Springer Monographs in Mathematics David N. Cheban Nonautonomous Dynamics Nonlinear Oscillations and Global Attractors Springer Monographs in Mathematics Editors-in-Chief Isabelle Gallagher, Paris, France Minhyong Kim, Oxford, UK Series Editors Sheldon Axler, San Francisco, USA Mark Braverman, Princeton, USA Maria Chudnovsky, Princeton, USA Tadahisa Funaki, Tokyo, Japan Sinan C. Güntürk, New York, USA Claude Le Bris, Marne la Vallée, France Pascal Massart, Orsay, France Alberto A. 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Cheban Nonautonomous Dynamics Nonlinear Oscillations and Global Attractors 123 DavidN.Cheban Faculty of Mathematics State University of Moldova Chisinau, Moldova ISSN 1439-7382 ISSN 2196-9922 (electronic) SpringerMonographs inMathematics ISBN978-3-030-34291-3 ISBN978-3-030-34292-0 (eBook) https://doi.org/10.1007/978-3-030-34292-0 Mathematics Subject Classification (2010): Primary: 34C12, 34C27, 34D25, 37B55, 37B35, 37B20, 37B25,Secondary:34K14,34K27,35A16,35B15,35B35,35B40,37L30,37N25 ©SpringerNatureSwitzerlandAG2020 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. 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ThisSpringerimprintispublishedbytheregisteredcompanySpringerNatureSwitzerlandAG Theregisteredcompanyaddressis:Gewerbestrasse11,6330Cham,Switzerland To my grandchildren Olivia and Nicholas Preface The present monograph is dedicated to the abstract theory of nonau- tonomous dynamical systems, which is a new branch of the theory of dynamical systems. In this monograph, I present the developments of the basic ideas and methods for nonautonomous dynamical systems and their applications over the past ten years. Ourmainapplicationsarenonautonomousordinarydifferential/difference equations, functional differential/difference equations and some classes of partial differential equations. In recent years, there seems to be a growing interest in nonautonomous differential/difference equations, both finite-dimensional (ordinary differential/difference equations) and infinite-dimensional (functional differential/differenceequationsandpartialdifferentialequations). Nonlocal problems concerning the conditions of existence of different classes of solutions play an important role in the qualitative theory of dif- ferential equations. Here we include the problems of boundedness, periodic- ity,Bohr/Levitanalmostperiodicity,almostautomorphy,almostrecurrence in the sense of Bebutov, recurrence in the sense of Birkhoff, stability in the sense of Poisson, problems of existence of limit regimes of different types, convergence, dissipativity, etc. The present work belongs to this direction, and it is devoted to the mathematical theory of nonautonomous dynamical systems and applica tions. The main goal of this book is to study Bohr/Levitan almost periodic and almost automorphic systems,different classes ofPoisson stable motions, and global attractors of Bohr/Levitan almost periodic systems with contin- uous and discrete time. Thus,therearetwokeyobjectsthatarethesubjectsofstudyinthisbook. These are various oscillatory regimes (Bohr/Levitan almost periodic and Poisson stable movements) and global attractors and application of the obtained general results (related to abstract nonautonomous dynamical vii viii Preface systems) to different classes of nonautonomous differential and difference equations. The problems that we consider in this book are mainly motivated by nonautonomous differential/difference equations. The monograph presents ideas and methods, developed by the author, to solve the problem of existence of Bohr/Levitan almost periodic (respectively, almostrecurrentinthesenseofBebutov,almostauthomorphic,Poissonstable) solutions, and global attractors of nonautonomous differential/difference equa- tions. Namely, the text provides answers to the following problems: 1. ProblemofexistenceofatleastoneBohr/Levitanalmostperiodicsolution for linear almost periodic differential/difference equations without Favard’s separation condition (Favard theory); 2. Problem of existence of Bohr/Levitan almost periodic solution for monotone differential/difference equations; 3. Problem of existence of at least one Bohr/Levitan almost periodic solution for uniformly stable and dissipative differential equations (I. U. Bronshtein’s conjecture, 1975); 4. Problem of description of the structure of the global attractor for holo- morphic and gradient-like nonautonomous dynamical systems. Chapters I and IV–VI are devoted to nonlinear oscillations, and global attractors are studied in Chaps. II, III, and VI. One fundamental question of the qualitative theory of nonautonomous differential/differenceequationsistheproblemofalmostperiodicity,ormore generally Poisson stability (in particular, Levitan almost periodicity, Bochner almost automorphy, almost recurrence in the sense of Bebutov, recurrence in the sense of Birkhoff, and so on) of solutions. Thetheoryofalmostperiodicfunctionswasmainlycreatedandpublished byBohr[42–45](inthisrelationseealsotheimportantresultsofBohl[40,41] and Esclangon [145–147]). Bohr’s theory was substantially extended by Bochner[35],Weyl[321],Besicovitch[22],Favard[151],vonNeumann[235], Stepanoff [308], Bogolyubov [37–39] and others. Levitan [212] introduced a new class offunctions (the so-called N-almost periodic or Levitan almost periodic functions) that includes all Bohr almost periodic functions, but does not coincide with the latter. The foundation of this type of function was created in the works of Levitan [214], Levin [210, 211], and Marchenko [220, 221]. A notion of almost automorphic function was introduced by Bochner [34–36] which also is an extension of Bohr almost periodicity. Substantial resultsaboutalmostautomorphicfunctionswereobtainedbyVeech[314–316]. ThedifferentclassesofPoissonstablefunctions(inparticular,recurrentin the sense of Birkhoff [25], almost recurrent in the sense of Bebutov [13], pseudo recurrent [283, 284, 302], and so on) were introduced and studied by Shcherbakov [283–296]. Preface ix The theory of Bohr/Levitan almost periodic, almost automorphic, and PoissonstablefunctionsiswidelypresentedinthemonographsofAmerioand Prouse [3], Bochner [35], Corduneanu [129, 130], Toka Diagana [140], Fink [155],Favard[151],Hinoetal.[179],Levitan[214],LevitanandZhikov [215], Pankov [244], N’Guerekata [236, 237], Shcherbakov [290, 294], Shen and Yi [297], Yoshizawa [324], Zaidman [325] and others (see also the references therein). Inthelast25–30years,thetheoryofBohr/Levitanalmostperiodic,almost automorphic and Poisson stable differential/difference equations has been developed in connection with problems of differential/difference equations, stability theory, dynamical systems, and so on. The main achievements are related to the application of ideas and methods of dynamic systems in the study of the above problems. Globalattractorsplayaveryimportantroleinthestudyoftheasymptotic behavior of dynamical systems (both autonomous and nonautonomous). In thelast20–25yearsmanyworksdedicatedtothestudyofglobalattractorsof dynamical systems (including the infinite-dimensional systems) have been published.See,forexample,BabinandVishik[9],Chueshov[109],Hale[171], Ladyzhenskaya [208], Robinson [255], Temam [310, 311] (for autonomous systems), Carvalho et al. [67], Cheban [84, 91], Chepyzhov and Vishik [106], Haraux [174], Kloeden and Rasmussen [197] (for nonautonomous systems), and references therein. In this book we study global attractors for a special class of nonau- tonomous dynamical systems, namely for the Bohr/Levitan almost periodic systems. We establish the structure of global attractors for this class of systems and the existence at least one almost periodic motion belonging to the Levinson center (maximal compact global invariant attractor). Our approach to the study of the problem of Bohr/Levitan almost peri- odicity of solutions of almost periodic differential equations and their com- pact global attractors consists in applying to the study of nonautonomous systemstheideasandmethodsdevelopedinthetheoryofabstractdynamical systems.Theideaofapplyingmethodsofthetheoryofdynamicalsystemsto the study of nonautonomous differential equations is not new. It has been successfully applied to the resolution of different problems in the theory of linear and nonlinear nonautonomous differential equations for more than 50 years. This approach to nonautonomous differential equations was first introduced in the works of Millionshchikov [226–228], Shcherbakov [290, 294], Deyseach and Sell [139], Miller [225], Seifert [273], Sell [275, 276], later inworksofZhikov[327],Bronshtein[48],Johnson[188,189],andmanyother authors. This approach consists in naturally associating with the equation x0 ¼fðt;xÞ ð1Þ x Preface apairofdynamicalsystemsandahomomorphismofthefirstontothesecond. One assigns the information about the right-hand side of Eq. (1) to one dynamicalsystem,andtheinformationaboutthesolutionsof(1)totheother. PlentyofworksarededicatedtothestudyoftheproblemofBohr/Levitan almost periodicity, almost automorphy, and different classes of Poisson sta- bility of solutions for differential/difference equations. We survey briefly some of these works in our book. Note that a bibliography of papers on Bohr/Levitan almost periodic, almost automorphic, and Poisson stable solutions and compact global attractors of almost periodic differential/difference equations contains over 300 items, i.e., it is still a very active area of research. The body of the book consists of six chapters. In the first chapter, on semigroup dynamical systems, different kinds of Poisson stability of motions and their comparability by character of recur- rence are introduced and studied: Bohr/Levitan almost periodicity, almost automorphy, Bebutov almost recurrence, Birkhoff recurrence, pseudorecur- rence, and other types of Poisson stability. Thesecondchapter isdedicatedtothestudyofcompactglobalattractors of dynamical systems (both autonomous and nonautonomous). For autono- mous systems, we study different kinds of dissipativity for autonomous dynamical systems: point, compact, local, and bounded. Criteria of point, compact, and local dissipativity are given. We show that for dynamical systems in locally compact spaces, any three types of dissipativity are equivalent.Wegiveexamplesshowingthatinthegeneralcase,thenotionsof point, compact, and local dissipativity are distinct. The notion of the Levinson center, which is an important characteristic of compact dissipative systems, is introduced. We study the dissipative nonautonomous dissipative dynamical systems with minimal (in particular, with Bohr almost periodic, almost automorphic, or recurrent in the sense of Birkhoff) base. We give a description of the Levinson center of nonautonomous systems satisfying the condition of uniform positive stability. We give series of conditions that are equivalent to dissipativity in finite-dimensional space, and we prove that for linear systems, dissipativity reduces to convergence. Also we give series of conditions equivalent to dissipativity of linear systems. The third chapter is dedicated to the study of one special class of nonautonomous dissipative dynamical systems that we call C-analytic. We prove that a C-analytic dissipative dynamical system has the property of uniform positive stability on compact subsets. A full description of the Levinson center of these systems is given. Finally, we study C-analytic dis- cretedynamicalsystemsoninfinite-dimensionalspaces.Apositiveanswerto the Belitskii-Lyubich conjecture (for C-analytic discrete dynamical systems and flows) is given. In the fourth chapter we present some new results about Bohr/Levitan almost periodic, almost automorphic, and Poisson stable solutions of linear differential equations that complement the classical theory of Favard. In