Asymptotically Almost Periodic Solutions of Differential Equations

203 Pages·2009·1.94 MB·English

by  David N. Cheban

Asymptotically
Almost Periodic
Solutions of
Differential Equations
David N. Cheban
Asymptotically Almost Periodic Solutions of
Differential Equations

Asymptotically Almost Periodic Solutions of
Differential Equations
DavidN.Cheban
HindawiPublishingCorporation
http://www.hindawi.com
HindawiPublishingCorporation
410ParkAvenue,15thFloor,#287pmb,NewYork,NY10022,USA
NasrCityFreeZone,Cairo11816,Egypt
Fax:+1-866-HINDAWI(USAToll-Free)
©2009HindawiPublishingCorporation
Allrightsreserved.Nopartofthematerialprotectedbythiscopyrightnoticemaybereproducedor
utilizedinanyformorbyanymeans,electronicormechanical,includingphotocopying,recording,
oranyinformationstorageandretrievalsystem,withoutwrittenpermissionfromthepublisher.
ISBN978-977-454-099-8
Dedication
DedicatedtoMyTeacher
B.A.Shcherbakov

Contents
Dedication v
Preface iii
Notation vii
1. AsymptoticallyAlmostPeriodicMotions 1
1.1. SomeNotionsandDenotations 1
1.2. PoissonAsymptoticallyStableMotions 4
1.3. CriterionofAsymptoticalAlmostPeriodicity 6
1.4. AsymptoticallyPeriodicMotions 10
1.5. AsymptoticallyAlmostPeriodicFunctions 13
1.6. AsymptoticallySpAlmostPeriodicFunctions 22
2. AsymptoticallyAlmostPeriodicSolutionsofOperatorEquations 27
2.1. ComparabilityofMotionsbytheCharacterofRecurrence 27
2.2. ComparabilityinLimitofAsymptoticallyPoissonStableMotions 30
2.3. AsymptoticallyPoissonStableSolutions 31
2.4. AsymptoticallyPeriodicSolutions 36
2.5. HomoclinicandHeteroclinicMotions 37
2.6. AsymptoticallyAlmostPeriodicSystemswithConvergence 40
2.7. SomeTestsofConvergence 44
3. AsymptoticallyAlmostPeriodicSolutionsofOrdinaryDifferentialEquations 49
3.1. SomeNonautonomousDynamicalSystems 49
3.2. CompatibleinLimitSolutions 51
3.3. LinearDifferentialEquations 54
3.4. SemilinearDifferentialEquations 92
3.5. AveragingPrincipleonSemiaxisforAsymptotically
AlmostPeriodicEquations 96
3.6. NonlinearDifferentialEquations 98
3.7. BilaterallyAsymptoticallyAlmostPeriodicSolutions 102
3.8. AsymptoticallyAlmostPeriodicEquationswithConvergence 110
4. AsymptoticallyAlmostPeriodicDistributionsandSolutionsofDifferential
Equations 117
4.1. BoundedonSemiaxisDistributions 117
4.2. AsymptoticallyAlmostPeriodicDistributions 120
4.3. AsymptoticallyAlmostPeriodicSolutionsofLinearDifferential
EquationswithDistributionPerturbations 123
4.4. AsymptoticallyAlmostPeriodicDistributions 125
viii Contents
4.5. SolvabilityoftheEquationx(cid:2) =A(t)x+f(t)intheClassofAsymptotically
AlmostPeriodicDistributionsAP(cid:2)m(R ) 127
+
4.6. DynamicalSystemsofShiftsintheSpacesofDistributionsand
AsymptoticallyAlmostPeriodicFunctionsintheSobolevSpaces 131
4.7. WeaklyAsymptoticallyAlmostPeriodicFunctions 136
4.8. LinearandSemilinearDifferentialEquationswithWeakly Asymptotically
AlmostPeriodicCoefficients 142
5. AsymptoticallyAlmostPeriodicSolutionsofFunctionallyDifferential,Integral,
andEvolutionaryEquations 147
5.1. FunctionalDifferentialEquations(FDEs)andDynamicalSystems 147
5.2. AsymptoticallyAlmostPeriodicSolutionsofFDEs 150
5.3. LinearFDEs 151
5.4. SemilinearFDEs 153
5.5. IntegralEquationsofVolterraandGeneratedbytheNonautonomous
DynamicalSystems 156
5.6. Asymptotically Almost Periodic Solutions for Integral Equations of
Volterra 159
5.7. ConvergenceofSomeEvolutionEquations 162
Bibliography 175
Index 183
Preface
Nonlocalproblemsconcerningtheconditionsoftheexistenceofdifferentclassesofsolu-
tions play an important role in the qualitative theory of differential equations. Here
belongtheproblemofboundedness,periodicity,almostperiodicity,stabilityinthesense
ofPoisson,andtheproblemoftheexistenceoflimitregimesofdifferenttypes,conver-
gence,dissipativity,andsoon.Thepresentworkbelongstothisdirectionandisdedicated
tothestudyofasymptoticallystableinthesenseofPoisson(inparticular,asymptotically
almostperiodic)motionsofdynamicalsystemsandsolutionsofdifferentialequations.
Thereisseriesofworksofknownauthorsdedicatedtotheproblemofasymptotically
stabilityinthesenseofPoisson.
Firstthenotionofasymptoticallyalmostperiodicityoffunctionsitwasintroduced
andstudiedintheworksofFre´chet[1,2].Latertheseresultsweregeneralizedforasymp-
totically almost periodic sequences in the works of Fan [3] and Precupanu [4] and for
abstractasymptoticallyalmostperiodicfunctionsintheworksofArarktsyan[5]andPre-
cupanu[4]andalsoKhaled[6],Cioranescu[7],Dontvi[8,9],MambrianiandManfredi
[10],andManfredi[11,12],Marchi[13,14],RuessandSummers[15–18],Seifert[19],
Vesentini[20]andseealsothebibliographytherein.
OtherseriesofworksAntonishin[21],ArendtandBatty[22],Barac[23],Barbalat
[24], Khaled [6], Bogdanowicz [25], Bus¸e [26], Casarino [27], Chen and Matano [28],
ChepyzhovandVishik[29],Coppel[30],Corduneanu[31],Dontvi[8,9,32],Draghichi
[33],Fink[34],Gerko[35],Gheorghiu[36],Grimmer[37],Guryanov[38],Nacer[39],
Henriquez [40], Hino, Murakami and Yoshizawa [41], Hino and Murakami [42, 43],
JordanandWheeler[44],Jordan,MadychandWheeler[45],Yao,Zhang,andWu[46],
Lovicar [47], Manfredi [48–51], Miller [52, 53], Muntean [54, 55], Puljaev and Caljuk
[56, 57], Risito [58], Ruess and Phong [59], Sandberg and Zyl [60], Seifert [19, 61–
63], Staffans [64], Shen and Yi [65], Taam [66], Tudor [67], Utz and Waltman [68],
Vuillermot[69],Yamaguchi[70],YamaguchiandNishihara[71],Yuan[72],Yoshizawa
[73],Zaidman[74,75],Zhang[76](seealsothebibliographytherein)isdedicatedtothe
problemofasymptoticallyalmostperiodicityofsolutionsofdifferential(bothODEsand
PDEs),functional-differentialandintegralequations.
Atlast,intheworksofKhaled[6],Bhatia[77],BhatiaandChow[78],Bronshteyn
and Cˇernii [79], Gerko [35, 80], Hino and Murakami [42], Millionshchikov [81, 82],
Nemytskii [83, 84], Ruess and Summers [85], Seifert [19, 63], Sibirskii [86] and oth-
ers they are studied motions of dynamical systems that are close by their properties to
asymptoticallyalmostperiodicones.
Fromtheabovesaiditfollowsthattheproblemofasymptoticallystabilityinthesense
of Poisson was studied earlier mainly for asymptotically periodicity and asymptotically
almost periodicity of motions of dynamical systems and solutions of differential and
integral equations. In this domain there were obtained important results, however the
problemwasnotstudiedthoroughly.