Table Of ContentGaston M. N'Guérékata
Almost Periodic
and Almost
Automorphic
Functions in
Abstract Spaces
Second Edition
Almost Periodic and Almost Automorphic
Functions in Abstract Spaces
Gaston M. N’Guérékata
Almost Periodic and Almost
Automorphic Functions
in Abstract Spaces
Second Edition
GastonM.N’Guérékata
DepartmentofMathematics
MorganStateUniversity
Baltimore,MD,USA
ISBN978-3-030-73717-7 ISBN978-3-030-73718-4 (eBook)
https://doi.org/10.1007/978-3-030-73718-4
Mathematics Subject Classification: 34G10, 34G20, 34M03, 34K13, 34K30, 35B15, 35D30, 35F10,
35F20,35F35,35F50,37C25,39A12,39A21,39A23,39A24,45M15,46A04,45D05
1stedition:©KluwerAcademicPublishers2001
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Thisbookisdedicatedtomybrother
thelate
MaîtreFrançoisN’Guérékata
Preface to the First Edition
Theaimofthismonographistopresentforthefirsttimeaunifiedandhomogeneous
expositionofthetheoryofalmostautomorphicfunctionsanditsapplicationtothe
fast growing field of differential equations in abstract spaces (Banach and Hilbert
spaces).
ItisbasedessentiallyontheworkofM.Zaki,S.Zaidmanandtheauthorduring
thelastthreedecades.
The concept of almost automorphy is a generalization of almost periodicity. It
has been introduced in the literature by S. Bochner in relation to some aspects
of differential equations [11–13], and [14]. Almost automorphic functions are
characterizedbythefollowingproperty:
(cid:2)
Givenanysequenceofrealnumbers(s ),wecanextractasubsequence(s )such
n n
that
lim lim f(t +s −s )=f(t)
n→∞m→∞ n m
for each real number t. The convergence is simply pointwise while one requires
uniformconvergenceforalmostperiodicity.
In his important publication [67], W.A. Veech has studied almost automorphic
functions on groups. We like to mention the contribution by M. Zaki [70] which
provides a clear presentation of the study of almost automorphic functions with
values in a Banach space. Zaki’s work has been done under the supervision of
ProfessorS.ZaidmanoftheUniversityofMontreal,Canada,andhassincestrongly
stimulatedinvestigationsinrelationtothefollowingproblem:
Whatisthestructureofboundedfunctionsofthedifferentialequationx(cid:2) =
Ax+f wheref isanalmostautomorphicfunction?
This equation was originally raised and solved by Bohr and Neugebauer for an
almostperiodicfunctionf inafinitedimensionalspace.Thegeneralizationofthis
result to the larger class of almost automorphic functions in infinite dimensional
spaces is not a trivial one. Indeed, it sometimes uses sophisticated techniques and
strongtoolsfromfunctionalanalysisandoperatortheory.
vii
viii PrefacetotheFirstEdition
Inthismonographwepresentseveralrecentresultsfromauthorswhocontributed
to solve the above problem and consider some nonlinear cases. We deal with
classicalsolutionsaswellastheso-calledmildsolutions.
The concept of weak almost automorphy as presented by M. Zaki [70] is also
discussed(Chap.2,Sect.2).
Also, continuous solutions on the non-negative semi-axis that approach almost
automorphic functions at infinity are studied in Sections 5 and 6 of Chap.2. In
particularsemi-groupsoflinearoperatorsareconsideredasanindependentsubject
in section of Chap.3 and discussed in the context of the so-called Nemytskii and
Stepanovtheoryofdynamicalsystems.
AwiderangeofsituationsispresentedinChaps.4through6.
In Chap.3, we present some results of the theory of almost periodic functions
taking values in a locally convex space. We use a definition introduced in the
literaturebyC.Corduneanuanddeveloped bytheauthorforthefirsttimein[54].
Applications to abstract differential equations are given in Chaps.7 and 8. At the
end of each chapter, we have included a Notes section that gives some comments
themainreferencesused.
Itisourhopethatthismonographwillconstituteausefulreferencetextbookfor
post-graduate students and researchers in analysis, ordinary differential equations,
partialdifferentialequations,anddynamicalsystems.
May it stimulate new developments of the theory of almost automotrphic and
almostperiodicfunctionsandenrichitsapplicationstootherfields.
It is a great pleasure to record our very sincere thanks to Professor Jerome
A. Goldstein, a friend and mentor for over two decades and Professor Georges
Anastassiou,whostronglyencouragedustocompletethisproject.
We express my warm gratitude to Professor Constantin Corduneanu and Pro-
fessor Joseph Auslander for their valuable comments and suggestions. Our thanks
toourfriendProfessorThomasSeidmanwhocorrectedsomeerrorsandStephanie
Smithforherextraordinaryskillandpatienceinsettingthistext.
WealsoexpressourappreciationtotheeditorialassistanceofKluwerAcademic
Publishers,especiallyfromAnaBozicevicandChrisCurcio.
Finally, we owe a great deal to Professor Samuel Zaidman, who introduced us
to the exciting world of mathematical research. His experience and outstanding
contributions to mathematics have been a great source of inspiration to several
youngmathematicians.
Preface to the Second Edition
Sincethepublicationofourbook[55]in2001,therehasbeenarealrebirthofthe
theory of almostautomorphic functions and applications toevolution equations as
weexpected.Anincrediblenumberofresearchershavebeenattractedbythistopic.
Thisleadstoafast-growingnumberofpublications.
We have received many helpful comments from colleagues and students, some
pointing out typographical errors, others asking for clarification and improvement
on some materials. In particular, Zheng, Ding, and N’Guérékata were able to
answerthelong-timeopenproblem:whatisthe“amount”ofalmostautomorphic
functions which are not almost periodic in the sense of Bohr? The answer is
thatthespaceofalmostperiodicfunctionsisasetoffirstcategoryinthespaceof
almostautomorphicfunctions(cf.Chap.1).Manyotherproblemsremainopen,for
instancethestudyofalmostperiodicfunctionstakingvaluesinnon-locallyconvex
spaces(cf.[30]).
Several generalizations were introduced in the literature including the study
of almost automorphic sequences. The interplay between almost automorphy and
almostperiodicityisbetterknown.
Researchersinthefieldoverwhelminglyencouragedustowriteasecondedition
includingsomeofthefreshandmostrelevantcontributionsandreferences.
Asinthefirstedition,wepresentthematerialsinasimplifiedandrigorousway.
Eachchapterisconcludedwithbibliographicalnotesshowingtheoriginalsources
oftheresultsandfurtherreading.
Wearemostgratefultoournumerousco-authorsandcolleagueswhomadesuch
great contributions to the theory of almost automorphy. We will not exhibit a list,
whichwouldbeanywayincomplete,butwehopeourfriendswillbesatisfiedwith
ourthanksandgratitude.
Finally, we thank our students Fatemeh Norouzi and Romario Gildas Foko
Tiomela and our friend and colleague Alexander Pankov for their careful proof-
readingandsuggestions.
Baltimore,MD,USA GastonM.N’Guérékata
October2020
ix
Contents
1 IntroductionandPreliminaries........................................... 1
1 BanachSpaces........................................................... 1
2 Lp Spaces................................................................ 3
3 LinearOperators......................................................... 5
4 FunctionswithValuesinaBanachSpace.............................. 6
5 SemigroupsofLinearOperators........................................ 8
6 TopologicalVectorSpaces .............................................. 9
7 TheExponentialofaBoundedLinearOperator ....................... 10
8 Non-locallyConvexSpaces............................................. 15
2 AlmostAutomorphicFunctions........................................... 17
1 AlmostAutomorphicFunctionsinaBanachSpace.................... 17
2 WeakAlmostAutomorphy.............................................. 27
3 AlmostAutomorphicSequences........................................ 30
3.1 KadetsTheorem ................................................... 30
4 AsymptoticallyAlmostAutomorphicFunctions....................... 31
3 AlmostAutomorphyoftheFunctionf(t,x)............................ 37
1 TheNemytskii’sOperator............................................... 37
4 DifferentiationandIntegration ........................................... 41
1 DifferentiationinAA(X)................................................ 41
2 IntegrationinAA(X).................................................... 41
3 DifferentiationinWAA(X)............................................. 51
4 IntegrationinAAA(X).................................................. 52
5 PseudoAlmostAutomorphy............................................... 55
1 PseudoAlmostAutomorphicFunctions................................ 55
2 μ-PseudoAlmostAutomorphicFunctions............................. 58
6 Stepanov-likeAlmostAutomorphicFunctions.......................... 65
1 DefinitionsandProperties............................................... 65
xi
xii Contents
7 DynamicalSystemsandC -Semigroups................................. 71
0
1 AbstractDynamicalSystems............................................ 71
2 CompleteTrajectories ................................................... 72
8 AlmostPeriodicFunctionswithValuesinaLocallyConvexSpace... 79
1 AlmostPeriodicFunctions.............................................. 79
2 WeaklyAlmostPeriodicFunctions..................................... 84
3 AlmostPeriodicityoftheFunctionf(t,x)............................. 97
4 Equi-AsymptoticallyAlmostPeriodicFunctions ...................... 97
9 Almost Periodic Functions withValues inaNon-locally
ConvexSpace................................................................ 103
1 DefinitionsandProperties............................................... 103
2 WeaklyAlmostPeriodicFunctions..................................... 106
3 Applications.............................................................. 109
10 TheEquationx’(t)=A(t)x(t)+f(t) .......................................... 111
1 TheEquationx’(t)=A(t)x(t)+f(t)........................................ 111
11 AlmostPeriodicSolutionsoftheDifferentialEquationin
LocallyConvexSpaces..................................................... 125
1 LinearEquations......................................................... 125
1.1 TheHomogeneousEquationx(cid:2) =Ax............................ 126
1.2 TheInhomogeneousCase......................................... 127
Appendix.......................................................................... 129
References......................................................................... 131
