Discrete Oscillation Theory (Contemporary Mathematics and Its Applications)

975 Pages·2005·5.63 MB·English

by  Ravi P. Agarwal

Discrete
Oscillation
Theory

Discrete
Oscillation
Theory
Ravi P. Agarwal
Martin Bohner
Said R. Grace
Donal O’Regan
HindawiPublishingCorporation
410ParkAvenue,15thFloor,#287pmb,NewYork,NY10022,USA
NasrCityFreeZone,Cairo11816,Egypt
Fax:+1-866-HINDAWI(USAtoll-free)
(cid:1)c 2005HindawiPublishingCorporation
Allrightsreserved.Nopartofthematerialprotectedbythiscopyrightnoticemaybereproducedor
utilizedinanyformorbyanymeans,electronicormechanical,includingphotocopying,recording,
oranyinformationstorageandretrievalsystem,withoutwrittenpermissionfromthepublisher.
ISBN977-5945-19-4
Contents
Preface xi
1. Oscillationtheoryforsecond-orderlineardifferenceequations 1
1.1. Introduction 1
1.2. Basicconcepts 3
1.3. Riccati-typetransformations 11
1.4. Reid’sroundabouttheoremandSturmiantheory 15
1.4.1. Thedisconjugacycharacterizationtheorem 15
1.4.2. Sturmiantheory 16
1.5. Disconjugacyanddisfocality 17
1.5.1. Dominantandrecessivesolutions 23
1.5.2. Moreondisconjugacycharacterization 26
1.5.3. Disfocality 31
1.6. Conjugacycriteria 33
1.7. Methodsoflineardiscreteoscillationtheory 40
1.7.1. Riccatitechnique 40
1.7.2. Variationalprinciple 40
1.8. Oscillationandnonoscillationcriteria 41
1.9. Comparisontheorems 69
1.10. Oscillation,nonoscillation,andmonotonesolutions 71
1.11. OscillationcriteriabyRiccatitechnique 75
1.12. Oscillationcriteriabyaveragingtechniques 87
1.13. Furthercriteriabyaveragingtechniques 102
1.14. Oscillationcriteriaforlineardampeddifferenceequations 117
1.14.1. Oscillationofareducibledampedequation 118
1.14.2. Oscillationcriteriaofsummationaveragingtype 121
1.14.3. Oscillationcriteriaofnonsummationtype 127
1.15. Oscillationandnonoscillationcriteriaforforcedequations 131
1.15.1. Implicit-typeresults 133
1.15.2. Someexplicitcriteria 137
1.16. Somequalitativepropertiesofsolutions 141
1.16.1. Somepropertiesofdominantandrecessivesolutions 141
1.16.2. Reciprocityprinciple 148
1.16.3. Behavioralpropertiesofdominantandrecessivesolutions148
1.16.4. Furtherclassificationofsolutions 156
1.16.5. Comparisontheorems 161
1.17. Moreonnonoscillationcriteria 163
1.18. Limitpointresults 168
1.19. Growthandsomeoscillationcriteria 173
vi Contents
1.19.1. Growthofsolutions 173
1.19.2. Oscillationcriteria 175
1.20. Mixeddifferenceequations 176
1.21. Notesandgeneraldiscussions 180
2. Oscillationtheoryforsystems
ofdifferenceequations 183
2.1. Introduction 183
2.2. Discretevariationalproblems 185
2.2.1. Problemandnotation 185
2.2.2. Inhomogeneousdiscretequadraticfunctionals 186
2.2.3. Solutionofdiscretevariationalproblems 189
2.3. LinearHamiltoniandifferencesystems 191
2.3.1. Notationandassumptions 191
2.3.2. SpecialsolutionsoftheHamiltoniansystem 194
2.3.3. Riccatimatrixdifferenceequations 198
2.4. ThediscretePiconeformula 203
2.4.1. Discretequadraticfunctionals 203
2.4.2. TheproofofthediscretePiconeidentity 207
2.5. Disconjugacyandcontrollability 212
2.5.1. Focalpoints 212
2.5.2. Generalizedzeros 215
2.5.3. Controllability 220
2.6. Positivityofdiscretequadraticfunctionals 226
2.6.1. TheReidroundabouttheorem 226
2.6.2. Sturm’sseparationandcomparisontheorems 227
2.6.3. TheextendedReidroundabouttheorem 231
2.6.4. Separatedboundaryconditions 235
2.6.5. SolvabilityofdiscreteRiccatiequations 239
2.7. Sturm-Liouvilledifferenceequations 244
2.7.1. Disconjugacy 244
2.7.2. TheReidroundabouttheorem 252
2.8. Moore-Penroseinverses 257
2.9. Notesandgeneraldiscussions 263
3. Oscillationtheoryforhalf-linear
differenceequations 265
3.1. Introduction 265
3.2. Preliminariesandbasicconcepts 266
3.3. Reid’sroundabouttheoremandSturmiantheory 273
3.3.1. Thedisconjugacycharacterizationtheorem 273
3.3.2. Sturmiantheory 282
3.4. Nonoscillationandconjugacycriteria 283
3.4.1. NonoscillationtheoremsbasedontheRiccatitechnique 283
3.4.2. Nonoscillationtheoremsbasedonthevariationalprinciple 294
3.4.3. Conjugacycriteria 299
Contents vii
3.5. Oscillationcriteria 304
3.5.1. OscillationcriteriabasedontheRiccatitechnique 304
3.5.2. FurtheroscillationcriteriabasedontheRiccatitechnique 319
3.5.3. Oscillationcriteriabasedonthevariationalprinciple 336
3.6. Comparisontheorems 342
3.6.1. Someresultsfromalgebra 342
3.6.2. Somecomparisontheorems 343
3.6.3. ComparisontheoremforgeneralizedRiccatidifference
equations 347
3.7. Moreonoscillation,nonoscillation,andpositivesolutions 347
3.7.1. Preparatorylemmas 348
3.7.2. Sufficientconditions 350
3.7.3. Necessaryconditions 353
3.8. Strongoscillation 361
3.8.1. Strong oscillation, nonoscillation, and conditional
oscillation 361
3.8.2. GeneralizedEulerdifferenceequation 364
3.9. Half-lineardifferenceequationswithdampingterm 365
3.10. Half-lineardifferenceequationswithforcingterm 372
3.11. Notesandgeneraldiscussions 375
4. Oscillationtheoryfornonlinear
differenceequationsI 377
4.1. Oscillationcriteria 377
4.2. Superlinearoscillation 391
4.3. Sublinearoscillation 399
4.4. Oscillationcharacterizations 412
4.5. Oscillationofdampednonlineardifferenceequations 426
4.6. Asymptoticbehaviorfornonlineardifferenceequations 435
4.7. OscillationcriteriaviaLiapunov’ssecondmethod 443
4.8. Notesandgeneraldiscussions 451
5. Oscillationtheoryfornonlinear
differenceequationsII 453
5.1. Oscillationcriteria 453
5.1.1. Oscillationcriteria—(I ) 454
1
5.1.2. Superlinearoscillation 463
5.1.3. Sublinearoscillation 474
5.1.4. Oscillationcriteria—(I ) 479
2
5.1.5. Oscillationcriteriafordampedequations 486
5.2. Monotonesolutionsofnonlineardifferenceequations 488
5.2.1. Existenceofpositivemonotonesolutionsof(5.2.2) 488
5.2.2. Existenceofmonotonesolutionsof(5.2.1) 491
5.3. Bounded,unbounded,andmonotoneproperties 492
5.3.1. Positivedecreasingsolutions 492
5.3.2. Positiveincreasingsolutions 503
viii Contents
5.4. Notesandgeneraldiscussions 514
6. Oscillationtheoryfordifferenceequationswithdeviatingarguments 515
6.1. Oscillationcriteria(I) 516
6.1.1. Oscillationofequation(6.1.1) 516
6.1.2. Comparisontheorems 541
6.1.3. Oscillationofequation(6.1.2) 545
6.2. Oscillationcriteria(II) 553
6.2.1. Oscillationofequation(6.2.1) 554
6.2.2. Oscillationofequation(6.2.2) 565
6.2.3. Asymptoticbehavior 572
6.2.4. Decayingnonoscillatorysolutionsofequation(6.2.130) 577
6.3. Oscillationviacharacteristicequations 584
6.3.1. Positivesolutionsofequation(6.3.1) 585
6.3.2. Oscillationofunboundedsolutionsofequation(6.3.2) 589
6.4. Oscillationofdampeddifferenceequations 594
6.4.1. Almostoscillation 600
6.5. Oscillationofforceddifferenceequations 615
6.6. Notesandgeneraldiscussions 624
7. Oscillationtheoryforneutral
differenceequations 627
7.1. Oscillationcriteriaviacomparison 627
7.1.1. Linearizationtheorems 635
7.1.2. Comparisonresultsforforcedneutralequations 643
7.2. Nonoscillationcriteria 648
7.3. Existenceofnonoscillatorysolutions 656
7.4. Classificationofnonoscillatorysolutions 667
7.4.1. Nonoscillatorysolutionsofequation(7.4.1+) 668
7.4.2. Nonoscillatorysolutionsofequation(7.4.1−) 674
7.4.3. Oscillationandasymptoticbehavior 678
7.5. Furtheroscillationcriteria 680
7.6. Oscillationcriteriaformixedneutraldifferenceequations 692
7.6.1. Oscillationofequationswithconstantcoefficients 693
7.6.2. Oscillationofequationswithperiodiccoefficients 707
7.6.3. Oscillationviacharacteristicequations 717
7.7. Notesandgeneraldiscussions 726
8. Stabilityandoscillationtheoryfordifferentialequationswithpiecewise
constantarguments 727
8.1. Oscillationoflinearfirst-orderdelayequations 727
8.1.1. Oscillationcriteria 728
8.1.2. First-orderequationsofalternatelyretardedandadvanced
type 743
8.1.3. Characteristicequations 748
8.2. Oscillationoflinearsecond-orderdelayequations 751
Contents ix
8.2.1. Existenceofsolutions 752
8.2.2. Thecaseω=0 756
8.2.3. Thecaseω(cid:1)=0 760
8.2.4. Second-order equations of alternately retarded and
advancedtype 766
8.2.5. Neutraldifferentialequationswithpiecewiseconstant
argument 770
8.3. Systemsofalternatelyretardedandadvancedtype 776
8.3.1. ThecaseA=0 777
8.3.2. ThecaseAnonsingular 781
8.4. Applications 784
8.5. Notesandgeneraldiscussions 787
9. Miscellaneoustopics 791
9.1. Generalizedcharacteristicequations 791
9.2. Differenceequationswithperiodiccoefficients 804
9.2.1. Oscillationofequation(9.2.1) 805
9.2.2. Delay differenceequationswith positive and negative
coefficients 820
9.2.3. Oscillationofequation(9.2.3) 825
9.2.4. Oscillationofequation(9.2.4) 832
9.3. Linearizedoscillationsfordifferenceequations 840
9.3.1. Linearizedoscillationsforequation(9.3.1) 840
9.3.2. Linearizedoscillationsforequation(9.3.2) 843
9.4. Oscillationofrecursivesequences 849
9.4.1. Oscillationofx(k+1)=ax(k)/[1+bx(k−τ)] 850
9.4.2. Oscillationofx(k+1)=[a+bx(k)]/[c+x(k−τ)] 854
9.4.3. Oscillationofthediscretedelaylogisticequation 863
9.4.4. Oscillationofcertainrecursivesequences 865
9.5. Globalbehaviorofcertaindifferenceequations 871
9.5.1. Globalattractivityinanonlinearsecond-orderdifference
equation 874
9.5.2. Globalasymptoticstabilityofasecond-orderdifference
equation 880
9.5.3. Arationalrecursivesequence 883
9.5.4. Ontherecursivesequencex(k+1)=α+(x(k−1)/x(k)) 889
9.5.5. Globalstabilityofacertainrecursivesequence 894
9.5.6. Globalattractivityinanonlineardelaydifferenceequation 899
9.5.7. Ontherecursivesequencex(k+1)=ax(k)+(b+cx(k−
1))e−x(k) 905
9.5.8. Globalattractivityinadifferentialequationwithpiecewise
constantarguments 914
9.6. Nonlineardifferenceequationswithcontinuousvariable 918
9.7. Oscillationforsystemsofdelaydifferenceequations 928
9.8. Oscillatorybehaviorofsolutionsoffunctionalequations 935