A Viscosity Approximation Method for Finding Common Solutions of Variational Inclusions, Equilibrium Problems, and Fixed Point Problems in Hilbert Spaces

Abstract

We introduce an iterative method for finding a common element of the set of common fixed points of a countable family of nonexpansive mappings, the set of solutions of a variational inclusion with set-valued maximal monotone mapping, and inverse strongly monotone mappings and the set of solutions of an equilibrium problem in Hilbert spaces. Under suitable conditions, some strong convergence theorems for approximating this common elements are proved. The results presented in the paper improve and extend the main results of J. W. Peng et al. 2008 and many others.

Strong Convergence Theorem for Equilibrium Problems and Fixed Points of a Nonspreading Mapping in Hilbert Spaces

“…Some methods have been proposed to solve the equilibrium problem and fixed point problem of nonexpansive mapping: see, for instance, 1,2,6,7,[17][18][19][20] and the references Fixed Point Theory and Applications 3 therein. In 1997, Combettes and Hirstoaga 3 introduced an iterative scheme of finding the best approximation to the initial data when EP F is nonempty and proved a strong convergence theorem.…”

Convergence Theorems for Common Solutions of Split Variational Inclusion and Systems of Equilibrium Problems

“…In the process of studying equilibrium problems and split inverse problems, not only techniques and methods for solving the respective problems have been proposed (see, for example, CQ-algorithm in Byrne [37,38], relaxed CQ-algorithm in Yang [39] and Gibali et al [40], self-adaptive algorithm in López et al [41], Moudafi and Thukur [42], and Gibali [43]), but also the common solution of equilibrium problems, split inverse problems, and other problems have been considered in many works (see, for example, Plubtieng and Sombut [44] considered the common solution of equilibrium problems and nonspreading mappings; Sobumt and Plubtieng [45] studied a common solution of equilibrium problems and split feasibility problems in Hilbert spaces; Sitthithakerngkiet et al [46] investigated a common solution of split monotone variational inclusion problems and fixed points problem of nonlinear operators; Eslamian and Fakhri [47] considered split equality monotone variational inclusion problems and fixed point problem of set-valued operators; Censor and Segal [48], Plubtieng and Sriprad [49] explored split common fixed point problems for directed operators). In particular, some applications to mathematical models for studying a common solution of convex optimizations and compressed sensing whose constraints can be presented as equilibrium problems and split variational inclusion problems, which stimulated our research on this kind of problem.…”

Weak Convergence Theorem for Finding Fixed Points and Solution of Split Feasibility and Systems of Equilibrium Problems

“…Numerous problems in physics, optimization, and economics reduce to find a solution of (1) in Hilbert spaces; see, for instance, Blum and Oettli [5], Flam and Antipin [6], and Moudafi [7]. Moreover, Flam and Antipin [6] introduced an iterative scheme of finding the best approximation to the solution of equilibrium problem, when EP( ) is nonempty, and proved a strong convergence theorem (see also in [8][9][10][11]). Let 1 , 2 : × → R be two-monotone bifunction and > 0 is a constant.…”