Iterative Algorithms for Solving the System of Mixed Equilibrium Problems, Fixed‐Point Problems, and Variational Inclusions with Application to Minimization Problem

Abstract

We introduce a new iterative algorithm for solving a common solution of the set of solutions of fixed point for an infinite family of nonexpansive mappings, the set of solution of a system of mixed equilibrium problems, and the set of solutions of the variational inclusion for aβ-inverse-strongly monotone mapping in a real Hilbert space. We prove that the sequence converges strongly to a common element of the above three sets under some mild conditions. Furthermore, we give a numerical example which supports our main theorem in the last part.

A viscosity iterative algorithm for the optimization problem system

“…Therefore , z = x 11r+1 , which yields T r (1) = x n 11r n +1 . By the same argument , for F 2 , one can conclude T r(2)…”

“…By using MATLAB software , we obtain the following table and figure of the result , with initial point x 1 = 1. n x n n x n n x n 1 1 11 2.287053908×10 −6 21 3.762255275×10 Example 5.2. Let H = R and C = [1,2] . For each x ∈ C , we know f (x) = x 2 4 is a contractive mapping, T(x) = 1 x is a nonexpansive mapping on C and F(T) = {1} , A(x) = 2x is a strongly positive linear bounded operator on H .…”

“…By intuition from [2], we suggest and analysis an iterative algorithm for finding a common element of the set of solution of a mixed equilibrium problem and the set of solutions of a variational inequality and all common fixed points of a nonexpansive semigroup in the framework of a Hilbert space.…”

A Viscosity Hybrid Steepest Descent Method for Equilibrium Problems, Variational Inequality Problems, and Fixed Point Problems of Infinite Family of Strictly Pseudocontractive Mappings and Nonexpansive Semigroup

“…In 2012, Chamnarnpan and Kumam [34] introduced the following explicit viscosity scheme with respect tomappings for an infinite family of nonexpansive mappings…”