Nonoscillation Theory of Functional Differential Equations with Applications

526 Pages·2012·2.4 MB·English

by  Ravi P. Agarwal

Nonoscillation Theory of Functional Differential
Equations with Applications
Ravi P. Agarwal (cid:2) Leonid Berezansky (cid:2)
Elena Braverman (cid:2) Alexander Domoshnitsky
Nonoscillation Theory
of Functional
Differential Equations
with Applications
RaviP.Agarwal ElenaBraverman
DepartmentofMathematics DepartmentofMathematics
TexasA&MUniversity—Kingsville UniversityofCalgary
Kingsville Calgary
USA Canada
AlexanderDomoshnitsky
LeonidBerezansky DepartmentofComputerSciences
DepartmentofMathematics andMathematics
Ben-GurionUniversityoftheNegev ArielUniversityCenterofSamaria
Beer-Sheva Ariel
Israel Israel
ISBN978-1-4614-3454-2 e-ISBN978-1-4614-3455-9
DOI10.1007/978-1-4614-3455-9
SpringerNewYorkDordrechtHeidelbergLondon
LibraryofCongressControlNumber:2012935965
MathematicsSubjectClassification(2010): 34K11,34K10,34K06,34K20,34K45,92D25
©SpringerScience+BusinessMedia,LLC2012
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Preface
Thewell-knownmonographsbyG.S.Ladde,V.LakshmikanthamandB.G.Zhang
[248],I.Gyo˝riandG.Ladas[192],L.H.Erbe,Q.KongandB.G.Zhang[154],R.P.
Agarwal, M. Bohner and W.-T. Li [3], R.P. Agarwal, S.R. Grace and D. O’Regan
[8] andD.D. BainovandD.P.Mishev[34] aredevotedtotheoscillationtheoryof
functionaldifferentialequations.Eachofthesemonographscontainsnonoscillation
tests,buttheirmainobjectivewastopresentmethodsandresultsconcerningoscil-
lationofallsolutionsforthefunctionaldifferentialequationsunderconsideration.
The main purpose of the present monograph is to consider nonoscillation and
existenceofpositivesolutionsforfunctionaldifferentialequationsandtodescribe
theirapplicationstomaximumprinciples,boundaryvalueproblemsandthestability
oftheseequations.
Inviewofthisobjective,weconsiderawideclassofequations:
1. scalar equations and systems of different types: linear and nonlinear first-order
functionaldifferentialequations,second-orderequationswithorwithoutdamp-
ingterms,high-orderequations,systemsoffunctionaldifferentialequations;
2. equationswithvariabletypesofdelays:delaydifferentialequations,integrodif-
ferentialequations,equationswithadistributeddelay,neutralequations;
3. equationswithvariabledeviationsoftheargument:advancedandmixed(includ-
ingbothdelayedandadvancedterms)differentialequations;
4. bothcontinuousandimpulsiveequations:first-andsecond-orderlinearandfirst-
ordernonlinearimpulsivedifferentialequations;
5. specific classes of linear and nonlinear equations, as well as linear differential
equationswithabstractVolterra(causal)operators;
6. bothinitialandboundaryvalueproblemsareconsideredforfunctionaldifferen-
tialequations.
Notethatwedonotusemethodsspecificonlytoequationswithcontinuouspa-
rameterssinceweconsidermodelswithmeasurablecoefficientsanddelays.
Nonoscillationresultsareapplied
• to nonlinear nonautonomous equations of mathematical biology with both con-
centratedanddistributeddelays;
v
vi Preface
• tostabilityproblems;and
• toboundaryvalueproblems.
Chapter1isabriefsurveyofintroductorynotionsandideasinnonoscillationthe-
ory: autonomous equations, characteristic equations, solution representations, dif-
ferentialandintegralequations,andinequalities.Thoughelementaryinitspresen-
tation (we believe it can easily be understood by senior undergraduate students),
this chapter incorporates many basic ideas that will be employed later: equiva-
lence of nonoscillation and existence of a nonnegative solution of the generalized
characteristicinequalityandtheapplicationofsolutionrepresentation,linearization
andtheapproachtoimpulsiveequations.Themainpopulationdynamicsequations
(Hutchinson’s, Lasota-Wazewska, Mackey-Glass, Nicholson’s blowflies) are also
introducedinChap.1.
Chapter 2 presents basic results for first-order linear delay equations with pos-
itive coefficients: nonoscillation criteria, comparison theorems, explicit nonoscil-
lation and oscillation results, sufficient conditions for positivity of solutions with
given initial conditions and slowly nonoscillating solutions. In Chap. 3, some of
theseresultsaregeneralizedtoequationswithpositiveandnegativecoefficients;it
isalsoillustratedthatsomeoftheresultscannotbeextended.Chapter4isconcerned
with a general linear equation with a distributed delay that is nonautonomous and
can include integral and concentrated delay terms. The case of positive kernels of
integralsandcoefficientsisconsidered,aswellastermsofdifferentsigns.
In Chap. 5, nonoscillation of linear equations of advanced and mixed types is
studied.Themainresultsofthischapterarebasedonvariousfixed-pointtheorems.
Chapter 6 is concerned with linear neutral equations of the first order that include
thederivativeoftheunknownfunctionbothwithandwithoutdelays.
In Chaps. 7 and 8, we consider linear second-order delay equations without
damping and with damping, respectively. Chapter 9 deals with linear systems of
delaydifferentialequationsandalsohigher-orderdifferentialequations.Inaddition
totheproblemsconsideredinthepreviouschapters,Chap.9includesanextensive
sectiononstabilityofnonoscillatorysystems.
Chapters10and11aredevotedtononlinearequations.InChap.10,thelineariza-
tion method is applied to various nonautonomous models of population dynamics
(inparticular,logistic,Lasota-WazewskaandNicholson’sblowfliesequations),and
all equations are considered with a distributed delay. In Chap. 11, some equations
thatcannotbehandledwiththelinearizationapproacharestudied(mostlydifferent
variationsofthelogisticmodel).
Chapters 12–14 are concerned with impulsive equations. Chapter 12 presents
nonoscillationresultsforfirst-orderlinearimpulsivedifferentialequationswithboth
concentrated and distributed delays. It is also demonstrated that nonoscillation of
an impulsive equation can be reduced to nonoscillation of a specially constructed
equation without impulses but with discontinuous coefficients. Chapter 13 deals
with second-order differential equations, and generally in the models considered
anylinearjumpsofboththesolutionandthefirstderivativecanoccur.InChap.14,
linearizationmethodsareappliedtofirst-ordernonlinearimpulsiveequations.
Preface vii
The study of many classical questions in the qualitative theory of linear n-th-
orderordinarydifferentialequations,suchasexistenceanduniquenessofsolutions
oftheinterpolationboundaryvalueproblems,positivity,oracorrespondingregular
behavior of their Green’s functions, maximum principles and stability, was con-
nectedwithandevenbasedonthenotionofnonoscillationintervalsofcorrespond-
ing linear ordinary differential equations. In Chaps. 15–17 we create a concept of
nonoscillationintervalsforfunctionaldifferentialequationsthatcanactuallybecon-
sideredasananalogueofnonoscillationtheoryforordinarydifferentialequations.
Variousrelationsbetweenthenotedpropertiesareobtainedforfunctionaldifferen-
tial equations on the basis of nonoscillation. Linear and nonlinear equations with
Volterra(causal)operatorswerepreviouslystudiedinthemonographs[29,98,239,
251].InChaps.15–17,weconsiderequationswithVolterraoperators.Itshouldbe
notedthatitisnotonlyageneralizationbutalsoanimportantinstrumentforstudy-
ingthebehaviorofacorrespondingcomponentx ofasolutionvector.Weconstruct
r
anequationforthiscomponentinChap.16.Eveninthecaseofsystemsofordinary
differentialequations,thisdifferentialequationforx isofquiteageneralformthat
r
includesVolterraoperators.Inthesechapters,wealsostudysuchquestionsasmaxi-
mumprinciples,existenceanduniquenessofsolutionstoboundaryvalueproblems,
regularbehavioroftheirGreen’sfunctions,andapplicationstostudystabilitythat
arenotconsideredinpreviouschapters.
All chapters conclude with a discussion, some open problems, and topics for
possiblefutureresearch.
Finally,AppendicesAandBincludesomereferencematerial.AppendixAcon-
tains all auxiliary notions and functional analysis results used in the monograph:
definitionsoffunctionalspaces,measuresandVolterraoperators,compactnesscon-
ditionsforsetsandlinearoperators,andfixed-pointtheoremsinBanachspaceswith
orwithoutorder.Theseresultsareappliedinthestudyofavarietyoftypesofequa-
tions: with several concentrated and distributed delays, with general Volterra and
non-Volterra equations and systems, linear and nonlinear, and continuous and im-
pulsive.AppendixBpresentsexistenceanduniquenessconditionsforallfunctional
differentialequationsconsideredinthismonograph;inaddition,solutionrepresen-
tationsaregivenforlinearequations.
Acknowledgements
Firstofall,wewouldliketoacknowledgetheilluminatinginfluenceofNikolaiVik-
torovichAzbelev(1922–2006)ontheresearchleadingtothepresentmonograph.As
aresearchadvisorofL.B.,E.B.,andA.D.,hesupportedandguidedtheauthorsfor
many years in the beginning of their research careers, and many of his ideas are
reflected in the book. L.B., E.B, and A.D. are also grateful to Michael Efimovich
Drakhlin(1936–2007)forhissupportiveandinteresteddiscussionsofourresearch.
We dedicate our monograph to the memory of our colleagues N.V. Azbelev and
M.E.Drakhlin.
Inaddition,theworksofI.T.Kiguradze,V.P.Maksimov,andL.F.Rakhmatullina
significantlycontributedtoourapproachtononoscillationandboundaryvalueprob-
lems.Asourmonographwastakingshape,ourcommunicationswithIstvánGyo˝ri,
JosefDiblík,YuriDomshlak,LevIdels,EduardoLiz,RobertHakl,AlexanderLom-
tatidze,andJiˇríŠremrwereveryhelpful.WearealsogratefultoMichaelGil’,who
wasthefirstreaderofourmonograph,andtoBas¸akKarpuzforhisusefulremarks
on the final version of the manuscript. It is our pleasure to acknowledge here the
contributionsofallwhoassistedus.
Wearegratefultoourdepartmentsanduniversitiesfortheirwonderfulworking
environments.OurresearchwaspartiallysupportedbyanNSERCDiscoveryGrant
(E.B.), and the Israeli Ministry of Absorption (L.B. and A.D.). Finally, we would
liketothankourfamiliesfortheirongoingsupport.
ix
Contents
1 IntroductiontoOscillationTheory . . . . . . . . . . . . . . . . . . . 1
1.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1
1.2 NonoscillationofAutonomousDelayEquationswithPositive
Coefficients . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1
1.3 NonlinearEquationsofMathematicalBiology . . . . . . . . . . 9
1.3.1 LinearizationofNonlinearDelayEquations. . . . . . . . 9
1.3.2 Hutchinson’sEquation . . . . . . . . . . . . . . . . . . . 10
1.3.3 Lasota-WazewskaEquation . . . . . . . . . . . . . . . . 12
1.3.4 Nicholson’sBlowfliesEquation . . . . . . . . . . . . . . 13
1.3.5 Mackey-GlassEquations . . . . . . . . . . . . . . . . . . 15
1.4 ImpulsiveEquations . . . . . . . . . . . . . . . . . . . . . . . . 16
1.5 SomeOtherClassesofEquations . . . . . . . . . . . . . . . . . 18
1.6 DiscussionandOpenProblems . . . . . . . . . . . . . . . . . . 20
2 ScalarDelayDifferentialEquationsonSemiaxes . . . . . . . . . . . 23
2.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23
2.2 Preliminaries . . . . . . . . . . . . . . . . . . . . . . . . . . . . 24
2.3 NonoscillationCriteria . . . . . . . . . . . . . . . . . . . . . . . 25
2.4 ComparisonTheorems . . . . . . . . . . . . . . . . . . . . . . . 28
2.5 NonoscillationConditions,Part1 . . . . . . . . . . . . . . . . . 32
2.6 NonoscillationConditions,Part2 . . . . . . . . . . . . . . . . . 37
2.7 OscillationConditions . . . . . . . . . . . . . . . . . . . . . . . 43
2.8 EstimationsofSolutions . . . . . . . . . . . . . . . . . . . . . . 46
2.9 PositivityofSolutions . . . . . . . . . . . . . . . . . . . . . . . 49
2.10 SlowlyOscillatingSolutionsforDelayDifferentialEquations . . 51
2.11 StabilityandNonoscillation . . . . . . . . . . . . . . . . . . . . 52
2.12 DiscussionandOpenProblems . . . . . . . . . . . . . . . . . . 52
3 ScalarDelayDifferentialEquationsonSemiaxiswithPositiveand
NegativeCoefficients . . . . . . . . . . . . . . . . . . . . . . . . . . . 59
3.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . 59
3.2 NonoscillationCriteria . . . . . . . . . . . . . . . . . . . . . . . 59
xi
xii Contents
3.3 NonoscillationConditions,Part1 . . . . . . . . . . . . . . . . . 66
3.4 NonoscillationConditions,Part2 . . . . . . . . . . . . . . . . . 71
3.5 EquationswithanOscillatoryCoefficient . . . . . . . . . . . . . 77
3.6 DiscussionandOpenProblems . . . . . . . . . . . . . . . . . . 79
4 OscillationofEquationswithDistributedDelays . . . . . . . . . . . 83
4.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . 83
4.2 Preliminaries . . . . . . . . . . . . . . . . . . . . . . . . . . . . 84
4.3 ExistenceofaPositiveSolution—GeneralResults . . . . . . . . 86
4.4 ComparisonTheorems . . . . . . . . . . . . . . . . . . . . . . . 92
4.5 NonoscillationCriteriaforSomeAutonomousIntegrodifferential
Equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 97
4.6 ExplicitNonoscillationandOscillationConditions . . . . . . . . 101
4.7 SlowlyOscillatingSolutions . . . . . . . . . . . . . . . . . . . . 107
4.8 EquationswithPositiveandNegativeCoefficients . . . . . . . . 108
4.9 DiscussionandOpenProblems . . . . . . . . . . . . . . . . . . 118
5 ScalarAdvancedandMixedDifferentialEquationsonSemiaxes . . 123
5.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . 123
5.2 AdvancedEquations . . . . . . . . . . . . . . . . . . . . . . . . 123
5.3 MixedEquationswithPositiveCoefficients . . . . . . . . . . . . 132
5.4 MixedEquationwithNegativeCoefficients . . . . . . . . . . . . 134
5.5 PositiveDelayTerm,NegativeAdvancedTerm . . . . . . . . . . 135
5.6 NegativeDelayTerm,PositiveAdvancedTerm . . . . . . . . . . 141
5.7 DiscussionandOpenProblems . . . . . . . . . . . . . . . . . . 144
6 NeutralDifferentialEquations . . . . . . . . . . . . . . . . . . . . . 149
6.1 IntroductionandPreliminaries . . . . . . . . . . . . . . . . . . . 149
6.2 NonoscillationCriteria . . . . . . . . . . . . . . . . . . . . . . . 151
6.3 EfficientNonoscillationConditions . . . . . . . . . . . . . . . . 156
6.4 ExplicitOscillationConditions . . . . . . . . . . . . . . . . . . 160
6.5 PositivityofSolutions . . . . . . . . . . . . . . . . . . . . . . . 164
6.6 SlowlyOscillatingSolutions . . . . . . . . . . . . . . . . . . . . 165
6.7 NeutralEquationswithPositiveandNegativeCoefficients . . . . 166
6.8 DiscussionandOpenProblems . . . . . . . . . . . . . . . . . . 168
7 Second-OrderDelayDifferentialEquations . . . . . . . . . . . . . . 171
7.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . 171
7.2 Preliminaries . . . . . . . . . . . . . . . . . . . . . . . . . . . . 171
7.3 NonoscillationCriteria . . . . . . . . . . . . . . . . . . . . . . . 172
7.4 ComparisonTheorems . . . . . . . . . . . . . . . . . . . . . . . 176
7.5 ExplicitNonoscillationandOscillationConditions . . . . . . . . 182
7.6 SlowlyOscillatingSolutions . . . . . . . . . . . . . . . . . . . . 187
7.7 ExistenceofaPositiveSolution . . . . . . . . . . . . . . . . . . 188
7.8 DiscussionandOpenProblems . . . . . . . . . . . . . . . . . . 190