Topics in Almost Automorphy

ISBN-13: 9781441935618 / Angielski / Miękka / 2010 / 182 str.

Gaston M. N'Guerekata

Opis

Since the publication of our first book 80], there has been a real resiu-gence of interest in the study of almost automorphic functions and their applications ( 16, 17, 28, 29, 30, 31, 32, 40, 41, 42, 46, 51, 58, 74, 75, 77, 78, 79]). New methods (method of invariant s- spaces, uniform spectrum), and new concepts (almost periodicity and almost automorphy in fuzzy settings) have been introduced in the literature. The range of applications include at present linear and nonlinear evolution equations, integro-differential and functional-differential equations, dynamical systems, etc...It has become imperative to take a bearing of the main steps of the the ory. That is the main purpose of this monograph. It is intended to inform the reader and pave the road to more research in the field. It is not a self contained book. In fact, 80] remains the basic reference and fimdamental source of information on these topics. Chapter 1 is an introductory one. However, it contains also some recent contributions to the theory of almost automorphic functions in abstract spaces. VIII Preface Chapter 2 is devoted to the existence of almost automorphic solutions to some Unear and nonUnear evolution equations. It con tains many new results. Chapter 3 introduces to almost periodicity in fuzzy settings with applications to differential equations in fuzzy settings. It is based on a work by B. Bede and S. G. Gal 40]."

Spis treści

1: Introduction and Preliminaries 1.1 Measurable Functions 1.2 Sobolev Spaces 1.3 Semigroups of Linear Operators 1.4 Fractional Powers of Operators 1.5 Evolution Equations 1.6 Almost Automorphic Functions 1.6.1 Asymptotically Almost Automorphic Functions 1.6.2 Applications to Abstract Dynamical Systems 1.7 Almost Periodic Functions 1.8 Bibliographical Remarks and Open Problems 2: Almost Automorphic Evolution Equations 2.1 Linear Equations 2.1.1 The inhomogeneous equation x’ = Ax + f 2.1.2 Method of Invariant Subspaces 2.1.3 Almost Automorphic Solutions to Some Second-Order Hyperbolic Equations 2.2 Nonlinear Equations 2.2.1 Existence of Almost Automorphic Mild Solutions-Case I 2.2.2 Existence of Almost Automorphic Mild Solutions-Case II 2.3 Optimal weak-almost periodic solutions 2.4 Existence of Weakly Almost Automorphic Solutions 2.5 A Correspondence Between Linear and Nonlinear Equations 3: Almost Periodicity in Fuzzy Setting 3.1 Fuzzy Sets 3.2 Almost Periodicity in Fuzzy Setting 3.3 Harmonics of Almost Periodic Functions in Fuzzy Setting 3.4 Applications to Fuzzy Differential Equations 3.5 Bibliographical Remarks and Open Problems 4: Almost Automorphy in Fuzzy Setting 4.1 Introduction 4.2 Preliminaries 4.3 Basic Definitions and Properties 4.4 Applications to Fuzzy Differential Equations 4.5 Bibliographical Remarks and Open Problems References Index