Halpern's Iteration in CAT(0) Spaces

Motivated by Halpern's result, we prove strong convergence theorem of an iterative sequence in CAT(0) spaces. We apply our result to find a common fixed point of a family of nonexpansive mappings. A convergence theorem for nonself mappings is also discussed.

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

[2]  W. G. Dotson,et al.  Approximating fixed points of nonexpansive mappings , 1974 .

[3]  D. R. Smart.  Fixed Point Theorems , 1974 .

[4]  Heinz H. Bauschke.  The Approximation of Fixed Points of Compositions of Nonexpansive Mappings in Hilbert Space , 1996 .

[5]  Sompong Dhompongsa,et al.  Common fixed points of a nonexpansive semigroup and a convergence theorem for Mann iterations in geodesic metric spaces , 2009 .

[6]  Ronald E. Bruck.  Properties of fixed-point sets of nonexpansive mappings in Banach spaces , 1973 .

[7]  R. Wittmann.  Approximation of fixed points of nonexpansive mappings , 1992 .

[8]  Wataru Takahashi,et al.  Approximation of common fixed points of a countable family of nonexpansive mappings in a Banach space , 2007 .

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

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

[11]  Wataru Takahashi,et al.  Strong convergence of approximated sequences for nonexpansive mappings in Banach spaces , 1997 .

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

[13]  William A. Kirk,et al.  FIXED POINT THEOREMS IN CAT(0) SPACES AND R-TREES , 2009 .

[14]  B. Halpern.  Fixed points of nonexpanding maps , 1967 .

[15]  W. A. Kirk,et al.  Iteration processes for nonexpansive mappings , 1983 .

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

[17]  Some notes on Bauschke’s condition , 2007 .

[18]  Huang Jianfeng,et al.  Strong convergence of approximated sequences for nonexpansive mappings in Banach spaces , 2007 .

[19]  Graham A. Niblo.  METRIC SPACES OF NON‐POSITIVE CURVATURE (Grundlehren der Mathematischen Wissenschaften 319) , 2001 .

[20]  William A. Kirk,et al.  Fixed point theorems in spaces and -trees , 2004 .

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

[22]  M. Birkner,et al.  Blow-up of semilinear PDE's at the critical dimension. A probabilistic approach , 2002 .