– 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.
TABLE OF CONTENTS
Acknowledgment i
Certication ii
Approval iii
Dedication v
Abstract vi
Introduction viii
CHAPTER ONE 1
1 Linear Functional Analysis 1
1.1 Hilbert Spaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1
1.2 Function spaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7
1.3 Sobolev spaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9
2 Variational Inequalities in RN 20
2.1 Basic Theorems and Denition about Fixed point . . . . . . . . . . . 20
2.2 First Theorem about variational inequalities . . . . . . . . . . . . . . 21
2.3 Some problems leading to variational inequality . . . . . . . . . . . . 24
3 Variational Inequality in Hilbert Spaces 30
3.1 Problem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30
3.2 Application . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34
4 CONCLUSION 38
INTRODUCTION
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 inequal- ities 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 sim- ple, 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 tech- nique, 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.
Projction 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.
BIBLIOGRAPHY
Blum, E. From Optimization and Variational Inequalities to Equilibrium Prob- lems Math.Student, pp. 123-145 Vol.63, 1994.
Brezis, H. Functional Analysis, Sobolev Spaces and Partial Differential Equa- tion Spring Science and Business Media, 2010.
Browder,F. E. Fixed Point Theory and Nonlinear Problems Proc. Sym. Pure. Math, pp. 49-88, Vol.39, 1983.
Chidume, C. E. Applicable Functional Analysis University of Ibadan, Press, 2014.
Cottle, R. W., Giannessi, F., and Lions, J. L. Variational Inequalities and Complementarity Problems: Theory and Applications John Wiley and Sons, 1980.
Ezzinbi, K. Lecture Notes on Distribution Theory, Sobolev Spaces and Elliptic Partial Differential Equation African University of Science and Technology, Abuja, 2018.
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