Strong Convergence Theorems for a Countable Family of Total Quasi-Φ-Asymptotically Nonexpansive Nonself Mappings

The purpose of this paper is to introduce a class of total quasi-ϕ-asymptotically nonexpansive-nonself mappings and to study the strong convergence under a limit condition only in the framework of Banach spaces. As an application, we utilize our results to study the approximation problem of solution to a system of equilibrium problems. The results presented in the paper extend and improve the corresponding results announced by some authors recently.

[1]  Hong-Kun Xu,et al.  Strong convergence theorems for two countable families of weak relatively nonexpansive mappings and applications , 2010 .

[2]  Yeol Je Cho,et al.  Strong and weak convergence theorems for nonself-asymptotically perturbed nonexpansive mappings , 2009 .

[3]  K. Deimling.  Nonlinear functional analysis , 1985 .

[4]  Isa Yildirim,et al.  A new iterative process for common fixed points of finite families of non-self-asymptotically non-expansive mappings , 2009 .

[5]  Yucai Dong,et al.  A modified Halpern-type iteration algorithm for a family of hemi-relatively nonexpansive mappings and systems of equilibrium problems in Banach spaces , 2011, J. Comput. Appl. Math..

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

[7]  Charles E. Chidume,et al.  Strong and weak convergence theorems for asymptotically nonexpansive mappings , 2003 .

[8]  Sornsak Thianwan,et al.  Common fixed points of new iterations for two asymptotically nonexpansive nonself-mappings in a Banach space , 2009 .

[9]  Hükmi Kiziltunç,et al.  Convergence theorems by a new iteration process for a finite family of nonself asymptotically nonexpansive mappings with errors in Banach spaces , 2011, Comput. Math. Appl..

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

[11]  Chi Kin Chan,et al.  A new hybrid method for solving a generalized equilibrium problem, solving a variational inequality problem and obtaining common fixed points in Banach spaces, with applications☆ , 2010 .

[12]  Wei Guo,et al.  Weak convergence theorems for asymptotically nonexpansive nonself-mappings , 2011, Appl. Math. Lett..

[13]  Poom Kumam,et al.  Convergence theorems for uniformly quasi-ϕ-asymptotically nonexpansive mappings, generalized equilibrium problems, and variational inequalities , 2011 .

[14]  Li Yang,et al.  Approximation theorems for total quasi-ϕ-asymptotically nonexpansive mappings with applications , 2011, Appl. Math. Comput..

[15]  Xiaolong Qin,et al.  Some Weak Convergence Theorems for a Family of Asymptotically Nonexpansive Nonself Mappings , 2009 .

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

[17]  S. Thianwan.  New iterations for approximation of common fixed points of two asymptotically nonexpansive nonself-mappings , 2009 .

[18]  Habtu Zegeye,et al.  Convergence theorems for equilibrium problem, variational inequality problem and countably infinite relatively quasi-nonexpansive mappings , 2010, Appl. Math. Comput..

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

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

[21]  Liping Yang,et al.  Weak and strong convergence theorems of three step iteration process with errors for nonself-asymptotically nonexpansive mappings , 2010, Math. Comput. Model..

[22]  Poom Kumam,et al.  A new modified block iterative algorithm for uniformly quasi-ϕ-asymptotically nonexpansive mappings and a system of generalized mixed equilibrium problems , 2011 .

[23]  Xu Su-mei.  A Strong Convergence Theorem for Nonexpansive Mapping , 2010 .

[24]  Chi Kin Chan,et al.  Modified block iterative algorithm for Quasi-ϕ-asymptotically nonexpansive mappings and equilibrium problem in banach spaces , 2011, Appl. Math. Comput..

[25]  Habtu Zegeye,et al.  Strong convergence theorems for monotone mappings and relatively weak nonexpansive mappings , 2009 .

[26]  Poom Kumam,et al.  The Modified Block Iterative Algorithms for Asymptotically Relatively Nonexpansive Mappings and the System of Generalized Mixed Equilibrium Problems , 2012, J. Appl. Math..

[27]  Shin Min Kang,et al.  Convergence of an implicit iterative process for asymptotically pseudocontractive nonself-mappings , 2011 .

[28]  Poom Kumam,et al.  Convergence theorems for mixed equilibrium problems, variational inequality problem and uniformly quasi-ϕ-asymptotically nonexpansive mappings , 2011, Appl. Math. Comput..

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

[30]  Poom Kumam,et al.  A Modified Halpern-TypeIterative Method of a System of Equilibrium Problems and a Fixed Point for a Totally Quasi-Φ-Asymptotically Nonexpansive Mapping in a Banach Space , 2012, J. Appl. Math..