A general iterative method for equilibrium problems and fixed point problems in Hilbert spaces

We introduce a general iterative scheme by the viscosity approximation method for finding a common element of the set of solutions of an equilibrium problem and the set of fixed points of a nonexpansive mapping in a Hilbert space. Our results improve and extend the corresponding ones announced by S. Takahashi and W. Takahashi in 2007, Marino and Xu in 2006, Combettes and Hirstoaga in 2005, and many others.

[1]  R. Wittmann.  Approximation of fixed points of nonexpansive mappings , 1992 .

[2]  I. Yamada,et al.  Minimizing certain convex functions over the intersection of the fixed point sets of nonexpansive mappings , 1998 .

[3]  Giuseppe Marino,et al.  A general iterative method for nonexpansive mappings in Hilbert spaces , 2006 .

[4]  Tie-Gang Fan,et al.  Viscosity approximation methods for nonexpansive mappings and monotone mappings , 2007 .

[5]  A. Moudafi.  Viscosity Approximation Methods for Fixed-Points Problems , 2000 .

[6]  I. Yamada.  The Hybrid Steepest Descent Method for the Variational Inequality Problem over the Intersection of Fixed Point Sets of Nonexpansive Mappings , 2001 .

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

[8]  Hong-Kun Xu.  An Iterative Approach to Quadratic Optimization , 2003 .

[9]  Benar Fux Svaiter,et al.  A Truly Globally Convergent Newton-Type Method for the Monotone Nonlinear Complementarity Problem , 1999, SIAM J. Optim..

[10]  Ram U. Verma,et al.  General convergence analysis for two-step projection methods and applications to variational problems , 2005, Appl. Math. Lett..

[11]  Ram U. Verma,et al.  Generalized System for Relaxed Cocoercive Variational Inequalities and Projection Methods , 2004 .

[12]  Hong-Kun Xu.  Iterative Algorithms for Nonlinear Operators , 2002 .

[13]  R. Rockafellar.  On the maximality of sums of nonlinear monotone operators , 1970 .

[14]  Wataru Takahashi,et al.  Strong convergence theorems for nonexpansive mappings and inverse-strongly monotone mappings , 2005 .

[15]  Wataru Takahashi,et al.  Viscosity approximation methods for equilibrium problems and fixed point problems in Hilbert spaces , 2007 .

[16]  Sjur Didrik Flåm,et al.  Equilibrium programming using proximal-like algorithms , 1997, Math. Program..

[17]  Z. Opial.  Weak convergence of the sequence of successive approximations for nonexpansive mappings , 1967 .

[18]  Wataru Takahashi,et al.  Weak Convergence Theorems for Nonexpansive Mappings and Monotone Mappings , 2003 .