Variational Inequality in Hilbert Spaces and their Applications

Description

– Variational Inequality in Hilbert Spaces and their Applications –

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Abstract

The study of variational inequalities frequently deals with a mapping F from a vector space X or a convex subset of X into its dual Xj .

Let H be a real Hilbert space and a(u, v) be a real bilinear form on H. Assume that the linear and continuous mapping A : H  Hj determines a bilinear form via the pairing a(u, v) = Au, v  .

Given  K  H  and  f   Hj. Then, Variational inequality(VI) is the problem of finding u K such that a(u, v  u)  f, v u , for all  v K.  In this work, we outline some results in theory of variational inequalities.

Their relationships with other problems of Nonlinear Analysis and some applications are also discussed.

Introduction

1.1 Background of the Study

In the study of variational inequalities, we are frequently concern with a mapping F from a vector space X or a convex subset of X into its dual Xj .

Variational inequalities and Complementary problems are of fundamental importance in a wide range of mathematical and applied problems, such as programming, traffic engineering, economics and equilibrium problems.

The idea and techniques of the variational inequalities are being applied in a variety of diverse areas in sciences and proved to be productive and innovative. It has been shown that this theory provides a simple, natural and unified framework for a general treatment of unrelated problems.

The fixed point theory has played an important role in the development of various algorithms for solving variational inequalities. Using the projection operator technique, one usually establishes an equivalence between the variational inequalities and the fixed point problem.

The alternative equivalent formulation was used by Lions and Stampacchia [8] to study the existence of a solution of the variational inequalities. Projection methods and its variant forms represent important tools for finding the approximate solution of variational inequalities.

In this work, we intend to present the element of variational inequalities and free boundary problems with several examples and their applications. The usual setting of the scalar variational inequality is the following: Let K be a nonempty subset of Rn and (., .) denote the scalar product in Rn. Let an operator F : K → Rn be given.

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