## TABLE OF CONTENTS

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 Dierential 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

7

## CHAPTER ONE

Preliminaries:

Notations, Elementary notions and Important facts.

1.1 Banach Spaces

Denition 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:

Denition 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:

1

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

Denition 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 toplogy on X such that the maps i; i 2 I are continous 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.

Denition 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

continous.

We write ! ? topology for the weak topology.

Denition 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 continous.

We write ! ? topology for the weak star topology.

2

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 reexive 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 bidual 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

denition.

Denition 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

Denition 1.2.1

A map : E E ?! C is sesquilinear 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 sesquilinear 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 denite Hermitian form is a Hermitian form such that

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

An inner product on E is a positive denite 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, specication

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

(E; h: ; :i):

Denition 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 :

Denition 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:

5

6

Dierential Calculus

in Banach spaces

1.3 Dierential Calculus in Banach spaces

In this section, we dene the derivative of a map dened between real Banach

spaces.

Denition 1.3.1 ( Directional Dierentiability)

Let f be a function dened from a real linear space X into a real normed linear

space Y and let x0 2 X and v 2 Xnf0g:

The function f is said to be dierentiable at x0 in the direction v if the function

t 7?! f(x0 + tv) is dierentiable at t = 0: i.e.

t 7?!

f(x0 + tv) ? f(x)

t

; t 6= 0;

has a limit in Y when t tends to 0: This limit, when it exists is denoted f0(x0; v)

or @f

@v (x0):

Denition 1.3.2 ( Gâteaux Dierentiability)

A function f dened from a real linear space X into a real normed linear space

Y is Gâteaux Dierentiable at a point x0 2 X if :

1) f is dierentiable at x0 in every direction v 2 Xnf0g and

2) there exists a bounded linear map A : X ?! Y such that f0(x0; v) = A(v); in

other words, the map

v 7?! f0(x0; v)

is a bounded linear map from X into Y:

In this case the map f0(x0; ? is called the Gâteaux dierential of f at x0 and is

denoted by DGf(x0; ? or f0G

(x0):

Denition 1.3.3 (Fréchet Dierentiability)

A map f : U X ?! Y whose domain U is an open set of a real Banach

space X and whose range is a real Banach space Y is ( Fréchet ) dierentiable

at x 2 U if there is a bounded linear map A : X ?! Y such that

lim

kuk?!0

kf(x + u) ? f(x) ? Auk

kuk

= 0;

or equivalently

f(x + u) ? f(x) ? Au = o(kuk) :

Proposition 1.3.4 If f : U X ?! Y is Fréchet Dierentiable, then f is

Gâteaux Dierentiable.

6

7

Dierential Calculus

in Banach spaces

Proof. Indeed by taking u = tv; in the denition of Fréchet Dierentiability we

have

f(x + tv) ? f(x)

t

=

A(v) +

o(ktvk)

ktvk

by the Fréchet Dierentiability. And since as t ?! 0; u ?! 0; so

lim

t?!0

f(x + tv) ? f(x)

t

= A(v)

and we are done.

Proposition 1.3.5 Let X be a real Banach space and Y be a real normed linear

space.Then

1) The set of Gâteaux dierentiable mappings from X into Y is a linear subspace

of the linear space of all the mappings dened from X into Y space is contained

in B(X; Y );

2) The set of Fréchet Dierentiable mappings from X into Y is also a subspace

of B(X; Y ):

Theorem 1.3.6 (Mean Value Theorem in Banach Spaces) Let X and Y be Banach

spaces, U X be open and let f : U ! Y be Gâteaux dierentiable. Then

for all x1 ; x2 2 X, we have

kf(x1) ? f(x2)k sup

t2[0;1]

kDGf(x1 + t(x2 ? x1)k kx1 ? x2k

provided that sup

t2[0;1]

kDGf(x1 + t(x2 ? x1)k is nite.

Proof. Suppose that the assumptions of Theorem 1.3.6 hold. Let g 2 Y (the

dual of Y ) such that jjgjj 1. Then the real-valued function ‘ : [0; 1] ?! R

dened by

‘(t) = g f(x1 + th) where h = x2 ? x1

is dierentiable on [0; 1] in the usual sense. Moreover we see that

‘0(t) = g?

DGf(x1 + th)(h)

; 8 t 2 (0; 1) :

It follows from the classical mean valued theorem that

j'(1) ? ‘(0)j sup

0

7

8

Dierential Calculus

in Banach spaces

that is

kg f(x1) ? g f(x2)k sup

0

Moreover for all t 2 (0; 1), we have

j’0(t)j =

g?

DGf(x1 + th)(h)

jjgjj kDGf(x1 + th)k khk

kDGf(x1 + th)k khk

And so

kg?

f(x1)?f(x2)

k = kgof(x1)?gf(x2)k

sup

0

khk :

But it is well known as a consequence of the Hahn-Banach theorem that

kyk = supfu(y) ; u 2 Y ; kuk 1 g:

Therefore we nally have

kf(x1) ? f(x2)k sup

t2[0;1]

kDGf(x1 + t(x2 ? x1)k kx1 ? x2k :

Remark 1.3.1 : The intereted reader is refered to [8] for another approach of

the proof.

Sucient conditions for the Fréchet Dierentiability is given by the following

Theorem 1.3.7 Suppose that f : U X ?! Y is a Gâteaux Dierentiable

function dened from an open subset of a real Banach space X into a real Banach

space Y: If the Gâteaux derivative f0G

: U X ?! B(X; Y ) is continous at

x 2 U; then f is Fréchet Dierentiable at x and f0(x) = f0G

(x):

Proof. Let x 2 U: Since U is open, there exixts > 0 such that B(x; ) U:

Now for h 2 B(x; ); we dene

r(h) = f(x + h) ? f(x) ? f0

G(x)h: (1.3.1)

8

9

Sobolev spaces and

Embeddings Theorems

The Gâteaux Dierentiability of f at x implies that r is also Gâteaux Dierentiable,

and

r0

G(h) = f0

G(x + h) ? f0

G(x):

Applying theorem 1.3.6 on the segment line connecting 0 and h; we have that

kr(h)k M(h)khk;

where

M(h) = sup

0t1

kr0

G(th)k:

The continuity of the Gâteaux Dierential of f at x implies that M(h) ! 0 as

h ! 0; so r(h) = o(h): Relation 1.3.1 assures that f is Fréchet Dierentiable at

x; and so f0(x) = f0G

(x):

1.4 Sobolev spaces and Embedding Theorems

We recall the following notations and basic results from Distridutions Theory.

Let

RN be an open subset of RN:

A multi-index is a vector (1 ; ; N ) 2 NN: The length of is

jj = 1 + + N :

Let u 2 L1

loc(

); where L1

loc(

) is the set of functions which are integrable on

every compact subset of

: If is a multi-index, we set

D :=

Djj

@x1

1 @x

N

N

We also recall that we denote by D(

) the set of C1 ?functions dened on

with compact support in

:

Denition 1.4.1

We say that the function v is the -th weak partial derivative of u if :

1) v 2 L1

loc(

);

2) v = Du in the sens of distribution , i.e.

Z

u(x)D (x)dx = (?1)jj

Z

v(x) (x)dx; 8 2 D(

):

9

10

Sobolev spaces and

Embeddings Theorems

Denition 1.4.2 Let f; g 2 L1

loc(RN): We dene the convolution product f g

of f and g by

(f g)(x) =

Z

RN

f(x ? y)g(y)dy

Theorem 1.4.3 Let (n)n be a sequence of functions such that :

n 2 D(RN); supp n = B(0;

1

n

);

Z

RN

n(x)dx = 1; n 0 on RN:

(Such a sequence of smooth functions is called Friedrich mollier ).

If f 2 L1

loc(RN) then the convolution product

f n(x) =

Z

RN

f(x ? y)n(y)dy

exists for each x 2 RN:

Moreover

1. f n 2 C1(RN);

2. If K is a compact set of points of continuity of f; then f n ?! f uniformly

on K as n ?! 1:

Proof. Since supp n = B(0; 1

n); ( which is compact ), and using f 2 L1l

oc(RN)

we get

jfn(x)j = j(fn)(x)j =

Z

B(0; 1

n )

f(x ? y)n(y)dy

=

Z

B(0; 1

n )

jf(x?y)jn(y)dy < 1:

Further, since

supp

@n

@xi

B(0;

1

n

) and

@

@xi

[f(y)n(x ? y)] =

@n(x ? y)

@xi

f(y);

we get

@n(x ? y)

@xi

f(y)

Mnjf(y)j

B(0; 1n

)

and using a corollary of Lebesgue dominated convergence theorem, we have :

@

@xi

Z

RN

f(y)n(x?y)dy =

Z

RN

@n(x ? y)

@xi

f(y)dy =

Z

RN

@n(y)

@xi

f(x?y)dy = f

@n

@xi

:

10

11

Sobolev spaces and

Embeddings Theorems

Let us prove now that fn ! f as n ! 1; uniformly on compact subsets of

RN:

Let K be a compact set of points of continuity of fn: So, for any > 0; there

exists > 0; such that for x; z 2 K

kx ? zk < =) kf(x) ? f(z)k < :

Now,

fn(x) ? f(x) =

Z

B(0; 1

n )

(f(x ? y) ? f(x))n(y)dy;

because

f(x) = f(x):1 = f(x)

Z

RN

n(y)dy =

Z

RN

f(x)n(y)dy and

Z

RN

n(y)dy = 1;

Hence, for n n0 with n0 =

1

+ 1,

jfn(x) ? f(x)j

Z

B(0; 1

n )

jf(x ? y) ? f(x)jn(y)dy

Z

RN

n(y)dy = for each x 2 K:

Indeed

n n0 =) n

1

=)

1

n

so that

k(x ? y) ? xk = kyk

1

n

and the result follows from the uniform continuity of fn:

We then conclude that f n ?! f uniformly on each compact.

Denition 1.4.4 Let 1 q +1; m 2 N: The Sobolev space Wm;p(

) is

dened by

Wm;p(

) = fu 2 Lp(

); jDu 2 Lp(

) for all jj mg:

Clearly, Wm;p(

) is a real vector space .

The case p = 2 will play a special role. The Sobolev spaces Wm;2(

) are denoted

by Hm(

); i.e.

Hm(

) := Wm;2(

):

11

12

Sobolev spaces and

Embeddings Theorems

The spaces Hm(

) have a natural inner-product dened by

hu; viHm =

jjm

Z

Du(x)Dv(x)dx; 8u; v 2 Hm(

)

and are Hilbert spaces with the inner-product dened above. We will be more

interested in our work by H1(

):

Concerning Sobolev spaces, we will give here two important results, Rellich-

Kondrachov compact embedding theorem (which is crucial in regularity

analysis) and the Poincaré Inequality .

Theorem 1.4.5 (Rellich-Kondrachov)

Let

be a C1?bounded open subset of RN; 1 p < 1 and p := Np

N?p :

The followings embeddings are compact:

a. If 1 p < N then W1;p(

) Lq(

); 8q 2 [1; p[;

b. If p = N then W1;p(

) Lq(

); 8q 2 [1;1[;

c. If p > N then W1;p(

) C(

):

We have D(

) Wm;p(

) 8m 2 N; 8p 1; and we dene Wm;p

0 (

) := D(

).

Proposition 1.4.6 (Poincaré Inequality)

Let 1 p < 1 and

a bounded open subset of RN: Then there exists a constant

C = C(p;

) such that

kuk

L

p

(

)

CkOuk

L

p

(

)

; 8u 2 W1;p

0 (

)

If

is connected and satises a C1 boundary condition, then there exists a constant

C = C(p;

) such that

ku ? uk

L

p

(

)

CkOuk

L

p

(

)

; 8u 2 W1;p(

)

where

u =

1

j

j

Z

u(x)dx:

12

13

Basic notions of

Convex analysis

1.5 Basic notions of Convex analysis

Denition 1.5.1 Let X be a real normed vector space, x0 2 X and

f : X ?! R = R [ f?1;+1g an extended real-valued function. One says

that f is lower semicontinuous (lsc) at x0 when for any real number r such

that r < f(x0); there exists some neighborhood V of x0 such that for all x 2 V;

r < f(x):

We next connect the lower semicontinuity to some geometric concept. For an

extended real-valued function f : X ?! R, we dene its epigraph epi f by

epi f := f(x; r) 2 X R : f(x) rg:

We also introduce the concept of lower level set for r 2 R by ff(:) rg

where for r 2 R,

ff(:) rg := fx 2 X : f(x) rg:

We therefore give the following characterisation ;

Theorem 1.5.2 Let X be a real normed vector space and f : X ?! R an

extended real-valued function. The following assertions are equivalent

a) f is lower semicontinous (lsc) ;

b) The epigraph epi f of f is closed in X R ;

c)For any r 2 R; the lower level set ff(:) rg is closed in X:

Denition 1.5.3 Let C be a nonempty subset of a real normed vector space X:

One says that the set C is convex provided that for x; y 2 C; and 2 [0; 1]; one

has x + (1 ? )y 2 C:

Through the epigraph of an extended real-valued function over a real vector

space, one can dene the concept of convex function as follow:

Denition 1.5.4 Let f : X ?! R an extended real valued function. Ones says

that the function f is convex provided that its epigraph is a convex set in XR:

13

14

Basic notions of

Convex analysis

We also give the following important results.

Proposition 1.5.5 Let X be a real normed vector space. If f : X ?! R is lsc

at x 2 X and fxng is a sequence in X which converges (strongly) to x then ,

lim inf

n!1

f(xn) f(x):

Proposition 1.5.6

Let f : X ?! R be any map.

Then , f is convex and lsc () f is convex and weakly lsc.

And we obtain the following corollary

Corollary 1.5.7 Let f : X ?! R be a convex and weakly lsc mapping. Suppose

fxng is a sequence in X which converges weakly to x: Then,

lim inf

n!1

f(xn) f(x):

Denition 1.5.8 Let X be a real normed vector space and C a nonempty convex

subset of X: A function f : C ?! R [ f+1g is said to be convex relative to C,

provided for all 2]0; 1[; x; y 2 C

f(x + (1 ? )y) f(x) + (1 ? )f(y);

and f is said to be strictly convex relative to C if for x; y 2 C with x 6= y and

f(x); f(y) nite, we have

f(x + (1 ? )y) < f(x) + (1 ? )f(y):

Lemma 1.5.9 (Slope inequality for convex functions)

Let I be an unterval of R and h : I ?! R [ f+1g be a proper convex function.

Let r1; r2; r3 2 I such that r1 < r2 < r3 and h(r1) and h(r2) are nite. Then

h(r2) ? h(r1)

r2 ? r1

h(r3) ? h(r1)

r3 ? r1

h(r3) ? h(r2)

r3 ? r2

:

Furthermore, these inequalities for all such r1; r2; r3 2 I characterizes the convexity

of f relative to I:

If we have

h(r2) ? h(r1)

r2 ? r1

<

h(r3) ? h(r1)

r3 ? r1

<

h(r3) ? h(r2)

r3 ? r2

for all r1; r2; r3 2 I such that r1 < r2 < r3 and h(r1); h(r2) and h(r3) are nite,

we obtain a characterisation of the strict convexity of f relative to C:

14

15

Basic notions of

Convex analysis

Through the above lemma, we can characterize the convexity of dierentiable

functions of one real variable as follows.

Proposition 1.5.10 Let I be an open interval of R and h : I ?! R be a realvalued

dierentiable function on I: The following assertions are equivalent :

(a) h is convex on I;

(b) the derivative function h0 is nondecreasing on I;

(c) h0(r)(s ? r) h(s) ? h(r) for all r; s 2 I:

Similarly, the following are equivalent

(a’) his strictly convex on I;

(b’) the derivative function h0 is increasing on I;

(c’) h0(r)(s ? r) < h(s) ? h(r) for all r; s 2 I with r 6= s:

Proof. (a) ) (b) Let r < t in I: According to the above lemma, we have

h0(r) = lim

s#r

h(s) ? h(r)

s ? r

h(t) ? h(r)

t ? r

lim

s”t

h(t) ? h(s)

t ? s

= lim

s”t

h(s) ? h(t)

s ? t

= h0(t);

which ensures the nondecreasing property of the derivative h0 on I:

(b) ) (c) Fix r 2 I and set ‘(s) := h(s) ? h(r) ? h0(r)(s ? r) for all s 2 I:

The function ‘ is dierentiable on I and ‘0(s) = h0(s)?h0(r): By the assumption

(b), taking s 2 I; we have that ‘0(s) 0 if s r and ‘0(s) 0 if s r: We

then deduce that ‘(s) ‘(r) = 0 for all s 2 I and we are done.

(c) ) (a) For s xed in (c), we

h(s) sup

r2I

[h0(r)(s ? r) + h(r)] h(s)

that is

h(s) = sup

r2I

[h0(r)(s ? r) + h(r)]

Further, setting H(s) = [h0(r)(s ? r) + h(r)] ; for s1; s2 2 I and 2 [0; 1]; we

have that

H(s1 + (1 ? )s2) = h0(r)(s1 + (1 ? )s2 ? r) + h(r)

= h0(r)(s1 + (1 ? )s2 ? r + (1 ? )r) + h(r) + (1 ? )h(r)

= [h0(r)(s1 ? r) + h(r)] + (1 ? ) [h0(r)(s2 ? r) + h(r)]

= H(s1) + (1 ? )H(s2)

15

16

Basic notions of

Convex analysis

that is, H is convex and hence, h is convex on I as the pointwise supremum of

a family of convex functions on I:

The case of the strict convexity of h follows the same arguments.

Proposition 1.5.11 Let I be an open interval of R and h : I ?! R be a realvalued

dierentiable function on I:

If the function h is twice dierentiable on I; then h is convex on I if and only if

h00(r) 0 for all r 2 I:

Similarly if h is twice dierentiable on I and h00(r) > 0 for all r 2 I; then h is

strictly convex on I: The converse does not hold, that is, the strict convexity of

a twice dierentiable function h on I does not entail the positivity of h00 on I:

Proof. Since h is twice derivable, we have

h00(r) 0 8r 2 I () h0 is nondecreasing () h is convex

and we are done.

The case of the strict convexity of h follows the same arguments.

We will consider now the more genaral case of dierentiable functions on an

open convex set of a normed vector space .

Theorem 1.5.12 Let U be an open set of a real normed space (X; k:k) and

f : U ?! R be a function which is (Fréchet) dierentiable on U: Then the following

assertions are equivalent:

(a) f is convex torelative U;

(b) hf0(y) ? f0(x); y ? xi 0 for all x; y 2 U;

(c) hf0(x); y ? xi f(y) ? f(x) for all x; y 2 U:

Similarly, the following are equivalent :

(a’) f is strictly convex relative to U;

(b’) hf0(y) ? f0(x); y ? xi > 0 for all x; y 2 U with x 6= y;

(c’) hf0(x); y ? xi < f(y) ? f(x) for all x; y 2 U with x 6= y.

Proof. For xed x; y 2 U with x 6= y; consider the open interval

I := fs 2 R : x + s(y ? x) 2 Ug

and set h(s) := f(x + s(y ? x) for all s 2 I: Observing that 0 2 I and 1 2 I

with h(0) = f(x) and h(1) = f(y): we have

f is convex relative to U if and only if the function h is convex relative to I:

16

17

Basic notions of

Convex analysis

Indeed, since 0 2 I and 1 2 I and I is an interval, then [0; 1] U;

so for all 2 [0; 1] U

f(y + (1 ? )x) = f(x + (y ? x))

= h()

= h(:1 + (1 ? ):0)

h(1) + (1 ? )h(0)

= f(y) + (1 ? )f(x)

We then apply proposition 1.5.10.

Theorem 1.5.13 Let U be an open set of a real normed space (X; k:k) and

f : U ?! R be a function which is (Fréchet) dierentiable on U:

If f is twice dierentiable on U; f is convex relative to U if and only if for each

x 2 U the bilinear form associated with f00(x) is positive semidenite, i.e.,

hf00(x):v; vi 0 for all v 2 X:

Similarly assuming the twice dierentiabilty of f on U; a sucient (but not necessary)

condition for the strict convexity of f on U is for each x 2 U the positive

deniteness of f00(x); i.e., hf00(x):v; vi > 0 for all v 2 X with v 6= 0X

Proof. It follows the same arguments as in the proof of the above theorem, but

in this case, we apply proposition 1.5.11

17

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