On a class of multivalued variational inequalities

Muhammad Aslam Noor

doi:10.1155/s1048953398000070

Abstract

In this paper, we introduce and study a new class of variational inequalities, which are called multivalued variational inequalities. These variational inequalities include as special cases, the previously known classes of variational inequalities. Using projection techniques, we show that multivalued variational inequalities are equivalent to fixed point problems and Wiener-Hopf equations. These alternate formulations are used to suggest a number of iterative algorithms for solving multivalued variational inequalities. We also consider the auxiliary principle technique to study the existence of a solution of multivalued variational inequalities and suggest a novel iterative algorithm. In addition, we have shown that the auxiliary principle technique can be used to find the equivalent differentiable optimization problems for multivalued variational inequalities. Convergence analysis is also discussed.

Highlights

It is well known that the general theory of the calculus of variations was developed by Euler and Lagrange
There are significant recent developments of variational inequalities related to multivalued operators, nonconvex optimization, iterative methods, Wiener-Hopf equations, and structural analysis
We show that the projection and auxiliary principle techniques can be extended and modified for multivalued variational inequalities

Summary

Introduction

Projection method represents an important computational tool for finding approximate solution of variational inequalities, which was developed in 1970 and 1980 This method has been extended and modified in various ways to other class of variational inequalities, see, for example, Noor [18, 19], for an account of the iterative methods. Using essentially the projection technique, Shi [34, 35] established the equivalence between the variational inequality problems and system of equations, known as Wiener-Hopf equations This equivalence was used to suggest an iterative algorithm. We consider the auxiliary variational principle technique to study the existence of a solution of multivalued variational inequalities and suggest a general iterative algorithm.

Preliminaries

Main Results

Extensions