Basic Topological Structures of Ordinary Differential Equations

ISBN-13: 9789048149957 / Angielski / Miękka / 2010 / 522 str.

V. V. Filippov

Opis

Traditionally, equations with discontinuities in space variables follow the ideology of the sliding mode'. This book contains the first account of the theory which allows the consideration of exact solutions for such equations. The difference between the two approaches is illustrated by scalar equations of the type y =f(y) and by equations arising under the synthesis of optimal control. A detailed study of topological effects related to limit passages in ordinary differential equations widens the theory for the case of equations with continuous right-hand sides, and makes it possible to work easily with equations with complicated discontinuities in their right-hand sides and with differential inclusions. Audience: This volume will be of interest to graduate students and researchers whose work involves ordinary differential equations, functional analysis and general topology."

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.