On certain classes of variational inequalities and related iterative algorithms

Muhammad Aslam Noor

doi:10.1155/s1048953396000056

Abstract

In this paper, we introduce and study some new classes of variational inequalities and Wiener-Hopf equations. Essentially using the projection technique, we establish the equivalence between the multivalued general quasi-variational inequalities and the multivalued implicit Wiener-Hopf equations. This equivalence enables us to suggest and analyze a number of iterative algorithms for solving multivalued general quasi-variational inequalities. We also consider the auxiliary principle technique to prove the existence of a unique solution of the variational-like inequalities. This technique is used to suggest a general and unified iterative algorithm for computing the approximate solution. Several special cases which can be obtained from our main results are also discussed. The results proved in this paper represent a significant refinement and improvement of the previously known results.

Highlights

The theory of variational inequalities emerged as an interesting and fascinating branch of applicable mathematics
We introduce and study some new classes of variational inequalities and Wiener-Hopf equations
The variational inequality theory provides us a natural, simple, general, and unified framework to study a wide class of unrelated linear and nonlinear problems arising in fluid mechanics, elasticity, and oceanography

Summary

Introduction

The theory of variational inequalities emerged as an interesting and fascinating branch of applicable mathematics. We prove that the multivalued general quasi-variational inequality is equivalent to a system of equations, known as the multivalued implicit Wiener-Hopf equations. By a suitable rearrangement of the implicit Wiener-Hopf equations, we suggest a number of iterative algorithms for solving quasi-variational inequalities. We consider another new class of variational inequalities, which is known as the strongly nonlinear mixed variational-like inequality. This equivalence is used to analyze some iterative algorithms for variational inequalities. We show that the auxiliary principle technique can be used to suggest a general iterative algorithm for variational inequalities

Preliminaries

Wiener-Hopf Equations Technique

Auxiliary Principle Technique