Handbook of Differential Equations: Ordinary Differential Equations, Volume 1 (Handbook of Differential Equations)

709 Pages·2004·3.16 MB·English

by  A. Canada

HANDBOOK
OF DIFFERENTIAL EQUATIONS
ORDINARY DIFFERENTIAL EQUATIONS
VOLUME I
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H
ANDBOOK
D E
OF IFFERENTIAL QUATIONS
O D
RDINARY IFFERENTIAL
E
QUATIONS
V I
OLUME
Editedby
A. CAÑADA
DepartmentofMathematicalAnalysis,FacultyofSciences,
UniversityofGranada,Granada,Spain
P. DRÁBEK
DepartmentofMathematics,FacultyofAppliedSciences,
UniversityofWestBohemia,Pilsen,CzechRepublic
A. FONDA
DepartmentofMathematicalSciences,FacultyofSciences,
UniversityofTrieste,Trieste,Italy
2004
NORTH
HOLLAND
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Handbookofdifferentialequations
Ordinarydifferentialequations:Vol.1
1.Differentialequations
I.Cañada,A.II.Drábek,P.III.Fonda,A.
515.3’5
ISBN0444511288
ISBN:0-444-51128-8(volume1)
0-444-51742-1(set)
(cid:2)∞ThepaperusedinthispublicationmeetstherequirementsofANSI/NISOZ39.48-1992(PermanenceofPaper).
PrintedinTheNetherlands.
Preface
Ordinarydifferentialequationsisawidemathematicaldisciplinewhichiscloselyrelatedto
bothpuremathematicalresearchandrealworldapplications.Mostmathematicalformula-
tionsofphysicallawsaredescribedintermsofordinaryandpartialdifferentialequations,
and this has been a great motivation for their study in the past. In the 20th century the
extremely fast developmentof Science led to applications in the fields of chemistry, bi-
ology,medicine,populationdynamics,geneticengineering,economy,socialsciencesand
others, as well. All these disciplines promoted to higher level and new discoveries were
made with the help of this kindof mathematicalmodeling.At the same time, realworld
problemshavebeenandcontinueto beagreatinspirationforpuremathematics,particu-
larlyconcerningordinarydifferentialequations:theyledtonewmathematicalmodelsand
challengedmathematicianstolookfornewmethodstosolvethem.
It should also be mentioned that an extremely fast developmentof computer sciences
tookplaceinthelastthreedecades:mathematicianshavebeenprovidedwithatoolwhich
hadnotbeenavailablebefore.Thisfactencouragedscientiststoformulatemorecomplex
mathematicalmodelswhich,inthepast,couldhardlyberesolvedorevenunderstood.Even
if computersrarely permita rigoroustreatmentof a problem,they are a very usefultool
togetconcretenumericalresultsortomakeinterestingnumericalexperiments.Inthefield
of ordinary differential equations this phenomenon led more and more mathematicians
to the study of nonlinear differential equations. This fact is reflected pretty well by the
contributionstothisvolume.
Theaimoftheeditorswastocollectsurveypapersinthetheoryofordinarydifferential
equationsshowingthe“stateoftheart”,presentingsomeofthemainresultsandmethods
tosolvevarioustypesofproblems.Thecontributors,besidesbeingwidelyacknowledged
expertsinthesubject,areknownfortheirabilityofclearlydivulgingtheirsubject.Weare
convincedthatpapersliketheonesinthisvolumeareveryuseful,bothfortheexpertsand
particularly for younger research fellows or beginners in the subject. The editors would
liketoexpresstheirdeepestgratitudetoallcontributorstothisvolumefortheeffortmade
inthisdirection.
The contributions to this volume are presented in alphabetical order according to the
nameofthefirstauthor.ThepaperbyAgarwalandO’Regandealswithsingularinitialand
boundary value problems (the nonlinear term may be singular in its dependent variable
and is allowed to change sign). Some old and new existence results are established and
the proofs are based on fixed point theorems, in particular, Schauder’s fixed point theo-
remandaLeray–Schauderalternative.ThepaperbyDeCosterandHabetsisdedicatedto
themethodofupperandlowersolutionsforboundaryvalueproblems.Thesecondorder
equationswithvariouskindsofboundaryconditionsareconsidered.Theemphasisisput
v
vi Preface
on well ordered and non-well ordered pairs of upper and lower solutions, connection to
the topologicaldegree and multiplicity of the solutions. The contributionof Došlý deals
withhalf-linearequationsofthesecondorder.Theprincipalpartoftheseequationsisrep-
resented by the one-dimensionalp-Laplacian and the author concentrates mainly on the
oscillatory theory. The paper by Jacobsen and Schmitt is devoted to the study of radial
solutionsforquasilinearellipticdifferentialequations.Thep-Laplacianservesagainasa
prototype of the main part in the equation and the domains as a ball, an annual region,
the exterior of a ball, or the entire space are under investigation.The paper by Llibre is
dedicatedtodifferentialsystemsorvectorfieldsdefinedontherealorcomplexplane.The
authorpresentsadeepandcompletestudyoftheexistenceoffirstintegralsforplanarpoly-
nomialvectorfieldsthroughtheDarbouxiantheoryofintegrability.ThepaperbyMawhin
takesthesimpleforcedpendulumequationasamodelfordescribingavarietyofnonlinear
phenomena: multiplicity of periodic solutions, subharmonics, almost periodic solutions,
stability,boundedness,Mathersets,KAMtheoryandchaoticdynamics.Itisareviewpa-
pertakingintoaccountmorethanahundredresearcharticlesappearedonthissubject.The
paperbySrzednickiisa reviewof themainresultsobtainedbytheWaz˙ewski methodin
the theory of ordinary differential equations and inclusions, and retarded functional dif-
ferentialequations,with some applicationsto boundaryvalue problemsand detection of
chaoticdynamics.ItisconcludedbyanintroductionoftheConleyindexwithexamplesof
possibleapplications.
Last, but not least, we thank the Editors at Elsevier, who gave us the opportunity of
making available a collection of articles that we hope will be useful to mathematicians
andscientistsinterestedintherecentresultsandmethodsinthetheoryandapplicationsof
ordinarydifferentialequations.
List of Contributors
Agarwal,R.P.,FloridaInstituteofTechnology,Melbourne,FL(Ch.1)
DeCoster,C.,UniversitéduLittoral,CalaisCédex,France(Ch.2)
Došlý,O.,MasarykUniversity,Brno,CzechRepublic(Ch.3)
Habets,P.,UniversitéCatholiquedeLouvain,Louvain-la-Neuve,Belgium(Ch.2)
Jacobsen,J.,HarveyMuddCollege,Claremont,CA(Ch.4)
Llibre,J.,UniversitatAutónomadeBarcelona,Bellaterra,Barcelona,Spain(Ch.5)
Mawhin,J.,UniversitéCatholiquedeLouvain,Louvain-la-Neuve,Belgium(Ch.6)
O’Regan,D.,NationalUniversityofIreland,Galway,Ireland(Ch.1)
Schmitt,K.,UniversityofUtah,SaltLakeCity,UT (Ch.4)
Srzednicki,R.,InstituteofMathematics,JagiellonianUniversity,Kraków,Poland(Ch.7)
vii
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Contents
Preface v
ListofContributors vii
1. Asurveyofrecentresultsforinitialandboundaryvalueproblemssingularinthe
dependentvariable 1
R.P.AgarwalandD.O’Regan
2. Theloweranduppersolutionsmethodforboundaryvalueproblems 69
C.DeCosterandP.Habets
3. Half-lineardifferentialequations 161
O.Došlý
4. Radialsolutionsofquasilinearellipticdifferentialequations 359
J.JacobsenandK.Schmitt
5. Integrabilityofpolynomialdifferentialsystems 437
J.Llibre
6. Globalresultsfortheforcedpendulumequation 533
J.Mawhin
7. Waz˙ewskimethodandConleyindex 591
R.Srzednicki
AuthorIndex 685
SubjectIndex 693
ix