Table Of ContentCISM COURSES AND LECTURES
Series Editors:
The Rectors of CISM
Sandor Kaliszky -Budapest
Mahir Sayir -Zurich
Wilhelm Schneider -Wien
The Secretary General of CISM
Giovanni Bianchi - Milan
Executive Editor
Carlo Tasso- Udine
The series presents lecture notes, monographs, edited works and
proceedings in the field of Mechanics, Engineering, Computer Science
and Applied Mathematics.
Purpose of the series is to make known in the international scientific
and technical community results obtained in some of the activities
organized by CISM, the International Centre for Mechanical Sciences.
INTERNATIONAL CENTRE FOR MECHANICAL SCIENCES
COURSES AND LECTURES-No. 371
NON LINEAR ANALYSIS
AND BOUNDARY VALUE PROBLEMS
FOR ORDINARY DIFFERENTIAL
EQUATIONS
EDITED BY
F. ZANOLIN
UNIVERSITY OF UDINE
~ Springer-Verlag Wien GmbH
Le spese di stampa di questo volume sono in parte coperte da
contributi del Consiglio Nazionale delle Ricerch\!.
This volume contains l illustrations
This work is subject to copyright.
All rights are reserved,
whether the whole or part of the material is concerned
specifically those of translation, reprinting, re-use of illustrations,
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or similar means, and storage in data banks.
© 1996 by Springer-Verlag Wien
Originally published by Springer-Verlag Wien New York in 1996
In order to make this volume available as economically and as
rapidly as possible the authors' typescripts have been
reproduced in their original forms. This method unfortunately
has its typographical limitations but it is hoped that they in no
way distract the reader.
ISBN 978-3-211-82811-3 ISBN 978-3-7091-2680-6 (eBook)
DOI 10.1007/978-3-7091-2680-6
PREFACE
The theory of ordinary differential equations has received a strong
impulse by the introduction of some methods from Nonlinear Analysis (like, e.g.,
fixed point theorems, degree theory, topological invariants, variational methods)
and the study of boundary value problems for ODEs is, at present, a rich area
full of significant achievements, both from the theoretical and the applied sides.
This volume concerns the study of some boundary value problems for nonlinear
ordinary differential equations, using topological or variational methods, with a
special emphasis also on the so-called qualitative (or geometric) approach in the
theory of ordinary differential equations, making use of a broad collection of
techniques coming from Analysis and Topology which are addressed to the study
of ODEs.
In this book, the study of more classical boundary value problems for
ordinary differential equations (like the periodic or the Sturm- Liouville ones)
which represent the main interest of a wide number of researchers in the world,
is accompanied by the presentation of some recent results dealing with other
interesting and more atypical BVPs. In particular, there are new contributions
to the study of bounded solutions and to the presentation of some very recent
approaches yielding to a rigorous proof of the presence of chaotic motions. The
aspect of numerical treatment of dynamical systems is discussed as well and
some new achievements about the validity of discretization metkods are
analyzed.
The volume is addressed to people interested in Nonlinear Analysis,
Ordinary Differential Equations, or Boundary Value Problems. In particular,
various items chosen from the following areas are included: Topological degree
and related topics; Fixed point theorems and applications; Bifurcation theory;
Application of variational methods; Qualitative theory of ODEs; Topological
invariants for the study of dynamical systems; Chaotic motions; Bounded
solutions; Numerical methods for the analysis of dynamical systems.
Thanks to CISM hospitality, it was possible to organize in Udine a
School on Nonlinear Analysis and Boundary Value Problems for Ordinary
Differential Equations. Financial support was given by various different
sources. In particular, we mention UNESCO-ROSTE which provided some
scholarships to partially support participants from developing countries and
Eastern Europe. We are also grateful to other international, national and local
institutions, like the European Community (through grants CII*-CT93-0323
and ERB CHRX-CT94-0555), the C.N.R. (Italy), the MURST (project
"Equazioni Differenziali Ordinarie e Applicazioni", by professor Roberto
Conti), the University of Udine with the Department of Mathematics and
Computer Science and the CRUP Bank.
The School consisted of a series of lectures delivered by some well-known
specialists. devoted to focus the basic facts and the recent advances of the theory
as well as to present interesting research problems. We had also a few one-hour
invited lectures which were addressed to more specific topics. Then, the rest of
the time was devoted to a number of communications and seminars concerning
the presentation of recent research achievements.
The success of the School was due to the kindness of all these
distinguished colleagues who accepted to share their mathematical expertise with
others and delivered some highly interesting lectures, as well as to the active
presence of all the participants. We are also indebted to professors A. Capietto,
A. Fonda, M. Gaudenzi, M. Marini and P. Omari who helped in various
important steps for the organization of the Conference. We finally thank
professor Carlo Tasso and the CISM staff for the publication of these
proceedings.
This volume collects six contributions, prepared by expert and active
mathematicians, corresponding to the content of their CISM lectures. Special
thanks are due to Jean Mawhin who found the right words to commemorate
Gilles Fournier, a friend and colleague who made brilliant contributions in this
area of research. This book is thus dedicated to the memory of Gilles.
F.Zanolin
IN MEMORIAM GILLES FOURNIER
Before starting a school devoted to topological methods in nonlinear
boundary value problems, we cannot avoid remembering the recent untimely
death of one of the most active and original contributors to this area of
mathematics.
Gilles Fournier died unexpectedly on August 14 1995, leaving his beloved
wife Reine and their three children Andre, Marie-Helene and Nadine in deep
sorrow.
A student at the University of Montreal, where he got his PhD in 1973
under the direction of Andrzej Granas, Gilles Fournier has made his whole
career at the University of Sherbrooke, being an indefatigable animator of its
Department of Mathematics and Computer Science, organizing several
international conferences and directing many PhD dissertations.
After some important work in algebraic topology, and specially in fixed
point theory, Gilles Fournier became interested in nonlinear differential
equations, using his talent in topology to develop new tools for proving the
existence and multiplicity of solutions.
An enthousiastic traveller, Gilles Fournier had in particular many
contacts with Italian mathematicians, spending several months as visiting
professor at the University of Calabria.
Gilles Fournier's extraordinary intuition and constant enthusiasm has
led him to joint work with many colleagues, some of them present here. Through
his important mathematical legacy and the memory of his sociable and friendly
personality, Gilles Fournier will stay in our mind and in our heart, always
remaining the gifted and cheerful companion we all remember.
JeanMawhin
CONTENTS
Page
Preface
In Memoriam Gilles Fournier
Upper and Lower Solutions in the Theory
of ODE Boundary Value Problems: Classical and Recent Results
by C. De Coster and P. Habets ........................................................................... 1
Boundary Value Problems for Quasilinear Second Order
Differential Equations
by R. Manasevich and K. Schmitt ...................................................................... 79
Bounded Solutions of Nonlinear Ordinary Differential Equations
by J. Mawhin ................................................................................................... 121
Hyperbolic Structures in ODE's and their Discretization
with an Appendix on
Differentiability Properties of the Inversion Operator
by B.M. Garay ................................................................................................ 149
The Conley Index and Rigorous Numerics
by M. Mrozek .................................................................................................. 175
On Geometric Detection of Periodic Solutions and Chaos
by R. Srzednicki ............................................................................................... 197
UPPER AND LOWER SOLUTIONS IN THE THEORY OF
ODE BOUNDARY VALUE PROBLEMS:
CLASSICAL AND RECENT RESULTS
C. De Coster and P. Habets
Catholic University of Louvain, Louvain-la-Neuve, Belgium
1 Introduction
The method of upper and lower solutions for ordinary differential equation was intro
duced in 1931 by G. Scorza Dragoni for a Dirichlet problem. Since then a large number
of contributions enriched the theory. Among others, one has to point out the pioneer
work of M. Nagumo who associated his name with derivative dependent right hand
side.
Basically, the method of upper and lower solutions deals with existence results
for boundary value problems. These upper and lower solutions can be thought of as
numerical approximations of solutions that satisfy the equations up to an error term
with constant sign. The method describes problems so that existence of a solution is
inferred from two such approximations with error'terms of opposite sign.
In this set of notes, we consider two such problems: the periodic one and Dirichlet
boundary value problem. We limit ourself to second order ordinary differential equa
tions and only consider problems with derivative independent right-hand side. This
last restriction wipes out the important problem of computing a-priori bounds for the
derivative of the solutions.
At first, the method of upper and lower solutions might look as being mainly of
theoretical interest since it takes for granted the existence of such functions. This
believe cuts off the large number of applications based on the method. It is true that
in practical situations it is often difficult to recognize that the assumptions at hand
imply the existence of such upper and lower solutions and we must admit that to
exhibit such functions is somewhat of an art. For this reason, a large part of this text
*Chargee de recherches du fonds national beige de Ia recherche scientifique.
2 C. De Coster and P. Habets
is devoted to applications and we believe that it is the practice of such problems which
really gives the ability and skill necessary to use the full power of the method.
The first section of these notes is of general character. ·we present the notion of
W2•1-upper and lower solutions which is adapted to prove existence of solutions in
the W2•1 sense. This notion allows also angles in the graph of these functions. This
aims to deal with applications such as singular perturbation problems. The Dirichlet
problem is studied for systems with singularities at the boundary points. Applications
to mechanical problems with singular forces and to Landesman-Lazer conditions are
worked out. Relation with degree theory is used to deal with multiplicity results and
is applied to the pendulum equation. A multiplicity result is also described using
the connection between the variational approach and the method of upper and lower
solutions. At last this introductory section ends with an introduction on monotone
iterative schemes related to upper and lower solutions.
Systems with singularities are investigated in the second section. The problem of
interest concerns existence of positive solutions for a Dirichlet problem which is singular
both at the end points of the time interval and at u = 0. Such situations appear in
applied mathematics problems such as the Emden-Fowler equation. In the second part
of this section, existence of pairs of positive solutions are considered.
An Ambrosetti-Prodi problem with two parameters is considered in Section 4.
The next section is devoted to upper and lower solutions in the reversed order. This
type of result is worked out for the periodic problem. A first result concerns upper
and lower solutions without ordering. In the second part, we consider systems with
asymmetric nonlinearities and assume the upper and lower solutions are ordered. The
last result concerns the use of monotone methods for systems with upper and lower
solutions in the reversed order.
Historical and bibliographical notes are given in the last section.
Throughout the text, some proofs are left as exercises, extensions are presented as
problems. The interested reader might find there the opportunity of further readings
and a chance to develop his ability to work upper and lower solutions.
The study made here is far from being complete. We can of course consider other
applications. \Ve can extend the class of boundary value problem we consider or deal
with partial differential equations. Upper and lower solutions can be used to obtain
bounded solutions or homoclinics, higher order equations and systems can be cousid
ered. Also, upper and lower solutions are efficient in other instances such as stability
problems. These subjects would be a natural complement of our work.
Definitions and notations
A function f(t, u) defined on E C [a, b] x lR is said to be a Caratheodory function or
to satisfy Caratlu?odory cond-itions on E if
(i) for almost every t E [a, b], f(t, ·) is continuous on its domain;
(i i) for any u E lR, the function f (· , u) is measurable on its domain.
A function f(t, u) defined onE C [a, b] x lR is said to be an Il'-Caratheodory function,
