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Stability, periodicity and boundedness in functional dynamical systems on time scales. (English) Zbl 1464.34001

Cham: Springer (ISBN 978-3-030-42116-8/hbk; 978-3-030-42117-5/ebook). xiv, 416 p. (2020).
This book contains recent developments of the qualitative theory of stability, periodicity and boundedness of functional dynamical systems on time scales. It consists of a Preface, 8 Chapters, a Bibliography and an Index. The aim of the book is presented in the Preface. Chapter 1 consists of the introduction to stability and boundedness in dynamical systems, where a brief introduction to time scale calculus and fundamental concepts needed throughout the book are provided. Chapter 2 is dealing with ordinary dynamical systems, Chapter 3 with functional dynamical systems, while Volterra integro-dymanic equations with or without delay are studied in Chapter 4. The construction of exotic Lyapunov functionals for the study of boundedness and stabilty is presented in Chapter 5, while Volterra integral dynamic equations are studied in Chapter 6. Chapter 7 is dealing with periodic solutions and in Chapter 8 periodic solutions of functional delay dynamic systems, where the periodic delays are expressed in terms of shift operators, are studied. The Bibliography consists of 169 entries. This book is organized in the classical mathematical style: definitions, lemmata, theorems, corollaries, proofs, examples, remarks, applications, etc. In each chapter, several examples are solved and open problems, which should be of great benefit to the researchers on the subject, are proposed. The book is well written and reads very well. It provides a useful reference in the field.

MSC:

34-02 Research exposition (monographs, survey articles) pertaining to ordinary differential equations
45-02 Research exposition (monographs, survey articles) pertaining to integral equations
34N05 Dynamic equations on time scales or measure chains
34D20 Stability of solutions to ordinary differential equations
34C25 Periodic solutions to ordinary differential equations
34C11 Growth and boundedness of solutions to ordinary differential equations
39-02 Research exposition (monographs, survey articles) pertaining to difference and functional equations
34K42 Functional-differential equations on time scales or measure chains
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