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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<t<1
j’0(t)j ;
7
8
Dierential Calculus
in Banach spaces
that is
kg f(x1) 􀀀 g f(x2)k sup
0<t<1
j’0(t)j :
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<t<1
kDGf(x1 + th)k

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
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