Handbook of differential equations. Ordinary differential equations

583 Pages·2005·2.812 MB·English

by  A. Canada, P. Drabek, A. Fonda

HANDBOOK
OF DIFFERENTIAL EQUATIONS
ORDINARY DIFFERENTIAL EQUATIONS
VOLUME II
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H
ANDBOOK
D E
OF IFFERENTIAL QUATIONS
O D
RDINARY IFFERENTIAL
E
QUATIONS
V II
OLUME
Editedby
A. CAÑADA
DepartmentofMathematicalAnalysis,FacultyofSciences,
UniversityofGranada,Granada,Spain
P. DRÁBEK
DepartmentofMathematics,FacultyofAppliedSciences,
UniversityofWestBohemia,Pilsen,CzechRepublic
A. FONDA
DepartmentofMathematicalSciences,FacultyofSciences,
UniversityofTrieste,Trieste,Italy
2005
NORTH
HOLLAND
Amsterdam Boston Heidelberg London NewYork Oxford
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Preface
This handbook is the second volume in a series devoted to self contained and up-to-date
surveys in the theory of ordinary differential equations, written by leadingresearchers in
the area. All contributors have made an additional effort to achieve readability for math-
ematicians and scientists from other related fields, in order to make the chapters of the
volumeaccessibletoawideaudience.Theseideasfaithfullyreflectthespiritofthismulti-
volume and the editors hope that it will become very useful for research, learning and
teaching.Weexpressourdeepestgratitudetoallcontributorstothisvolumefortheirclearly
writtenandelegantarticles.
Thisvolumeconsistsofsixchapterscoveringavarietyofproblemsinordinarydiffer-
entialequations.Both,puremathematicalresearchandrealwordapplicationsarereflected
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 Barbu and Lefter is dedicated to
thediscussionofthefirstordernecessaryandsufficientconditionsofoptimalityincontrol
problemsgovernedbyordinarydifferentialsystems.Theauthorsprovideacompleteanaly-
sisofthePontriaghinmaximumprincipleanddynamicprogrammingequation.Thepaper
byBartschandSzulkinisasurveyonthemostrecentadvancesinthesearchofperiodic
andhomoclinicsolutionsforHamiltoniansystemsbytheuseofvariationalmethods.After
developingsomebasicprinciplesofcriticalpointtheory,theauthorsconsideravarietyof
situations where periodic solutions appear, and they show how to detect homoclinic so-
lutions,includingtheso-called“multibump”solutions,aswell.ThecontributionofCârja˘
and Vrabie deals with differential equations on closed sets. After some preliminaries on
Brezis–BrowderorderingprincipleandClarke’stangentcone,theauthorsconcentrateon
problemsofviabilityandproblemsofinvariance.Moreover,thecaseofCarathéodorysolu-
tionsanddifferentialinclusionsareconsidered.ThepaperbyHirschandSmithisdedicated
tothetheoryofmonotonedynamicalsystemswhichoccurinmanybiological,chemical,
physicalandeconomicmodels.Theauthorsgiveaunifiedpresentationandabroadrange
oftheapplicabilityofthistheorylikedifferentialequationswithdelay,secondorderqua-
silinearparabolicproblems,etc.ThepaperbyLópez-Gómezanalyzesthedynamicsofthe
positivesolutionsofageneralclassofplanarperiodicsystems,includingthoseofLotka–
Volterratypeandamoregeneralclassofmodelssimulatingsymbioticinteractionswithin
global competitive environments. The mathematical analysis is focused on the study of
coexistencestatesandtheproblemofascertainingthestructure,multiplicityandstability
ofthesecoexistencestatesinpurelysymbioticandcompetitiveenvironments.Finally,the
paperbyNtouyasisasurveyonnonlocalinitialandboundaryvalueproblems.Here,some
oldandnewresultsareestablishedandtheauthorshowshowthenonlocalinitialorbound-
v
vi Preface
aryconditionsgeneralizetheclassicalones,havingmanyapplicationsinphysicsandother
areasofappliedmathematics.
WethankagaintheEditorsatElsevierforefficientcollaboration.
List of Contributors
Barbu,V.,“Al.I.Cuza”University,Ias¸i,Romania,and“OctavMayer”InstituteofMath-
ematics,RomanianAcademy,Ias¸i,Romania(Ch.1)
Bartsch,T.,UniversitätGiessen,Giessen,Germany(Ch.2)
Cârja˘,O.,“Al.I.Cuza”University,Ias¸i,Romania(Ch.3)
Hirsch,M.W.,UniversityofCalifornia,Berkeley,CA(Ch.4)
Lefter,C.,“Al.I.Cuza”University,Ias¸i,Romania,and“OctavMayer”InstituteofMath-
ematics,RomanianAcademy,Ias¸i,Romania(Ch.1)
López-Gómez,J.,UniversidadComplutensedeMadrid,Madrid,Spain(Ch.5)
Ntouyas,S.K.,UniversityofIoannina,Ioannina,Greece(Ch.6)
Smith,H.,ArizonaStateUniversity,Tempe,AZ(Ch.4)
Szulkin,A.,StockholmUniversity,Stockholm,Sweden(Ch.2)
Vrabie,I.I.,“Al.I.Cuza”University,Ias¸i,Romania,and“OctavMayer”InstituteofMath-
ematics,RomanianAcademy,Ias¸i,Romania(Ch.3)
vii
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Contents
Preface v
ListofContributors vii
ContentsofVolume1 xi
1. Optimalcontrolofordinarydifferentialequations 1
V.BarbuandC.Lefter
2. Hamiltoniansystems:periodicandhomoclinicsolutionsbyvariationalmethods 77
T.BartschandA.Szulkin
3. Differentialequationsonclosedsets 147
O.Cârja˘ andI.I.Vrabie
4. Monotonedynamicalsystems 239
M.W.HirschandH.Smith
5. Planarperiodicsystemsofpopulationdynamics 359
J.López-Gómez
6. Nonlocalinitialandboundaryvalueproblems:asurvey 461
S.K.Ntouyas
Authorindex 559
Subjectindex 565
ix