552)
  • [5] R. Huff, Banach spaces which are nearly uniformly convex, Rocky Mountain J. Math. 10 (1980), 743-749. MR 595102 (82b:46016)
  • [6] S. Ishikawa, Fixed points by a new iteration method, Proc. Amer. Math. Soc. 44 (1974), 147-150. MR 0336469 (49:1243)
  • [7] Z. Opial, Weak convergence of the sequence of successive approximations for nonexpansive mappings, Bull. Amer. Math. Soc. 73 (1967), 595-597. MR 0211301 (35:2183)
  • [8] G. B. Passty, Construction of fixed points for asymptotically nonexpansive mappings, Proc. Amer. Math. Soc. 84 (1982), 213-216. MR 637171 (83a:47065)
  • [9] S. Reich, Weak convergence theorems for nonexpansive mappings in Banach spaces, J. Math. Anal. Appl. 67 (1979), 274-276. MR 528688 (80d:47090)
  • [10] J. Schu, Weak and strong convergence to fixed points of asymptotically nonexpansive mappings, Bull. Austral. Math. Soc. 43 (1991), 153-159. MR 1086729 (91k:47136)
  • [11] -, Iterative construction of fixed points of asymptotically nonexpansive, J. Math. Anal. Appl. 158 (1991), 407-413. MR 1117571 (92d:47072)
  • [12] K. K. Tan and H. K. Xu, The nonlinear ergodic theorem for asymptotically nonexpansive mappings in Banach spaces, Proc. Amer. Math. Soc. 114 (1992), 399-404. MR 1068133 (92e:47100)
  • [13] -, A nonlinear ergodic theorem for asymptotically nonexpansive mappings, Bull. Austral. Math. Soc. 45 (1992), 25-36. MR 1147241 (93c:47068)
  • [14] H. K. Xu, Existence and convergence for fixed points of mappings of asymptotically nonexpansive type, Nonlinear Anal. 16 (1991), 1139-1146. MR 1111624 (92h:47089)

Similar Articles

Retrieve articles in Proceedings of the American Mathematical Society with MSC: 47H17, 47H09, 47H10

Retrieve articles in all journals with MSC: 47H17, 47H09, 47H10


Additional Information

DOI: https://doi.org/10.1090/S0002-9939-1994-1203993-5
Keywords: Fixed point, asymptotically nonexpansive mapping, fixed point iteration process, uniformly convex Banach space, Fréchet differentiable norm, Opial's condition
Article copyright: © Copyright 1994 American Mathematical Society

American Mathematical Society