Approximating fixed points of asymptotically nonexpansive mappings in Banach spaces by metric projections

Abstract In this paper, a strong convergence theorem for asymptotically nonexpansive mappings in a uniformly convex and smooth Banach space is proved by using metric projections. This theorem extends and improves the recent strong convergence theorem due to Matsushita and Takahashi [S. Matsushita, W. Takahashi, Approximating fixed points of nonexpansive mappings in a Banach space by metric projections, Appl. Math. Comput. 196 (2008) 422–425] which was established for nonexpansive mappings.

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

[2]  H. Dehghan,et al.  Approximating Fixed Points of Non-Lipschitzian Mappings by Metric Projections , 2011 .

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

[4]  Wataru Takahashi,et al.  Approximating fixed points of nonexpansive mappings in a Banach space by metric projections , 2008, Appl. Math. Comput..

[5]  Hong-Kun Xu,et al.  Strong Convergence of Approximating Fixed Point Sequences for Nonexpansive Mappings , 2006, Bulletin of the Australian Mathematical Society.

[6]  Dennis F. Cudia.  The geometry of Banach spaces , 1964 .

[7]  J. Schu,et al.  Weak and strong convergence to fixed points of asymptotically nonexpansive mappings , 1991, Bulletin of the Australian Mathematical Society.

[8]  Yeol Je Cho,et al.  Weak and strong convergence theorems for three-step iterations with errors for asymptotically nonexpansive mappings☆ , 2004 .

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

[10]  F. Browder.  Nonlinear functional analysis , 1970 .

[11]  Ronald E. Bruck.  On the convex approximation property and the asymptotic behavior of nonlinear contractions in Banach spaces , 1981 .

[12]  Muhammad Aslam Noor,et al.  Fixed-point iterations for asymptotically nonexpansive mappings in Banach spaces , 2002 .

[13]  William A. Kirk,et al.  A FIXED POINT THEOREM FOR ASYMPTOTICALLY NONEXPANSIVE MAPPINGS , 1972 .