Topics in Almost Automorphy

1. Auflage 2007
ISBN: 978-0-387-27439-3
Verlag: Springer-Verlag
Format: PDF
Kopierschutz: Adobe DRM (»Systemvoraussetzungen)

E-Book, Englisch, 168 Seiten

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].

Autoren/Hrsg.

Inhaltsverzeichnis

1;Preface;6
2;Contents;9
3;Introduction and Preliminaries;11
3.1;1.1 Measurable Functions;11
3.2;1.2 Sobolev Spaces;15
3.3;1.3 Semigroups of Linear Operators;17
3.4;1.4 Fractional Powers of Operators;18
3.5;1.5 Evolution Equations;19
3.6;1.6 Almost Automorphic Functions;22
3.7;1.7 Almost Periodic Functions;44
3.8;1.8 Bibliographical Remarks and Open Problems;49
4;Almost Automorphic Evolution Equations;51
4.1;2.1 Linear Equations;51
4.2;2.2 Nonlinear Equations;66
4.3;2.3 Optimal weak- almost periodic solutions;83
4.4;2.4 Existence of Weakly Almost Automorphic Solutions;92
4.5;2.5 A Correspondence Between Linear and Nonlinear Equations;98
4.6;Bibliographical Remarks and Open Problems;103
5;Almost Periodicity in Fuzzy Setting;105
5.1;3.1 Fuzzy Sets;105
5.2;3.2 Almost Periodicity in Fuzzy Setting;108
5.3;3.3 Harmonics of Almost Periodic Functions in Fuzzy Setting;115
5.4;3.4 Applications to Fuzzy Differential Equations;126
5.5;3.5 Bibliographical Remarks and Open Problems;131
6;Almost Automorphy in Fuzzy Setting;132
6.1;4.1 Introduction;132
6.2;4.2 Preliminaries;133
6.3;4.3 Basic Definitions and Properties;139
6.4;4.4 Applications to Fuzzy Differential Equations;161
6.5;4.5 Bibliographical Remarks and Open Problems;166
7;References;167
8;Index;175