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**Isoperimetric Variational Techniques and Applications**

Epigraph 2

0 Introduction and Motivations 8

1 Preliminaries:

Notations, Elementary notions and Important facts. 1

1.1 Banach Spaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1

1.2 Hilbert Spaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4

1.3 Differential Calculus in Banach spaces . . . . . . . . . . . . . . 6

1.4 Sobolev spaces and Embedding Theorems . . . . . . . . . . . 9

1.5 Basic notions of Convex analysis . . . . . . . . . . . . . . . . . 13

2 Minimization and Variational methods 18

3 Existence Results of Periodic Solutions of some Dynamical Systems.

28

Bibliography 49

**CHAPTER ONE**

**Preliminaries:**

Notations, Elementary notions and Important facts.

**1.1 Banach Spaces**

Definition 1.1.1 Let X be a real linear space, and k:kX a norm on X and dX the corresponding metric dened by dX(x; y) = kx ykX 8x; y 2 X: The normed linear space (X; k:kX) is a real Banach space if the metric space (X; dX) is complete, i.e., if any Cauchy sequence of elements of space (X; k:kX) converges in (X; k:kX). That is, every sequence satisfying the following Cauchy criterion:

8″ > 0; 9n0 2 N : p; q n0 ) dX(xp; xq) ” converges in X:

Definition 1.1.2 Given any vector space V over a eld F ( where F = R or C), the topological dual space (or simply) dual space of V is the linear space of all bounded linear functionals. We shall denote it by V :

V := f’ : ‘ : V ! F; ‘ linear and bounded g

Remark 1.1.1

1)The topological dual space of V is sometimes denoted V 0:

2 Banach Spaces

2)The dual space V has a canonical norm dened by

kfkV = sup

x2V;kxk6=0

jf(x)j

kxk

; 8f 2 V :

3)The dual of every real normed linear space, endowed with its canonical norm is a Banach space.

In order to dene other useful topologies on dual spaces, we recall the following

**Definition 1.1.3 (Initial topology)**

Let X be a nonempty set, fYigi2I a family of topological spaces (where I is an arbitrary index set) and i : X ! Y ; i 2 I; a family of maps.

The smallest topology on X such that the maps i; i 2 I are continuous is called the initial topology.

Next, we dene the weak topology of a normed vector space X and the weak star topology of its dual space X which are special initial topologies.

Definition 1.1.4 (weak topology)

Let X be a real normed linear space, and let us associate to each f 2 X the map f : X ! R given by f (x) = f(x) 8x 2 X:

The weak topology on X is the smallest topology on X for which all the f are continuous.

We write ! topology for the weak topology.

Definition 1.1.5 (weak star topology)

Let X be a real normed linear space and X its dual. Let us associate to each

x 2 X the map

x : X ! R

given by

x(f) = f(x) 8f 2 X:

The weak star topology on X is the smallest topology on X for which all the

x are continuous.

We write ! topology for the weak star topology.

3 Banach Spaces

Proposition 1.1.6 Let X be a real normed linear space and X its dual space.

Then, there exists on X three standard topologies, the strong topology given by the canonical norm k:kX on X; the weak topology (! topology) and the weak star topology ! topology such that :

(X; !) ,! (X; !) ,! (X; k:kX ) :

The following part of this section is devoted to revive spaces.

For any normed real linear space X; the space X of all bounded linear functionals on X is a real Banach space and as a linear space, it has its own corresponding dual space which we denote by (X) or simply by X and often refer to as the the second conjugate of X or double dual or the bi-dual of X:

There exists a natural mapping J : X ! X dened , for each x 2 X by

J(x) = x

where

x : X ! R

is given by

x(f) = f(x)

for each f 2 X:

Thus

hJ(x); fi f(x) for each f 2 X:

J is linear and kJxk = kxk for all x 2 X; (i.e.) J is an isometry embedding .

In general, the map J needs not to be onto. Since an isometry is injective, we always identify X to a subspace of X:

The mapping J is called canonical embedding. This leads to the following definition.

Definition 1.1.7 Let X be a real Banach space and let J be the canonical embedding of X into X: If J is onto, then X is said to be reexive. Thus, a reexive real Banach space is one for which the canonical embedding is onto.

We now state the following important theorem.

Theorem 1.1.8 (Eberlein-Smul’yan theorem)

A real Banach space X is reexive if and only if every ( norm ) bounded sequence

in X has a subsequence which converges weakly to an element of X:

3

4 Hilbert spaces

1.2 Hilbert Spaces

Definition 1.2.1

A map : E E ! C is sesqui linear if:

1) (x + y; z + w) = (x; z) + (x;w) + (y; z) + (y;w)

2) (ax; by) = ab(x; y) where the bar indicates the complex conjugation

for all x; y; z;w 2 E and all a; b 2 C:

A Hermitian form is a sesqui linear form : E E ! C such that

3) (x; y) = (y; x) ;

A positive Hermitian form is a Hermitian form such that

4) (x; x) 0 for all x 2 E ;

A definite Hermitian form is a Hermitian form such that

5) (x; x) = 0 =) x = 0 :

An inner product on E is a positive definite Hermitian form and will be denoted h: ; :i := (: ; :). The pair (E; h: ; :i) is called an inner product space.

We shall simply write E for the inner product space (E; h: ; : i) when the inner product h: ; : i is known.

In the case where we are using more than one inner product spaces, specification

will be made by writing h: ; :iE when talking about the inner product space

(E; h: ; :i):

Definition 1.2.2 Two vectors x and y in an inner product space E are said to

be orthogonal and we write x ? y if hx; yi = 0: For a subset F of E; then we

write x ? F if x ? y for every y 2 F:

Proposition 1.2.3 Let E be an inner product space and x; y 2 E:

Then

jhx; yij2 hx; xi:hy; yi :

4

5 Hilbert spaces

For an inner product space (E; h: ; :i); the function k:kE : E ! R dened by

kxkE =

p

hx; xiE

is a norm on E.

Thus, (E; k:kE) is a normed vector space, hence a metric space endowed with the distance dE : E E ! R dened by dE(x; y) = kx ykE :

Definition 1.2.4 (Hilbert Space)

An inner product space (E; h: ; :i) is called a Hilbert space if the metric space (E; dE) is complete.

Remark 1.2.1

1)Hilbert spaces are thus a special class of Banach spaces.

2)Every nite dimension inner product space is complete and simply called

Euclidian Space.

Proposition 1.2.5

Let H be a Hilbert space. Then, for all u 2 H; Tu(v) := hu; vi denes a bounded linear functional, i.e. Tu 2 H. Furthermore kukH = kTukH:

Theorem 1.2.6 (Riesz Representation theorem)

Let H be a Hilbert space and let f be a bounded linear functional on H: Then,

(i) There exists a unique vector y0 2 H such that f(x) = hx; y0i for each x 2 H;

(ii) Moreover, kfk = ky0k:

Remark 1.2.2 The map T : H ! H dened by T(u) = Tu is linear,(antilinear in the complex case) and isometric. Therefore the canonical embedding is an isometry showing that any Hilbert space is reexive .

At the end of this part, we state this important proposition which is just a corollary of Eberlein-Smul’yan theorem.

Proposition 1.2.7 Let H be a Hilbert space, then any bounded sequence in H has a subsequence which converges weakly to an element of H:

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