An iterative algorithm for solving fixed point problems, variational inequality problems and mixed equilibrium problems

Abstract In this paper, we introduce an iterative algorithm for finding a common element of the set of solutions of a mixed equilibrium problem, the set of fixed points of an infinite family of nonexpansive mappings and the set of solutions of some variational inequality. Furthermore, we prove that the proposed iterative algorithm converges strongly to a common element of the above three sets. Our results extend and improve the corresponding results of Zeng and Yao [L.C. Zeng and J.C. Yao, A hybrid iterative scheme for mixed equilibrium problems and fixed point problems, J. Comput. Appl. Math. 214 (2008) 186–201], Takahashi and Takahashi [S. Takahashi and W. Takahashi, Viscosity approximation methods for equilibrium problems and fixed point problems in Hilbert spaces, J. Math. Anal. Appl., 331(2007), 506–515] and many others.

[1]  W. Takahashi,et al.  STRONG CONVERGENCE TO COMMON FIXED POINTS OF INFINITE NONEXPANSIVE MAPPINGS AND APPLICATIONS , 2001 .

[2]  Jen-Chih Yao,et al.  Convergence Theorem for Equilibrium Problems and Fixed Point Problems of Infinite Family of Nonexpansive Mappings , 2007 .

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

[4]  Jen-Chih Yao,et al.  Descent methods for equilibriumproblems in a Banach space , 2004 .

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

[6]  Jen-Chih Yao,et al.  GENERALIZED KKM THEOREM WITH APPLICATIONS TO GENERALIZED MINIMAX INEQUALITIES AND GENERALIZED EQUILIBRIUM PROBLEMS , 2006 .

[7]  Jen-Chih Yao,et al.  A hybrid iterative scheme for mixed equilibrium problems and fixed point problems , 2008 .

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

[9]  Somyot Plubtieng,et al.  A general iterative method for equilibrium problems and fixed point problems in Hilbert spaces , 2007 .

[10]  Jen-Chih Yao,et al.  Combined Relaxation Method for Mixed Equilibrium Problems , 2005 .

[11]  Chi Kin Chan,et al.  Generalized system for relaxed cocoercive variational inequalities in Hilbert spaces , 2007, Appl. Math. Lett..

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

[13]  P. L. Combettes,et al.  Hilbertian convex feasibility problem: Convergence of projection methods , 1997 .

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

[15]  Heinz H. Bauschke,et al.  On Projection Algorithms for Solving Convex Feasibility Problems , 1996, SIAM Rev..

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

[17]  Jen-Chih Yao,et al.  Equilibrium Problems with Applications to Eigenvalue Problems , 2003 .

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

[19]  丁协平,et al.  PREDICTOR-CORRECTOR ALGORITHMS FOR SOLVING GENERALIZED MIXED IMPLICIT QUASI-EQUILIBRIUM PROBLEMS , 2006 .

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

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

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

[23]  Tomonari Suzuki.  Strong convergence of Krasnoselskii and Mann's type sequences for one-parameter nonexpansive semigroups without Bochner integrals , 2005 .

[24]  Hong-Kun Xu.  VISCOSITY APPROXIMATION METHODS FOR NONEXPANSIVE MAPPINGS , 2004 .

[25]  Jen-Chih Yao,et al.  Regularized Equilibrium Problems with Application to Noncoercive Hemivariational Inequalities , 2004 .