Playing around resonance

Birkhäuser Advanced Texts
Basler Lehrbücher

Alessandro Fonda

Playing
Around
Resonance
An Invitation to the Search of
Periodic Solutions for Second Order
Ordinary Differential Equations



Birkhäuser Advanced Texts Basler Lehrbücher

Series editors
Steven G. Krantz, Washington University, St. Louis, USA
Shrawan Kumar, University of North Carolina at Chapel Hill, Chapel Hill, USA
Jan Nekováˇr, Université Pierre et Marie Curie, Paris, France

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Alessandro Fonda

Playing Around Resonance
An Invitation to the Search of Periodic
Solutions for Second Order Ordinary
Differential Equations

Alessandro Fonda
Dipartimento di Matematica e Geoscienze
UniversitJa degli Studi di Trieste
Trieste, Italy

ISSN 1019-6242
ISSN 2296-4894 (electronic)
BirkhRauser Advanced Texts Basler LehrbRucher
ISBN 978-3-319-47089-4
ISBN 978-3-319-47090-0 (eBook)
DOI 10.1007/978-3-319-47090-0
Library of Congress Control Number: 2016958441
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To Rodica


Contents

1

Preliminaries on Hilbert Spaces . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . .
1.1 The Hilbert Space Structure . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . .
1.2 Some Examples of Hilbert Spaces. . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . .
1.3 Fundamental Properties . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . .
1.4 Subspaces .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . .
1.5 Orthogonal Subspaces.. . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . .
1.6 The Orthogonal Projection .. . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . .
1.7 Basis in a Hilbert Space .. . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . .
1.8 Linear Functions.. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . .
1.9 Weak Convergence . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . .
1.10 Concluding Remarks .. . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . .

1
1
3
6
8
10
13
14
19
25

28

2

Operators in Hilbert Spaces . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . .
2.1 First Definitions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . .
2.2 The Adjoint Operator . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . .
2.3 Resolvent Set and Spectrum . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . .
2.4 Selfadjoint Operators.. . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . .
2.5 Operators in Real Hilbert Spaces .. . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . .
2.6 Concluding Remarks .. . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . .

31
31
33
36
39
42
45

3

The Semilinear Problem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . .
3.1 The Main Problem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . .
3.2 Properties of the Differential Operator . . . . . . . . . .. . . . . . . . . . . . . . . . . . . .
3.3 The Linear Equation . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . .
3.3.1 The Case > 0 . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . .
3.3.2 The Case < 0 . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . .
3.3.3 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . .
3.4 The Contraction Theorem.. . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . .

3.5 Nonresonance: Existence and Uniqueness . . . . . .. . . . . . . . . . . . . . . . . . . .
3.6 Equations in Hilbert Spaces. . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . .
3.7 Concluding Remarks .. . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . .

47
47
49
52
53
56
57
57
59
61
68

vii


viii

Contents

4

The Topological Degree . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . .
4.1 The Brouwer Degree . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . .
4.2 Further Considerations on the Brouwer Degree . . . . . . . . . . . . . . . . . . . .
4.3 The Leray–Schauder Degree.. . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . .
4.4 Concluding Remarks .. . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . .

71
71
87
90
98

5

Nonresonance and Topological Degree .. . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . .
5.1 The Use of Schauder Theorem . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . .
5.2 Lower and Upper Solutions .. . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . .
5.3 The Continuation Principle . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . .
5.4 Asymmetric Oscillators . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . .
5.5 Nonlinear Nonresonance .. . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . .
5.6 Non-bilateral Conditions .. . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . .
5.7 The Ambrosetti–Prodi Problem . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . .
5.8 Concluding Remarks .. . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . .

101
101
104
108
111
112
119
129
133

6

Playing Around Resonance .. . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . .
6.1 Some Useful Inequalities . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . .
6.2 Resonance at the First Eigenvalue .. . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . .
6.3 Landesman–Lazer: Resonance at Higher Eigenvalues . . . . . . . . . . . . .
6.4 The Lazer–Leach Condition . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . .
6.5 Landesman–Lazer Conditions: The Asymmetric Case . . . . . . . . . . . . .
6.6 Lazer–Leach Conditions for the Asymmetric Oscillator . . . . . . . . . . .
6.7 More Subtle Nonresonance Conditions . . . . . . . . .. . . . . . . . . . . . . . . . . . . .
6.8 Concluding Remarks .. . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . .

137
137
139
141
144
145
149
151
155

7

The Variational Method . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . .
7.1 Definition of the Functional . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . .
7.2 Minimization . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . .
7.3 The Ekeland Principle.. . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . .
7.4 The Search of Saddle Points . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . .
7.5 Concluding Remarks .. . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . .

157
157
161
164
165
171

8

At Resonance, Again . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . .
8.1 Resonance at the First Eigenvalue .. . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . .
8.2 Subharmonic Solutions . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . .
8.3 Ahmad–Lazer–Paul: Resonance at Higher Eigenvalues .. . . . . . . . . . .
8.4 Landesman–Lazer vs Ahmad–Lazer–Paul . . . . . .. . . . . . . . . . . . . . . . . . . .
8.5 Periodic Nonlinearities .. . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . .
8.6 Concluding Remarks .. . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . .

173
174
176
181
184
187
190

9

Lusternik–Schnirelmann Theory . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . .
9.1 The Periodic Problem for Systems . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . .
9.2 An Equivalent Functional .. . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . .

9.3 Some Hints on Differential Equations.. . . . . . . . . .. . . . . . . . . . . . . . . . . . . .
9.4 Lusternik–Schnirelmann Category . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . .
9.5 Multiplicity of Critical Points . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . .

193
193
194
197
199
201


Contents

ix

9.6
9.7

Relative Category . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 206
Concluding Remarks .. . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 211

10 The Poincaré–Birkhoff Theorem . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . .
10.1 The Multiplicity Result . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . .
10.2 A Modified System .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . .
10.3 The Variational Setting .. . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . .
10.4 Finite Dimensional Reduction .. . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . .
10.5 Periodic Solutions of the Original System . . . . . .. . . . . . . . . . . . . . . . . . . .
10.6 The Poincaré–Birkhoff Theorem on an Annulus . . . . . . . . . . . . . . . . . . .
10.7 Concluding Remarks .. . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . .

213
214
215
218
221
223
225
227

11 A Myriad of Periodic Solutions . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . .
11.1 Equations Depending on a Parameter . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . .
11.2 Superlinear Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . .
11.3 Forced Superlinear Equations . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . .
11.4 Concluding Remarks .. . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . .

231
231
243
250
253

A

Spaces of Continuous Functions . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . .
A.1 Uniform Convergence .. . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . .
A.2 Continuous Functions with Compact Domains .. . . . . . . . . . . . . . . . . . . .
A.3 Uniformly Continuous Functions.. . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . .
A.4 The Ascoli–Arzelà Theorem .. . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . .
A.5 The Stone–Weierstrass Theorem . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . .

255
255
257
258
259
261

B

Differential Calculus in Normed Spaces . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . .
B.1 The Fréchet Differential.. . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . .
B.2 Some Computational Rules . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . .
B.3 The Mean Value Theorem . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . .
B.4 The Gateaux Differential.. . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . .
B.5 Partial Differentials.. . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . .
B.6 The Implicit Function Theorem . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . .
B.7 Higher Order Differentials .. . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . .

265
265
267
270
272
273
276
282

C

A Brief Account on Differential Forms . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . .
C.1 Preliminary Definitions . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . .
C.2 The External Differential . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . .
C.3 Pull-Back Functions .. . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . .
C.4 Integrating M-Differential Forms Over M-Surfaces .. . . . . . . . . . . . . . .
C.5 Differentiable Manifolds .. . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . .
C.6 Orientation .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . .
C.7 The Stokes–Cartan Theorem.. . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . .

287
287
289
290
291
292
293
294

Bibliography . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 297
Index . . . . . . . . .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 307


Introduction

This book is an introduction to the problem of the existence of solutions to some
type of semilinear boundary value problems. It arises from a series of courses which
I have given to undergraduate and graduate students in the last few years.
The aim of the book is to give the possibility to any good student to reach
a research level in this field, starting from the basic knowledge of mathematical
analysis which is usually acquired before graduation. To this aim, I will develop

some tools which could be used to attack many different boundary value problems,
arising from ordinary or partial differential equations. However, I have chosen
to deal mainly with the periodic problem for a second-order scalar ordinary
differential equation. One reason for this choice is that this apparently simple model
already shows so many different aspects, and can be approached by such different
techniques, that it seems the ideal starting point to the further understanding of
more technical boundary value problems. Another reason comes, of course, from
its intrinsic importance in the applications.
So, I will be concerned with an equation of the type
x00 C g.t; x/ D 0 ;

(1)

where g W R R ! R is a continuous function, which is T-periodic in its first
variable. The main problem will be to find some conditions on the function g which
guarantee the existence of T-periodic solutions of Eq. (1).
More generally, we will deal with the problem
.P/

x00 C g.t; x/ D 0 ;
x.0/ D x.T/ ; x0 .0/ D x0 .T/ ;

where g W Œ0; T R ! R is continuous. Indeed, if g.t; x/ is defined on R R, and
T-periodic in its first variable, it is easy to see that any solution x.t/ of problem (P)
can be extended to the whole R as a T-periodic solution of Eq. (1).

xi


xii

Introduction

What about the word resonance appearing in the title? When does resonance
appear? Can it be precisely identified? Can it be related to the existence or
nonexistence of solutions for problem .P/? Answering these questions is not a
simple task, but I will say a few words, to clarify.
It is easier to explain the idea of resonance in the particular case when g.t; x/ D
g.x/ e.t/, so that Eq. (1) can be written as
x00 C g.x/ D e.t/ :

(2)

This equation can be seen as a model for the motion of a particle when subjected
to a restoring force g.x/ and an external forcing e.t/, which we assume to be Tperiodic. The simplest situation is when g.x/ D x, the linear case, where is a
positive constant. The equation x00 C x D e.t/ then models a linear oscillator, and
it is well known what resonance means in this case: if the frequency of a Fourier
component of the external forcing coincides with the natural frequency of the free
oscillator, then the solutions will grow in amplitude as time goes on, without any
possible bound.1 This is the so-called linear resonance: we notice that, in this case,
there are no periodic solutions.
When the restoring force g.x/ is not linear any more, the situation can be much
more complicated. Even if it is not clear what resonance (or perhaps nonlinear
resonance) would mean in the general case, one can expect that a phenomenon
similar to linear resonance may appear in situations when the function g.x/ gives rise
to free periodic oscillations whose frequencies interfere with those of the external
forcing e.t/. To be more precise, assume, for example, that we are in a case when
all the solutions of the differential equation x00 C g.x/ D 0 are periodic, but not
necessarily of the same period. Then, it is intuitive that, if we want to avoid a
nonlinear resonance phenomenon to appear, the frequencies of e.t/ should not be

approached too much by those of the large amplitude periodic solutions of the free
oscillator. Otherwise, indeed, the solutions of the forced oscillator (2) could enter
into a resonance-like situation, become larger and larger, and, in particular, there
might be no periodic solutions.
So, it seems that the existence of periodic solutions to Eq. (2) can be a way
to stay away from resonance: at least, this could be a starting point. However,
it is known that nonlinear phenomena can be much more intricate: for example,
in some situations, Eq. (2) may have a T-periodic solution, but at the same time
some solutions may be unbounded in the future or in the past. Of course, similar
considerations can be made for Eq. (1), as well.
In this book, I have not tried to give a general definition of resonance, except
for the well-known linear case. On the other hand, I have used this word, or its
counterpart, nonresonance, several times. They are often used to put in evidence the
kind of behavior we expect for the solutions of the differential equation: is there

1

This can be a very undesirable situation in mechanical systems, since large oscillations can lead to
a collapse. On the other hand, it could be used on purpose, to amplify some physical phenomena.


Introduction

xiii

some control in the oscillations, leading to the existence of periodic solutions? Are
we near a resonance-type situation or far away from it? Following the tradition, I
have thus adopted the practical use of the word, trying, however, not to mistreat it
too much.
The methods introduced in the book are developed in full details. Each method

has its own advantages in the applications to the existence or multiplicity of
solutions to problem (P). However, I will not search the greatest generality in
the applications. On the contrary, I will try not to make the computations too
cumbersome, even if the results I have chosen represent some of the most advanced
achievements in the field. Moreover, when necessary, further results and remarks
are added at the end of the chapters, as a guide to the most recent references.
I will now briefly formally describe the contents of the book.
The first two chapters are devoted to an introduction to the theory of linear
operators in Hilbert spaces. Indeed, problem (P) can be transformed into a fixed
point problem in a function space, e.g., the Hilbert space L2 .0; T/. In Chap. 3, the
main properties of the differential operator are analyzed, and the fixed point problem
associated to (P) is attacked by the use of the contraction theorem, assuming some
nonresonance conditions upon the function g.t; x/. The same approach then leads to
the study of more general nonresonance conditions for abstract semilinear equations
in Hilbert spaces.
A fundamental technique which has been extensively used to solve problem (P)
is the topological degree. In Chap. 4, we develop the theory of both the Brouwer
degree and the Leray–Schauder degree which will be needed in the subsequent two
chapters, where they will be applied to problem (P). In Chap. 5, starting with the
use of the Schauder fixed point theorem, we at first introduce the method of lower
and upper solutions. Then, we develop the Leray–Schauder continuation principle,
which will be the main tool to deal with many different kinds of symmetric or
asymmetric-type nonlinearities. Chapter 5 mainly deals with nonresonant situations,
with respect to the spectrum of the differential operator or, more generally, to the
associated Fuˇcík spectrum. Also, a multiplicity result of the Ambrosetti–Prodi type
is presented. In Chap. 6, the more subtle conditions of Landesman–Lazer type are
introduced, which permit a closer approach to resonance. Most of these problems
can be reduced to a situation where the associated topological degree is equal to 1.
At the end of the chapter, a different situation is analyzed, where the degree can also
be an arbitrary negative number.

In Chap. 7, an introduction to variational methods is provided. Problem (P) is
shown to be equivalent to the search of critical points of a functional defined on a
well-chosen Hilbert space. In particular, we will be interested in finding minimum
points or saddlelike points. The Ambrosetti–Rabinowitz mountain pass theorem
and the Rabinowitz saddle point theorem are presented, as particular cases of a
more general situation. The proof is based on the Ekeland variational principle. In
Chap. 8, we will show how to apply these methods to deal with functions g.t; x/
satisfying the Ahmad–Lazer–Paul conditions, a still closer approach to resonance.
As a final result, we present a multiplicity result for periodic solutions of pendulumlike equations due to Mawhin and Willem.


xiv

Introduction

In Chap. 9, we explain the theory by Lusternik and Schnirelmann in the simple
case of functionals defined on the product of a torus and a Hilbert space. Since we do
not assume the reader to be familiar with calculus on manifolds, we tried to maintain
the exposition at an as elementary as possible level. The notions of category and
relative category are introduced, leading to some theorems on multiplicity of critical
points.
In Chap. 10, we propose a version of the Poincaré–Birkhoff theorem which is
well suited for Hamiltonian systems in the plane. This is a very recent result I
have obtained in collaboration with Antonio J. Ureña [107], which extends also to
higher dimensions. However, we only deal here with the planar case, for simplicity.
In Chap. 11, we show the far-reaching consequences of the Poincaré–Birkhoff
theorem and obtain the multiplicity of periodic solutions for equations either with
asymmetric nonlinearities or with nonlinearities having a superlinear growth.
The remaining part of the book consists of three appendices.
In Appendix A, we recall the main properties of spaces of continuous functions

which are used in the book. In particular, we state and prove the Ascoli–Arzelà
theorem and the Stone–Weierstrass theorem.
In Appendix B, we provide the needed background for differential calculus in
infinite dimensions. The Fréchet differential is introduced, and its main properties
are analyzed. In particular, the implicit function theorem is reported here.
Since for the construction of the topological degree we use some properties
of differential forms, Appendix C is meant to briefly collect some of their main
features, including the Stokes–Cartan theorem. This appendix could also be useful
for clarifying the notations used in the text. For a more complete treatment, we refer
to the nice book by Spivak [209].
The choice of the results contained in this book has been greatly influenced by
my own research interests. I hope that the reader will share my enthusiasm for the
beauty of this theory, which in recent years shows a still growing interest, as can be
seen from the large number of recent publications in specialized journals.
The list of references is by no means complete, and I apologize for this. However,
I have included some very recent papers, and the references therein will help the
interested reader to find an up-to-date picture of the present situation.
I wish to warmly thank all the students who, following my courses, have often
given me hints on how to clarify the exposition of the arguments contained in the
book. Without them I would not even have found the motivation to write it.


List of Symbols

N
Z
R
C
K
xC

x
hv ; wi
kvk
d.x ; y/
dist.x ; U/
B.x; r/
B.x; r/
Br
U
U?
z
A
det A
L.X; Y/
L.X/
I
J
L
.L/
.L/
k k2

Natural numbers
Integer numbers
Real numbers
Complex numbers
Scalars (real or complex)
Positive part of x, i.e., xC D maxfx; 0g
Negative part of x, i.e., x D maxf x; 0g
Scalar product of v and w

Norm of v
Distance from x to y
Distance from a point x to a set U
Open ball centered at x with radius r
Closed ball centered at x with radius r
Open ball centered at 0 with radius r
Closure of the set U
Orthogonal to the set U
Complex conjugate of z
Adjoint of the matrix A
Determinant of A
The set of linear bounded functions from X to Y
The set L.X; X/
Identity matrix or function
Standard symplectic matrix
Adjoint of the operator L
The spectrum of L
The resolvent set of L
Norm in L2

xv


xvi

k k1
k kW m;2
d. f ; /
df .x/, f 0 .x/
rf .x/

Jf .x/
d!
fn * f
fn ! f

List of Symbols

The norm of uniform convergence
Norm in W m;2
Degree of f in
Differential of f at x
Gradient of f at x
Jacobian matrix of f at x
Exterior differential of the differential form !
. fn /n weakly converges to f
. fn /n converges to f


Chapter 1

Preliminaries on Hilbert Spaces

In this first chapter, we provide the definition and the basic properties of a Hilbert
space H, together with some examples of spaces which will be needed in the next
chapters.

1.1 The Hilbert Space Structure
Let H be a vector space on K, the field of scalars. We will always assume K to be
either the field of the reals R, or that of the complex numbers C. A scalar product
H H ! K is defined, which we will denote by h ; i. It is such that, for every

f ; g; h 2 H and ˛ 2 K, the following properties are satisfied1 :
a/ h f ; f i

0I

b/ h f ; f i D 0 ” f D 0 I
c/ h f C g; hi D h f ; hi C hg; hi I
d/ h˛f ; gi D ˛h f ; gi I
e/ h f ; gi D hg; f i :
Notice that
h f ; ˛gi D ˛ h f ; gi :

1

Here, z is the complex conjugate of z (so z D z if z 2 R).

© Springer International Publishing AG 2016
A. Fonda, Playing Around Resonance, BirkhRauser Advanced Texts Basler
LehrbRucher, DOI 10.1007/978-3-319-47090-0_1

1


2

1 Preliminaries on Hilbert Spaces

For any f 2 H, let us set
k f k D h f ; f i1=2 :
Theorem 1.1.1 For every f ; g 2 H, one has

jh f ; gij Ä k f k kgk :
(Schwarz inequality).
Proof The inequality is surely true if g D 0, since in that case h f ; gi D 0 and
kgk D 0. Let us then suppose that g ¤ 0. For every ˛ 2 K, we have
0 Ä kf

˛gk2 D h f

˛g; f

˛gi D k f k2

˛ h f ; gi

˛hg; f i C j˛j2 kgk2 :

Taking ˛ D h f ; gi=kgk2 , we obtain
0 Ä k f k2

2

jh f ; gij2
jh f ; gij2
C
kgk2 D k f k2
kgk2
kgk4

jh f ; gij2
;

kgk2

and the conclusion follows.
We have that k k is a norm on H:
a/ k f k

0I

b/ k f k D 0 ” f D 0 I
c/ k˛f k D j˛j k f k I
d/ k f C gk Ä k f k C kgk :
The first three properties are straightforward. Let us prove the last inequality2 :
k f C gk2 D h f C g; f C gi
D h f ; f i C h f ; gi C hg; f i C hg; gi
D k f k2 C 2<.h f ; gi/ C kgk2
Ä k f k2 C 2jh f ; gij C kgk2
Ä k f k2 C 2k f k kgk C kgk2
D .k f k C kgk/2 :

2

Here, <.z/ denotes the real part of the complex number z.


1.2 Some Examples of Hilbert Spaces

3

For f ; g 2 H, let us set
d. f ; g/ D k f

gk :

We have that d. ; / is a distance, so that H becomes a metric space:
a/ d. f ; g/

0I

b/ d. f ; g/ D 0 ” f D g I
c/ d. f ; g/ D d.g; f / I
d/ d. f ; h/ Ä d. f ; g/ C d.g; h/ :
We will say that H is a Hilbert space if, with respect to the above distance, H is
complete. In the following, H will always denote a Hilbert space.

1.2 Some Examples of Hilbert Spaces
We illustrate here four examples of Hilbert spaces, which we will need in the sequel
of the book.
1) Consider the set KN , and define, for ˛ D .˛1 ; : : : ; ˛N / and ˇ D .ˇ1 ; : : : ; ˇN / in
it,
h˛; ˇi D

N
X

˛k ˇk :

kD1

It is easily seen that all the properties of the scalar product are satisfied.
Notice that, if K D R, we have the usual Euclidean scalar product, which

defines the Euclidean norm and distance. We know that, in this case, RN is a
complete metric space, hence a Hilbert space. When K D C, writing every ˛k
in the form ˛k D ak C ibk , the norm associated with the above defined scalar
product is
k˛k D

N
X
kD1

j˛k j2

Á1=2

D

N
X

a2k C b2k

Á1=2

:

kD1

From a metric point of view, we can thus identify CN with R2N , so that, even in
this case, we have indeed a Hilbert space.
Let us emphasize that, when N D 1, the scalar product is given by h˛; ˇi D

˛ˇ .


4

1 Preliminaries on Hilbert Spaces

2) Consider the set `2 .K/, whose elements are the sequences .˛k /k
that
1
X

1

in K, such

j˛k j2 < C1 :

kD1

It is possible to verify that, for ˛ D .˛k /k
h˛; ˇi D

and ˇ D .ˇk /k 1 , setting

1

1
X

˛k ˇk ;

kD1

we have a scalar product which makes `2 .K/ a Hilbert space.
3) Consider the set L2 .Œa; b; K/, made up of the functions f W Œa; b ! K, which are
Lebesgue integrable, such that j f j2 is integrable, as well. It is possible to verify
that, for f ; g 2 L2 .Œa; b; K/, it makes sense to define
Z

b

f .t/g.t/ dt ;

h f ; gi D
a

an element of K. (In some applications, it could be useful to multiply the integral
by a suitable constant.) The scalar product properties are verified, provided that,
as usual, two functions are identified whenever they coincide almost everywhere.
It can be proved, moreover, that L2 .Œa; b; K/, with this scalar product, is a
Hilbert space. The corresponding norm is
ÂZ

b

k f k2 D

2

j f .t/j dt

Ã1=2
:

a

When there will be no ambiguity, we will simply denote this space by L2 .a; b/.
Let us remark that the use of the Lebesgue integral is crucial here. If, for
instance, we were considering the functions which are only Riemann integrable,
we would loose the completeness of the space.
4) In this example we introduce the set W 1;2 .Œa; b; R/. A function f W Œa; b ! R
belongs to this set if it is in L2 .Œa; b; R/, and it has a weak derivative in
L2 .Œa; b; R/, as well.
Let us recall what weak derivative means. It is said that a function f 0 W Œa; b ! R
is a weak derivative of f if, for every C1 -smooth function ' W Œa; b ! R having a
compact support in a; bŒ , one has that
Z

b
a

0

Z

b

f .t/' .t/ dt D
a

f 0 .t/'.t/ dt :

(1.1)


1.2 Some Examples of Hilbert Spaces

5

Notice that, if f 2 W 1;2 .Œa; b; R/, both f and its weak derivative f 0 are in principle
defined only almost everywhere on Œa; b. Moreover, it can be seen that the weak
derivative is unique, with the usual agreement that two functions are identified
whenever they coincide almost everywhere.
One could wonder why does the notation chosen for the weak derivative coincide
with that of the usual derivative. Well, assume f to be differentiable on Œa; b, with
a not too ugly derivative f 0 , so to be integrable. Then, (1.1) is easily obtained,
integrating by parts, for every C1 -smooth function ' W Œa; b ! R such that
'.a/ D '.b/ D 0. So, the usual derivative is a weak derivative, in this case. On
the other hand, clearly enough, a function f could have a weak derivative without
being differentiable in the usual sense.
It can be seen that W 1;2 .Œa; b; R/, with the operations of sum and multiplication
by scalars, is a vector space of functions. Moreover, setting
Z

b

h f ; gi D

Á

f .t/g.t/ C f 0 .t/g0 .t/ dt ;

a

one can verify that this is a scalar product. The associated norm is
k f kW 1;2 D .k f k22 C k f 0 k22 /1=2 ;
and it is possible to prove that this normed space is complete, hence W 1;2 .Œa; b; R/
is a Hilbert space. We will often denote such a space simply by W 1;2 .a; b/. It is
sometimes also denoted by H 1 .a; b/.
The theory of Sobolev spaces (see, e.g., [30]) tells us that each function f 2
W 1;2 .a; b/ coincides almost everywhere with a continuous function. We can thus
identify the two, and finally write W 1;2 .a; b/  C.Œa; b/. Moreover, the embedding
is continuous, meaning that, if . fn /n is a sequence in W 1;2 .a; b/ which converges to
some f in W 1;2 .a; b/, then the convergence is uniform over Œa; b.
Another important property we will need is that the inclusion W 1;2 .a; b/ Â
C.Œa; b/ is compact. This means that, if . fn /n is a sequence in W 1;2 .a; b/, and it is a
bounded sequence in that space, then there is a subsequence . fnk /k which converges
uniformly over Œa; b.
As a variant of the above, we will particularly need the space
HT1 D ff 2 W 1;2 .Œ0; T; R/ W f .0/ D f .T/g ;
with the same scalar product as in W 1;2 .0; T/. Since HT1 is closed in W 1;2 .0; T/, it is
complete, hence itself a Hilbert space. We will see in Sect. 1.7 how this space could
also be defined by the use of Fourier series.
5) The Sobolev space W 2;2 .Œa; b; R/ is the space of those functions f 2
L2 .Œa; b; R/ which have first and second weak derivatives in L2 .Œa; b; R/.


6

1 Preliminaries on Hilbert Spaces

The second weak derivative of f is just the weak derivative of f 0 , and is denoted
by f 00 . Setting
Z

b

h f ; gi D

Á
f .t/g.t/ C f 0 .t/g0 .t/ C f 00 .t/g00 .t/ dt ;

a

we have a scalar product, and W 2;2 .Œa; b; R/ is indeed a Hilbert space. We will
often denote such a space simply by W 2;2 .a; b/. It is sometimes also denoted by
H 2 .a; b/. Its norm is given by
k f kW 2;2 D .k f k22 C k f 0 k22 C k f 00 k22 /1=2 :
The theory of Sobolev spaces (see again [30]) tells us that each function
f 2 W 2;2 .a; b/ coincides almost everywhere with a continuously differentiable
function: we can thus write W 2;2 .a; b/  C1 .Œa; b/. Moreover, the embedding
is continuous: if . fn /n is a sequence in W 2;2 .a; b/ which converges to some f in
W 2;2 .a; b/, then also limn fn D f in C1 .Œa; b/. Moreover, this inclusion is also
compact: if . fn /n is a sequence in W 2;2 .a; b/, which is bounded in that space,
then there is a subsequence . fnk /k which converges in C1 .Œa; b/.
With respect to the three last examples above, let us mention that we will
sometimes deal with functions of the type f W Œa; b ! RM . Writing f .x/ D
. f1 .x/; : : : ; fM .x//, for every x 2 Œa; b, we thus define the components fk W Œa; b !
R, with k D 1; : : : ; M. We have that f 2 L2 .Œa; b; RM / if fk 2 L2 .Œa; b; R/, for every
k D 1; : : : ; M. Analogously, we say that f 2 W m;2 .Œa; b; RM /, with m 2 f1; 2g, if

fk 2 W m;2 .Œa; b; R/, for every k D 1; : : : ; M. The scalar products in these spaces
will be the same as the ones defined above, replacing the product in R with the scalar
product in RM .

1.3 Fundamental Properties
The following inequality can be easily verified:
ˇ
ˇ
ˇk f k

ˇ
ˇ
kgkˇ Ä k f

gk :

As a consequence, one sees that the norm is a continuous function. Moreover, from
the Schwarz inequality, it follows that the scalar product is continuous in its single
components, as well. If . fn /n is a sequence in H such that limn fn D f , we can then
write
k f k D lim k fn k ;
n

h f ; gi D limh fn ; gi :
n


1.3 Fundamental Properties

Moreover, if the series

*

1
X

+

*

fk ; g D lim

kD1

n

7

P1

kD1 fk

n
X

converges, we have
*

+
fk ; g D lim
n

kD1

n
X

+
fk ; g D lim
n

kD1

n
X

h fk ; gi D

kD1

1
X
h fk ; gi :
kD1

Let us now state the following.
Theorem 1.3.1 For every f ; g 2 H, it is
k f C gk2 C k f

gk2 D 2k f k2 C 2kgk2 :

(parallelogram identity).
Proof Being
k f C gk2 D k f k2 C 2<.h f ; gi/ C kgk2 ;
kf

gk2 D k f k2

2<.h f ; gi/ C kgk2 ;

summing the two we get the identity we are looking for.
Given f ; g 2 H, if h f ; gi D 0, we say that f and g are orthogonal. More generally,
we say that a (finite or infinite) family . fk /k in H is orthogonal if h fj ; fk i D 0, for
every j ¤ k.
Theorem 1.3.2 (Pythagorean Theorem) We have three statements, in increasing
order of generality.
I) If f and g are orthogonal, then
k f C gk2 D k f k2 C kgk2 :
II) If . f1 ; f2 ; : : : ; fn / is an orthogonal family, then
n
X

2

D

fk

kD1

n

X

k fk k2 :

kD1

III) if . fk /k is an orthogonal sequence, then
1
X

fk converges

”

kD1

1
X

k fk k2 converges I

kD1

in that case,
1
X
kD1

2

fk

D

1
X
kD1

k fk k2 :


8

1 Preliminaries on Hilbert Spaces

Proof
I) Being h f ; gi D 0, one has
k f C gk2 D k f k2 C 2<.h f ; gi/ C kgk2 D k f k2 C kgk2 :
II) Just proceed recursively, using I) : for every j D 1; 2; : : : ; n
k f1 C f2 C

C fjC1 k2 D k. f1 C f2 C

1,

C fj / C fjC1 k2
C fj k2 C k fjC1 k2

D k f1 C f2 C

D .k f1 k2 C k f2 k2 C

C k fj k2 / C k fjC1 k2 :

III) Using II), for every m; n one has
n
X

2

fk

kDm

D

n
X

k fk k2 :

kDm

Since H is complete,
it is possible to use the
criterion to establish that,
P
PCauchy
1
2

if the series 1
k
f
k
converges,
then
f
converges,
and vice versa.
k
k
kD1
kD1
Assume that both series converge. Then, by the continuity of the norm, using
II),
1
X

2

fk

D lim

kD1

n

n
X

2

fk

kD1

D lim
n

n
X
kD1

2

fk

D lim
n

n
X
kD1

2

k fk k D

1

X

k fk k2 ;

kD1

thus proving the theorem.

1.4 Subspaces
A subset M of H is a linear manifold if, taken any f ; g 2 M and ˛; ˇ 2 K, one
has that ˛f C ˇg 2 M. In general, a linear manifold M of a Hilbert space is not
itself a Hilbert space, since the completeness is preserved only if M is a closed
subset (we recall that a subset is closed if and only if it contains the limits of all its
convergent sequences). We will then say that M is a subspace of H if it is a closed
linear manifold.3
An example of a linear manifold which is not closed is, in L2 .Œa; b; K/, the set
C.Œa; b; K/ of continuous functions. It can be proved, indeed, that its closure is the
whole space L2 .Œa; b; K/.
3

Some authors prefer calling subspace and closed subspace what we have called linear manifold
and subspace, respectively.


1.4 Subspaces

9

Theorem 1.4.1 If M is a linear manifold, its closure M is still a linear manifold,
hence a subspace.

Proof Let f ; g 2 M and ˛; ˇ 2 K be given. We can find two sequences . fn /n
and .gn /n such that fn ; gn 2 M for every n, and limn fn D f , limn gn D g. Then,
˛fn C ˇgn 2 M, and limn .˛fn C ˇgn / D ˛f C ˇg, so that ˛f C ˇg 2 M.
Theorem 1.4.2 If .Mk /k is a family (not necessarily a countable family) of
subspaces, then the intersection of all of them is still a subspace.
Proof This follows from the fact that the intersection of linear manifolds is a linear
manifold, and the intersection of closed sets is a closed set.
Corollary 1.4.3 Given any set U Â H, there is a unique subspace M such that
a) M contains U ,
b) if M0 is a subspace which contains U, then M0 contains M .
Proof Define M as the intersection of all subspaces containing U. Then, M is a
subspace, and properties a/ and b/ follow immediately.
The set M, whose existence has been established in the previous theorem, is the
smallest subspace containing U; we say that M is the subspace generated by U.
Given two subspaces M1 and M2 , the set
M1 C M2 D ff C g W f 2 M1 ; g 2 M2 g
is a linear manifold, but we cannot say in general that it is a subspace. Let us
compare it with the subspace generated by the union M1 [ M2 , which we denote
by M1 _ M2 .
Theorem 1.4.4 One has
M1 _ M2 D M1 C M2 :
Proof Both M1 and M2 are contained in M1 _ M2 . Since this last set is a linear
manifold, also M1 C M2 is contained in it. Moreover, since M1 _ M2 is closed,
we have that M1 C M2 Â M1 _ M2 .
Vice versa, both M1 and M2 are contained in M1 C M2 , hence also their
union is. Since, by Theorem 1.4.1, M1 C M2 is a subspace, it must contain
M1 _ M2 .
If .Mk /k is a sequence of subspaces, we define
1
X

Mk

kD1

P
as the set whose elements are the sum of a convergent series 1
kD1 fk , with fk 2
Mk . It is easy to see that it is a linear manifold. As in the case of the sum of two


10

1 Preliminaries on Hilbert Spaces

1
subspaces, we compare
W1it now with the subspace generated by their union [kD1 Mk ,
which we denote by kD1 Mk .

Theorem 1.4.5 One has
1
_

Mk D

kD1

1
X

Mk :

kD1

W
Proof Every Mk is contained in 1
kD1 Mk . Since this last set is a subspace,
P1 it also
contains those elements P
which are the sum of a convergent
series
kD1 fk , with
W1
fk 2 Mk , so it contains 1
M
.
Moreover,
since
M
is
closed,
we have
k
k
kD1
kD1
W1
P1
that kD1 Mk  kD1 Mk .

P
Vice versa, every Mk is contained in 1
kD1 Mk , hence also their union is. Since,
P1
W
by Theorem 1.4.1, kD1 Mk is a subspace, it must contain 1
kD1 Mk .

1.5 Orthogonal Subspaces
We say that two sets U and V are orthogonal if
h f ; gi D 0 ;

for every . f ; g/ 2 U

V:

Moreover, we denote by U ? the set of those elements of H which are orthogonal to
all elements of U.
?

Theorem 1.5.1 The set U ? is a subspace of H. Moreover, U D U ? .
Proof Choose f ; g 2 U ? and ˛; ˇ 2 K. Then, for every u 2 U, we have
h˛f C ˇg; ui D ˛h f ; ui C ˇhg; ui D 0 ;
so that ˛f C ˇg 2 U ? . Hence, U ? is a linear manifold. Let us see that it is a closed
set. Let . fn /n be a sequence in U ? such that limn fn D f . Then, for every u 2 U,
h f ; ui D limh fn ; ui D 0 ;
n

so that f 2 U ? . Therefore, U ? is a subspace.
?

Since U Â U, we have that U Â U ? . On the other hand, if f 2 U ? and g 2 U,
there is a sequence .gn /n in U such that limn gn D g, and
h f ; gi D limh f ; gn i D 0 :
n

?

?

So, f 2 U , proving that U ? Â U .