Spectral Theory for Bounded Functions and Applications to Evolution Equations

Gaston Mandata N’Guerekata
University Distinguished Professor of Mathematics, The World Academy of Sciences (TWAS) Research Professor, School of Computer, Mathematical and Natural Sciences, Morgan State University Baltimore, MD, USA

Series: Mathematics Research Developments
BISAC: MAT007000

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One of the central questions in the qualitative theory of difference and differential equations is to find the conditions of existence and asymptotic behavior of bounded solutions. For equations with almost periodic coefficients, the problem goes back to Favard and Perron. A remarkable theory has been developed in harmonic analysis with outstanding contributions by Loomis, Arendt, Batty, Lyubic, Phong, Naito, Minh and many others, when the Carleman spectrum of the functions is countable. Uniform continuity in this case plays a key role. In the absence of this condition, the theory does not apply. This lead to the introduction over the last decade of new types of spectrum of functions which helped solve the problem, especially in the case of almost automorphic functions, using the theory of commutating operators.

This monograph presents in a unique and unified manner recent developments in the theory of spectra of bounded continuous functions including the space of (Bohr) almost periodic functions and some of their generalizations, and the spaces of (Bochner) almost automorphic functions and almost automorphic sequences. Classical concepts from harmonic analysis such as the Bohr spectrum, Beurling spectrum and Carleman spectrum are also presented with some examples. A special attention is devoted to the recently introduced concepts of uniform spectrum and circular spectrum of bounded functions derived from the study of the existence of solutions of linear differential equations whose forcing terms are not necessarily uniformly continuous. Connections between these various types of spectrum are also investigated. As applications the book provides a semigroup-free study of the existence and asymptotic behavior of mild solutions of evolution equations of the first and second order, and difference equations. Bibliographical and historical notes complete the major chapters. An appendix on basic results on the theory of commutating operators is given. The content is presented in a way that is easily accessible to readers who are working in differential equations but are not familiar with harmonic analysis and advanced functional analysis. It’s our hope that this first monograph ever on this topic will attract more researchers. (Imprint: Novinka)

Preface

Chapter 1. Preliminaries

Chapter 2. Almost Periodic Functions

Chapter 3. Almost Automorphic Functions and Sequences

Chapter 4. Spectrum of Bounded Functions

Chapter 5. Applications to Differential Equations

Appendix

References

Author's Contact Information

Index

“This monograph is devoted to the theory of spectra of bounded functions in abstractspaces. Spectral theory is very important as it gives important qualitative properties ofsolutions to evolution equations. It is particularly useful in the study of the asymptoticbehavior of mild solutions of evolution equations. The monograph is divided into fivechapters. Chapter 1 is about some basic defnitions and notations. Chapter 2 is devotedto the theory of almost periodic functions. These functions have important applicationsin celestial mechanics, control theory and other fields. In chapter 3, the author discussesthe concepts of almost automorphic functions and sequences, which are generalizationsof almost periodic functions and sequences. After these chapters, the author moves onto defining the spectrum of bounded functions, in chapter 4. The concepts of Carleman,Beurling, uniform and circular spectra are defined. The last chapter is devoted to thestudy of difference and first- and second-order differential equations.Overall, this monograph is useful for researchers working in the field of spectraltheory of bounded functions and the qualitative theory of differential equations inabstract spaces. The monograph is written for graduate students, and to read it oneneeds to first go through the basics of abstract spaces and functional analysis.” - Dr. Syed Abbas, Professor, Indian Institute of Technology Mandi, India

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