Handbook of Differential Equations: Ordinary Differential Equations, Volume 3

753 Pages·2006·11.96 MB·English

by  A. Canada

HANDBOOK
OF DIFFERENTIAL EQUATIONS
ORDINARY DIFFERENTIAL EQUATIONS
VOLUME III
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H
ANDBOOK
D E
OF IFFERENTIAL QUATIONS
O D
RDINARY IFFERENTIAL
E
QUATIONS
V III
OLUME
Editedby
A. CAÑADA
DepartmentofMathematicalAnalysis,FacultyofSciences,
UniversityofGranada,Granada,Spain
P. DRÁBEK
DepartmentofMathematics,FacultyofAppliedSciences,
UniversityofWestBohemia,Pilsen,CzechRepublic
A. FONDA
DepartmentofMathematicalSciences,FacultyofSciences,
UniversityofTrieste,Trieste,Italy
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Preface
Thisisthethirdvolumeinaseriesdevotedtoselfcontainedandup-to-datesurveysinthe
theory of ordinary differential equations, written by leading researchers in the area. All
contributorshavemadeanadditionalefforttoachievereadabilityformathematiciansand
scientistsfromotherrelatedfields,inordertomakethechaptersofthevolumeaccessible
to a wide audience. These ideas faithfully reflect the spirit of this multi-volume and the
editorshopethatitwillbecomeveryusefulforresearch,learningandteaching.Weexpress
ourdeepestgratitudetoallcontributorstothisvolumefortheirclearlywrittenandelegant
articles.
Thisvolumeconsistsofsevenchapterscoveringavarietyofproblemsinordinarydiffer-
entialequations.Both,puremathematicalresearchandrealwordapplicationsarereflected
pretty well by the contributions to this volume. They are presented in alphabetical order
accordingtothenameofthefirstauthor.ThepaperbyAndresprovidesacomprehensive
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-
tionalstructure.Itisdividedin2parts:thefirstonedealswithsecondorderequationsand
systems,whilethesecondonedescribesrecentresultsonfourthorderequations.Thecon-
tributionbyDeCoster,ObersnelandOmaridealswithqualitativepropertiesofsolutions
oftwokindsofscalardifferentialequations:firstorderODEs,andsecondorderparabolic
PDEs. Their setting is very general, so that neither uniqueness for the initial value prob-
lemsnorcomparisonprinciplesareguaranteed.Theyparticularlyconcentrateonperiodic
solutions, their localization and possible stability. The paper by Han is dedicated to the
theoryoflimitcyclesofplanardifferentialsystemsandtheirbifurcations.Itisstructured
inthreemainparts:generalpropertiesoflimitcycles,Hopfbifurcationsandperturbations
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,WaltherandWureportsaboutthemorerecentworkonstate-dependentdelayed
functionaldifferentialequations.Theseequationsappearinanaturalwayinthemodelling
ofevolutionprocessesinverydifferentfields:physics,automaticcontrol,neuralnetworks,
infectiousdiseases,populationgrowth,cellbiology,epidemiology,etc.Theauthorsempha-
size on particular models and on the emerging theory from the dynamical systems point
v
vi Preface
ofview.ThepaperbyKormanisdevotedtotwopointnonlinearboundaryvalueproblems
dependingonaparameterλ.Themainquestionistheprecisenumberofsolutionsofthe
problem and how these solutions change with the parameter. To study the problem, the
authorusesbifurcationtheorybasedontheimplicitfunctiontheorem(inBanachspaces)
and on a well known theoremby Crandall and Rabinowitz.Othertopicshe discusses in-
volvepitchforkbifurcationandsymmetrybreaking,signchangingsolutions,etc.Finally,
thepaperbyRachu˚nková,StaneˇkandTvrdýisasurveyonthesolvabilityofvariousnon-
linearsingularboundaryvalueproblemsforordinarydifferentialequationsonthecompact
interval.Thenonlinearitiesindifferentialequationsmaybesingularbothinthetimeand
spacevariables.Locationofallsingularpointsneednotbeknown.
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.,PalackýUniversity,Olomouc-Hejcˇín,CzechRepublic(Ch.1)
Bonheure,D.,UniversitéCatholiquedeLouvain,Louvain-La-Neuve,Belgium(Ch.2)
DeCoster,C.,UniversitéduLittoral-Côted’Opale,CalaisCédex,France(Ch.3)
Han,M.,ShanghaiNormalUniversity,Shanghai,China(Ch.4)
Hartung,F.,UniversityofVeszprém,Veszprém,Hungary(Ch.5)
Korman,P.,UniversityofCincinnati,Cincinnati,OH,USA(Ch.6)
Krisztin,T.,UniversityofSzeged,Szeged,Hungary(Ch.5)
Obersnel,F.,UniversitàdegliStudidiTrieste,Trieste,Italy(Ch.3)
Omari,P.,UniversitàdegliStudidiTrieste,Trieste,Italy(Ch.3)
Rachu˚nková,I.,PalackýUniversity,Olomouc,CzechRepublic(Ch.7)
Sanchez,L.,UniversidadedeLisboa,Lisboa,Portugal(Ch.2)
Staneˇk,S.,PalackýUniversity,Olomouc,CzechRepublic(Ch.7)
Tvrdý, M., Mathematical Institute, Academy of Sciences of the Czech Republic, Praha,
CzechRepublic(Ch.7)
Walther,H.-O.,UniversitätGießen,Gießen,Germany(Ch.5)
Wu,J.,YorkUniversity,Toronto,Canada(Ch.5)
vii
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Contents
Preface v
ListofContributors vii
ContentsofVolume1 xi
ContentsofVolume2 xiii
1. Topologicalprinciplesforordinarydifferentialequations 1
J.Andres
2. Heteroclinicorbitsforsomeclassesofsecondandfourthorderdifferentialequa-
tions 103
D.BonheureandL.Sanchez
3. Aqualitativeanalysis,vialoweranduppersolutions,offirstorderperiodicevo-
lutionaryequationswithlackofuniqueness 203
C.DeCoster,F.ObersnelandP.Omari
4. Bifurcationtheoryoflimitcyclesofplanarsystems 341
M.Han
5. Functionaldifferentialequationswithstate-dependentdelays:Theoryandappli-
cations 435
F.Hartung,T.Krisztin,H.-O.WaltherandJ.Wu
6. Globalsolutionbranchesandexactmultiplicityofsolutionsfortwopointbound-
aryvalueproblems 547
P.Korman
7. SingularitiesandLaplaciansinboundaryvalueproblemsfornonlinearordinary
differentialequations 607
I.Rachu˚nková,S.StaneˇkandM.Tvrdý
Authorindex 725
Subjectindex 735
ix