A general composite iterative method for generalized mixed equilibrium problems, variational inequality problems and optimization problems

Abstract

In this article, we introduce a new general composite iterative scheme for finding a common element of the set of solutions of a generalized mixed equilibrium problem, the set of fixed points of an infinite family of nonexpansive mappings and the set of solutions of a variational inequality problem for an inverse-strongly monotone mapping in Hilbert spaces. It is shown that the sequence generated by the proposed iterative scheme converges strongly to a common element of the above three sets under suitable control conditions, which solves a certain optimization problem. The results of this article substantially improve, develop, and complement the previous well-known results in this area.

Weak Convergence Theorems for Generalized Mixed Equilibrium Problems, Monotone Mappings and Pseudocontractive Mappings

“…Fixed point problems for pseudocontractive mappings and strictly pseudocontractive mappings were studied by many authors, see, for example, [1,9,14,16,19,22,23,24,35] and the references therein. Recently, many authors have introduced some iterative algorithms for finding a common element of the set of the solutions of the GMEP, the GEP, the MEP, the EP, and the VIP and the set of fixed points of a countable family of nonexpansive mappings, and have proved strong convergence of the sequences generated by the proposed iterative algorithms; see [6,13,15,17,25,26,27,29,30,31,32] and the references therein. Also we refer to [4,5,7,18,21] for the GMEP, the GEP, the EP, and the VIP combined with the fixed point problem for nonexpansive semigroups and strictly pseudocontractrive mappings.…”