Strong Convergence to Common Fixed Points of Countable Relatively Quasi-Nonexpansive Mappings

Correspondence should be addressed to Satit Saejung,saejung@kku.ac.thReceived 30 August 2007; Accepted 24 December 2007Recommended by Simeon ReichWe prove that a sequence generated by the monotone CQ-method converges strongly to a commonfixed point of a countable family of relatively quasi-nonexpansive mappings in a uniformly convexand uniformly smooth Banach space. Our result is applicable to a wide class of mappings.Copyrightq2008 W. Nilsrakoo and S. Saejung. This is an open access article distributed underthe Creative Commons Attribution License, which permits unrestricted use, distribution, andreproduction in any medium, provided the original work is properly cited.

[1]  W. Takahashi,et al.  Strong convergence theorems for nonexpansive mappings and nonexpansive semigroups , 2003 .

[2]  Kasamsuk Ungchittrakool,et al.  Strong convergence theorems for a common fixed point of two relatively nonexpansive mappings in a Banach space , 2007, J. Approx. Theory.

[3]  Heinz H. Bauschke,et al.  Projection and proximal point methods: convergence results and counterexamples , 2004 .

[4]  Meijuan Shang,et al.  Strong Convergence of Monotone Hybrid Algorithm for Hemi-Relatively Nonexpansive Mappings , 2007 .

[5]  Yongfu Su,et al.  Monotone CQ iteration processes for nonexpansive semigroups and maximal monotone operators , 2008 .

[6]  Y. Alber.  Metric and Generalized Projection Operators in Banach Spaces: Properties and Applications , 1993, funct-an/9311001.

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

[8]  Wataru Takahashi,et al.  Block Iterative Methods for a Finite Family of Relatively Nonexpansive Mappings in Banach Spaces , 2007 .

[9]  I. Ciorǎnescu.  Geometry of banach spaces, duality mappings, and nonlinear problems , 1990 .

[10]  W. R. Mann,et al.  Mean value methods in iteration , 1953 .

[11]  S. Reich.  Book Review: Geometry of Banach spaces, duality mappings and nonlinear problems , 1992 .

[12]  J. Lindenstrauss,et al.  An example concerning fixed points , 1975 .

[13]  Wataru Takahashi,et al.  A strong convergence theorem for relatively nonexpansive mappings in a Banach space , 2005, J. Approx. Theory.

[14]  W. Takahashi,et al.  Weak and strong convergence theorems for relatively nonexpansive mappings in Banach spaces , 2004 .

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

[16]  Wataru Takahashi,et al.  Strong Convergence of a Proximal-Type Algorithm in a Banach Space , 2002, SIAM J. Optim..

[17]  Benar Fux Svaiter,et al.  Forcing strong convergence of proximal point iterations in a Hilbert space , 2000, Math. Program..

[18]  S. Reich.  Weak convergence theorems for nonexpansive mappings in Banach spaces , 1979 .

[19]  Yongfu Su,et al.  Strong convergence theorems for relatively nonexpansive mappings in a Banach space , 2007 .

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