Basic Topological Structures of Ordinary Differential Equations

ISBN-13: 9780792349518 / Angielski / Twarda / 1998 / 522 str.

V. V. Filippov

Opis

The aim of this book is a detailed study of topological effects related to continuity of the dependence of solutions on initial values and parameters. This allows us to develop cheaply a theory which deals easily with equations having singularities and with equations with multivalued right hand sides (differential inclusions). An explicit description of corresponding topological structures expands the theory in the case of equations with continuous right hand sides also. In reality, this is a new science where Ordinary Differential Equations, General Topology, Integration theory and Functional Analysis meet. In what concerns equations with discontinuities and differential inclu sions, we do not restrict the consideration to the Cauchy problem, but we show how to develop an advanced theory whose volume is commensurable with the volume of the existing theory of Ordinary Differential Equations. The level of the account rises in the book step by step from second year student to working scientist."

Spis treści

Preface. 1. Topological and Metric Spaces. 2. Some Properties of Topological, Metric and Euclidean Spaces. 3. Spaces of Mappings and Spaces of Compact Subsets. 4. Derivation and Integration. 5. Weak Topology on the Space L1 and Derivation of Convergent Sequences. 6. Basic Properties of Solution Spaces. 7. Convergent Sequences of Solution Spaces. 8. Peano, Caratheodory and Davy Conditions. 9. Comparison Theorem. 10. Changes of Variables, Morphisms and Maximal Extensions. 11. Some Methods of Investigation of Equations. 12. Equations and Inclusions with Complicated Discontinuities in the Space Variables. 13. Equations and Inclusions of Second Order. Cauchy Problem Theory. 14. Equations and Inclusions of Second Order. Periodic Solutions, Dirichlet Problem. 15. Behavior of Solutions. 16. Two-Dimensional Systems. References. Index. Notation.