Nonautonomous dynamics. Nonlinear oscillations and global attractors 9783030342913, 9783030342920

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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. Pinto, Porto, Portugal Gabriella Pinzari, Padova, Italy Ken Ribet, Berkeley, USA René Schilling, Dresden, Germany Panagiotis Souganidis, Chicago, USA Endre Süli, Oxford, UK Shmuel Weinberger, Chicago, USA Boris Zilber, Oxford, UK

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David N. Cheban

Nonautonomous Dynamics Nonlinear Oscillations and Global Attractors

123

David N. Cheban Faculty of Mathematics State University of Moldova Chisinau, Moldova

ISSN 1439-7382 ISSN 2196-9922 (electronic) Springer Monographs in Mathematics ISBN 978-3-030-34291-3 ISBN 978-3-030-34292-0 (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 © Springer Nature Switzerland AG 2020 This work is subject to copyright. All rights are reserved by the Publisher, whether the whole or part 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 or information storage and retrieval, electronic adaptation, computer software, or by similar or dissimilar methodology now known or hereafter developed. The use of general descriptive names, registered names, trademarks, service marks, etc. in this publication does not imply, even in the absence of a specific statement, that such names are exempt from the relevant protective laws and regulations and therefore free for general use. 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 authors or the editors give a warranty, expressed or implied, with respect to the material contained herein or for any errors or omissions that may have been made. The publisher remains neutral with regard to jurisdictional claims in published maps and institutional affiliations. This Springer imprint is published by the registered company Springer Nature Switzerland AG The registered company address is: Gewerbestrasse 11, 6330 Cham, Switzerland

To my grandchildren Olivia and Nicholas

Preface

The present monograph is dedicated to the abstract theory of nonautonomous 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. Our main applications are nonautonomous ordinary differential/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/difference equations and partial differential equations). Nonlocal problems concerning the conditions of existence of different classes of solutions play an important role in the qualitative theory of differential equations. Here we include the problems of boundedness, periodicity, Bohr/Levitan almost periodicity, almost automorphy, almost recurrence 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 of Poisson stable motions, and global attractors of Bohr/Levitan almost periodic systems with continuous and discrete time. Thus, there are two key objects that are the subjects of study in this book. 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

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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, almost recurrent in the sense of Bebutov, almost authomorphic, Poisson stable) solutions, and global attractors of nonautonomous differential/difference equations. Namely, the text provides answers to the following problems: 1. Problem of existence of at least one Bohr/Levitan almost periodic solution 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 holomorphic 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/difference equations is the problem of almost periodicity, or more 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. The theory of almost periodic functions was mainly created and published by Bohr [42–45] (in this relation see also the important results of Bohl [40, 41] and Esclangon [145–147]). Bohr’s theory was substantially extended by Bochner [35], Weyl [321], Besicovitch [22], Favard [151], von Neumann [235], Stepanoff [308], Bogolyubov [37–39] and others. Levitan [212] introduced a new class of functions (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 results about almost automorphic functions were obtained by Veech [314–316]. The different classes of Poisson stable functions (in particular, recurrent in 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].

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The theory of Bohr/Levitan almost periodic, almost automorphic, and Poisson stable functions is widely presented in the monographs of Amerio and Prouse [3], Bochner [35], Corduneanu [129, 130], Toka Diagana [140], Fink [155], Favard [151], Hino et al. [179], Levitan [214], Levitan and Zhikov [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). In the last 25–30 years, the theory of Bohr/Levitan almost periodic, 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. Global attractors play a very important role in the study of the asymptotic behavior of dynamical systems (both autonomous and nonautonomous). In the last 20–25 years many works dedicated to the study of global attractors of dynamical systems (including the infinite-dimensional systems) have been published. See, for example, Babin and Vishik [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 nonautonomous 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 periodicity of solutions of almost periodic differential equations and their compact global attractors consists in applying to the study of nonautonomous systems the ideas and methods developed in the theory of abstract dynamical systems. The idea of applying methods of the theory of dynamical systems to 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 in works of Zhikov [327], Bronshtein [48], Johnson [188, 189], and many other authors. This approach consists in naturally associating with the equation x0 ¼ fðt; xÞ

ð1Þ

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Preface

a pair of dynamical systems and a homomorphism of the first onto the second. One assigns the information about the right-hand side of Eq. (1) to one dynamical system, and the information about the solutions of (1) to the other. Plenty of works are dedicated to the study of the problem of Bohr/Levitan almost periodicity, almost automorphy, and different classes of Poisson stability 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 recurrence are introduced and studied: Bohr/Levitan almost periodicity, almost automorphy, Bebutov almost recurrence, Birkhoff recurrence, pseudorecurrence, and other types of Poisson stability. The second chapter is dedicated to the study of compact global attractors of dynamical systems (both autonomous and nonautonomous). For autonomous 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. We give examples showing that in the general case, the notions of 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 discrete dynamical systems on infinite-dimensional spaces. A positive answer to 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

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conclusion, we give conditions that guarantee the dissipativity of semilinear systems of differential equations, which is a condition that ensures the existence of almost periodic solutions of semilinear systems with almost periodic coefficients in the Levinson center. The fifth chapter is dedicated to the study of order-preserving nonautonomous dynamical systems. We give some criteria for the existence of a fixed point for a semigroup of transformations. The problem of existence of Bohr/Levitan almost periodic, almost periodic, and Poisson stable solutions for different classes of monotone differential equations (first- and second-order finite-dimensional equations and also for some class of parabolic equations) is solved. In the sixth chapter we give conditions for the existence of Bohr/Levitan almost periodic, almost automorphic Poisson stable solutions of nonautonomous perturbed gradient-like autonomous differential equations. We present here also the description of the Levinson center for gradients and gradient-like dynamical systems with a finite number of fixed points. We establish a relationship between the Levinson center, chain recurrent sets, and the Birkhoff center for compact dissipative dynamical systems. Some of the results of this book are contained in the courses of lectures which the author has given for many years gives to graduate students of the State University of Moldova. I hope that the ideas and methods presented in this book will be interesting to the mathematical community working in the field of dynamical systems and their applications (control theory, economic dynamics, mathematical theory of climate, population dynamics, oscillation theory, etc.). The book should be accessible to graduate students who have taken courses in real analysis (including the elements of functional analysis, general topology) and have a general background in dynamical systems and the qualitative theory of differential/difference equations. The book contains a number of new, original results. Many of the results presented here belong to the author [68–97]. The results of Chap. 1 (Sects. 1.6 and 1.7.3), Chap. 4 (Sects. 4.4 and 4.5), and Chap. 5 (Sect. 5.2) were jointly obtained with Caraballo [58–64], Chap. 2 (Sect. 2.10) was obtained jointly with Kloeden and Schmalfuss [99], Chap. 5 (Sect. 5.3) was obtained jointly with Mammana [101], and Chap. 5 (Sect. 5.4) was obtained jointly with Zenxin Liu [100]. Chisinau, Moldova August 2019

David N. Cheban

Contents

1 Almost Periodic Motions of Dynamical Systems . . . . . . . . . . . . 1.1 Some Notions, Notation, and Facts from the Theory of Dynamical Systems . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.2 Minimal Sets and Recurrent Motions . . . . . . . . . . . . . . . . 1.2.1 Minimal Sets . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.2.2 Almost Recurrent Motions . . . . . . . . . . . . . . . . . . 1.2.3 Recurrent Motions . . . . . . . . . . . . . . . . . . . . . . . . 1.3 Lyapunov Stable Sets . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.4 Almost Periodic Motions . . . . . . . . . . . . . . . . . . . . . . . . . 1.5 Invariant Measures . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.5.1 Adjoint Dynamical System . . . . . . . . . . . . . . . . . 1.5.2 Measure on the Compact Space . . . . . . . . . . . . . . 1.5.3 Ergodicity of Almost Periodic Systems . . . . . . . . . 1.5.4 Existence a Unique Invariant Measure for Almost Periodic Minimal Dynamical Systems . . . . . . . . . 1.6 Two Definitions of Almost Periodic Motions in Semigroup Dynamical Systems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.7 Comparability by Character of Recurrence of Motions for Dynamical Systems . . . . . . . . . . . . . . . . . . . . . . . . . . 1.7.1 B. A. Shcherbakov’s Principle of Comparability of Motions by Their Character of Recurrence . . . . 1.7.2 Comparability by Character of Recurrence of Almost Periodic Motions . . . . . . . . . . . . . . . . . 1.7.3 Some Generalization of B. A. Shcherbakov’s Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.7.4 Strongly Comparability of Motions by Their Character of Recurrence . . . . . . . . . . . . . . . . . . .

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1.9

Asymptotically Almost Periodic Motions . . . . . . . . . . . 1.8.1 Poisson Asymptotically Stable Motions . . . . . . 1.8.2 Criterion of Asymptotical Almost Periodicity . . 1.8.3 Asymptotically Periodic Motions . . . . . . . . . . . 1.8.4 Comparability by the Character of Recurrence in the Limit of Asymptotically Poisson Stable Motions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Almost Periodic and Asymptotically Almost Periodic Functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.9.1 Bohr Almost Periodic Functions . . . . . . . . . . . 1.9.2 S p -Almost Periodic Functions . . . . . . . . . . . . . 1.9.3 Fréchet Asymptotically Almost Periodic Functions . . . . . . . . . . . . . . . . . . . . . . 1.9.4 Asymptotically S p Almost Periodic Functions .

2 Compact Global Attractors . . . . . . . . . . . . . . . . . . . . . . 2.1 Limit Properties of Dynamical Systems . . . . . . . . 2.2 Levinson Center . . . . . . . . . . . . . . . . . . . . . . . . . 2.3 Dissipative Systems on Local Compact Spaces . . . 2.4 Criteria for Compact Dissipativity . . . . . . . . . . . . 2.5 Local Dissipative Dynamical Systems . . . . . . . . . . 2.6 Global Attractors . . . . . . . . . . . . . . . . . . . . . . . . 2.7 Global Attractors of Nonautonomous Dynamical Systems . . . . . . . . . . . . . . . . . . . . . . . 2.8 Global Attractors of Cocycles . . . . . . . . . . . . . . . 2.9 Global Attractors of Nonautonomous Dynamical Systems with Minimal Base . . . . . . . . . . . . . . . . . 2.10 The Relationship Between Pullback, Forward, and Global Attractors . . . . . . . . . . . . . . . . . . . . . 2.10.1 Pullback, Forward, and Global Attractors 2.10.2 Asymptotic Stability in a-Condensing Semidynamical Systems . . . . . . . . . . . . . . 2.10.3 Uniform Pullback and Global Attractors . 2.10.4 Examples of Uniform Pullback Attractors

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3 Analytical Dissipative Systems . . . . . . . . . . . . . . . . . . . . . 3.1 Skew-Product Dynamical Systems and Cocycles . . . . 3.2 Positively Stable Systems . . . . . . . . . . . . . . . . . . . . 3.3 C-Analytic Systems . . . . . . . . . . . . . . . . . . . . . . . . . 3.4 Pullback Attractors of Holomorphic Systems . . . . . . 3.4.1 C-Analytic Cocycles . . . . . . . . . . . . . . . . . . 3.4.2 Some General Facts About Nonautonomous Dynamical Systems . . . . . . . . . . . . . . . . . . . 3.4.3 Positively Uniformly Stable Cocycles . . . . . .

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The Compact Global Pullback Attractors of C-Analytic Cocycles with Compact Base . . . . 3.4.5 Uniform Dissipative Cocycles with Noncompact Base . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.4.6 Compact and Local Dissipative Cocycles with Noncompact Base . . . . . . . . . . . . . . . . . . . 3.4.7 Applications . . . . . . . . . . . . . . . . . . . . . . . . . . . Infinite-Dimensional Holomorphic Systems . . . . . . . . . . . 3.5.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.5.2 The Belitskii–Lyubich Conjecture . . . . . . . . . . . 3.5.3 Belitskii–Lyubich Conjecture for Holomorphic Flows . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.5.4 Holomorphic Dissipative Dynamical Systems . . . 3.5.5 Some Applications . . . . . . . . . . . . . . . . . . . . . . .

4 Almost Periodic Solutions of Linear Differential Equations . . 4.1 Birkhoff’s Theorem on Nonautonomous Dynamical Systems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.1.1 Nonautonomous Version of Birkhoff’s Theorem 4.1.2 Strongly Poisson Stable Motions . . . . . . . . . . . 4.2 Semigroup Dynamical Systems . . . . . . . . . . . . . . . . . . 4.3 Invariant and Minimal Sets for Some Abstract Semigroups of Transformations . . . . . . . . . . . . . . . . . . 4.4 Some Tests of Comparability and Uniform Comparability of Motions . . . . . . . . . . . . . . . . . . . . . . 4.5 Compatible and Strongly Compatible Solutions . . . . . . 4.5.1 Linear Differential Equations . . . . . . . . . . . . . . 4.5.2 Linear Difference Equations . . . . . . . . . . . . . . . 4.5.3 Linear Functional Differential/Difference Equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.6 Almost Periodic Solutions of Linear Almost Periodic Systems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.6.1 Bounded Motions of Linear Systems . . . . . . . . 4.6.2 Linear Contractive Systems . . . . . . . . . . . . . . . 4.6.3 Applications . . . . . . . . . . . . . . . . . . . . . . . . . . 4.7 Almost Periodic Solutions of Semilinear Dissipative Systems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

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5 Almost Periodic Solutions of Monotone Differential Equations . . . . 275 5.1 Bohr/Levitan Almost Periodic Solutions of Scalar Differential Equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 277 5.1.1 Cocycles, Skew-Product Dynamical Systems and Nonautonomous Dynamical Systems Generated by Differential Equations . . . . . . . . . . . . . 279

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Some Criteria for the Existence of Fixed Points for a Semigroup of Transformations . . . . . . . . . . . 5.1.3 One-Dimensional Nonautonomous Dynamical Systems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.1.4 Scalar Differential Equations . . . . . . . . . . . . . . . . Levitan/Bohr Almost Periodic Solutions of Second-Order Differential Equations . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.2.1 Some Nonautonomous Dynamical Systems Generated by Second-Order Differential Equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.2.2 Levitan Almost Periodic and Almost Automorphic Solutions of Second-Order Differential Equations . . . . . . . . . . . . . . . . . . . . . 5.2.3 Quasiperiodic, Bohr Almost Periodic, Almost Automorphic, and Recurrent Solutions of Second-Order Differential Equations . . . . . . . . . 5.2.4 Some Generalizations . . . . . . . . . . . . . . . . . . . . . . Monotone Second-Order Differential Equations in Hilbert Spaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.3.1 Some Nonautonomous Dynamical Systems Generated by Differential Equations in Banach Spaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.3.2 Invariant Sections (Manifolds) of Second-Order Differential Equations . . . . . . . . . . . . . . . . . . . . . 5.3.3 Almost Automorphic Solutions of Monotone Second-Order Differential Equations . . . . . . . . . . Monotone Nonautonomous Dynamical Systems . . . . . . . . 5.4.1 Structure of the x-Limit Set . . . . . . . . . . . . . . . . 5.4.2 Comparable Motions by the Nature of Their Recurrence for Monotone Nonautonomous Dynamical Systems . . . . . . . . . . . . . . . . . . . . . . . 5.4.3 Applications . . . . . . . . . . . . . . . . . . . . . . . . . . . . I. U. Bronshtein’s Conjecture for Monotone Nonautonomous Dynamical Systems . . . . . . . . . . . . . . . . 5.5.1 Global Attractors of Cocycles . . . . . . . . . . . . . . . 5.5.2 I. U. Bronshtein’s Conjecture for Cocycles . . . . . . 5.5.3 Applications . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.5.4 Ordinary Differential Equations . . . . . . . . . . . . . . 5.5.5 Difference Equations . . . . . . . . . . . . . . . . . . . . . .

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6 Gradientlike Dynamical Systems . . . . . . . . . . . . . . . . . . . . . . . . . . 351 6.1 Compact Global Attractors of Nonautonomous Gradientlike Dynamical Systems . . . . . . . . . . . . . . . . . . . . . 352

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6.7

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Gradient Systems with Bounded Set of Fixed Points . . 6.2.1 Gradient Systems with Finite Number of Fixed Points . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.2.2 Chain Recurrent Motions of Gradient Systems . 6.2.3 Nonautonomous Gradientlike Dynamical Systems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Relation Between Levinson Center, Chain Recurrent Set, and Birkhoff Center . . . . . . . . . . . . . . . . . . . . . . . Chain Recurrent Motions of Compact Dissipative Dynamical Systems . . . . . . . . . . . . . . . . . . . . . . . . . . . Linear Systems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.5.1 Linear Nonautonomous Dynamical Systems with Exponential Dichotomy . . . . . . . . . . . . . . 6.5.2 Linear Inhomogeneous (Affine) Dynamical Systems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.5.3 Invariant Sections (Manifolds) . . . . . . . . . . . . . 6.5.4 Dependence on Parameter of Green’s Function 6.5.5 Relationship Between Different Definitions of Hyperbolicity . . . . . . . . . . . . . . . . . . . . . . . . . . 6.5.6 Uniformly Compatible Solutions . . . . . . . . . . . Semilinear Differential Equations . . . . . . . . . . . . . . . . . 6.6.1 Local Existence and Uniqueness . . . . . . . . . . . . 6.6.2 Global Existence and Uniqueness . . . . . . . . . . . 6.6.3 Invariant Sections (Manifolds) of Semilinear Equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . Perturbation of Gradient Systems . . . . . . . . . . . . . . . .

. . . . 354 . . . . 356 . . . . 359 . . . . 360 . . . . 365 . . . . 375 . . . . 381 . . . . 381 . . . . 385 . . . . 387 . . . . 388 . . . . .

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. . . . .

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390 395 397 397 400

. . . . 402 . . . . 406

References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 415 Index . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 429

Notation

8 9 :¼ 0 N Z Q R C S S þ ðS Þ XY Mn En fxn g x2X @X X SY X Y XnY T X Y ; ðX; qÞ M f 1 fðMÞ f g

for every exists equals (coincides) by definition zero, and also the zero element of any additive group (semigroup) is the set of all natural numbers is the set of all integers is the set of all rational numbers is the set of all real numbers is the set of all complex numbers is one of the sets R, Z is the set of all non-negative (non-positive) numbers from S is the Cartesian product of two sets is the direct product of n copies of the set M is the real or complex n-dimensional Euclidian space is a sequence is an element of the set X is the boundary of the set X the set X is a proper subset of the set Y or coincides with it is the union of the sets X and Y is the complement of the set Y in X is the intersection of the sets X and Y the empty set is a complete metric space with the metric q is the closure of the set M is the mapping inverse to f is the image of the set M  X in the mapping f : X ! Y , i.e. fy 2 Y : y ¼ fðxÞ; x 2 Mg is the composition of the mappings f and g, i.e., ðf  gÞðxÞ ¼ fðgðxÞÞ

xix

xx

fjM fð; xÞ IdX ImðfÞ DðfÞ jxj or jjxjj ðx; yÞ CðX; Y Þ C k ðU; MÞ f:X!Y BðM; "Þ B½M; " fx; y; . . .; zg 1; n fx 2 Xj