Perturbed projection and iterative algorithms for a system of general regularized nonconvex variational inequalities

Javad Balooee,Yeol Je Cho

doi:10.1186/1029-242x-2012-141

Abstract

The purpose of this paper is to introduce a new system of general nonlinear regularized nonconvex variational inequalities and verify the equivalence between the proposed system and fixed point problems. By using the equivalent formulation, the existence and uniqueness theorems for solutions of the system are established. Applying two nearly uniformly Lipschitzian mappings S1 and S2 and using the equivalent alternative formulation, we suggest and analyze a new perturbed p-step projection iterative algorithm with mixed errors for finding an element of the set of the fixed points of the nearly uniformly Lipschitzian mapping Q = (S1,S2) which is the unique solution of the system of general nonlinear regularized nonconvex variational inequalities. We also discuss the convergence analysis of the proposed iterative algorithm under some suitable conditions. MSC : Primary 47H05; Secondary 47J20, 49J40, 90C33.

Highlights

The theory of variational inequalities introduced by Stampacchia [1] in the early 1960s have enjoyed vigorous growth for the last 30 years
By using two nearly uniformly Lipschitzian mappings S1 and S2 and the equivalent alternative formulation, we suggest and analyze a new perturbed p-step iterative algorithm with mixed errors for finding an element of the set of the fixed points of the nearly uniformly Lipschitzian mapping Q = (S1, S2) which is the unique solution of the system of general nonlinear regularized nonconvex variational inequalities
3 System of general regularized nonconvex variational inequalities we introduce a new system of general nonlinear regularized nonconvex variational inequalities and establish the existence and uniqueness theorem for a solution of the mentioned system

Summary

Introduction

The theory of variational inequalities introduced by Stampacchia [1] in the early 1960s have enjoyed vigorous growth for the last 30 years. By using two nearly uniformly Lipschitzian mappings S1 and S2 and the equivalent alternative formulation, we suggest and analyze a new perturbed p-step iterative algorithm with mixed errors for finding an element of the set of the fixed points of the nearly uniformly Lipschitzian mapping Q = (S1, S2) which is the unique solution of the system of general nonlinear regularized nonconvex variational inequalities. We prove the existence and uniqueness theorem for a solution of the system of general nonlinear regularized nonconvex variational inequalities (3.1) For this end, we need the following lemma in which by using the projection operator technique, we verify the equivalence between the system of general nonlinear regularized nonconvex variational inequalities (3.1) and a fixed point problem.

Perturbed projection and iterative algorithms

Conclusion