Variational Inequality in Hilbert Spaces and Their Applications

Variational Inequality in Hilbert Spaces and Their Applications

Chapter One

PREAMBLE 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 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 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. 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.

CHAPTER TWO

Variational Inequalities in RN

Given K ⊂ RN and F : K −→ RN , a continuous mapping. Then, the Variational inequalities(VI) is the problem of finding a point u ∈ K such that

(F (u), v − u) ≥ 0, v ∈ K. (2.0.1)

Variational inequalities(VI) are closely related with many general problems of non- linear Analysis such as complementary, fixed point and optimization problem. The simplest examples of variational inequalites is the problem of solving a system of equation. Here, we intend to discuss variational inequalities in RN , fixed point and some elementary problem that are associated to variational inequality. In particular,we discuss the connection between variational inequalities and convex funtions

Basic Theorems and Definition about Fixed point

Definition 2.1.1 Let S be a metric space with metric d. A mapping F : S −→ S

is said to be a strictly contraction map if there exists α ∈ [0, 1[

d(F (x), F (y)) ≤ αd(x, y) , for all x, y ∈ S.

Remark 2.1.2 if α = 1, then F is nonexpansive.

Theorem 2.1.3 [3] (Banach’s fixed point Theorem) Let S be a complete met- ric space and let F : S S be a strict contraction mapping. Then, there exist a unique fixed point of F.

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Theorem 2.1.4 [3] (Brouwer’s fixed point Theorem) Let F be a continuous mapping from a closed ball G RN into itself. Then, F admit at least one fixed point in G.

Theorem 2.1.5 [3] (Schauder’s fixed point Theorem) Let G be a compact convex subset of RN and F be a continuous mapping from G into itself. Then, F admits a fixed point in G.

First Theorem about variational inequalities

Theorem 2.2.1 [8] Let K be compact and convex set in RN and let F : K −→ RN

be continuous. Then, there exists x ∈ K such that

(F (x), y − x) ≥ 0, for all y ∈ K.

Proof. Let Π : RN −→ RN be the identication and (., .) be the scalar product on RN . Let P_K(I − ΠF ) : K −→ K be continuous, where Ix = x. Then by Schauder fixed point Theorem, P_K(I ΠF ) admits a fixed point. Thus there exists x K such that

P_K(I − ΠF )x = x.

By the characterisation of projection Theorem we obtain that

(x, y − x) ≥ ((I − ΠF )x, y − x), for allx, y ∈ k

= (x − ΠF (x), y − x)

= (x, y − x) − Π(F (x), y − x), for allx, y ∈ K.

Then, namely

Π(F (x), y − x) ≥ (x, y − x) − (x, y − x) = 0, for all x, y ∈ K,

(F (x), y − x) ≥ 0, for all y ∈ K.

Therefore, there exists x ∈ K such that

(F (x), y − x) ≥ 0, for all y ∈ K.

Applications

Variational Inequality theory provides us with a tool for: formulating a variety of equilibrium problems; qualitatively analysing the problem in terms of existence and uniquness of solutions and stability. Many of the applications explored to date that have been formulated, studied and solved as variational inequality problems are in fact, network problems. Indeed, many mathematical problems can be formulated as variational inequality problems and several examples applicable to equilibrium analysis follows thus

Systems Equations

Many classical economic equilibrium problems have been formulated as systems of equation, since market clearing conditions necessarily equate the total supply with the total demand. In terms of variational inequality problem, the formulation of a system of equation is as follow.

Proposition 2.2.2 [9] Let F : RN RN be a mapping. Then for any x RN

we have that if and only if F (x) = 0.

CHAPTER THREE

Variational Inequality in Hilbert Spaces

Here, we study variational inequalities in Hilbert space. Some basic theorems and proofs are presented in this chapter. This will be used in obtaining our main exis- tence and uniqueness theorem. The study of variational inequalites started being considered around nintheenth century. Many differential equations that arise from different kind of application are solved by a very simple calculation. This approach does not give the existence and uniqueness of classical and weak solutions. Hence, the concept of Variational approach is paramount.

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

Problem

a(u, v) = (Au, v).

Let H be a real Hilbert space and f ∈ Hj . Let K ⊂ H be closed and convex. Find

u ∈ K such that

a(u, v − u) ≥ (f, v − u), for all v ∈ K. (3.1.1)

Theorem 3.1.1 [2](Stampacchia Theorem) Let a(u, v) be a continuous coercive bilinear form on H. Let K H be a nonempty closed and convex with f Hj . Then there exists a unique solution to problem (3.1.1).

Moreover, if u₁, u₂ K are solutions to problem (3.1.1) corresponding to f₁, f₂ Hj , then

CHAPTER FOUR

CONCLUSION

In this work, we studied variational inequalities in Hilbert space. Some basic theo- rems and proofs were presented. We studied and obtained existence and uniqueness theorems for variational inequalities. Many differential equations that arise from different kind of application were solved by a very simple calculation. We discov- ered that this approach does not give the existence and uniqueness of classical and weak solutions. Hence, the concept of Variational approach is paramount. we es- tablished the existence and uniqueness of solutions of variational inequalities. This was achieved through the use of Stampacchia theorem and Lax-Milgram theorem. And its applications.

We Considered the following Problem

−u^jj + u = f on I = (0, 1), u(0) = α, u(1) = β.

with α, β ∈ R given and f ∈ L₂(I) given.

(4.0.1

And obtained its solution using variational approach via Stammpacchia Theorem.

We also looked at its application in Rn and more generally in Hilbert Space. We also considered the problem of the form

−∆u + u = f on Ω, f ∈ L₂(Ω) u = g on Γ.

(4.0.2)

We obtained its solution using variational approach by applying Stammpacchia theorem

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