Stability Theory and the Existence of Periodic Solutions and Almost Periodic Solutions

ISBN-13: 9780387901121 / Angielski / Miękka / 1975 / 233 str.

Taro Yoshizawa

Opis

Since there are several excellent books on stability theory, the author selected some recent topics in stability theory which are related to existence theorems for periodic solutions and for almost periodic solutions. The author hopes that these notes will also serve as an introduction to stability theory. These notes contain stability theory by Liapunov's second method and somewhat extended discussion of stability properties in almost periodic systems, and the existence of a periodic solution in a periodic system is discussed in connection with the boundedness of solutions, and the existence of an almost periodic solution in an almost periodic system is considered in con- nection with some stability property of a bounded solution. In the theory of almost periodic systems, one has to consider almost periodic functions depending on parameters, but most of text books on almost periodic functions do not contain this case. Therefore, as mathemati- cal preliminaries, the first chapter is intended to provide a guide for some properties of almost periodic functions with parameters as well as for properties of asymptotically almost periodic functions. These notes originate from a seminar on stability theory given by the author at the Mathematics Department of Michigan State Univer- sity during the academic year 1972-1973. The author is very grateful to Professor Pui-Kei Wong and members of the Department for their warm hospitality and many helpful conversations. The author wishes to thank Mrs.

Spis treści

I. Preliminaries.- 1. Liapunov Functions.- 2. Almost Periodic Functions.- 3. Asymptotically Almost Periodic Functions.- 4. Quasi-Periodic Functions.- 5. Boundary Value Problem.- II. Stability and Boundedness.- 6. Stability of a Solution.- 7. Asymptotic Stability of a Solution.- 8. Boundedness of Solutions.- 9. Asymptotic Stability in the Large.- 10. Asymptotic Behavior of Solutions.- 11. Converse Theorems.- 12. Total Stability.- 13. Inherited Properties in Almost Periodic Systems.- 14. Uniformly Asymptotic Stability in Almost Periodic Systems.- III. Existence Theorems for Periodic Solutions and Almost Periodic Solutions.- 15. Existence Theorems for Periodic Solutions.- 16. Existence Theorems for Almost Periodic Solutions.- 17. Separation Condition in Almost Periodic Systems.- 18. Uniform Stability and Existence of Almost Periodic Solutions.- 19. Existence of Almost Periodic Solutions by Liapunov Functions.- References.