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Synthesis Lectures on Mathematics & Statistics Chao Wang · Ravi P. Agarwal Dynamic Equations and Almost Periodic Fuzzy Functions on Time Scales Synthesis Lectures on Mathematics & Statistics SeriesEditor StevenG.Krantz,DepartmentofMathematics,WashingtonUniversity,SaintLouis,MO,USA This series includes titles in applied mathematics and statistics for cross-disciplinary STEMprofessionals,educators,researchers,andstudents.Theseriesfocusesonnewand traditionaltechniquestodevelopmathematicalknowledgeandskills,anunderstandingof core mathematical reasoning, and the ability to utilize data in specific applications. 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 ©TheEditor(s)(ifapplicable)andTheAuthor(s),underexclusivelicensetoSpringerNatureSwitzerlandAG 2022 Thisworkissubjecttocopyright.AllrightsaresolelyandexclusivelylicensedbythePublisher,whetherthewhole orpartofthematerialisconcerned,specificallytherightsoftranslation,reprinting,reuseofillustrations,recitation, broadcasting,reproductiononmicrofilmsorinanyotherphysicalway,andtransmissionorinformationstorage andretrieval,electronicadaptation,computersoftware,orbysimilarordissimilarmethodologynowknownor hereafterdeveloped. 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ThisSpringerimprintispublishedbytheregisteredcompanySpringerNatureSwitzerlandAG Theregisteredcompanyaddressis:Gewerbestrasse11,6330Cham,Switzerland WededicatethisbooktoourwivesWeiDuand SadhnaAgarwal 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