Modified block iterative method for solving convex feasibility problem, equilibrium problems and variational inequality problems

The purpose of this paper is by using the modified block iterative method to propose an algorithm for finding a common element in the intersection of the set of common fixed points of an infinite family of quasi-ϕ-asymptotically nonexpansive and the set of solutions to an equilibrium problem and the set of solutions to a variational inequality. Under suitable conditions some strong convergence theorems are established in 2-uniformly convex and uniformly smooth Banach spaces. As applications we utilize the results presented in the paper to solving the convex feasibility problem (CFP) and zero point problem of maximal monotone mappings in Banach spaces. The results presented in the paper improve and extend the corresponding results announced by many authors.

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

[2]  Li Yang,et al.  Iterative Approximation to Convex Feasibility Problems in Banach Space , 2007 .

[3]  Poom Kumam,et al.  Strong convergence theorems by a new hybrid projection algorithm for fixed point problems and equilibrium problems of two relatively quasi-nonexpansive mappings , 2009 .

[4]  Kasamsuk Ungchittrakool,et al.  Hybrid Iterative Methods for Convex Feasibility Problems and Fixed Point Problems of Relatively Nonexpansive Mappings in Banach Spaces , 2008 .

[5]  Poom Kumam,et al.  Corrigendum to: “Strong convergence theorems by a new hybrid projection algorithm for fixed point problems and equilibrium problems of two relatively quasi-nonexpansive mappings” [Nonlinear Anal. Hybrid Syst. 3 (2009) 11–20] , 2009 .

[6]  Wataru Takahashi,et al.  STRONG AND WEAK CONVERGENCE THEOREMS FOR EQUILIBRIUM PROBLEMS AND RELATIVELY NONEXPANSIVE MAPPINGS IN BANACH SPACES , 2009 .

[7]  Wataru Takahashi,et al.  Image recovery by convex combinations of sunny nonexpansive retractions , 1993 .

[8]  Shi-sheng Zhang,et al.  Generalized mixed equilibrium problem in Banach spaces , 2009 .

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

[10]  P. L. Combettes,et al.  Equilibrium programming in Hilbert spaces , 2005 .

[11]  Chi Kin Chan,et al.  A new method for solving equilibrium problem fixed point problem and variational inequality problem with application to optimization , 2009 .

[12]  Paranjothi Pillay,et al.  Iterative approaches to convex feasibility problems in Banach spaces , 2006 .

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

[14]  Simeon Reich,et al.  Block-iterative algorithms for solving convex feasibility problems in Hilbert and in Banach spaces , 2008 .

[15]  Zhang Shi-sheng.  On the Generalized Mixed Equilibrium Problem in Banach Spaces , 2009 .

[16]  L. Hongyu,et al.  Strong convergence theorem by a new hybrid method for equilibrium problems and variational inequality problems , 2010 .

[17]  Haiyun Zhou,et al.  Convergence theorems of a modified hybrid algorithm for a family of quasi-φ-asymptotically nonexpansive mappings , 2010 .

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

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

[20]  P. L. Combettes,et al.  The Convex Feasibility Problem in Image Recovery , 1996 .