Approximating fixed points of α-nonexpansive mappings in uniformly convex Banach spaces and CAT(0) spaces

An existence theorem for a fixed point of an α-nonexpansive mapping of a nonempty bounded, closed and convex subset of a uniformly convex Banach space has been recently established by Aoyama and Kohsaka with a non-constructive argument. In this paper, we show that appropriate Ishikawa iterate algorithms ensure weak and strong convergence to a fixed point of such a mapping. Our theorems are also extended to CAT(0) spaces.AMS Subject Classification:54E40, 54H25, 47H10, 37C25.

[1]  W. A. Kirk,et al.  Topics in Metric Fixed Point Theory , 1990 .

[2]  William A. Kirk,et al.  Fixed points of uniformly lipschitzian mappings , 2006 .

[3]  Jean-Pierre Gossez,et al.  Some geometric properties related to the fixed point theory for nonexpansive mappings. , 1972 .

[4]  W. Takahashi.  Nonlinear Functional Analysis , 2000 .

[5]  S. Stević.  APPROXIMATING FIXED POINTS OF NONEXPANSIVE MAPPINGS , 2006 .

[6]  Hong-Kun Xu.  Inequalities in Banach spaces with applications , 1991 .

[7]  Teck-Cheong Lim,et al.  Remarks on some fixed point theorems , 1976 .

[8]  K. Brown,et al.  Graduate Texts in Mathematics , 1982 .

[9]  Fixed point theorem for a -nonexpansive mappings in Banach spaces , 2011 .

[10]  A. Phon-on,et al.  A note on fixed point sets in CAT(0) spaces , 2006 .

[11]  S. Dhompongsa,et al.  On -convergence theorems in CAT(0) spaces , 2008, Comput. Math. Appl..

[12]  S. Reich,et al.  Uniform Convexity, Hyperbolic Geometry, and Nonexpansive Mappings , 1984 .

[13]  William A. Kirk,et al.  A concept of convergence in geodesic spaces , 2008 .

[14]  Z. Opial.  Weak convergence of the sequence of successive approximations for nonexpansive mappings , 1967 .

[15]  D. Burago,et al.  A Course in Metric Geometry , 2001 .

[16]  Peter Abramenko,et al.  Buildings: Theory and Applications , 2008 .

[17]  Bancha Panyanak,et al.  Approximating Fixed Points of Nonexpansive Mappings in CAT(0) Spaces , 2009 .

[18]  高橋 渉.  Nonlinear functional analysis : fixed point theory and its applications , 2000 .

[19]  M. Gromov.  Metric Structures for Riemannian and Non-Riemannian Spaces , 1999 .

[20]  M. Bridson,et al.  Metric Spaces of Non-Positive Curvature , 1999 .

[21]  Wataru Takahashi,et al.  APPROXIMATING FIXED POINTS OF NONEXPANSIVE MAPPINGS IN BANACH SPACES , 1998 .

[22]  Jiang-Hua Lu,et al.  Progress in Mathematics , 2013 .

[23]  Withun Phuengrattana,et al.  Fixed point theorems and convergence theorems for Suzuki-generalized nonexpansive mappings in CAT(0) spaces , 2010 .

[24]  D. Dulst.  Equivalent Norms and the Fixed Point Property for Nonexpansive Mappings , 1982 .