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 e-mail: [email protected] 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 withahighquality.Theintentistocoverthetheory,applications,andperspectives on the state of the art and future developments relevant to systems, decision making,control,complexprocessesandrelatedareas, asembeddedinthefieldsof engineering,computerscience,physics,economics,socialandlifesciences,aswell astheparadigmsandmethodologiesbehindthem.Theseriescontainsmonographs, textbooks, lecture notes and edited volumes in systems, decision making and control spanning the areas of Cyber-Physical Systems, Autonomous Systems, Sensor Networks, Control Systems, Energy Systems, Automotive Systems, Biological Systems, Vehicular Networking and Connected Vehicles, Aerospace Systems, Automation, Manufacturing, Smart Grids, Nonlinear Systems, Power Systems, Robotics, Social Systems, Economic Systems and other. Of particular valuetoboththecontributorsandthereadershiparetheshortpublicationtimeframe and the world-wide distribution and exposure which enable both a wide and rapid dissemination of research output. More information about this series at http://www.springer.com/series/13304 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 LibraryofCongressControlNumber:2018937682 ©SpringerInternationalPublishingAG,partofSpringerNature2018 Thisworkissubjecttocopyright.AllrightsarereservedbythePublisher,whetherthewholeorpart of the material is concerned, specifically the rights of translation, reprinting, reuse of illustrations, recitation, broadcasting, reproduction on microfilms or in any other physical way, and transmission orinformationstorageandretrieval,electronicadaptation,computersoftware,orbysimilarordissimilar methodologynowknownorhereafterdeveloped. The use of general descriptive names, registered names, trademarks, service marks, etc. in this publicationdoesnotimply,evenintheabsenceofaspecificstatement,thatsuchnamesareexemptfrom therelevantprotectivelawsandregulationsandthereforefreeforgeneraluse. The publisher, the authors and the editors are safe to assume that the advice and information in this book are believed to be true and accurate at the date of publication. Neither the publisher nor the authorsortheeditorsgiveawarranty,expressorimplied,withrespecttothematerialcontainedhereinor for any errors or omissions that may have been made. The publisher remains neutral with regard to jurisdictionalclaimsinpublishedmapsandinstitutionalaffiliations. Printedonacid-freepaper ThisSpringerimprintispublishedbytheregisteredcompanySpringerInternationalPublishingAG partofSpringerNature Theregisteredcompanyaddressis:Gewerbestrasse11,6330Cham,Switzerland Dedicated to My Family. 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
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