Relaxed Extragradient Methods with Regularization for General System of Variational Inequalities with Constraints of Split Feasibility and Fixed Point Problems

L C Ceng,J C Yao,A Petruşel

doi:10.1155/2013/891232

Abstract

We suggest and analyze relaxed extragradient iterative algorithms with regularization for finding a common element of the solution set of a general system of variational inequalities, the solution set of a split feasibility problem, and the fixed point set of a strictly pseudocontractive mapping defined on a real Hilbert space. Here the relaxed extragradient methods with regularization are based on the well-known successive approximation method, extragradient method, viscosity approximation method, regularization method, and so on. Strong convergence of the proposed algorithms under some mild conditions is established. Our results represent the supplementation, improvement, extension, and development of the corresponding results in the very recent literature.

Highlights

Let H be a real Hilbert space, whose inner product and norm are denoted by ⟨⋅, ⋅⟩ and ‖⋅‖, respectively
Motivated and inspired by the research going on this area, we propose and analyze the following relaxed extragradient iterative algorithms with regularization for finding a common element of the solution set of the general system of variational inequalities (GSVI) (14), the solution set of the SFP (1), and the fixed point set of a strictly pseudocontractive mapping S : C → C
For given x0 ∈ C arbitrarily, let {xn}, {un}, {ũn} be the sequences generated by the following relaxed extragradient iterative scheme with regularization: un = PC [PC − μ1B1PC], ũn = PC (un − λn∇fαn), (17)

Summary

Introduction

Let H be a real Hilbert space, whose inner product and norm are denoted by ⟨⋅, ⋅⟩ and ‖⋅‖, respectively. For given x0 ∈ C arbitrarily, let {xn}, {un}, {ũn} be the sequences generated by the following relaxed extragradient iterative scheme with regularization: un = PC [PC (xn − μ2B2xn) − μ1B1PC (xn − μ2B2xn)] , ũn = PC (un − λn∇fαn (un)) , (17)

Results

Conclusion