Variational Inequality in Hilbert Spaces and their Applications

 – 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.

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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

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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.

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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.