Modified Hybrid Block Iterative Algorithm for Convex Feasibility Problems and Generalized Equilibrium Problems for Uniformly Quasi--Asymptotically Nonexpansive Mappings

We introduce a modified block hybrid projection algorithm for solving the convex feasibility problems for an infinite family of closed and uniformly quasi--asymptotically nonexpansive mappings and the set of solutions of the generalized equilibrium problems. We obtain a strong convergence theorem for the sequences generated by this process in a uniformly smooth and strictly convex Banach space with Kadec-Klee property. The results presented in this paper improve and extend some recent results.

[1]  P. Kumam,et al.  Hybrid iterative scheme by a relaxed extragradient method for solutions of equilibrium problems and a general system of variational inequalities with application to optimization , 2009 .

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

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

[4]  Yair Censor,et al.  Parallel application of block-iterative methods in medical imaging and radiation therapy , 1988, Math. Program..

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

[6]  Wataru Takahashi,et al.  Strong Convergence Theorem by a New Hybrid Method for Equilibrium Problems and Relatively Nonexpansive Mappings , 2008 .

[7]  Poom Kumam,et al.  A hybrid projection method for generalized mixed equilibrium problems and fixed point problems in Banach spaces , 2010 .

[8]  Meijuan Shang,et al.  Strong Convergence of Monotone Hybrid Algorithm for Hemi-Relatively Nonexpansive Mappings , 2007 .

[9]  Poom Kumam,et al.  A general iterative method for solving equilibrium problems, variational inequality problems and fixed point problems of an infinite family of nonexpansive mappings , 2010 .

[10]  Poom Kumam,et al.  A hybrid approximation method for equilibrium and fixed point problems for a monotone mapping and a nonexpansive mapping , 2008 .

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

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

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

[14]  Habtu Zegeye,et al.  A hybrid iteration scheme for equilibrium problems, variational inequality problems and common fixed point problems in Banach spaces , 2010 .

[15]  Rabian Wangkeeree,et al.  An iterative approximation method for solving a general system of variational inequality problems and mixed equilibrium problems , 2009 .

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

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

[18]  Poom Kumam,et al.  A new hybrid iterative method for solution of equilibrium problems and fixed point problems for an inverse strongly monotone operator and a nonexpansive mapping , 2009 .

[19]  A. Moudafi,et al.  SECOND-ORDER DIFFERENTIAL PROXIMAL METHODS FOR EQUILIBRIUM PROBLEMS , 2003 .

[20]  K. Ball,et al.  Sharp uniform convexity and smoothness inequalities for trace norms , 1994 .

[21]  P. Kumam,et al.  A new hybrid iterative method for mixed equilibrium problems and variational inequality problem for relaxed cocoercive mappings with application to optimization problems , 2009 .

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

[23]  Y. Censor,et al.  Iterations of paracontractions and firmaly nonexpansive operators with applications to feasibility and optimization , 1996 .

[24]  Poom Kumam,et al.  A new iterative algorithm of solution for equilibrium problems, variational inequalities and fixed point problems in a Hilbert space , 2010 .

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

[26]  Kasamsuk Ungchittrakool,et al.  Strong convergence theorems for a common fixed point of two relatively nonexpansive mappings in a Banach space , 2007, J. Approx. Theory.

[27]  Yair Censor,et al.  On a Class of Bargaining Schemes for Points in the Core of a Cooperative N-Person Game , 1991 .

[28]  Phayap Katchang,et al.  A viscosity of extragradient approximation method for finding equilibrium problems, variational inequalities and fixed point problems for nonexpansive mappings , 2009 .

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

[30]  Poom Kumam,et al.  Convergence theorems of a hybrid algorithm for equilibrium problems , 2009 .

[31]  Gang Cai,et al.  A hybrid approximation method for equilibrium and fixed point problems for a family of infinitely nonexpansive mappings and a monotone mapping , 2009 .

[32]  S. Stević.  APPROXIMATING FIXED POINTS OF NONEXPANSIVE MAPPINGS , 2006 .

[33]  Rabian Wangkeeree,et al.  Strong convergence of the iterative scheme based on the extragradient method for mixed equilibrium problems and fixed point problems of an infinite family of nonexpansive mappings , 2009 .

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

[35]  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 .

[36]  Suthep Suantai,et al.  Hybrid iterative scheme for generalized equilibrium problems and fixed point problems of finite family of nonexpansive mappings , 2009 .

[37]  Poom Kumam,et al.  A convergence theorem based on a hybrid relaxed extragradient method for generalized equilibrium problems and fixed point problems of a finite family of nonexpansive mappings , 2010 .

[38]  Wataru Takahashi,et al.  Block Iterative Methods for a Finite Family of Relatively Nonexpansive Mappings in Banach Spaces , 2007 .

[39]  Poom Kumam,et al.  Convergence Theorem Based on a New Hybrid Projection Method for Finding a Common Solution of Generalized Equilibrium and Variational Inequality Problems in Banach Spaces , 2010 .

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

[41]  Qiao-Li Dong,et al.  Strong convergence theorem by hybrid method for equilibrium problems, variational inequality problems and maximal monotone operators , 2010 .

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

[43]  Shin Min Kang,et al.  On hybrid projection methods for asymptotically quasi-phi-nonexpansive mappings , 2010, Appl. Math. Comput..

[44]  Satit Saejung,et al.  Strong Convergence to Common Fixed Points of Countable Relatively Quasi-Nonexpansive Mappings , 2008 .

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

[46]  Shih-Sen Chang,et al.  Modified Block Iterative Algorithm for Solving Convex Feasibility Problems in Banach Spaces , 2010 .

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