Playing Around Resonance: An Invitation to the Search of Periodic Solutions for Second Order Ordinary Differential Equations

314 Pages·2016·3.21 MB·English

by  Alessandro Fonda (auth.)

Birkhäuser Advanced Texts
Basler Lehrbücher
Alessandro Fonda
Playing
Around
Resonance
An Invitation to the Search of
Periodic Solutions for Second Order
Ordinary Differential Equations

BirkhäuserAdvancedTextsBaslerLehrbücher
Serieseditors
StevenG.Krantz,WashingtonUniversity,St.Louis,USA
ShrawanKumar,UniversityofNorthCarolinaatChapelHill,ChapelHill,USA
JanNekováˇr,UniversitéPierreetMarieCurie,Paris,France
Moreinformationaboutthisseriesat:http://www.springer.com/series/4842
Alessandro Fonda
Playing Around Resonance
An Invitation to the Search of Periodic
Solutions for Second Order Ordinary
Differential Equations
AlessandroFonda
DipartimentodiMatematicaeGeoscienze
UniversitaJdegliStudidiTrieste
Trieste,Italy
ISSN1019-6242 ISSN2296-4894 (electronic)
BirkhaRuserAdvancedTextsBaslerLehrbuRcher
ISBN978-3-319-47089-4 ISBN978-3-319-47090-0 (eBook)
DOI10.1007/978-3-319-47090-0
LibraryofCongressControlNumber:2016958441
©SpringerInternationalPublishingAG2016
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To Rodica
Contents
1 PreliminariesonHilbertSpaces .......................................... 1
1.1 TheHilbertSpaceStructure ......................................... 1
1.2 SomeExamplesofHilbertSpaces................................... 3
1.3 FundamentalProperties.............................................. 6
1.4 Subspaces............................................................. 8
1.5 OrthogonalSubspaces................................................ 10
1.6 TheOrthogonalProjection........................................... 13
1.7 BasisinaHilbertSpace.............................................. 14
1.8 LinearFunctions...................................................... 19
1.9 WeakConvergence................................................... 25
1.10 ConcludingRemarks................................................. 28
2 OperatorsinHilbertSpaces............................................... 31
2.1 FirstDefinitions ...................................................... 31
2.2 TheAdjointOperator ................................................ 33
2.3 ResolventSetandSpectrum......................................... 36
2.4 SelfadjointOperators................................................. 39
2.5 OperatorsinRealHilbertSpaces.................................... 42
2.6 ConcludingRemarks................................................. 45
3 TheSemilinearProblem................................................... 47
3.1 TheMainProblem ................................................... 47
3.2 PropertiesoftheDifferentialOperator.............................. 49
3.3 TheLinearEquation ................................................. 52
3.3.1 TheCase(cid:2)>0 .............................................. 53
3.3.2 TheCase(cid:2)<0 .............................................. 56
3.3.3 Conclusions .................................................. 57
3.4 TheContractionTheorem............................................ 57
3.5 Nonresonance:ExistenceandUniqueness.......................... 59
3.6 EquationsinHilbertSpaces.......................................... 61
3.7 ConcludingRemarks................................................. 68
vii
viii Contents
4 TheTopologicalDegree.................................................... 71
4.1 TheBrouwerDegree................................................. 71
4.2 FurtherConsiderationsontheBrouwerDegree .................... 87
4.3 TheLeray–SchauderDegree......................................... 90
4.4 ConcludingRemarks................................................. 98
5 NonresonanceandTopologicalDegree................................... 101
5.1 TheUseofSchauderTheorem ...................................... 101
5.2 LowerandUpperSolutions.......................................... 104
5.3 TheContinuationPrinciple .......................................... 108
5.4 AsymmetricOscillators.............................................. 111
5.5 NonlinearNonresonance............................................. 112
5.6 Non-bilateralConditions............................................. 119
5.7 TheAmbrosetti–ProdiProblem ..................................... 129
5.8 ConcludingRemarks................................................. 133
6 PlayingAroundResonance................................................ 137
6.1 SomeUsefulInequalities ............................................ 137
6.2 ResonanceattheFirstEigenvalue................................... 139
6.3 Landesman–Lazer:ResonanceatHigherEigenvalues ............. 141
6.4 TheLazer–LeachCondition......................................... 144
6.5 Landesman–LazerConditions:TheAsymmetricCase............. 145
6.6 Lazer–LeachConditionsfortheAsymmetricOscillator........... 149
6.7 MoreSubtleNonresonanceConditions............................. 151
6.8 ConcludingRemarks................................................. 155
7 TheVariationalMethod ................................................... 157
7.1 DefinitionoftheFunctional ......................................... 157
7.2 Minimization ......................................................... 161
7.3 TheEkelandPrinciple................................................ 164
7.4 TheSearchofSaddlePoints......................................... 165
7.5 ConcludingRemarks................................................. 171
8 AtResonance,Again ....................................................... 173
8.1 ResonanceattheFirstEigenvalue................................... 174
8.2 SubharmonicSolutions .............................................. 176
8.3 Ahmad–Lazer–Paul:ResonanceatHigherEigenvalues............ 181
8.4 Landesman–LazervsAhmad–Lazer–Paul.......................... 184
8.5 PeriodicNonlinearities............................................... 187
8.6 ConcludingRemarks................................................. 190
9 Lusternik–SchnirelmannTheory......................................... 193
9.1 ThePeriodicProblemforSystems .................................. 193
9.2 AnEquivalentFunctional............................................ 194
9.3 SomeHintsonDifferentialEquations............................... 197
9.4 Lusternik–SchnirelmannCategory.................................. 199
9.5 MultiplicityofCriticalPoints ....................................... 201
Contents ix
9.6 RelativeCategory .................................................... 206
9.7 ConcludingRemarks................................................. 211
10 ThePoincaré–BirkhoffTheorem ......................................... 213
10.1 TheMultiplicityResult .............................................. 214
10.2 AModifiedSystem................................................... 215
10.3 TheVariationalSetting............................................... 218
10.4 FiniteDimensionalReduction....................................... 221
10.5 PeriodicSolutionsoftheOriginalSystem.......................... 223
10.6 ThePoincaré–BirkhoffTheoremonanAnnulus ................... 225
10.7 ConcludingRemarks................................................. 227
11 AMyriadofPeriodicSolutions ........................................... 231
11.1 EquationsDependingonaParameter............................... 231
11.2 SuperlinearProblems ................................................ 243
11.3 ForcedSuperlinearEquations ....................................... 250
11.4 ConcludingRemarks................................................. 253
A SpacesofContinuousFunctions.......................................... 255
A.1 UniformConvergence................................................ 255
A.2 ContinuousFunctionswithCompactDomains..................... 257
A.3 UniformlyContinuousFunctions.................................... 258
A.4 TheAscoli–ArzelàTheorem......................................... 259
A.5 TheStone–WeierstrassTheorem .................................... 261
B DifferentialCalculusinNormedSpaces ................................. 265
B.1 TheFréchetDifferential.............................................. 265
B.2 SomeComputationalRules.......................................... 267
B.3 TheMeanValueTheorem ........................................... 270
B.4 TheGateauxDifferential............................................. 272
B.5 PartialDifferentials................................................... 273
B.6 TheImplicitFunctionTheorem ..................................... 276
B.7 HigherOrderDifferentials........................................... 282
C ABriefAccountonDifferentialForms .................................. 287
C.1 PreliminaryDefinitions .............................................. 287
C.2 TheExternalDifferential ............................................ 289
C.3 Pull-BackFunctions.................................................. 290
C.4 IntegratingM-DifferentialFormsOverM-Surfaces................ 291
C.5 DifferentiableManifolds............................................. 292
C.6 Orientation............................................................ 293
C.7 TheStokes–CartanTheorem......................................... 294
Bibliography...................................................................... 297
Index............................................................................... 307
Introduction
This book is an introductionto the problem of the existence of solutions to some
typeofsemilinearboundaryvalueproblems.Itarisesfromaseriesofcourseswhich
Ihavegiventoundergraduateandgraduatestudentsinthelastfewyears.
The aim of the book is to give the possibility to any good student to reach
a research level in this field, starting from the basic knowledge of mathematical
analysis which is usually acquired before graduation. To this aim, I will develop
sometoolswhichcouldbeusedtoattackmanydifferentboundaryvalueproblems,
arising from ordinary or partial differential equations. However, I have chosen
to deal mainly with the periodic problem for a second-order scalar ordinary
differentialequation.Onereasonforthischoiceisthatthisapparentlysimplemodel
alreadyshowsso manydifferentaspects, andcanbe approachedbysuchdifferent
techniques, that it seems the ideal starting point to the further understanding of
more technical boundary value problems. Another reason comes, of course, from
itsintrinsicimportanceintheapplications.
So,Iwillbeconcernedwithanequationofthetype
x00Cg.t;x/D0; (1)
where g W R (cid:2)R ! R is a continuous function, which is T-periodic in its first
variable.Themainproblemwillbetofindsomeconditionsonthefunctiongwhich
guaranteetheexistenceofT-periodicsolutionsofEq.(1).
Moregenerally,wewilldealwiththeproblem
(cid:2)
x00Cg.t;x/D0;
.P/
x.0/Dx.T/; x0.0/Dx0.T/;
whereg WŒ0;T(cid:3)(cid:2)R! Riscontinuous.Indeed,ifg.t;x/isdefinedonR(cid:2)R,and
T-periodicinitsfirstvariable,itiseasytoseethatanysolutionx.t/ofproblem(P)
canbeextendedtothewholeRasaT-periodicsolutionofEq.(1).
xi