Table Of ContentSynthesis Lectures on
Mathematics & Statistics
Chao Wang · Ravi P. Agarwal
Dynamic Equations and
Almost Periodic Fuzzy
Functions on Time
Scales
Synthesis Lectures on Mathematics &
Statistics
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This series includes titles in applied mathematics and statistics for cross-disciplinary
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Chao Wang · Ravi P. Agarwal
Dynamic Equations
and Almost Periodic Fuzzy
Functions on Time Scales
ChaoWang RaviP.Agarwal
DepartmentofMathematics DepartmentofMathematics
YunnanUniversity TexasA&MUniversity-Kingsville
Yunnan,China Kingsville,TX,USA
ISSN1938-1743 ISSN1938-1751 (electronic)
SynthesisLecturesonMathematics&Statistics
ISBN978-3-031-11235-5 ISBN978-3-031-11236-2 (eBook)
https://doi.org/10.1007/978-3-031-11236-2
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Preface
Stefan Hilger introduced the concept of time scales in 1988 which can be applied to
unify the study of differential equations for continuous case and difference equations for
discretecase(see[1]).Sincethen,timescaletheoryhasdevelopedrapidly(see[2])andit
wasusedtodevelopfunctioncalculusinvariousmathematicalfieldssuchasthecalculus
for real-valued functions (see [2, 4]), quaternion dynamic equations (see [5, 6]), measure
theory (see [7]), set-valued functions (see [8, 9]) and fuzzy-valued functions (see [10,
11, 12]), etc. When studying the dynamical behavior of an object on time scales it is
significant to unify a larger range of time scales to obtain more comprehensive results
(see [13, 14, 15]).
Foramoreaccuratedescriptionoftherealworldphenomena,itisnecessarytoconsider
anumberofuncertainfactorsandthisleadsnaturallytofuzzydynamicalmodels(see[16,
17,18,19,20,21,22,23,24])andset-valuedfunctions,fuzzy-valuedfunctionsandtheir
related applications to dynamic equations on time scales (see [10, 11, 12]).
Almost periodic theory was initiated by Bohr during the period 1923–1925 (see [25,
26]) which is an important theory to precisely describe almost periodic approximation
phenomena in engineering, life sciences, information sciences and control theory (see
[27,28,29,30,31,32,33])andinparticularitarisesincelestialmechanics,bioengineer-
ing and electronic circuits (see [28, 30]). The study of almost periodicity of solutions on
time scales was considered in [34, 35, 36]. In 2001, Park, Jung and Lee considered the
existence and stability of almost periodic solutions for fuzzy functional differentialequa-
tions (see [37]), then in 2004, Bede and Gal developed a theory of almost periodic fuzzy
number-valued functions and studied a class of almost periodic fuzzy dynamical systems
(see [38]). In 2016, Hong and Peng extended the concept of almost periodic functions
to set-valued functions on periodic time scales and established the existence of almost
periodicsolutionsforset-valueddynamicequations(see[39]).Unfortunately,thevalidity
of all results involved above must be restricted to traditional periodic time scales (i.e.,
periodic time scale under a translation) which have a nice closedness under translations.
It is worth noting that these results cannot be applied to some important irregular time
scales like (−q)Z = {(−q)n : q > 1,n ∈ Z}∪{0} (which has important applications in
quantumtheory),±N21 andT=Tn,thespaceofharmonicnumbers,becausetheconcept
vii
viii Preface
of almost periodic functions in these works become unsuitable and inapplicablefor these
irregular time scales which are without closedness under translations. To overcome this
difficulty,in2017,inspiredbyAdvar’swork[40],WangandAgarwalintroducedthecon-
cept of relatively dense set under shift operators and proposed almost periodic stochastic
processes.Thisdevelopmentmakesitpossibletostudyalmostperiodicproblemsonmany
irregulartimescalesinvolvingq-differenceequationsandmore generaltypesofdynamic
equations (see [15, 41]).
Ontheotherhand,fuzzyarithmeticisalsoacomplexandintractableproblemincluding
themultiplicationanddivisionoffuzzynumbersandfuzzyvectors.Fortunately,Stefanini
etal.,openedafeasibleavenuetofuzzyarithmeticandintroducedageneralizedHukuhara
differenceanddivisionforintervalandfuzzyarithmetic(see[42,24]).Moreover,theyalso
initiatedtheunidimensionalandmultidimensionalboxeswhichmotivatedustointroduce
fuzzy (box) vectors and establish the calculus of fuzzy vector-valued functions on time
scales (see [43]). Further, in [43], we introduced a new multiplication determined by a
determinantalgorithm,basedonwhichtheclassicalfundamentalformulasofcalculuscan
be derived under fuzzy background, and this provides a powerful calculus tool to solve
classicalproblemsofmultidimensionalfuzzydynamicsystemsinvariousresearchfields.
We organize this book into six chapters.
InChap.1,somenecessaryknowledgeofintervalandfuzzyarithmeticispresented.A
generalizationoftheHukuharadifferenceisintroduced.First,thecaseofcompactconvex
sets is investigated which are applied to generalize the Hukuhara difference of fuzzy
numbers by using their compact and convex level-cuts. Moreover, a similar approach is
presented to propose a generalization of division for real intervals and fuzzy numbers.
Some applications are provided to solve interval and fuzzy linear equations and fuzzy
differential equations.
InChap.2,anembeddingtheoremforfuzzymultidimensionalspaceisestablishedand
two new types of multiplication of fuzzy vectors are introduced and studied.
InChap.3,weintroducethebasicnotionsofgH-(cid:2)-derivativesoffuzzyvector-valued
functionsontimescalesandobtaintheirfundamentalproperties.Moreover,the(cid:2)-integral
of fuzzy vector-valued functions is introduced and studied. Some basic results related to
calculus of fuzzy vector-valued functions are established on time scales.
In Chap. 4, some necessary knowledge of shift operators and a generalized periodic
timescalesispresented.Anotionofshiftalmostperiodicfuzzyvector-valuedfunctionsis
addressedandstudiedoncomplete-closedtimescalesundernon-translationalshifts,some
fundamentalresultsofshiftalmostperiodicfuzzyvector-valuedfunctionsareestablished.
InChap.5,somebasicresultsoffuzzymultidimensionalspacesaredemonstratedand
a new division of multidimensional intervals and fuzzy vectors induced by a determinant
algorithm is introduced and studied.
In Chap. 6, we develop a theory of almost periodic fuzzy multidimensional dynamic
systems on time scales and several applications are provided. In particular, a new type
Preface ix
of fuzzy dynamic systems called fuzzy q-dynamic systems (i.e., fuzzy quantum dynamic
systems) is proposed and studied.
Thisbookwillestablishanalmostperiodictheoryofmultidimensionalfuzzydynamic
equations and fuzzy vector-valued functions on complete-closed time scales under non-
translational shifts including some commonly irregular time scales, and it involves an
almost periodic theory of fuzzy functions on quantum-like time scales. Our results are
not only effective on periodic time scales (i.e., T = Z,R or hZ, etc.) but also are valid
for irregular time scales qZ,−qZ,±N21,N13,(−q)Z, etc.
Thebookiswrittenatagraduatelevelandisintendedforuniversitylibraries.Graduate
students and researchers working in the field of fuzzy dynamic equations on time scales
will be able to stimulate further research. The book is also a good reference material for
those undergraduates who are interested in fuzzy dynamic equations and functions on
time scales and familiar with fuzzy sets and systems and ordinary differential equations.
WeacknowledgewithgratitudethesupportofNationalNaturalScienceFoundationof
China(11961077),CAS“LightofWestChina”ProgramofChineseAcademyofSciences
and Educational Reform Research Project of Yunnan University (No. 2021Y10).
Kunming, China Chao Wang
Kingsville, USA Ravi P. Agarwal
Contents
1 GeneralizedHukuharaDifferenceandDivisionforIntervalandFuzzy
Arithmetic ........................................................... 1
1.1 Generalized Hukuhara Difference and Properties ..................... 1
1.2 The Case of Compact Intervals in Rn ............................... 7
1.3 gH-Difference of Fuzzy Numbers .................................. 10
1.3.1 Support Functions and Fuzzy gH-Difference .................. 11
1.3.2 A Decomposition of Fuzzy Numbers and gH-Difference ....... 15
1.3.3 Approximated Fuzzy gH-Difference ......................... 19
1.4 Generalized Division ............................................. 20
1.4.1 The Fuzzy Case ........................................... 22
1.4.2 Approximated Fuzzy g-Division ............................. 23
1.5 Applications of gH-Difference and g-Division ....................... 24
1.5.1 Interval and Fuzzy Algebraic Equations ...................... 25
1.5.2 Interval and Fuzzy Differential Equations ..................... 28
2 AnEmbeddingTheoremandMultiplicationofFuzzyVectors ............ 29
2.1 An Embedding Theorem for Fuzzy Multidimensional Space ........... 36
2.2 Multiplication of Fuzzy Vectors in Fuzzy Multidimensional Space ...... 41
3 CalculusofFuzzyVector-ValuedFunctionsonTimeScales ............... 57
3.1 gH-Δ-Derivative of Fuzzy Vector-Valued Functions on Time Scales .... 58
3.2 Δ-Integral of Fuzzy Vector-Valued Functions on Time Scales .......... 79
4 ShiftAlmostPeriodicFuzzyVector-ValuedFunctions .................... 85
4.1 Shift Operators on Time Scales .................................... 85
4.1.1 Shift Operators ............................................ 85
4.1.2 Periodicity of Time Scales .................................. 89
4.2 Complete-Closed Time Scales Under Non-translational Shifts .......... 94
4.3 Shift Almost Periodic Fuzzy Vector-Valued Functions ................. 97
xi
