Fixed Point Methods for Solving Solutions of a Generalized Equilibrium Problem

Abstract

In this paper, a generalized equilibrium problem is investigated based on fixed point methods. Strong convergence theorems of solutions are established in the framework of Hilbert spaces.

A new iterative algorithm for solving a system of generalized equilibrium problems

“…This shows that u ∈ C is a solution of the variational inequality if and only if u is a fixed point of P C (I − λ A); where λ > 0 is a constant, and I is the identity operator. Recently, many authors have studied problem (1.2) and its two special cases via fixed-point methods; see [13,14,15,16,17,18] and the references therein.…”

“…In [19], Shimoji and Takahashi introduced a W n -mapping induced by an infinite family of nonexpansive mappings {S i : C → C } and a real number sequence of {γ i } with the restriction 0 ≤ γ i < 1, ∀ i ≥ 1 (see next section). In 2016, Zhang and Hao [13], based on the W n -mapping, studied the convergence analysis of the following iterative sequence, which is defined as following:…”

“…In this paper, inspired by [13] and [14], we introduce a new iterative algorithm and prove a strong convergence theorem for generalized equilibrium problem (1.2) and common fixed-point problem of a family of nonexpasnive mappings. Comparing with the algorithm in [13], we here consider a system of generalized equilibrium problems (instead of one bifunction and one α-inverse strongly monotone operator). It also deserves mentioning that the number of nonexpansive mappings in our algorithm is infinite.…”

Strong convergence of hybrid Halpern processes in a Banach space

“…Recently, many author studied fixed points of nonexpansive mappings and equilibrium (1.1); see [6], [13], [15]- [17], [24], [27]- [33] and the references therein. The aim of this paper is to investigate convergence of a hybrid Halpern process for fixed point and the equilibrium problem.…”

Construction of a common element for the set of solutions of fixed point problems and generalized equilibrium problems in Hilbert spaces

“…It is worth mentioning that the problem of finding a common element in the set of solutions of an equilibrium problem and the set of fixed points of a nonlinear mapping is a fascinating field of research. Therefore, numerous iterative algorithms have been proposed and analyzed to solve such problems, see, for example, [4,8,10,11,13,21,22,23] and the references cited therein. Approximation of a common solution for a pair of equilibrium problems has relevant important applications in various branches of applied mathematics.…”