Almost Periodic and Almost Automorphic Functions in Abstract Spaces

This book presents the foundation of the theory of almost automorphic functions in abstract spaces and the theory of almost periodic functions in locally and non-locally convex spaces and their applications in differential equations. Since the publication of Almost automorphic and almost periodic functions in abstract spaces (Kluwer Academic/Plenum, 2001), there has been a surge of interest in the theory of almost automorphic functions and applications to evolution equations. Several generalizations have since been introduced in the literature, including the study of almost automorphic sequences, and the interplay between almost periodicity and almost automorphic has been exposed for the first time in light of operator theory, complex variable functions and harmonic analysis methods. As such, the time has come for a second edition to this work, which was one of the most cited books of the year 2001. This new edition clarifies and improves upon earlier materials, includes many relevant contributions and references in new and generalized concepts and methods, and answers the longtime open problem, "What is the number of almost automorphic functions that are not almost periodic in the sense of Bohr?" Open problems in non-locally convex valued almost periodic and almost automorphic functions are also indicated.
As in the first edition, materials are presented in a simplified and rigorous way. Each chapter is concluded with bibliographical notes showing the original sources of the results and further reading.

Discusses basic properties of almost automorphic functions in Banach spaces and their generalizations Presents open problems for almost periodicity in nonlocally convex spaces Examines the interplay between almost automorphy and almost periodicity Contains a rich and up-to-date bibliography

Résumé
"This book is a nice text for those interested in the topics of almost automorphy in abstract spaces and almost periodicity in locally and non-locally convex spaces. ... The author, a noted specialist in the area, develops, with examples and applications, these new concepts and results, which are now receiving the attention of an increasing number of researchers. Thus the book will serve as a good reference for specialists in this subject." (Mohamed Zitane, zbMATH 1479.43001, 2022)

Contenu

1. Introduction and Preliminaries.- 2. Almost Automorphic Functions.- 3. Almost Automorphy of the Function f(t,x).- 4. Differentiation and Integration.- 5. Pseudo Almost Automorphy.- 6. Stepanov-like Almost Automorphic Functions.- 7. Dynamical Systems and C0-Semigroups.- 8. Almost Periodic Functions with Values in a Locally Convex Space.- 9. Almost Period Functions with Values in a Non-Locally Convex Space.- 10. The Equation x'(t)=A(t)x(t)+f(t).- 11. Almost Periodic Solutions of the Differential Equation in Locally Convex Spaces.- Bibliography.