Almost Automorphic Type and Almost Periodic Type Functions in Abstract Spaces

312 Pages·2013·1.89 MB·English

by  Toka Diagana (auth.)

Toka Diagana
Almost Automorphic
Type and Almost
Periodic Type
Functions in Abstract
Spaces
Almost Automorphic Type and Almost Periodic
Type Functions in Abstract Spaces

Toka Diagana
Almost Automorphic Type
and Almost Periodic Type
Functions in Abstract Spaces
123
TokaDiagana
DepartmentofMathematics
HowardUniversity
Washington,DC,USA
ISBN978-3-319-00848-6 ISBN978-3-319-00849-3(eBook)
DOI10.1007/978-3-319-00849-3
SpringerChamHeidelbergNewYorkDordrechtLondon
LibraryofCongressControlNumber:2013942386
Mathematics Subject Classification (2010): 34K14, 35B15, 34C27, 35L10, 35L25, 43A60, 47A56,
47A62,47-XX,47D06,58D25,65J08,58J35,46B70,46M35,12H20,35R20
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InMemoryofMySisterCoumbaDiagana
andMyBrotherDembaDiagana

Preface
AlmostAutomorphicTypeandAlmostPeriodicTypeFunctionsinAbstractSpaces
presentsacomprehensiveandreader-friendlyintroductiontotheconceptsofalmost
periodicity, asymptotic almost periodicity, almost automorphy, asymptotic almost
automorphy, pseudo-almost periodicity, and pseudo-almost automorphy as well as
theirrecentgeneralizations.Further,itpresentsawiderangeofsufficientconditions
for the boundedness, existence, uniqueness, and stability of solutions to various
classes of first-, second-, third-, and higher-order abstract differential equations,
difference equations, and integrodifferential equations whose coefficients belong
to the above-mentioned classes of functions. It also offers various applications to
partialdifferentialequationsincludingthebeamboundary-valueproblem,theheat
boundary-valueproblem,andsomepartialneutralintegrodifferentialequationswith
delayarisingincontrolsystems.
Some of the results presented are either new or else cannot be easily found
in the mathematical literature. In order to establish the main results of the book,
one makes extensive use of a myriad of tools ranging from functional analysis to
operatortheory.
Noteworthy progress has been made during the past 15 years in studying
almostperiodic,almostautomorphic,andpseudo-almostperiodicfunctionsaswell
as their applications to abstract differential equations. In recent years, various
generalizationsoftheaboveclassesoffunctionshavebeenintroducedinthemath-
ematical literature including, but are not limited to, the concepts of Stepanov-like
almost automorphy, weighted pseudo-almost periodicity, pseudo-almost automor-
phy,weightedpseudo-almostautomorphy,Stepanov-likepseudo-almostperiodicity,
Stepanov-like pseudo-almost automorphy, and many more. Despite the noticeable
and rapid progress made on these important topics, the only standard references
that currently exist on those new classes of functions and their applications are
stillscatteredresearcharticles.Oneofthemainobjectivesofthisbookconsistsof
closingthatgap.Namely,thebookcollectsandpresentsinaunifiedmannermostof
therecentmaterialintroducedintheliteratureupontheabove-mentionedclassesof
functionsaswellastheirapplicationstodifferentialequations,differenceequations,
integrodifferentialequations,andpartialdifferentialequations.
vii
viii Preface
Thebookisdividedinto12mainchaptersandanappendix.Italsoofferssome
usefulbibliographicalnotesattheendofeachchapter.
The first chapter collects the basic background on metric, Banach, and Hilbert
spaces. Proofs of some of the most important results presented will be given
includingthatoftheBanachfixed-pointtheorem.
Chapter 2 collects basic material in functional analysis and operator theory
needed in the sequel including compact operators, self-adjoint operators, oper-
ators with compact resolvents, sectorial operators, analytic semigroups, abstract
interpolation spaces, and various preliminary results on evolution families under
Acquistapace–Terreniconditionsandexponentialdichotomy.
Chapter 3 gathers basic properties on almost periodic, Stepanov-like almost
periodic, and asymptotically almost periodic functions, and then introduces and
studiesdifferentiablealmostperiodicandStepanov-likealmostperiodicfunctions.
Chapter 4 collects basic results on almost automorphic and asymptotically almost
automorphicfunctions,andthenintroducesandstudiesStepanov-likealmostauto-
morphic,differentiablealmostautomorphicandStepanov-likealmostautomorphic
functions. The main objective of Chap.5 consists of reviewing basic properties of
pseudo-almost periodic functions including some composition theorems as well
as some of their recent extensions such as weighted pseudo-almost periodic,
differentiable pseudo-almost periodic and Stepanov-like pseudo-almost periodic
functions. Chapter 6reviews basic properties of pseudo-almost automorphic func-
tions as well as some of their recent extensions such as weighted pseudo-almost
automorphic,differentiablepseudo-almostautomorphic,andStepanov-likepseudo-
almostautomorphicfunctions.
Chapter7studiestheexistenceofalmostperiodic(respectively,almostautomor-
phic) solutions to the classes of nonautonomous damped second-order differential
equations
d2u du
+b(t)B +a(t)Au=h(t,u),
dt2 dt
whereAisaself-adjointlinearoperatorwithcompactresolvent,Bisaclosedlinear
operatorsatisfyingsomeadditionalconditions,andthefunctionsa,b,harealmost
periodic(respectively,almostautomorphic).
Chapter 8 studies the existence of asymptotically almost automorphic mild
solutionstotheabstractpartialneutralintegrodifferentialequations
(cid:2)
d t
D(t,u)=AD(t,u)+ B(t−s)D(s,u )ds+g(t,u), t∈[σ,σ+a),
t t s t
dt
0
uσ =ϕ∈B,
where A,B(t) are densely defined closed linear operators with a common domain
D(A),whichisindependentoft,thehistoryu,isthefunctiondefinedbyu(θ):=
t t
u(t+θ),whichbelongstoanabstractphasespaceBdefinedaxiomatically, f,gare
functionssubjecttosomeadditionalconditions,and
Preface ix
D(t,ϕ):=ϕ(0)+f(t,ϕ).
Chapter 9 studies the existence of differentiable pseudo-almost automorphic
solutions to some general higher-order differential equations involving operator
coefficientsgivenby
dnu+n∑−1A dku+Au= f,
dtn kdtk
k=1
whereAandA (k=1,2,...,n−1)aredenselydefinedclosedlinearoperatorson
k
a Banach space X, and f :R(cid:3)→D (α+m,p) with D (α+m,p) being a real
A A
interpolationspace.OurstudyconcernsonlythespecialcaseAk=Aαk (ifitexists)
withtheexponents0≤α <1fork=1,2,...,n−1.Anonautonomousversionof
k
theabovehigher-orderdifferentialequationwillalsobeconsidered.
InChap.10,usingtheSchauderandBanachfixed-pointtheorems,andexponen-
tialdichotomytechniques,westudyandobtain,invariouscontexts,theexistenceof
pseudo-almost periodic (respectively, weighted pseudo-almost periodic) solutions
tothenonautonomousthird-orderdifferentialequations
(cid:3) (cid:4)
d d2u
+g(t,Bu) =w(t)Au+f(t,Cu), t∈R
dt dt2
whereAisaself-adjointlinearoperator,BandCarelinearoperatorssuchthattheir
algebraicsumB+Cisnontrivial,thefunctionwgivenbyw(t)=−ρ(t)isassumed
tobealmostperiodic,thereexisttwoconstantsρ,ρ >0satisfyingthefollowing
0 1
conditions ρ ≤ ρ(t) ≤ ρ, and the functions f,g are pseudo-almost periodic
0 1
(respectively, weighted pseudo-almost periodic) in the first variable uniformly in
thesecondone.
Chapter11studiestheexistenceofpseudo-almostautomorphicsolutionstothe
classofSobolev-typedifferentialequationsgivenby
d
[u+f(t,u)]=A(t)u+g(t,u),
dt
where A(t) is a family of densely defined closed linear operators on a domain D,
independentoft,and f,gareStepanov-likepseudo-almostautomorphicfunctions.
In Chap.12, one studies and obtains the existence of a globally asymptotically
stablealmostperiodic(respectively,globallyasymptoticallystablealmostautomor-
phic)solutionstothehigher-orderdifferenceequations
n−1
x(t+n)+ ∑ A (t)x(t+m)+A (t)x(t)= f(t,x(t)),
m 0
m=1