Table Of ContentStudies in Systems, Decision and Control 441
George A. Anastassiou
Fractional
Inequalities
In Banach
Algebras
Studies in Systems, Decision and Control
Volume 441
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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
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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(sequen-
tial)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
