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: Nova)
Mathematical Reviews Clippings
- Reviewed by Dr. Syed Abbas, Professor, Indian Institute of Technology Mandi, India