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Studies in Systems, Decision and Control 441 George A. Anastassiou Fractional Inequalities In Banach Algebras Studies in Systems, Decision and Control Volume 441 SeriesEditor JanuszKacprzyk,SystemsResearchInstitute,PolishAcademyofSciences, Warsaw,Poland The series “Studies in Systems, Decision and Control” (SSDC) covers both new developments and advances, as well as the state of the art, in the various areas of broadly perceived systems, decision making and control–quickly, up to date and with a high quality. The intent is to cover the theory, applications, and perspec- tives on the state of the art and future developments relevant to systems, decision making, control, complex processes and related areas, as embedded in the fields of engineering, computer science, physics, economics, social and life sciences, as wellastheparadigmsandmethodologiesbehindthem.Theseriescontainsmono- graphs, textbooks, lecture notes and edited volumes in systems, decision making and control spanning the areas of Cyber-Physical Systems, Autonomous Systems, SensorNetworks,ControlSystems,EnergySystems,AutomotiveSystems,Biolog- ical Systems, Vehicular Networking and Connected Vehicles, Aerospace Systems, Automation, Manufacturing, Smart Grids, Nonlinear Systems, Power Systems, Robotics,SocialSystems,EconomicSystemsandother.Ofparticularvaluetoboth thecontributorsandthereadershiparetheshortpublicationtimeframeandtheworld- widedistributionandexposurewhichenablebothawideandrapiddisseminationof researchoutput. IndexedbySCOPUS,DBLP,WTIFrankfurteG,zbMATH,SCImago. AllbookspublishedintheseriesaresubmittedforconsiderationinWebofScience. Moreinformationaboutthisseriesathttps://link.springer.com/bookseries/13304 George A. Anastassiou Fractional Inequalities In Banach Algebras GeorgeA.Anastassiou DepartmentofMathematicalSciences UniversityofMemphis Memphis,TN,USA ISSN 2198-4182 ISSN 2198-4190 (electronic) StudiesinSystems,DecisionandControl ISBN 978-3-031-05147-0 ISBN 978-3-031-05148-7 (eBook) https://doi.org/10.1007/978-3-031-05148-7 ©TheEditor(s)(ifapplicable)andTheAuthor(s),underexclusivelicensetoSpringerNature SwitzerlandAG2022 Thisworkissubjecttocopyright.AllrightsaresolelyandexclusivelylicensedbythePublisher,whether thewholeorpartofthematerialisconcerned,specificallytherightsoftranslation,reprinting,reuse ofillustrations,recitation,broadcasting,reproductiononmicrofilmsorinanyotherphysicalway,and transmissionorinformationstorageandretrieval,electronicadaptation,computersoftware,orbysimilar ordissimilarmethodologynowknownorhereafterdeveloped. Theuseofgeneraldescriptivenames,registerednames,trademarks,servicemarks,etc.inthispublication doesnotimply,evenintheabsenceofaspecificstatement,thatsuchnamesareexemptfromtherelevant protectivelawsandregulationsandthereforefreeforgeneraluse. Thepublisher,theauthorsandtheeditorsaresafetoassumethattheadviceandinformationinthisbook arebelievedtobetrueandaccurateatthedateofpublication.Neitherthepublishernortheauthorsor theeditorsgiveawarranty,expressedorimplied,withrespecttothematerialcontainedhereinorforany errorsoromissionsthatmayhavebeenmade.Thepublisherremainsneutralwithregardtojurisdictional claimsinpublishedmapsandinstitutionalaffiliations. ThisSpringerimprintispublishedbytheregisteredcompanySpringerNatureSwitzerlandAG Theregisteredcompanyaddressis:Gewerbestrasse11,6330Cham,Switzerland DedicatedtoCovid-19vaccinecreators Preface FractionalCalculusanditslonghistorystartedin1695byL’HospitalandLeibnitz. Overtheyears,manydifferentkindsofitemergedaccordingtotheneedsofscience givenbymanyfamousmathematicians.Inthelast50years,fractionalcalculusdue toitswideapplicationstomanyappliedscienceshasgainedextremepopularity. By employing regular and iterated (sequential) generalized Caputo and Cana- vatifractionalleftandrightvectorialTaylorformulae,weestablishagreatvariety of mixed regular and iterated generalized fractional inequalities. In particular, we derive generalized Caputo Fractional Ostrowski and Grüss type inequalities involvingseveralBanachalgebravaluedfunctions,theregularanditerated(sequential)versions.Also,weestablishgeneralizedCanavatifractionalOstrowski,Opial, Grüss and Hilbert-Pachpatte type inequalities for multiple Banach algebra valued functions.Thesamekindoftheabovetypesofinequalitiesarestudiedatunivariate and multivariate settings for ordinary vectorial derivatives and vectorial partial derivatives, respectively. The counterpart studies continue, extensively, by using the p-Schatten norms over the von Neumann-Schatten classes (∗-ideals) B (H), p p >_1,andweproducealltheanalogousinequalities.Weprovideagreatvarietyof applicationsperchapter. Sointhismonograph,allthatispresentedistheoriginalworkbytheauthorgiven ataverygeneralleveltocoveramaximumnumberofdifferentfractionalcalculus applications. Asaresult,thismonographisthenaturalandexpectedevolutionoftheauthor’s recentresearchworkputinabookformforthefirsttime.Thepresentedapproaches are original, and chapters are self-contained and can be read independently. This monographissuitabletobeusedinrelatedgraduateclassesandresearchprojects. The motivation to write this monograph came by the following: various issues relatedtothemodelingandanalysisofordinaryandfractionalordersystemshave gainedincreasedpopularity,aswitnessesbymanybooksandvolumesinSpringer’s program: http://www.springer.com/gp/search?query=fractional&submit=Prze%C5%9Blij vii viii Preface and the purpose of our book is to capture at a very general level a deeper formal analysisonsomeissuesthatarerelevanttomanyareas,forinstance:decisionmaking, complexprocesses,systemsmodelingandcontrol,andrelatedareas. The above are deeply embedded in the fields of mathematics, engineering, computerscience,physics,economics,socialandlifesciences. Next is listed the author’s recent monographs in Fractional Analysis and applications: • George Anastassiou, “Intelligent Computations: Abstract Fractional Calculus, Inequalities, Approximations”, Springer, Studies in Computational Intelligence 734,Heidelberg,NewYork,2018. • GeorgeAnastassiou,“Nonlinearity:OrdinaryandFractionalApproximationsby Sublinear and Max-product Operators”, Springer, Studies in Systems, Decision andControl147,Heidelberg,NewYork,2018. • George Anastassiou, “Ordinary and Fractional approximation by non-additive integrals: Choquet, Shilkret and Sugeno integral approximators”, Springer, StudiesinSystems,DecisionandControl190,Heidelberg,NewYork,2019. • GeorgeAnastassiou,“IntelligentAnalysis:FractionalInequalitiesandApproxi- mationsExpanded”,Springer,StudiesinComputationalIntelligence886,Heidel- berg,NewYork,2020. • George Anastassiou, “Generalized Fractional Calculus: New Advancements and Applications”, Springer, Studies in Systems, Decision and Control 305, Heidelberg,NewYork,2021. • George Anastassiou, “Constructive Fractional Analysis with Applications”, Springer,StudiesinSystems,DecisionandControl362,Heidelberg,NewYork, 2021. • George Anastassiou, “Unification of Fractional Calculi with Applications”, Springer,StudiesinSystems,DecisionandControl398,Heidelberg,NewYork, 2022. Thecompletelistofpresentedtopicsherefollows: • Generalized Fractional Ostrowski and Grüss type inequalities involving several Banachalgebravaluedfunctions, • Sequential Generalized Fractional Ostrowski and Grüss type inequalities for severalBanachalgebravaluedfunctions, • GeneralizedCanavatiFractionalOstrowski,OpialandGrüsstypeinequalitiesfor Banachalgebravaluedfunctions, • Generalized Canavati Fractional Hilbert-Pachpatte type inequalities for Banach algebravaluedfunctions, • GeneralizedOstrowski,OpialandHilbert-PachpattetypeinequalitiesforBanach algebravaluedfunctionsinvolvingintegervectorialderivatives, • Multivariate Ostrowski type inequalities for several Banach algebra valued functions, • p-Schatten norm generalized fractional Ostrowski and Grüss type inequalities involvingseveralfunctions, Preface ix • p-Schatten norm sequential generalized fractional Ostrowski and Grüss type inequalitiesforseveralfunctions, • p-SchattennormgeneralizedCanavatifractionalOstrowski,OpialandGrüsstype inequalitiesinvolvingseveralfunctions, • p-SchattennormGeneralizedCanavatiFractionalHilbert-Pachpattetypeinequal- itiesforvonNeumann-SchattenclassB (H)valuedfunctions, p • p-Schatten norm Generalized Ostrowski, Opial and Hilbert Pachpatte type inequalitiesforvonNeumann-SchattenclassB (H)valuedfunctionswithinteger p vectorialderivatives, • p-SchattennormMultivariateOstrowskitypeinequalitiesforseveralNeumann- SchattenclassB (H)valuedfunctions. p The book’s results are expected to find applications in many areas of pure and appliedmathematics,especiallyinfractionalinequalitiesandfractionaldifferential equations. Other possible applications can be in applied sciences like geophysics, physics, chemistry, economics, engineering, etc. All in all, what is presented here is a valuable tool for a large range of applications. Therefore, this monograph is suitableforresearchers,graduatestudents,practitionersandseminarsoftheabove disciplines,andalsotobeinallscienceandengineeringlibraries. The preparation of the book took place during 2021–22 at the University of Memphis,duringtheauthor’shomestayduringtheCovid-19andOmicronoutbreak keepinghimalive,happyandincontrol! The author likes tothank Prof.Alina Lupas of University of Oradea, Romania, forcheckingandreadingthemanuscript. Memphis,TN,USA GeorgeA.Anastassiou March2022 Contents 1 GeneralizedFractionalOstrowskiandGrüssInequalities withMultipleBanachAlgebraValuedFunctions ................. 1 1.1 Introduction ............................................. 1 1.2 VectorialBackgroundFractionalCalculus ................... 2 1.3 BanachAlgebrasBackground .............................. 4 1.4 MainResults ............................................ 6 1.5 Applications ............................................. 17 References .................................................... 19 2 Iterated Generalized Fractional Ostrowski and Grüss InequalitieswithMultipleBanachAlgebraValuedFunctions ...... 21 2.1 Introduction ............................................. 21 2.2 Vectorial Sequential Generalized Fractional Calculus Background ............................................. 22 2.3 BanachAlgebrasBackground .............................. 26 2.4 MainResults ............................................ 27 2.5 Applications ............................................. 42 References .................................................... 44 3 GeneralizedCanavatiFractionalOstrowski,OpialandGrüss InequalitieswithMultipleBanachAlgebraValuedFunctions ...... 45 3.1 Introduction ............................................. 45 3.2 BackgroundonVectorialGeneralizedCanavatiFractional Calculus ................................................ 46 3.3 BanachAlgebrasBackground .............................. 50 3.4 MainResults ............................................ 51 3.5 Applications ............................................. 61 3.6 Addendum .............................................. 62 References .................................................... 66 xi