Table Of ContentUniversitext
Pablo Amster
Topological
Methods in
the Study of
Boundary Value
Problems
Universitext
Universitext
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Pablo Amster
Topological Methods in the
Study of Boundary Value
Problems
123
PabloAmster
DepartamentodeMatema´tica
FCEN-UniversidaddeBuenosAires
andIMAS-CONICET
BuenosAires,Argentina
ISSN0172-5939 ISSN2191-6675(electronic)
ISBN978-1-4614-8892-7 ISBN978-1-4614-8893-4(eBook)
DOI10.1007/978-1-4614-8893-4
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Preface
This bookconstitutes an elementaryintroductionto the applicationof topological
techniquesinnonlinearanalysis.Forthereader’sconvenience,onlyboundaryvalue
problemsforordinarydifferentialequationsaretreated,althoughmostoftheideas
canbegeneralizedforpartialdifferentialequationsandsomeotherareasofmathe-
matics.Thisapproachwillallowthestudenttoavoidmanyofthetechnicaldifficul-
tiesrelatedtononlinearproblemsandfocusontheapplicationoftopologicalmeth-
ods.Onlybasicknowledgeofthemaintopicsinanalysisisneeded,makingthebook
easytounderstandfornonspecialists,too.Wheneverpossible,justelementarytools
areemployed;inparticular,inmanysituationswearriveatimportantandnontrivial
resultsbymeansofelementarytechniques.Despiteitssimplicity,themainingredi-
entsofnonlinearanalysisarepresent,soreaderswithsomeexperienceinfunctional
analysisordifferentialequationsmayalsofindsomeelementsthatcomplementand
enrichtheirtoolsforsolvingnonlinearproblemsinmanydifferentfields.Thestyle
throughoutthebookisconciseandinformal,allowingstudentsofalllevelstohave
afirstglimpseatthisinterestingandbeautifulbranchofmathematics.
This book could never have existed without the supportof my colleagues, stu-
dents, and friends (most of them belong to at least two of the listed categories):
Rafael Ortega, Mo´nica Clapp, Colin Rogers, Man Kam Kwong, Alfonso Castro,
LevIdels,JorgeCossio,Julia´nHaddad,PabloDeNa´poli,JuanPabloPinasco,Diego
Rial,PaulaKuna,AlbertoDe´boli,ManuelMaurette,andRoc´ıoBalderrama.Allthat
I’velearnedaboutmathematicsisduetothem.Also,I’mgratefulforthesupportof
myfamilyandtheteamof“nonmathematical”friendsfromwhomI’velearnedall
thatIknowaboutlife.IwanttogivespecialthankstoDonnaChernykfortrusting
in this project and for all her help and patience and to the anonymous reviewers
forallthecorrectionsandcommentsthathelpedtoimprovethefirstversionofthis
manuscript.Allremainingmistakesandflawsofthetextaremysoleresponsibility.
BuenosAires,Argentina PabloAmster
v
Introduction
Nonlinearanalysisisafieldwithalargenumberofapplicationsinvarioussciences.
In particular, the study of boundary value problems for differential equations has
been the subject of intense research in recent decades. Many different techniques
havebeendevelopedforthestudyofnonlinearproblems;amongthem,oneofthe
most effectivetechniquesconsists in the use of diverse topologicalmethods,such
astheshootingmethod,fixedpointtheorems,upperandlowersolutions,ordegree
theory. In particular, topological methods have proved to be successful for many
problemsthat have no variational structure. A first basic result in the direction of
thefixedpointmethodsisthenowwell-knownBanachfixedpointtheorem,which
generalizesthe methodof successive approximationsproposedby Picard. Despite
its simplicity (and the fact that it is almost 100 years old), the Banach theorem is
still an efficient and populartoolfor provingmanydifferentexistence-uniqueness
results. However, in some cases the conditionsof this theorem are too restrictive,
andmorepowerfultechniquesarerequired.AnexampleistheSchauderfixedpoint
theorem,whichcanberegardedasanextension,foracompactoperatorinaBanach
space, of the well-knownBrouwer theoremand is especially usefulin the field of
differential equations, where the associated operators usually have a compact in-
verseinsomeappropriatespace.TheSchaudertheoremcanalsobeusedtodevelop
themethodofupperandlowersolutions,whichmakesitpossibletoprovetheexis-
tenceandlocationofsolutionsforaverygeneralclassofequations.Also,thereare
somefixedpointtheoremsinconesoftheKrasnoselskiitypethathaveapplications
tosomespecificproblems,forinstance,manyproofsofexistenceofpositivesolu-
tionsofsomedifferentialequations,frequentlyfoundinreal-worldapplications,are
basedonthesekindsofresults.
All the aforementioned techniques can be introduced within the more general
scopeoftopologicaldegreetheory,which,roughlyspeaking,isanalgebraiccount
of the zeros of a continuous function defined over a bounded subset of a normed
space. In the finite-dimensional case, it was defined by Brouwer and provides a
straightforwardproofofhisfixedpointtheorem,amongotherresults.Theextension
toageneralBanachspaceisduetoLerayandSchauderandassumesthatthefunc-
tionisacompactperturbationoftheidentity,namely,anoperatorofthetypeI−K,
vii
viii Introduction
with K compact.Again,this settingcan be regardedas“natural”in the contextof
manyboundaryvalueproblemsfordifferentialequations.Besidesthesolutionprop-
erty,whichensuresthatamappingwithnonzerodegreehasazero,oneofthemost
powerfulpropertiesofthedegreeisitshomotopyinvariancethatmakesitpossible,
underappropriateconditions,totransformaproblemintoasimpleroneforwhich
the degree is easy to compute. The equation under study is thus embeddedinto a
one-parameterfamilyofequations;theexistenceofaprioriboundsofthesolutions
guaranteesthat the degree will be constant over the deformation.The topological
degreeisparticularlyusefulinso-calledresonantproblems,thoseinwhichtheas-
sociated linear operator is noninvertible, and hence it is not always clear how to
convertthemintoafixedpointequation.
The book is self-contained,in the sense that only basic notions of analysis are
neededtounderstandmostofthecontents.Theexamplesmainlyconcernboundary
valueproblemsforordinarydifferentialequations.Inmostcases,weshalltakeasa
modelequationthesecond-orderproblem
u(cid:2)(cid:2)(t)= f(t,u(t),u(cid:2)(t)), 0<t<1,
with f continuous,undertheboundaryconditions
u(0)=u(1)=0 (Dirichlet),
u(cid:2)(0)=u(cid:2)(1)=0 (Neumann),
or
u(0)=u(1), u(cid:2)(0)=u(cid:2)(1) (periodic),
among others. The latter conditions can be interpreted as truly periodic when
f(t+1,u,v)= f(t,u,v)forallu,v,andt∈R:inthiscase,asolutionu:[0,1]→R
canbeextendedperiodicallytoaC2functiondefinedinthewholerealline.Thisset-
tingwillbeparticularlyusefulwhendealingwithsomedelaydifferentialequations,
in which the nonlinear term f also depends on u(t−τ) for some τ>0. In some
cases, ourmodelequationwillinfactbeasystemof nequations:aswe shallsee,
thisextensionisnotalwaystrivialandmayinvolvesomegeometricalortopological
difficulties.
Theresultspresentedherearenotthebestofallpossibleresults,inthesensethat,
in mostcases, we prioritizedgivingan intuitiveandeasy approachoverobtaining
betterorsharpertheorems.Wefocusallthetimeonthemethodsandonthespecific
problems;inparticular,thisisoneofthereasonsforwhichallexamplesreferonlyto
ordinarydifferentialequations,whichmakesitpossibletoavoidsometechnicalities.
Also, we havechosento workalwaysin the spacesof continuousor continuously
differentiablefunctions,althoughbetterresultscanbeobtainedusing,forexample,
Sobolevspaces. In many cases, the same problemis studied using differenttools,
soitmayhappensometimesthataresultpresentedinonechapterisimprovedina
laterone.Thereadermayalsofeelthatsomeofthecomputationsrequiredforthe
differentmethodsareunnecessarilyrepeatedindifferentchapters,butthiswasdone
forthesolepurposeofpreservingthe“self-containedness”spiritofthetext.
Introduction ix
Thebookisorganizedasfollows.InChap.1,we introduceoneofthesimplest
topologicalmethods,usuallyknownas theshootingmethod,whichbasicallycon-
sistsinreducingaproblemtoafinite-dimensionalequationforacertainparameter
λ.Then,appropriatetoolscanbeused,suchastheBrouwerfixedpointtheoremor
equivalentresults.Thechapterwasdesignedtobeself-containedandemploysonly
conceptsfrombasiccalculus;forsimplicity,ourstudyofsystemsisrestrictedhere
to the two-dimensionalcase, for which we presenta veryelementaryproofof the
fixed point theorems we shall be using. There are many extensions and improve-
mentsofthebasicresults,whichrequireslightlymoreadvancedtopics(e.g.,Stone–
Weierstrasstheorems);forthisreason,theywereincludedinstarredsections.This
does not mean, of course, they are extremely difficult: the idea was just to show
that most of the topics—the nonstarred sections—could be understood within the
contextofafirstcourseincalculus.
The nextchapter is devoted to the Banach fixed pointtheorem and some of its
immediateconsequences.Inparticular,weshallprovetheusualversionoftheim-
plicitfunctiontheoreminBanachspacesandpresentsomeapplicationstoboundary
valueproblems.Thisrequiresaknowledgeofthebasicnotionsofdifferentiationin
Banachspaces,whichforthesakeofcompletenessarepresentedhere.
In Chap. 3 we developsome iterativemethodssuch as the monotoneiterations
methodandtheNewtonmethodandsomeofitsvariants.Applicationsaregivento
someboundaryvalueproblems.Also,weintroduceaCantordiagonalizationargu-
ment,whichmakesitpossibletodealwithproblemsinunboundeddomains.
In Chap. 4 we prove the general version of the Brouwer theorem and a well-
knownextensiontoBanachspaces:theSchaudertheorem.Amongotheruses,this
latterresultallowsustogiveacompleteversionofthemethodofupperandlower
solutions introduced in the previous chapter, with applications to many different
problems.As a corollary,we obtainthe so-calledSchaefertheorem,which canbe
regardedasaparticularcaseoftheLeray–Schaudercontinuationtechniques.
These techniques require a more sophisticated topological tool: the aforemen-
tioned topological degree, constructed with some detail in Chap. 5. As usual, the
Brouwerdegreeisintroducedfirstandthenextendedforcompactperturbationsof
the identity in a Banach space, namely, the Leray–Schauder degree. The specific
difficultiesofthe constructionarenotessentialfortheapplications,soreadersnot
particularlyinterestedincertaintopologicalissuesmayavoidmostofthecontents
inthischapterandkeepinmindonlythemainpropertiesofthedegreemapping.
Finally,inChap.6wepresentapplicationstovariousboundaryvalueproblems.
Startingwithspecificexamples,weobtainsomegeneralcontinuationtheoremsthat
can be applied in many situations. In particular, most of the sections are devoted
tothestudyofresonantproblems,forwhichwediscusssomeclassicalresultsand
differentextensions.
Forthereader’sconvenience,weincludeabriefreviewofthemainresultsfrom
the theory of ordinary differentialequations used in the book. The list reduces to
some fundamentaltheorems such as existence and uniqueness, continuousdepen-
dence,andafewspecificfactsconcerningsecond-orderequations.Also,wegivean
accountofsomedefinitionsandelementaryaspectsofthegeneraltheoryofmetric
