A new mapping for finding common solutions of equilibrium problems and fixed point problems of finite family of nonexpansive mappings

Abstract

In this paper, we introduce and study a new mapping generated by a finite family of nonexpansive mappings and finite real numbers and introduce a general iterative method concerning the new mappings for finding a common element of the set of solutions of an equilibrium problem and of the set of common fixed points of a finite family of nonexpansive mappings in a Hilbert space. Then, we prove a strong convergence theorem of the proposed iterative method for a finite family of nonexpansive mappings to the unique solution of variational inequality which is the optimality condition for a minimization problem. Our main result can be applied to obtain strong convergence of the general iterative methods which are modifications of those in [G. Marino, H.K. Xu, A general iterative method for nonexpansive mappings in Hilbert spaces, J. Math. Anal. Appl. 318 (1) (2006) 43-52; S. Plubtieng, R. Punpaeng, A general iterative method for equilibrium problems and fixed point problems in Hilbert spaces, J. Math. Anal. Appl. 336 (1) (2007) 455-469; S. Takahashi, W. Takahashi, Viscosity approximation methods for equilibrium problems and fixed point problems in Hilbert spaces, J. Math. Anal. Appl. 331 (1) (2007) 506-515] to a common element of the set of solutions of an equilibrium problem and the set of fixed points of a nonexpansive mapping. © 2009 Elsevier Ltd. All rights reserved.