Theory of Translation Closedness for Time Scales: With Applications in Translation Functions and Dynamic Equations (Developments in Mathematics (62))

586 Pages·2020·2.972 MB·English

by  Chao Wang, Ravi P. Agarwal, Donal O' Regan, Rathinasamy Sakthivel

Developments in Mathematics
Chao Wang
Ravi P. Agarwal
Donal O’ Regan
Rathinasamy Sakthivel
Theory of
Translation
Closedness for
Time Scales

With Applications in Translation
Functions and Dynamic Equations
Developments in Mathematics
Volume 62
SeriesEditors
KrishnaswamiAlladi,DepartmentofMathematics,UniversityofFlorida,
Gainesville,FL,USA
PhamHuuTiep,DepartmentofMathematics,RutgersUniversity,Piscataway,NJ,
USA
LoringW.Tu,DepartmentofMathematics,TuftsUniversity,Medford,MA,USA
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Chao Wang • Ravi P. Agarwal • Donal O’Regan
Rathinasamy Sakthivel
Theory of Translation
Closedness for Time Scales
With Applications in Translation Functions
and Dynamic Equations
ChaoWang RaviP.Agarwal
DepartmentofMathematics DepartmentofMathematics
YunnanUniversity TexasA&MUniversity–Kingsville
Kunming,Yunnan,China Kingsville,TX,USA
DonalO’Regan RathinasamySakthivel
SchoolofMathematics DepartmentofAppliedMathematics
StatisticsandAppliedMathematics BharathiarUniversity
NationalUniversityofIreland Coimbatore,TamilNadu,India
Galway,Ireland
ISSN1389-2177 ISSN2197-795X (electronic)
DevelopmentsinMathematics
ISBN978-3-030-38643-6 ISBN978-3-030-38644-3 (eBook)
https://doi.org/10.1007/978-3-030-38644-3
MathematicsSubjectClassification(2020):34N05,43A60,42A75,93A30
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Preface
The theory of time scales was initiated by S. Hilger in his PhD thesis [141] in
1988 in order to unify continuous and discrete analysis. This new and exciting
type of mathematics is more general and versatile than the traditional theories of
differentialanddifferenceequationsasitcan,underoneframework,mathematically
describecontinuous-discretehybridprocessesandhenceistheoptimalwayforward
for accurate mathematical modeling in physics, chemical technology, population
dynamics, biotechnology and economics, neural networks, and social sciences. In
fact, the progressive field of dynamic equations on time scales contains links and
extends the classical theory of differential and difference equations. For instance,
if T = Z, we have a result for difference equations, if T = R, we obtain a result
for differential equations. This theory represents a powerful tool for applications
to economics, population models, quantum physics among others. Not only does
thenewtheoryoftheso-calleddynamicequationsunifythetheoriesofdifferential
equations and difference equations but also extends these classical cases to cases
“in between,” e.g., to the so-called q-difference equations when T = qN0 :=
{qt : t ∈ N forq > 1} ∪ {0} or T = qZ := qZ ∪ {0} (which has important
0
applications in quantum theory) and can be applied on different types of time
scales like T = hN, T = N2, and T = T the space of the harmonic numbers.
n
Therefore,dealingwithproblemsofdifferentialequationsontimescalesbecomes
veryimportantandmeaningfulintheresearchfieldofdynamicsystems.
Since a time scale is an arbitrary nonempty closed subset of real numbers, its
irregulardistributiononthereallineleadstomanydifficultiesinstudyingfunctions
on time scales, especially in investigating functions defined by the translations
of arguments such as periodic functions, almost periodic functions, and almost
automorphic functions, etc. The classes of functions defined by the translations of
argumentsarereferredtoasTranslationFunctions.Theconceptofalmostperiodic
functions was first proposed by H. Bohr, and such a type of real-valued functions
isapproximatelyperiodicandcanbetowithinanydesiredlevelofprecisionifwe
endowthefunctionwithsuitablylong,well-distributed“almost-periods”[59].This
concept was generalized by V. Stepanov, H. Weyl, and A.S. Besicovitch, among
vii
viii Preface
others [57]. Moreover, J.V. Neumann also introduced and studied the notion of
almost periodic functions on locally compact abelian groups [60, 189]. Almost
periodicity is a property of dynamical systems that seems to be repeated in their
paths through phase space, but not precisely. For example, consider a planetary
system,withplanetsinorbitsmovingwithperiodsthatarenotcommensurable(i.e.,
with a period vector that is disproportional to a vector of integers). A theorem of
KroneckerfromDiophantineapproximationcanbeappliedtodemonstratethatany
particularconfigurationthatoccursoncewillrecurtowithinanyspecifiedaccuracy:
if we wait long enough we can observe the planets all return to within a second
of arc to the positions they once were in. In 1955–1962, S. Bochner observed in
various contexts that a certain property enjoyed by the almost periodic functions
on the group G can be applied in obtaining more concise and logical proofs of
certaintheoremsintermsofthesefunctions;S.Bochnercalledhisproperty“almost
automorphy” since it first arouse in work on differential geometry (see [61–63]).
Basedonwell-knownalmostperiodicandalmostautomorphicfunctionsproposed
byBohrandBochner,manynewgeneralizedconceptswereintroducedandstudied
by several researchers on the real line. However, these theories do not work on
timescalessincetheclassicalconceptsofalmostperiodicandalmostautomorphic
functionspurelydependonthetranslationoffunctions.Forexample,theclassical
definitionofalmostperiodicfunctionsonthereallineisgivenasfollows:
For any bounded complex function f and ε > 0, we define T(f,ε) = {τ :
|f(t +τ)−f(t)|<εforallt},T(f,ε)iscalledtheε-translationsetoff.Wesay
f isBohralmostperiodicifforanyε >0,T(f,ε)isrelativelydense.
However, the above definition will not be true on time scales. In fact, for an
arbitrary time scale T, there may be no fixed τ ∈ R such that t +τ ∈ T for all
t ∈ T. This problem is so complex that it will change the classical concept of
relatively dense set on the real line, the convergence of function sequences, the
completeness of function spaces, and an almost periodicity of the variable limit
integrals,etc.Therefore,itisextremelyinsufficientjusttoassumet+τ :=τ(t)∈T
for all t ∈ T when we consider almost periodic problems. In fact, this general
assumption also has some other serious deficiencies, for instance, (1) it is to the
disadvantageoftheanalysisofnumericalcomputationincludingthesimulationof
almostperiodicfunctionssincetheconstructionofthesubsetofRwhereτ isfrom
isunknown;(2)therelativelydensepropertyinthesenseoftimescalescannotbe
considered under this general and abstract assumption; (3) there is no translation
closednessforthevastmajorityoftimescales,muchlessthetranslationinvariance,
that is, there is no τ ∈ R (i.e., there is no subset of R where τ is from) such that
t +τ ∈ Tforallt ∈ T,whichleadstotheassumptionτ(t) ∈ Tmeaningless(we
have provided several representative examples in Chap.2). For the same reasons,
the study of almost automorphic problems on time scales is also a difficult task.
To overcome these difficulties, it is important to study the classification of time
scales under translations and the translation closedness of time scales. Depending
onthereferencesystemoftherealline,wefindthatanarbitrarytimescalewitha
boundedgraininessfunctionμmaypossessawelllocalcompleteclosednesswhich
is more general than translation invariance, it provides an essential condition to
Preface ix
consideralmostperiodicproblems,almostautomorphicproblems,andotherrelated
generalized problems on arbitrary time scales. For our accurate discussion in the
book, we say a time scale is a translation time scale if a mathematical problem
arisingandsolvingmustbebasedonatranslationofthetimescale.
Inthismonograph,weestablishatheoryofclassificationandtranslationclosed-
ness of time scales and based on this we develop a theory of translation functions
on time scales which contains (piecewise) almost periodic functions, (piecewise)
almostautomorphicfunctions,andtheirrelatedgeneralizedfunctions(e.g.,pseudo
almost periodic functions, weighted pseudo almost automorphic functions, etc.).
Underthebackgroundofdynamicequations,thesefunctiontheoriesontimescales
are applied to study the dynamical behavior of solutions for various types of
dynamicequationsonhybriddomainsincludingevolutionequations,discontinuous
equations,impulsiveintegro-differentialequations.Also,thebookprovidesseveral
applications of dynamic equations on mathematical models which cover neural
networks,Nicholson’sblowfliesmodel,Lasota–Wazewskamodel,KeynesianCross
model, those realistic dynamical models with more complex hybrid domain are
consideredunderdifferenttypesoftranslationtimescales.
Thismonographisorganizedin9chapters:
In Chap.1, we present the preliminaries and basic concepts of calculus and
measuretheoryontimescales.
In Chap.2, we classify time scales by translation and develop a theory of
translationclosednessfortimescales.Weintroducetheconceptsofcomplete-closed
translation time scales (CCTS for short), almost-complete closed translation time
scales (ACCTS for short), and changing-periodic time scales. As the particular
cases,thepropertiesofthetranslationandalmosttranslationinvarianceofperiodic
andalmostperiodictimescalesareinvestigated.Moreover,theconceptoftimescale
spaces is introduced and the embedding theorems of time scales are established.
Based on it, almost automorphic time scales are introduced and studied. For
changing-periodic time scales, some basic theorems such as the Decomposition
Theorem of Time Scales and the Periodic Coverage Theorem of Time Scales
areproposedandproved.Inaddition,weinitiatethemethodsofdelayclassification
analysisofdelaydynamicequationsontranslationtimescales.
InChap.3,atheoryofalmostperiodicfunctionsandtheirgeneralizationssuchas
pseudoalmostperiodicfunctionsandweightedpseudoalmostperiodicfunctionsis
establishedonCCTSandchanging-periodictimescales.Also,theBohr-Transform
andMean-Valueofalmostperiodicfunctions,Π-semigroupandmoving-operators
are proposed and discussed, which are the effective tools of investigating almost
periodic and almost automorphic solutions of dynamic equations on CCTS and
changing-periodictimescales.
InChap.4,anotionofpiecewisealmostperiodicfunctionsandthecorresponding
generalizations are introduced and studied on different types of time scales.
Moreover,theconceptsofdouble-almostperiodicfunctionsandweightedpiecewise
pseudodouble-almostperiodicfunctionsareintroducedanddiscussedonACCTS.
In Chap.5, we develop a theory of generalized almost automorphic functions
on translation time scales. The Bochner and Bohr almost automorphic functions
x Preface
on semigroups induced by CCTS are proposed and studied. As their generaliza-
tions, the weighted (piecewise) pseudo almost periodic functions on CCTS are
investigated. Moreover, a notion of local pseudo almost periodic functions on
changing-periodictimescalesisintroducedandsomebasicpropertiesareobtained.
In Chap.6, we mainly discuss nonlinear dynamic equations on translation time
scales.Thealmostperiodicgeneralized solutionsfordynamicequationsonCCTS
areinvestigated.UnderΠ-semigroupsontimescales,theweightedpseudoalmost
periodicsolutionsfornonlinearabstractdynamicequationsarestudied.Inaddition,
based on the C space on changing-periodic time scales, the existence of local
lh
periodicsolutionsforfunctionaldynamicequationswithinfinitedelayisestablished
throughKrasnosel’ski˘ı’sFixedPointTheorem.
In Chap.7, we discuss some related problems of impulsive dynamic equations
on translation time scales. The Cauchy matrix and Liouville’s formula on time
scales for impulsive dynamic equations are derived and their almost periodicity is
analyzed. Based on it, the almost periodic solutions of impulsive delay dynamic
equations are investigated and several applications are provided. In addition, ε-
equivalent impulsive functional dynamic equations are proposed and studied.
Moreover, the existence and exponential stability of weighted piecewise pseudo
double-almost periodic mild solutions of impulsive evolution equations are dis-
cussedonACCTS.
InChap.8,ondifferenttypesoftranslationtimescales,thealmostautomorphic
problems of different types of dynamic equations on time scales including several
representativeclassesofΔand∇-dynamicequationsarediscussed.Wemainlydis-
cusstherelatedproblemsonCCTS,changing-periodictimescales,andsemigroups
inducedbyCCTS.
In Chap.9, we focus on analyzing dynamical system models on translation
timescales.Theexponentialdichotomiesofsomerepresentativetypesofdynamic
equations on time scales are discussed and some new mean-value criteria for
exponential dichotomy are given and proved and applied to analyze the almost
periodic problems of several real dynamic systems models. Moreover, the matrix
measure on time scales is introduced to analyze a class of impulsive Lasota–
WazewskamodelonACCTSandtheexistenceandexponentialstabilityofalmost
periodic solutions of the model are obtained. Finally, a class of double-almost
periodic high-order Hopfield neural networks is proposed and some sufficient
conditions for the existence and ψ-exponential stability of double-almost periodic
solutionswithslightvibrationintimevariablesareestablished.
Thisisamonographdevotedtodevelopingatheoryoftranslationtimescalesand
applicationstotranslationfunctionsanddynamicequations.Thestudyoftranslation
closednessoftimescaleswillnotonlycontributetostudyingtranslationfunctions
such as periodic functions, almost periodic functions, and almost automorphic
functionsandtheirgeneralizationsbutalsowillcontributetoanalyzingthedelaysin
delaydynamicequationsonarbitrarytimescales.Theserelatedtopicsondynamic
equations have become a major research field in pure and applied mathematics.
In particular, this book will cover related results in the discrete and continuous
cases.Moreover,somenewnotionsoftimescalesareintroducedanddiscussedin