Nonlinearity: Ordinary and Fractional Approximations by Sublinear and Max-Product Operators

298 Pages·2018·3.23 MB·English

by  George A. Anastassiou

Studies in Systems, Decision and Control 147
George A. Anastassiou
Nonlinearity: Ordinary
and Fractional
Approximations by
Sublinear and Max-
Product Operators
Studies in Systems, Decision and Control
Volume 147
Series editor
Janusz Kacprzyk, Polish Academy of Sciences, Warsaw, Poland
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George A. Anastassiou
Nonlinearity: Ordinary
and Fractional
Approximations by Sublinear
and Max-Product Operators
123
George A.Anastassiou
Department ofMathematical Sciences
University of Memphis
Memphis,TN
USA
ISSN 2198-4182 ISSN 2198-4190 (electronic)
Studies in Systems,DecisionandControl
ISBN978-3-319-89508-6 ISBN978-3-319-89509-3 (eBook)
https://doi.org/10.1007/978-3-319-89509-3
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Preface
Nonlinear mathematics extend the linear mathematics to meet various needs and
demandsofpureandappliedmathematics,amongotherstocoveragreatvarietyof
applications in the real world. Approximation by sublinear operators with appli-
cations to max-product operators is a new trend in approximation theory. These
operators are nonlinear and rational giving very fast and flexible approximations
based on limited data.
In this book, we focus more in approximations under the presence of ordinary
and various kinds of fractional smoothness, deriving better approximations than
withoutsmoothness.Wepresentboththeunivariateandmultivariatecases.Thelast
three chapters contain approximations under the influence of convexity, there the
estimatesaremoreelegantandcompactwithsmallconstants,andtheconvergence
ofhighspeeds.Thismonographisthenaturalevolutionofrecentauthor’sresearch
work put in a book form for the first time. The presented approaches are original,
and chapters are self-contained and can be read independently. This monograph is
suitable to be used in related graduate classes and research projects. We exhibit to
the maximum our approximation methods to all possible directions.
The motivation to write this monograph came by the following: various issues
relatedtothemodellingandanalysisofordinaryandfractional-ordersystemshave
gained an increased popularity, as witnesses by many books and volumes in
Springer’s program:
http://www.springer.com/gp/search?query=fractional&submit=Prze%C5%
9Blij
and thepurpose ofour book istoprovide adeeper formal analysis onsome issues
that are relevant to many areas, for instance, decision-making, complex processes,
systems modelling and control, and related areas. The above are deeply embedded
in the fields of mathematics, engineering, computer science, physics, economics,
social and life sciences.
The list of presented topics follows:
approximationbysublinearoperators,approximationbymax-productoperators,
conformable fractional approximation by max-product operators,
vii
viii Preface
Caputo fractional approximation by sublinear operators,
Canavati fractional approximation by max-product operators,
iterated fractional approximation by max-product operators,
mixed conformable fractional approximation by sublinear operators,
approximation offuzzy numbers by max-product operators,
approximation by multivariate sublinear and max-product operators,
approximation by sublinear and max-product operators using convexity,
conformable fractional approximations by max-product operators using
convexity,
and approximations by multivariate sublinear and max-product operators under
convexity.
An extensive list of references is given per chapter.
The book’s results are expected to find applications in many areas of pure and
appliedmathematics,especially in approximationtheory and numerical analysisin
both ordinary and fractional sense. As such this monograph is suitable for
researchers, graduate students and seminars of the above disciplines, also to be in
all science and engineering libraries.
The preparation of the book took place during 2017 at the University of
Memphis.
The author likes to thank Prof. Alina Alb Lupas of University of Oradea,
Romania, for checking and reading the manuscript.
Memphis, USA George A. Anastassiou
January 2018
Contents
1 Approximation by Positive Sublinear Operators. . . . . . . . . . . . . . . 1
1.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1
1.2 Main Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3
1.3 Applications . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8
References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17
2 High Order Approximation by Max-Product Operators. . . . . . . . . 19
2.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19
2.2 Main Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21
References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41
3 Conformable Fractional Approximations Using Max-Product
Operators . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 43
3.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 43
3.2 Background . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 45
3.3 Main Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 49
3.4 Applications . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 57
References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 65
4 Caputo Fractional Approximation Using Positive Sublinear
Operators . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 67
4.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 67
4.2 Main Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 69
4.3 Applications . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 75
References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 94
ix
x Contents
5 Canavati Fractional Approximations Using Max-Product
Operators . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 95
5.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 95
5.2 Main Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 96
5.3 Applications . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 102
References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 118
6 Iterated Fractional Approximations Using Max-Product
Operators . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 119
6.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 119
6.2 Main Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 120
6.3 Applications, Part A . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 126
6.4 Applications, Part B . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 143
References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 149
7 Mixed Conformable Fractional Approximation Using Positive
Sublinear Operators . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 151
7.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 151
7.2 Background . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 152
7.3 Main Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 156
7.4 Applications, Part A . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 160
7.5 Applications, Part B . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 171
References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 177
8 Approximation of Fuzzy Numbers Using Max-Product
Operators . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 179
8.1 Background . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 179
8.2 Main Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 182
References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 187
9 High Order Approximation by Multivariate Sublinear
and Max-Product Operators. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 189
9.1 Background . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 189
9.2 Main Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 195
References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 227
10 High Order Approximation by Sublinear and Max-Product
Operators Using Convexity. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 229
10.1 Background . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 229
10.2 Main Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 233
References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 241