Handbook of differential equations. Ordinary differential equations

753 Pages·2006·3.84 MB·English

by  A. Canada, P. Drabek, A. Fonda

HANDBOOK
OF DIFFERENTIAL EQUATIONS
ORDINARY DIFFERENTIAL EQUATIONS
VOLUME III
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HANDBOOK
OF DIFFERENTIAL EQUATIONS
ORDINARY DIFFERENTIAL
EQUATIONS
VOLUME III
Edited by
A. CAÑADA
Department of Mathematical Analysis, Faculty of Sciences,
University of Granada, Granada, Spain
P. DRÁBEK
Department of Mathematics, Faculty of Applied Sciences,
University of West Bohemia, Pilsen, Czech Republic
A. FONDA
Department of Mathematical Sciences, Faculty of Sciences,
University of Trieste, Trieste, Italy
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First edition 2006
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Preface
This is the third volume in a series devoted to self contained and up-to-date surveys in the
theory of ordinary differential equations, written by leading researchers in the area. All
contributors have made an additional effort to achieve readability for mathematicians and
scientists from other related fields, in order to make the chapters of the volume accessible
to a wide audience. These ideas faithfully reflect the spirit of this multi-volume and the
editors hope that it will become very useful for research, learning and teaching. We express
our deepest gratitude to all contributors to this volume for their clearly written and elegant
articles.
This volume consists of seven chapters covering a variety of problems in ordinary differ-
ential equations. Both, pure mathematical research and real word applications are reflected
pretty well by the contributions to this volume. They are presented in alphabetical order
according to the name of the first author. The paper by Andres provides a comprehensive
survey on topological methods based on topological index, Lefschetz and Nielsen num-
bers. Both single and multivalued cases are investigated. Ordinary differential equations
are studied both on finite and infinite dimensions, and also on compact and noncompact
intervals. There are derived existence and multiplicity results. Topological structures of
solution sets are investigated as well. The paper by Bonheure and Sanchez is dedicated
to show how variational methods have been used in the last 20 years to prove existence
of heteroclinic orbits for second and fourth order differential equations having a varia-
tional structure. It is divided in 2 parts: the first one deals with second order equations and
systems, while the second one describes recent results on fourth order equations. The con-
tribution by De Coster, Obersnel and Omari deals with qualitative properties of solutions
of two kinds of scalar differential equations: first order ODEs, and second order parabolic
PDEs. Their setting is very general, so that neither uniqueness for the initial value prob-
lems nor comparison principles are guaranteed. They particularly concentrate on periodic
solutions, their localization and possible stability. The paper by Han is dedicated to the
theory of limit cycles of planar differential systems and their bifurcations. It is structured
in three main parts: general properties of limit cycles, Hopf bifurcations and perturbations
of Hamiltonian systems. Many results are closely related to the second part of Hilbert’s
16th problem which concerns with the number and location of limit cycles of a planar
polynomial vector field of degree n posed in 1901 by Hilbert. The survey by Hartung,
Krisztin, Walther and Wu reports about the more recent work on state-dependent delayed
functional differential equations. These equations appear in a natural way in the modelling
of evolution processes in very different fields: physics, automatic control, neural networks,
infectious diseases, population growth, cell biology, epidemiology, etc. The authors empha-
size on particular models and on the emerging theory from the dynamical systems point
v
vi Preface
of view. The paper by Korman is devoted to two point nonlinear boundary value problems
depending on a parameter λ. The main question is the precise number of solutions of the
problem and how these solutions change with the parameter. To study the problem, the
author uses bifurcation theory based on the implicit function theorem (in Banach spaces)
and on a well known theorem by Crandall and Rabinowitz. Other topics he discusses in-
volve pitchfork bifurcation and symmetry breaking, sign changing solutions, etc. Finally,
the paper by Rachu˚nková, Staneˇk and Tvrdý is a survey on the solvability of various non-
linear singular boundary value problems for ordinary differential equations on the compact
interval. The nonlinearities in differential equations may be singular both in the time and
space variables. Location of all singular points need not be known.
With this volume we end our contribution as editors of the Handbook of Differential
Equations. We thank the staff at Elsevier for efficient collaboration during the last three
years.
List of Contributors
Andres, J., Palacký University, Olomouc-Hejcˇín, Czech Republic (Ch. 1)
Bonheure, D., Université Catholique de Louvain, Louvain-La-Neuve, Belgium (Ch. 2)
De Coster, C., Université du Littoral-Côte d’Opale, Calais Cédex, France (Ch. 3)
Han, M., Shanghai Normal University, Shanghai, China (Ch. 4)
Hartung, F., University of Veszprém, Veszprém, Hungary (Ch. 5)
Korman, P., University of Cincinnati, Cincinnati, OH, USA (Ch. 6)
Krisztin, T., University of Szeged, Szeged, Hungary (Ch. 5)
Obersnel, F., Università degli Studi di Trieste, Trieste, Italy (Ch. 3)
Omari, P., Università degli Studi di Trieste, Trieste, Italy (Ch. 3)
Rachu˚nková, I., Palacký University, Olomouc, Czech Republic (Ch. 7)
Sanchez, L., Universidade de Lisboa, Lisboa, Portugal (Ch. 2)
Staneˇk, S., Palacký University, Olomouc, Czech Republic (Ch. 7)
Tvrdý, M., Mathematical Institute, Academy of Sciences of the Czech Republic, Praha,
Czech Republic (Ch. 7)
Walther, H.-O., Universität Gießen, Gießen, Germany (Ch. 5)
Wu, J., York University, Toronto, Canada (Ch. 5)
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Contents
Preface v
List of Contributors vii
Contents of Volume 1 xi
Contents of Volume 2 xiii
1. Topological principles for ordinary differential equations 1
J. Andres
2. Heteroclinic orbits for some classes of second and fourth order differential equa-
tions 103
D. Bonheure and L. Sanchez
3. A qualitative analysis, via lower and upper solutions, of first order periodic evo-
lutionary equations with lack of uniqueness 203
C. De Coster, F. Obersnel and P. Omari
4. Bifurcation theory of limit cycles of planar systems 341
M. Han
5. Functional differential equations with state-dependent delays: Theory and appli-
cations 435
F. Hartung, T. Krisztin, H.-O. Walther and J. Wu
6. Global solution branches and exact multiplicity of solutions for two point bound-
ary value problems 547
P. Korman
7. Singularities and Laplacians in boundary value problems for nonlinear ordinary
differential equations 607
I. Rachu˚nková, S. Staneˇk and M. Tvrdý
Author index 725
Subject index 735
ix