Convergence of a modified Halpern-type iteration algorithm for quasi-phi-nonexpansive mappings

The purpose of this work is to modify the Halpern-type iteration algorithm to have strong convergence under a limit condition only in the framework of Banach spaces. The results presented in this work improve on the corresponding ones announced by many others.

[1]  Hong-Ku N Xu.  Another control condition in an iterative method for nonexpansive mappings , 2002, Bulletin of the Australian Mathematical Society.

[2]  S. Reich,et al.  Asymptotic Behavior of Relatively Nonexpansive Operators in Banach Spaces , 2001 .

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

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

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

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

[7]  Hong-Kun Xu,et al.  Strong convergence of the CQ method for fixed point iteration processes , 2006 .

[8]  F. Browder,et al.  FIXED-POINT THEOREMS FOR NONCOMPACT MAPPINGS IN HILBERT SPACE. , 1965, Proceedings of the National Academy of Sciences of the United States of America.

[9]  Dan Butnariu,et al.  Weak Convergence of Orbits of Nonlinear Operators in Reflexive Banach Spaces , 2003 .

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

[11]  Nenad Ujević,et al.  An iterative method for solving nonlinear equations , 2007 .

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

[13]  Yeol Je Cho,et al.  Convergence theorems of common elements for equilibrium problems and fixed point problems in Banach spaces , 2009 .

[14]  A. Kartsatos.  Theory and applications of nonlinear operators of accretive and monotone type , 1996 .