Strong Convergence Theorems for a Finite Family of -Strict Pseudocontractions in 2-Uniformly Smooth Banach Spaces by Metric Projections

A new hybrid projection algorithm is considered for a finite family of -strict pseudocontractions. Using the metric projection, some strong convergence theorems of common elements are obtained in a uniformly convex and 2-uniformly smooth Banach space. The results presented in this paper improve and extend the corresponding results of Matsushita and Takahshi, 2008, Kang and Wang, 2011, and many others.

[1]  Chi Kin Chan,et al.  On the convergence of implicit iteration process with error for a finite family of asymptotically nonexpansive mappings , 2006 .

[2]  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.

[3]  Klaus Deimling,et al.  Zeros of accretive operators , 1974 .

[4]  Weerayuth Nilsrakoo,et al.  Halpern-type iterations for strongly relatively nonexpansive mappings in Banach spaces , 2011, Comput. Math. Appl..

[5]  Chi Kin Chan,et al.  A modified halpern-type iteration algorithm for totally quasi-ϕ-asymptotically nonexpansive mappings with applications , 2012, Appl. Math. Comput..

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

[7]  Baoqing Liu,et al.  Strong convergence theorems for equilibrium problems and relatively nonexpansive mappings in Banach spaces , 2011, Comput. Math. Appl..

[8]  F. Browder,et al.  Construction of fixed points of nonlinear mappings in Hilbert space , 1967 .

[9]  Habtu Zegeye,et al.  Convergence of Mann's type iteration method for generalized asymptotically nonexpansive mappings , 2011, Comput. Math. Appl..

[10]  Felix E. Browder,et al.  Convergence of approximants to fixed points of nonexpansive nonlinear mappings in banach spaces , 1967 .

[11]  Ronald E. Bruck.  Nonexpansive projections on subsets of Banach spaces. , 1973 .

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

[13]  Xiaolong Qin,et al.  On the convergence of hybrid projection algorithms for asymptotically quasi-ϕ-nonexpansive mappings , 2011, Comput. Math. Appl..

[14]  Satit Saejung,et al.  Strong convergence theorems by Halpern-Mann iterations for relatively nonexpansive mappings in Banach spaces , 2011, Appl. Math. Comput..

[15]  Suthep Suantai,et al.  An implicit iteration process for solving a fixed point problem of a finite family of multi-valued mappings in Banach spaces , 2012, Appl. Math. Lett..

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

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

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

[19]  Shin Min Kang,et al.  Strong convergence of shrinking projection methods for quasi-ϕ-nonexpansive mappings and equilibrium problems , 2010, J. Comput. Appl. Math..

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

[21]  Shin Min Kang,et al.  New Hybrid Algorithms for Nonexpansive Mappings in Banach Spaces , 2011 .

[22]  Lin Wang,et al.  Strong convergence theorems for a countable family of quasi-ϕ-asymptotically nonexpansive nonself mappings , 2012, Appl. Math. Comput..

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

[24]  Hong-Kun Xu,et al.  Strong convergence of modified Mann iterations , 2005 .

[25]  D. Pascali,et al.  Nonlinear mappings of monotone type , 1979 .

[26]  Hong Zhang,et al.  New monotone hybrid algorithm for hemi-relatively nonexpansive mappings and maximal monotone operators , 2011, Appl. Math. Comput..

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

[28]  Chi Kin Chan,et al.  Approximating solutions of variational inequalities for asymptotically nonexpansive mappings , 2009, Appl. Math. Comput..

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

[30]  F. Browder.  Convergence theorems for sequences of nonlinear operators in Banach spaces , 1967 .

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

[32]  Haiyun Zhou,et al.  Convergence theorems for λ-strict pseudo-contractions in 2-uniformly smooth Banach spaces , 2008 .

[33]  Haiyun Zhou,et al.  CONVERGENCE THEOREMS OF COMMON FIXED POINTS FOR A FINITE FAMILY OF LIPSCHITZ PSEUDOCONTRACTIONS IN BANACH SPACES , 2008 .

[34]  Chi Kin Chan,et al.  Strong convergence theorems for countable families of asymptotically relatively nonexpansive mappings with applications , 2011, Appl. Math. Comput..

[35]  C. E. Chidume,et al.  An example on the Mann iteration method for Lipschitz pseudocontractions , 2001 .

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

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

[38]  Shih-Sen Chang,et al.  Approximation of common fixed points for a family of finite nonexpansive mappings in Banach space , 2005 .