AdvancedDifferential Equations
YoussefN.Raffoul ProfessorofMathematics
UniversityofDayton
Dayton,OH,UnitedStates
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CHAPTER1PreliminariesandBanachspaces
1.4 MetricsandBanachspaces
1.5 Variationofparameters.............................18
1.6 Specialdifferentialequations
CHAPTER2Existenceanduniqueness
2.1 Existenceanduniquenessofsolutions
2.2 ExistenceonBanachspaces
2.3 Existencetheoremforlinearequations
2.4 Continuationofsolutions............................45
2.5 Dependenceoninitialconditions..
2.6 Exercises........................................53
CHAPTER3Systemsofordinarydifferentialequations
3.1 Existenceanduniqueness
3.2
3.2.1Fundamentalmatrix
3.5 Exercises........................................98
CHAPTER4Stabilityoflinearsystems
4.1 Definitionsandexamples............................103
4.2
4.3 Floquettheory
4.3.1Mathieu’sequation
4.3.2ApplicationstoMathieu’sequation
4.4 Exercises........................................126
CHAPTER5Qualitativeanalysisoflinearsystems
5.1 Preliminarytheorems.. .............................129
5.2 Near-constantsystems. .............................133
5.3 Perturbedlinearsystems ............................137
5.4 Autonomoussystemsintheplane.
5.5 Hamiltonianandgradientsystems.....................145
5.6 Exercises........................................155
CHAPTER6Nonlinearsystems
6.1 Bifurcationsinscalarsystems........................159
6.2 Stabilityofsystemsbylinearization
6.3 AnSIRepidemicmodel
6.4 Limitcycle......................................181
6.5 Lotka–Volterracompetitionmodel.
6.6 Bifurcationinplanarsystems
6.7 ManifoldsandHartman–Grobmantheorem...
6.7.1Thestablemanifoldtheorem
6.7.2Globalmanifolds
6.7.3Centermanifold.............................211
6.7.4Centermanifoldandreducedsystems..
6.7.5Hartman–Grobmantheorem
6.8 Exercises........................................224
CHAPTER7Lyapunovfunctions
7.1.1Stabilityofautonomoussystems
7.2 Globalasymptoticstability
7.3 Instability.
7.4 ω -limitset.......................................260
7.5 ConnectionbetweeneigenvaluesandLyapunovfunctions...267 7.6 Exponentialstability..
8.5 Stabilityusingfixedpointtheory..
8.5.1Neutraldifferentialequations
8.6 Exponentialstability..
8.7 Existenceofpositiveperiodicsolutions
8.8 Exercises........................................326
9.1 Applicationstoordinarydifferentialequations.
9.1.1Periodicsolutions............................334
9.2 Applicationstodelaydifferentialequations... ...........335
9.2.1Themaininversion...........................338
9.2.2Variabletimedelay...........................340
9.3 Exercises........................................341
Bibliography..................................................343 Index.......................................................347
StudentResources
ForthePartialSolutionsManual,visitthecompanionsite: https://www.elsevier.com/books-and-journals/book-companion/9780323992800
Preface
Differentialequations arewidelyusedbymathematicians,physicists,engineers,biologists,chemists,andscientistswhoworkinrelevantfields.Theyencountertheuseof differentialequationsinthestudyofNewton’slawofcooling,Maxwell’sequations, Newton’slawsofmotion,fluiddynamicsequations,equationsinplasmadynamics, equationsinstellardynamics,Hook’slaw,Schrödinger’sequation,acousticwave equation,equationsinchemicalkinetics,equationsinthermodynamics,Einstein’s equationsforgeneralrelativity,populationmodels,epidemics,andsoon.Formany years,theauthorhasbeenencouragedbythegraduatestudentsattheUniversityof Daytontowriteaconciseandreader-friendlybookonthesubjectofadvanceddifferentialequations.Sothisbookgrewoutoflecturenotesthattheauthorhasbeen constantlyrevisingandusingforagraduatecourseindifferentialequations.The bookshouldserveasatwo-semestergraduatetextbookinexploringthetheoryand applicationsofordinarydifferentialequationsanddifferentialequationswithdelays. Itisintendedforstudentswhohavebasicknowledgeofordinarydifferentialequationsandrealanalysis.Whilewritingthisbook,theauthortriedtobalancerigor andpresentingthemostdifficultmaterialinanelementaryformatbyadoptingeasier andfriendliernotationsthatmakethebookaccessibletoawideaudience.Itwasthe author’smainintentiontoprovidemanyexamplestoillustratethetheoryconveyed inthetheorems.Theauthormadeeveryefforttoincludecontemporarytopicssuch astheuseof fixedpointtheory inseveralplacestoprovetheexistenceanduniqueness,variousnotionsofstability,andtheexistenceofpositiveperiodicsolutionson Banachspaces.Whatmakesthebookappealinganddistinguishedfromotherbooks istheadditionofChapters 8 and 9 ondelaydifferentialequationswithadvanced topics.Theauthorisconvincedthatanystudentwhocompletesthewholebook,especiallyChapters 8 and 9,shouldbereadytocarryonwithmeaningfulresearchin delaydifferentialsystems.
Muchofthepedagogicalandmathematicaldevelopmentofthisbookisinfluencedbytheauthor’sstyleofpresentation.Theliteratureondifferentialequationsis vastandwellestablished,andsomeoftheideasfoundtheirwayintothisbook.
Sincestabilityisthecentralpartofthisbook,namelybytheLyapunovmethod, wemustmentionsomehistory.LyapunovfunctionsarenamedafterAlexander Lyapunov,aRussianmathematician,whoin1892publishedhisbook TheGeneral ProblemofStabilityofMotion.Lyapunovwasthefirsttoconsiderthemodifications necessaryinnonlinearsystemstothelineartheoryofstabilitybasedonlinearizingnearapointofequilibrium.Hiswork,initiallypublishedinRussianandthen translatedtoFrench,receivedlittleattentionformanyyears.InterestinLyapunov stabilitystartedsuddenlyduringtheColdWarperiodwhenhismethodwasfoundto beapplicabletothestabilityofaerospaceguidancesystems,whichtypicallycontain
strongnonlinearitiesnottreatablebyothermethods.Morerecently,theconceptof theLyapunovexponentrelatedtoLyapunov’sfirstmethodofdiscussingstabilityhas receivedwideinterestinconnectionwithchaostheory.
Chapter 1 dealswithvariousintroductorytopics,includingvariationofparametersformula,metricspaces,andBanachspaces.
InChapter 2 weintroduceGronwall’sinequalitythatwemakeuseoftoprovethe uniquenessofsolutions.Weintroducetheoremsontheexistenceanduniquenessof solutions,theirdependenceoninitialdata,andtheircontinuationonmaximalinterval.
InChapter 3 weintroducesystemsofdifferentialequations.Webrieflydiscuss howtheexistenceanduniquenesstheoremsofChapter 2 areextendedtosuitsystems. Thenwedevelopthenotionofthefundamentalmatrixasasolutionandutilizeitto writesolutionsofnon-homogeneoussystemssothattheycanbeanalyzed.
Stabilitytheoryisthecentralpartofthisbook.Chapters 4–8 aretotallydevoted tostability.InChapter 4 wearemainlyconcernedwiththestabilityoflinearsystems viathevariationofparameters.ThechapteralsoincludesanicesectionofFloquet theorywithitsapplicationtoMathieu’sequation.
Chapters 5 and 6 aredeeplydevotedtothestudyofthestabilityoflinearsystems, near-linearsystems,perturbedsystems,autonomoussystemsintheplane,andstabilitybylinearization.Chapter 5 isendedwiththestudyofHamiltoniansandgradient systems.WebeginChapter 6 bylookingatstabilitydiagramsinscalarequationsand moveintothestudyofbifurcationsasitnaturallyariseswhilelookingatstability. Bifurcationoccurswhenthedynamicsabruptlychangeascertainparametersmove acrosscertainvalues.WeendChapter 6 byconsideringstableandunstablemanifolds,whichthendelvesintotheHartman–Grobmantheorem.Thetheoremsays thatthebehaviorofadynamicalsysteminthedomainnearahyperbolicequilibrium pointisqualitativelythesameasthebehaviorofitslinearizationnearthisequilibrium point,wherehyperbolicitymeansthatnoeigenvalueofthelinearizationhasazero realpart.
Chapter 7 delvesdeeplyintothestabilityofgeneralsystemsusingLyapunov functions.WeprovegeneraltheoremsregardingthestabilityofautonomousandnonautonomoussystemsbyassumingtheexistenceofsuchLyapunovfunction.Wetouch onthenotionof ω -limitsetanditscorrelationtoLyapunovfunctions.Thechapteris concludedwithadetaileddiscussiononexponentialstability.
Chapter 8 issolelydevotedtothestudyofdelaydifferentialequations.Itcontainsrecentdevelopmentintheresearchofdelaydifferentialequations.Webegin thechapterbypointingouthowbasicresultsfromordinarydifferentialequations areeasilyextendedtodelaydifferentialequations.Weintroducethe methodofsteps andshowhowtopiecetogetherasolution.Thetransitionofmovingfromordinary differentialequationstodelaydifferentialequationswasmadesimplethroughthe extensionofLyapunovfunctionstoLyapunovfunctionals.Thenwemoveonto
awholenewconcept, fixedpointtheory.Theuseoffixedpointtheoryalleviates someofthedifficultiesthatarisefromtheuseofLyapunovfunctionalswhenstudyingstability.Lateron,weapplyfixedpointtheorytothestudyofstabilityandthe existenceofpositiveperiodicsolutionsofneutraldifferentialequationsandneutral Volterraintegro-differentialequations,respectively.Weendthechapterwiththeuse ofLyapunovfunctionalstoobtainnecessaryconditionsfortheexponentialstability ofVolterraintegro-differentialequationswithfinitedelay.
Chapter 9 dealswithcurrentresearchconcerningtheuseofa newvariationof parametersformula.Theobjectiveistointroduceanewmethodforinvertingfirstorderordinarydifferentialequationswithtime-delaytermstoobtainanewvariation ofparametersformulathatweusetostudythestability,boundedness,andperiodicity ofgeneralequationsinordinaryanddelaydifferentialequations.
AcombinationofChapters 1–3, 6,and 7 canbeusedtodeliveracourseonnonlinearsystemsforengineers.
Theauthorhasnotattemptedtogivethehistoricaloriginofthetheory,except inveryrarecases.Thisresultedinthesituationthatnoteveryreferencelistedin Referencesismentionedinthetextorthebodyofthebook.
Exercisesplayanessentiallearningtoolofthecourseandaccompanyeachchapter.Theyrangefromroutinecalculationstosolvingmoredifficultproblemstoopenendedones.Studentsmustreadtherelevantmaterialbeforeattemptingtodothe exercises.
IamindebtedtoDr.MohamedAburakhis,whofullydevelopedallthecodes forallthefiguresinthebook.Iliketothankthehundredsofgraduatestudents attheUniversityofDaytonwhomtheauthortaughtforthelast20yearsandwho helpedthepolishingandrefiningofthelecturenotes,mostofwhichhavebecome thisbook.
AheartfeltappreciationtoJeffHemmelgarnfromtheUniversityofDaytonfor carefullyreadingthewholebookandpointingoutmanytypos.
YoussefN.Raffoul UniversityofDayton Dayton,Ohio June2021
PreliminariesandBanach spaces
Webrieflydiscussbasictopicsofordinarydifferentialequationsandprovideexamplesthatillustratetheneedforacomprehensiveandsystematictheoryofdifferential equations.Inaddition,weintroducemetricsandBanachspaces,whichwewilluse throughoutthebook.
1.1 Preliminaries
Let R denotethesetofrealnumbers,let I beanintervalin R,andconsiderafunction x : I → R.Wesaythefunction x hasaderivativeatapoint t ∗ ∈ I ifthelimit
existsasafinitenumber.Inthiscase,weadoptthenotation
(t ∗ ) =
where x (t ∗ ) istheinstantaneousrateofchangeofthefunction x at t ∗ .If t ∗ isoneof theendpointsoftheinterval I ,thentheabovedefinitionofderivativebecomesthat ofaone-sidedderivative.If x (t ∗ ) existsateverypoint t ∗ ∈ I ,thenwesaythat x is differentiableon I andwrite x (t).Throughoutthebook,wemightuse dx dt toindicate x (t).Similarly,if x (t) hasaderivativefunction,thenwecallitthesecondderivative ofthefunction x(t) anddenoteitby x (t).Forhigher-orderderivatives,weusethe notations x (t),x (4) (t),...,x (n) (t), or d n x dt n .
The orderofadifferentialequation isdefinedbythehighestderivativepresentinthe equation.An nth-orderordinarydifferentialequation isafunctionalrelationofthe form
F t,x, dx dt , d 2 x dt 2 , d 3 x dt 3 ,..., d n x dt n = 0,t ∈ R (1.1.1)
https://doi.org/10.1016/B978-0-32-399280-0.00007-3
betweentheindependentvariable t andthedependentvariable x ,anditsderivatives
dx dt , d 2 x dt 2 , d 3 x dt 3 ,..., d n x dt n
Looselyspeaking,byasolutionof(1.1.1)onaninterval I ,wemeanafunction x(t) = ϕ(t) suchthat
F t,ϕ(t),ϕ (t),...,ϕ (n) (t)
isdefinedforall t ∈ I and
F t,ϕ(t),ϕ (t),...,ϕ (n) (t) = 0
forall t ∈ I .Ifwerequire,forsomeinitialtime t0 ∈ R,asolution x(t) tosatisfythe initialconditions
x(t0 ) = a0 , dx dt (t0 ) = a1 , d 2 x dt 2 (t0 ) = a2 ,...,
forconstants ai , i = 0, 1, 2,...,n 1,then(1.1.1)alongwith(1.1.2)iscalledaninitial valueproblem(IVP).
Followingthenotationof(1.1.1),afirst-orderdifferentialequationtakestheform
F t,x, dx dt = 0.
Hence,ifweassumethatwecansolvefor dx dt ,thenwehave
x (t) = f(t,x)
forsomefunction f thatsatisfiescertaincontinuityconditions.
Let x : I → R beafunction.Adifferentiablefunction z : I → R iscalledan antiderivative ofthefunction x ontheinterval I if
z (t) = x(t) forall t ∈ I.
Thesetofallantiderivativesof x isdenotedby x(t)dt andcalledtheindefiniteintegralofthefunction x .Whenwecalculatetheindefiniteintegral t 2 dt ,weinfactsolvethefirst-orderdifferentialequation x (t) = t 2 Thefamilyofitssolutionsisgivenby t 3 /3 + c ,where c isanyconstant.Thuswe maywritethesolutionas x(t) = t 3 /3 + c ,whichisaone-parameterfamilyofsolutions,thesameasthefamilyofalltheantiderivativesof t 3 .Now,ifweimpose aninitialconditiononthedifferentialequation,say x(t0 ) = x0 ,forsomeinitialtime
t0 andrealnumber x0 ,thentheconstant c isuniquelydeterminedbytherelation x0 = t 3 0 /3 + c .Inthiscasethedifferentialequationhastheuniquesolutiongivenby x(t) = t 3 /3 + x0 t 3 0 /3.However,withoutimposingthecondition x(t0 ) = x0 ,the differentialequationwouldhaveinfinitelymanysolutionsgivenby x(t) = t 3 /3 + c .
Differentialequationsplayanimportantroleinmodelingthebehaviorofphysical systemssuchasfallingbodies,vibrationofamassonaspring,andswingingpendulum.Toillustratetheneedforthetheoreticalstudyofdifferentialequationsandin particularnonlinearones,weexamineafewexamples.
Considerthefirst-orderdifferentialequation
x (t) = h(t)g(x),x(t0 ) = x0 ,t ≥ t0 ,
where h,g : R → R arecontinuous.
1. If g(x0 ) = 0,then x(t) = x0 isasolution.
2. Inaregionwhere g(x) = 0,wecandivideby g(x) sothat
x (t)
g(x(t)) = h(t).
Separatingthevariablesandthenintegratingbothsidesfrom t0 to t give
t t0 x (s)ds g(x(s)) = t t0 h(s)ds.
Usingthetransformation u = x(s) with x(t0 ) = x0 ,wearriveat
x(t) x0 du g(u) = t t0 h(s)ds.
Ifforsomefunction G,wehave dG dx = 1 g ,thentheaboveexpressionimpliesthat
G(x(t)) G(x(t0 )) = t t0 h(s)ds or x(t) = G 1 G(x0 ) + t t0 h(s)ds ,
providedthattheinverseof G exists.Notethattherightsideoftheaboveexpressiondependsontheinitialtime t0 andtheinitialvalue x0 .Thereforetoemphasize thedependenceofsolutionsontheinitialdata,wemaywriteasolution x(t) in theform x(t) = x(t,t0 ,x0 ).
FIGURE1.1
Thisexampleshowsinfinitelymanysolutions.
Inthenextexample,weillustratetheexistenceofmorethanonesolution.Consider thedifferentialequation
x (t) = 3 2 x 1/3 (t),x(0) = 0,t ∈ R
Itisclearthat x(t) = 0isasolution.Hencewemayconsiderasolution x1 (t) = 0and let
x2 (t) =
0for t ≤ 0, t 3/2 for t> 0, whichisalsoacontinuousanddifferentiablesolution.Likewise,for t1 > 0,wehave x3 (t) = 0for t ≤ t1 , (t t1 )3/2 for t>t1
Continuinginthisway,weseethatthedifferentialequationhasinfinitelymanysolutions.Similarly,if x isasolution,then x isalsoasolution(seeFig. 1.1).
Inthenextexample,weshowthatsolutionsmayescape(becomeunbounded)in finitetime.Thedifferentialequation
x (t) = x 3 (t),x(t0 ) = x0 > 0,t ≥ t0
hasthesolution
x(t) = 1 1 x 2 0 + 2(t0 t)
Wecaneasilyseethatthesolutionisonlyvalidfor t<t0 + 1 2x 2 0 andbecomesunbounded(escapes)as t approaches t0 + 1 2x 2 0 fromtheleft.
ThemostimportantapplicationinengineeringproblemsisNewton’slaw
fortheposition x(t) ofaparticlewithmass m actedonbyaforce F ,whichmaybe afunctiontotime t ,theposition x(t),andthevelocity dx(t) dt .Forexample,iftheforce isonlyduetogravity,thenwehavethesecond-orderdifferentialequation
whichhasthesolution
where c1 and c2 areconstantsthatcanbeuniquelydeterminedbyspecifyingthe positionandvelocityoftheparticleatsomeinstantoftime.
Inthenextexample,weconsidertheproblemof leakybucket
Example 1.1 Wehaveabucketwithnowaterflowingintoitandhavingaholeinthe bottom.If Q isthevolumetricflowrate,then Qin Qout = Qstored .Since Qin = 0, weobtain Qstored =−Qout .Let h(t) betheheightofthewaterinthebucketattime t , andassumethattheinitialheightattime t = 0is h0 .Let v(t) bethevelocityof theleakedwater(flowvelocity).Thevolumetricflowrate Qstored canbecalculated bymultiplyingthevelocitybytheareaofthebucket,thatis, Qstored = Abucket dh(t) dt . Similarly, Qout istheflowvelocitymultipliedbytheareaofthehole Ahole .Itfollows that Qout = Ahole v(t).Forfluidsofheight h(t),thevelocityofwatercomingoutat thebottomis v(t) = √2gh(t).Byrearrangingtheterms,wearriveatthefirst-order differentialequationin h givenby
where k = Ahole Abucket 2g> 0.Byseparatingthevariablesin(1.1.3)andthenintegrating,wearriveatthesolution
Notethatthesolution h(t) givenby(1.1.4)decreasesfromtheinitialheight h0 .Moreover,attime t ∗ = 2√h0 k ,thewateriscompletelydrained(bygravity), h(t ∗ ) = 0,and
thebucketwillremainemptyortheheightwillremainzeroafter
.Thereforewemaywritethesolutionas
Anotherinterestingapplicationistheprojectileproblemthatweanalyzeinthe nextsection.
1.2 Escapevelocity
Let M and R bethemassandradiusoftheEarth,respectively.Weareinterested infindingthesmallestinitialvelocityforamass m toexittheEarth’sgravitational field,theso-called escapevelocity.Weassumethatnoexternalforcesareactingon thesystemotherthanthegravitationalforce.Newton’suniversalgravitationallaw statesthattheforcebetweentwomassivebodiesisproportionaltotheproductofthe massesandinverselyproportionaltothesquareofthedistancebetweenthem,where themassofeachbodycanbeconsideredasconcentratedatitscenter.Foramass m withposition x abovethesurfaceoftheEarth,theforce F onthemassisgivenby
where G istheproportionalityconstantintheuniversalgravitationallaw.Theminussignmeanstheforceonthemass m pointsinthedirectionofdecreasing x .By Newton’ssecondlaw,force F = m d 2 x dt 2 (masstimesacceleration).When x = 0,that is,attheEarth’ssurface,thegravitationalforceequals mg ,where g isthegravitationalconstant.Therefore
wheretheradiusoftheEarthisknowntobe R ≈ 6350km.WetransformEq.(1.2.1) intoafirst-orderdifferentialequationintermsofthevelocity v bynotingthat d 2 x dt 2 = dv dt .Ifwewrite v(t) = v(x(t)),thatis,consideringthevelocityofthemass m asa
functionofitsdistanceabovetheEarth.Thenusingthechainrule,wehave dv dt = dv dx dx dt = v dv dx , since v = dx dt .Asaconsequence,(1.2.1)becomesthefirst-orderdifferentialequation
SupposethemassisshotverticallyfromtheEarth’ssurfacewithinitialvelocity v(x = 0) = v0 .Separatingthevariablesandintegratingbothsides,weobtain
=−g
whichgives
forsomeconstant c .Usingtheinitialvelocitycondition,wefind c =−gR + v 2 0 2
Substituting c backintothesolutionandsimplifying,wearriveatthesolution v 2 = v 2 0 gRx R + x .
Theescapevelocityisdefinedastheminimuminitialvelocity v0 ,suchthatthemass canescapetoinfinity.Therefore v0 = vescape when v → 0as x →∞.Takingthe limit,wehave 0 = v 2 0 lim x →∞ gRx R + x = v 2 0 2gR, or v 2 escape = 2gR.
With R ≈ 6350kmand g = 127,008km/h2 ,weget vescape = √2gR ≈ 40,000km/h. Incontrast,themuzzlevelocityofamodernhigh-performancerifleis4300km/h, whichisnotenoughforabulletshotintotheskytoescapefromEarth’sgravity.
Nowweformallyattempttoqualitativelyanalyzedifferentialequations.
Definition1.2.1. Let D beanopensubsetof R2 ,andlet f : D → R beacontinuous function.Let (t0 ,x0 ) ∈ D .Wesaythat x(t) = x(t,t0 ,x0 ) isasolutionof x = f(t,x),x(t0 ) = x0 , (1.2.2)
onaninterval I if t0 ∈ I , x : I → R isdifferentiable, (t,x(t)) ∈ D for t ∈ I , x (t) = f(t,x(t)) for t ∈ I ,and x(t0 ) = x0 .
Definition1.2.2. Asolution x(t) of(1.2.2)issaidtobeboundedontheinterval I =[0, ∞) ifforany t0 ∈[0, ∞) and r> 0,thereexistsa positivenumber α(t0 ,r) dependingon t0 and r suchthat |x(t,t0 ,x0 )|≤ α(t0 ,r) forall t ≥ t0 and x0 suchthat |x0 | <r .Itisuniformlyboundedif α isindependentoftheinitialtime t0 .
Definition1.2.3. Let x(t) and y(t) besolutionsof(1.2.2)withrespecttoinitial conditions x0 and y0 ,respectively.Thesolution x(t) isthensaidtobestableiffor every ε> 0,thereexists δ = δ(ε,t0 )> 0suchthat
|x(t) y(t)| <ε whenever |x0 y0 | <δ.
Considerthelineardifferentialequation
x (t) = 1,x(t0 ) = x0 . (1.2.3)
Itiseasytocheckthat x(t) = x0 + (t t0 ) isthesolutionof(1.2.3).If y(t) isanother solutionwith y(t0 ) = y0 ,thenwehave y(t) = y0 + (t t0 ).Forany ε> 0,let δ = ε . Then
|x(t) y(t)|=|x0 + (t t0 ) y0 (t t0 )|=|x0 y0 | <ε
whenever |x0 y0 | <δ .Hencethesolution x(t) isstablebutclearlyunbounded. Thissimpleexampleshowsthatthenotionofasolutionbeingunboundeddoesnot automaticallyimplythatthesamesolutionisunstablewithrespecttoanothersolution startingatadifferentinitialpoint.InChapters 6 and 7 wewilldiscussboundedness andstabilityinmoredetail.Thepreviousexamplesillustratetheneedforacoherent theoryforaddressingthefollowingissues:
1. Existenceanduniqueness.
2. Boundednessandstability.
3. Thedependenceofsolutionsontheinitialdata.
1.3 Applicationstoepidemics
The lawofmassaction isausefulconceptthatdescribesthebehaviorofasystem thatconsistsofmanyinteractingparts,suchasmolecules,thatreactwitheachother, orvirusesthatarepassedalongfromapopulationofinfectedindividualstononimmuneones.Thelawofmassactionwasfirstderivedforchemicalsystemsbut
subsequentlyfoundwideuseinepidemiologyandecology.Todescribethelawof massaction,weassumethat m substances s1 ,s2 ,...,sm togetherformaproduct withconcentration p .Thenthelawofmassactionstatesthat dp dt isproportionalto theproductofthe m concentrations si , i = 1,...,m,thatis,
= ks1 s2 ...sm
Supposewehaveahomogeneouspopulationoffixedsizedividedintotwogroups. Thosewhohavethediseasearecalledinfective,andthosewhodonothavethediseasearecalledsusceptible.Let S = S(t) bethesusceptibleportionofthepopulation, andlet I = I(t) betheinfectiveportion.Thenbyassumption,wemaynormalizethe populationandhave S + I = 1.Wefurtherassumethatthedynamicsofthisepidemic satisfiesthelawofmassaction.Hence,forpositiveconstant λ,wehavethenonlinear differentialequation
I (t) = λSI.
(1.3.1)
Let I(0) = I0 ,0 <I(0)< 1,beagiveninitialcondition.Bysubstituting S = 1 I into(1.3.1)itfollowsthat
I (t) = λI(1 I),I(0) = I0
(1.3.2)
Ifwecansolve(1.3.2)for I(t),then S(t) canbefoundfromtherelation I + S = 1. Weseparatethevariablesin(1.3.2)andobtain
dI I(1 I) = λdt.
Usingpartialfractionontheleftsideoftheequationandthenintegratingbothsides yield
+ c, orforpositiveconstant c1 ,wehave
Applying I(0) = I0 givesthesolution
Nowfor0 <I(0)< 1,thesolutiongivenby(1.3.3)isincreasingwithtimeasexpected.Moreover,usingL’Hospital’srule,wehave
Hencetheinfectionwillgrow,andeveryoneinthepopulationwilleventuallyget infected.
1.4 MetricsandBanachspaces
ThissectionisdevotedtointroductorymaterialsrelatedtoCauchysequences,metric spaces,contraction,compactness,contractionmappingprinciple,andBanachspaces. Materialsinthissectionwillbeofuseinseveralplacesofthebook,especiallyin Chapters 2, 8,and 9.Throughoutthebook,by C(I, Rn ),wedenotethespaceofall continuousfunctions f : I → Rn onaninterval I ,possiblyinfinite.
Definition1.4.1. Apair (E,ρ) isametricspaceif E isasetand ρ : E × E →[0, ∞) suchthatforall y , z,and u in E ,wehave
(a) ρ(y,z) ≥ 0, ρ(y,y) = 0,and ρ(y,z) = 0implies y = z; (b) ρ(y,z) = ρ(z,y);and (c) ρ(y,z) ≤ ρ(y,u) + ρ(u,z).
ThenextdefinitionisconcernedwithCauchysequences.
Definition1.4.2. (Cauchysequence)Asequence {xn }⊆ E isaCauchysequenceif foreach ε> 0,thereexists N ∈ N suchthat n,m>N =⇒ ρ(xn ,xm )<ε
Completemetricspacesplayamajorrolewhenshowingthatafixedpointbelongs tothemetricspaceofinterest.
Definition1.4.3. (Completenessofmetricspace)Ametricspace (E,ρ) issaidtobe completeifeveryCauchysequencein E convergestoapointin E .
Definition1.4.4. Aset L inametricspace (E,ρ) iscompactifeachsequencein L hasasubsequencewithalimitin L.
Definition1.4.5. Let {fn } beasequenceofrealfunctionswith fn :[a,b ]→ R.
1. {fn } isuniformlyboundedon [a,b ] ifthereexists M> 0suchthat |fn (t)|≤ M forall n ∈ N and t ∈[a,b ]
2. {fn } isequicontinuousat t0 ifforeach ε> 0,thereexists δ> 0suchthatfor all n ∈ N,if t ∈[a,b ] and |t0 t | <δ ,then |fn (t0 ) fn (t)| <ε .Also, {fn } is equicontinuousif {fn } isequicontinuousateach t0 ∈[a,b ].
3. {fn } isuniformlyequicontinuousifforeach ε> 0,thereexists δ> 0suchthatfor all n ∈ N,if t1 ,t2 ∈[a,b ] and |t1 t2 | <δ ,then |fn (t1 ) fn (t2 )| <ε .
Itiseasytoseethat {fn }={x n } isnotanequicontinuoussequenceoffunctions on [0, 1] buteach fn isuniformlycontinuous.
Proposition1.1. [Cauchycriterionforuniformconvergence]If {Fn } isasequence ofboundedfunctionsthatisCauchyintheuniformnorm,then {Fn } convergesuniformly.
Definition1.4.6. Areal-valuedfunction f definedon E ⊆ R issaidtobeLipschitz continuouswithLipschitzconstant K if |f(x) f(y)|≤ K |x y | forall x,y ∈ E
Itiseasytoseethatthefunction f(x) = x 2 isnotLipschitzon R.Thisisduetothe factthatforany x and y in R,wehavethat f(x) f(y) =|x 2 y 2 |=|x + y ||x y |, andsothereisnoconstant K suchthat |x 2 y 2 |≤ K |x y |.Definition 1.4.6 implies that f isgloballyLipschitzsincetheconstant K isuniformforall x and y in R.
Remark 1.1 ItisaneasyexercisethataLipschitzcontinuousfunctionisuniformly continuous.Also,ifeach fn inasequenceoffunctions {fn } hasthesameLipschitz constant,thenthesequenceisuniformlyequicontinuous.
Lemma1.1. If {fn } isanequicontinuoussequenceoffunctionsonaclosedbounded interval,then {fn } isuniformlyequicontinuous.
Proof. Suppose {fn } isequicontinuouson [a,b ].Let ε> 0.Foreach x ∈ K ,let δx > 0besuchthat |y x | <δx =⇒|fn (x) fn (y)| <ε/2forall n ∈ N.The collection {B(x,δx /2) : x ∈[a,b ]} isanopencoverof [a,b ],soithasafinite subcover {B(xi ,δxi /2) : i = 1,...,k }.Let δ = min{δxi /2 : i = 1,...,k }.Then,if x,y ∈[a,b ] with |x y | <δ ,thenthereissome i with x ∈ B(xi ,δxi /2).Since |x y | <δ ≤ δxi /2,wehave |xi y |≤|xi x |+|x y | <δxi /2 + δxi /2 = δxi Hence |xi y | <δxi and |xi x | <δxi .So,forany n ∈ N,wehave |fn (x) fn (y)|≤ |fn (x) fn (xi )|+|fn (xi ) fn (y)| <ε/2 + ε/2 = ε .So {fn } isuniformlyequicontinuous.
Thenexttheoremgivesusthemainmethodofprovingcompactnessinthespaces weareinterestedin.
Theorem1.4.1. [Ascoli–Arzelà]If {fn (t)} isauniformlyboundedandequicontinuoussequenceofreal-valuedfunctionsonaninterval [a,b ],thenthereisasubsequencethatconvergesuniformlyon [a,b ] toacontinuousfunction.
Proof. Since {fn (t)} isequicontinuouson [a,b ],byLemma(1.1) {fn (t)} isuniformlyequicontinuous.Let t1 ,t2 ,... bealistingoftherationalnumbersin [a,b ] (notethatthesetofrationalnumbersiscountable,sothisenumerationispossible).Thesequence {fn (t1 )}∞ n=1 isaboundedsequenceofrealnumbers(since {fn } isuniformlybounded),soithasasubsequence {fnk (t1 )} convergingtoanumber,whichwedenote φ(t1 ).Itwillbemoreconvenienttorepresentthissubsequencewithoutsubsubscripts,sowewrite f 1 k for fnk andswitchtheindexfrom k to n.Sothesubsequenceiswrittenas {f 1 n (t1 )}∞ n=1 .Now,thesequence {f 1 n (t2 )} is bounded,soithasaconvergentsubsequence,say {f 2 n (t2 )},withlimit φ(t2 ).Wecontinueinthiswayobtainingasequenceofsequences {f m n (t)}∞ n=1 (onesequencefor each m),eachofwhichisasubsequenceofthepreviousone.Furthermore,wehave f m n (tm ) → φ(tm ) as n →∞ foreach m ∈ N.Now,considerthe“diagonal”functions defined Fk (t) = f k k (t).Since f m n (tm ) → φ(tm ),itfollowsthat Fr (tm ) → φ(tm ) as r →∞ foreach m ∈ N (inotherwords,thesequence {Fr (t)} convergespointwiseat
each tm ).Wenowshowthat {Fk (t)} convergesuniformlyon [a,b ] byshowingthat itisCauchyintheuniformnorm.Let ε> 0,andlet δ> 0beasinthedefinitionof uniformlyequicontinuousfor {fn (t)} appliedwith ε/3.Divide [a,b ] into p intervals, where p> b a δ .Let ξj bearationalnumberinthe j thintervalfor j = 1,...,p .Recallthat {Fr (t)} convergesateachofthepoints ξj ,sincetheyarerationalnumbers.
So,foreach j ,thereis Mj ∈ N suchthat |Fr (ξj ) Fs (ξj )| <ε/3whenever r,s>Mj . Let M = max{Mj : j = 1,...,p }.If t ∈[a,b ],thenitisinoneofthe p intervals,say the j th.So |t ξj | <δ ,andthus |f r r (t) f r r (ξj )|=|Fr (t) Fr (ξj )| <ε/3forevery r .Also,if r,s>M ,then |Fr (ξj ) Fs (ξj )| <ε/3(since M isthemaximumof the Mi ).Sofor r,s>M ,wehave |Fr (t) Fs (t)|=|Fr (t) Fr (ξj ) + Fr (ξj ) Fs (ξj ) + Fs (ξj ) Fs (t)| ≤|Fr (t) Fr (ξj )|+|Fr (ξj ) Fs (ξj )|+|Fs (ξj ) Fs (t)|
BytheCauchycriterionforconvergence,thesequence {Fr (t)} convergesuniformly on [a,b ].Sinceeach Fr (t) iscontinuous,thelimitfunction φ(t) isalsocontinuous.
Remark 1.2. TheAscoli–Arzelàtheoremcanbegeneralizedtoasequenceoffunctionsfrom [a,b ] to Rn .ApplytheAscoli–Arzelàtheoremtothefirstcoordinate functiontogetauniformlyconvergentsubsequence.Thenapplythetheoremagain, thistimetothecorrespondingsubsequenceoffunctionsrestrictedtothesecondcoordinate,gettingasubsubsequence,andsoon.
Thenextcriterion,knownastheWeierstrassM-testplaysanimportantrolein showingtheexistenceofsolutions.
Lemma1.2. (WeierstrassM-test)Let {fn } beasequenceoffunctionsdefinedona set E .Supposethatforall n = 1,... ,thereisaconstant Mn suchthat |fn (t)|≤ Mn forall t ∈ E .If
n=1 Mn < ∞, then
n=1 fn (t) convergesabsolutelyanduniformlyonthe E
WeremarkthattheWeierstrassM-testcanbeeasilygeneralizedifthedomainof thesequenceoffunctionsisasubsetofBanachspaceendowedwithanappropriate norm.
HereisanexampleoftheWeierstrassM-test.
Example 1.2 For n = 1, 2,... ,definethesequenceoffunctions {fn } on R by fn (t) = 1 t 2 + n2 .Then |fn (t)|=| 1 t 2 + n2 |≤ 1 n2 := Mn forall t ∈ R and n ≥ 1.Since
theseries ∞ n=1 1 n2 converges,bytheWeierstrassM-testtheseries
n=1 1 t 2 + n2 convergesuniformlyon R.Moreover,aseachtermoftheseriesiscontinuousandthe convergenceisuniform,thesumfunctionisalsocontinuous.(Astheuniformlimit ofcontinuousfunctionsiscontinuous.)
Hereisanotherexamplewithasimpletwisttoit.
Example 1.3 Weprovethattheseries
convergestoacontinuousfunction f : R → R. Let c beapositiveconstant.Thenforall x ∈[−c,c ],wehavethat
Ontheotherhand,theseries
converges,soWeierstrassM-testimpliesthattheseriesconvergesuniformlyto afunction f ontheboundedinterval [−c,c ].Eachtermintheseriesiscontinuous andsincetheuniformlimitofcontinuousfunctionsiscontinuous,thelimitfunction f iscontinuouson [−c,c ] forevery c> 0.Nowsinceevery x ∈ R liesinsuch anintervalforsufficientlylarge c ,itfollowsthat f iscontinuouson R.Notethatthe seriesdoesnotconvergeuniformlyon R,sowecannotusetheargumentthatthesum iscontinuouson R becausetheseriesconvergesuniformlyon R. Banachspacesformanimportantclassofmetricspaces.WenowdefineBanach spacesinseveralsteps.
Definition1.4.7. Atriple (V, +, ·) issaidtobealinear(orvector)spaceover afield F if V isasetandthefollowingaretrue.
1. Propertiesof +
a. + isafunctionfrom V × V to V .Outputsaredenoted x + y .
b. forall x,y ∈ V , x + y = y + x (+ iscommutative).
c. forall x,y,w ∈ V , x + (y + w) = (x + y) + w (+ isassociative).
d. thereisauniqueelementof V ,whichwedenote0,suchthatforall x ∈ V , 0 + x = x + 0 = x (additiveidentity).
e. foreach x ∈ V ,thereisauniqueelementof V ,whichwedenote x ,such that x + ( x) =−x + x = 0(additiveinverse).
2. Scalarmultiplication
a. isafunctionfrom F × V to V .Outputsaredenoted α x or αx
b. forall α,β ∈ F and x ∈ V , α(βx) = (αβ)x .
c. forall x ∈ V ,1 · x = x .
d. forall α,β ∈ F and x ∈ V , (α + β)x = αx + βx .
e. forall α ∈ F and x,y ∈ V , α(x + y) = αx + αy
Commonly,therealnumbersandcomplexnumbersarefieldsintheabovedefinition.Forourpurposes,weonlyconsiderthefieldofrealnumbers F = R
Definition1.4.8. (Normedspaces)Avectorspace (V, +, ·) isa normedspace iffor each x ∈ V ,thereisanonnegativerealnumber x ,calledthe norm of x ,suchthat forall x,y ∈ V and α ∈ R,
1. x = 0ifandonlyif x = 0,
2. αx =|α | x ,
3. x + y ≤ x + y
Remark 1.3. Anormonavectorspacealwaysdefinesametric ρ(x,y) = x y on thevectorspace.Givenametric ρ definedonavectorspace,itistemptingtodefine v = ρ(v, 0).Butthisisnotalwaysanorm.
Definition1.4.9. ABanachspaceisacompletenormedvectorspace,thatis,avector space (X, +, ·) withnorm · forwhichthemetric ρ(x,y) = x y iscomplete.
Example 1.4. Thespace (Rn , +, ·) overthefield R isavectorspace(withtheusual vectoraddition + andscalarmultiplication ·),andtherearemanysuitablenormsfor it.Forexample,if x = (x1 ,x2 ,...,xn ),then
1. x = max 1≤i ≤n |xi |,
2. x = n i =1 x 2 i ,
3. x = n i =1 |xi |,
4. x p = n i =1 |xi |p 1/p , p ≥ 1
areallsuitablenorms.Norm2istheEuclideannorm:thenormofavectorisits Euclideandistancetothezerovector,andthemetricdefinedfromthisnormisthe usualEuclideanmetric.Norm3generatesthe“taxi-cab”metricon R2 ,andNorm4 isthe l p norm.
Throughoutthebook,itshouldcausenoconfusiontouse |·| insteadof ||·|| to denoteaparticularnorm.
Remark 1.4 Considerthevectorspace (Rn , +, ) asametricspacewithitsmetric definedby ρ(x,y) = x y ,where · isanyofthenormsinExample 1.4.The
completenessofthismetricspacecomesdirectlyfromthecompletenessof R,and hence (Rn , · ) isaBanachspace.
Remark 1.5. IntheEuclideanspace Rn ,compactnessisequivalenttoclosedness andboundedness(Heine–Boreltheorem).Infact,themetricsgeneratedfromanyof thenormsinExample 1.4 areequivalentinthesensethattheygeneratethesame topologies.Moreover,compactnessisequivalenttoclosednessandboundednessin eachofthosemetrics.
Example 1.5 Let C([a,b ], Rn ) denotethespaceofallcontinuousfunctions f :[a,b ]→ Rn .
1. C([a,b ], Rn ) isavectorspaceover R.
2. If f = max a ≤t ≤b |f(t)|,where |·| isanormon Rn ,then (C([a,b ], Rn ), · ) is aBanachspace.
3. Let M and K betwopositiveconstantsanddefine
L ={f ∈ C([a,b ], Rn ) : f ≤ M ;|f(u) f(v)|≤ K |u v |}.
Then L iscompact.
Proof(ofpart3). Let {fn } beanysequencein L.Thefunctionsareuniformly boundedby M andhavethesameLipschitzconstant, K .Sothesequenceisuniformlyequicontinuous.BytheAscoli–Arzelàtheoremthereisasubsequence {fnk } thatconvergesuniformlytoacontinuousfunction f :[a,b ]→ Rn .Wenowshowthat f ∈ L.Well, |fn (t)|≤ M foreach t ∈[a,b ],so |f(t)|≤ M foreach t ∈[a,b ] and hence f ≤ M .Nowfix u,v ∈[a,b ] and ε> 0.Since {fnk } convergesuniformly to f ,thereis N ∈ N suchthat |fnk (t) f(t)| <ε/2forall t ∈[a,b ] and k ≥ N .So fixingany k ≥ N ,wehave
|f(u) f(v)|=|f(u) fnk (u) + fnk (u) fnk (v) + fnk (v) f(v)| ≤|f(u) fnk (u)|+|fnk (u) fnk (v)|+|fnk (v) f(v)| <ε/2 + K |u v |+ ε/2 = K |u v |+ ε.
Since ε> 0wasarbitrary, |f(u) f(v)|≤ K |u v |.Hence f ∈ L.Wehavedemonstratedthat {fn } hasasubsequenceconvergingtoanelementof L.Hence L is compact.
Example 1.6 Consider R asavectorspaceover R anddefinethemetric d(x,y) = |x y | 1 +|x y | .Foreach x ∈ R,wedefine x = d(x, 0).Explainwhy · isnota normon R.
Example 1.7 Let φ :[a,b ]→ Rn becontinuous,andlet S bethesetofcontinuous functions f :[a,c ]→ Rn with c>b and f(t) = φ(t) for a ≤ t ≤ b .Define ρ(f,g) = f g = sup a ≤t ≤c< |f(t) g(t)| for f,g ∈ S .Then (S,ρ) isacompletemetricspace butnotaBanachspacesince f + g isnotin S .
Example 1.8 Let (S,ρ) bethespaceofcontinuousboundedfunctions f : (−∞, 0]→ R with ρ(f,g) = f g = sup −∞<t ≤0 |f(t) g(t)|.Then:
1. (S,ρ) isaBanachspace.
2. Theset L ={f ∈ S : f ≤ 1, |f(u)f(v)|≤|u v |} isnotcompactin (S,ρ)
Proof(of2). Considerthesequenceoffunctionsdefined fn (t) = 0if t ≤−n, t n + 1if n<t ≤ 0
Thenthesequenceconvergespointwiseto f = 1,but ρ(fn ,f) = 1forall n ∈ N
Sothereisnosubsequenceof {fn } converginginthenorm · (i.e.,converging uniformly)to f
Example 1.9. Let (S,ρ) bethespaceofcontinuousfunctions f : (−∞, 0]→ Rn with
ρ(f,g) = ∞ n=1 2 n ρn (f,g)/{1 + ρn (f,g)}, where
(f,g) = max
|f(s) g(s)|, and |·| istheEuclideannormon Rn .Then:
1. (S,ρ) isacompletemetricspace.Thedistancebetweenallfunctionsisbounded by1.
2. (S, +, ·) isavectorspaceover R.
3. (S,ρ) isnotaBanachspacebecause ρ doesnotdefineanorm,since ρ(x, 0) = x doesnotsatisfy αx =|α | x .
4. Let M and K begivenpositiveconstants.Thentheset
L ={f ∈ S : f ≤ M on (−∞, 0], |f(u) f(v)|≤ K |u v |}
iscompactin (S,ρ)
Proof(of4). Let {fn } beasequencein L.Itisclearthatif fn → f uniformlyon compactsubsetsof (−∞, 0],thenwehave ρ(fn ,f) → 0as n →∞.Letusbeginbyconsidering {fn } on [−1, 0].Thenthesequenceisuniformlyboundedand equicontinuous,andsothereisasubsequence,say {f 1 n },converginguniformlyto somecontinuous f on [−1, 0].Moreover,theargumentofExample 1.5 showsthat |f(t)|≤ M and |f(u) f(v)|≤ K |u v |.Next,weconsider {f 1 n } on [−2, 0]. Thenthesequenceisuniformlyboundedandequicontinuous,andsothereisasubsequence,say {f 2 n },converginguniformly,say,tosomecontinuous f on [−2, 0].
