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2000 | Buch

Differential Inclusions in a Banach Space

verfasst von: Alexander Tolstonogov

Verlag: Springer Netherlands

Buchreihe : Mathematics and Its Applications

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Über dieses Buch

Preface to the English Edition The present monograph is a revised and enlarged alternative of the author's monograph [19] which was devoted to the development of a unified approach to studying differential inclusions, whose values of the right hand sides are compact, not necessarily convex subsets of a Banach space. This approach relies on ideas and methods of modem functional analysis, general topology, the theory of multi-valued mappings and continuous selectors. Although the basic content of the previous monograph has been remained the same this monograph has been partly re-organized and the author's recent results have been added. The contents of the present book are divided into five Chapters and an Appendix. The first Chapter of the J>ook has been left without changes and deals with multi-valued differential equations generated by a differential inclusion. The second Chapter has been significantly revised and extended. Here the au­ thor's recent results concerning extreme continuous selectors of multi-functions with decomposable values, multi-valued selectors ofmulti-functions generated by a differential inclusion, the existence of solutions of a differential inclusion, whose right hand side has different properties of semicontinuity at different points, have been included. Some of these results made it possible to simplify schemes for proofs concerning the existence of solutions of differential inclu­ sions with semicontinuous right hand side a.nd to obtain new results. In this Chapter the existence of solutions of different types are considered.

Inhaltsverzeichnis

Frontmatter
Chapter 1. Multi-Valued Differential Equation Generated by a Differential Inclusion
Abstract
In this Chapter a multi-valued differential equation generated by a differential inclusion, is introduced in a semi-linear space of all convex compact sets of an initial Banach space. The solution of this equation is a multi-function of time having convex compact sets as its values. Questions of the existence of both local and global solutions of this equation are examined. These questions are studied in terms of ideas and methods of the theory of limitedly compact and condensing operators and of the method of vector Lyapunov functions modified with applications to a multi-valued case. In terms of vector Lyapunov functions we give uniform, in both form and content, conditions for the existence of local solutions of a multi-valued differential equation and for their extendibility. A multi-valued differential equation generated by a differential inclusion is being used here to prove the existence of solutions of a differential inclusion with uniform point of view.
Alexander Tolstonogov
Chapter 2. Differential Inclusions. Existence of Solutions
Abstract
In this Chapter questions of the existence of classical, regular, and Caratheodory type of solutions of a differential inclusion with non-convex right hand side are considered. The solutions of the differential inclusion are sought as continuous selectors of a solution of a multi-valued differential equation generated by a differential inclusion, and the interval of their existence concides with that of a multi-valued differential equation. The proof of the existence of solutions of a differential inclusion with non-convex right hand side is based on theorems about continuous selectors with certain properties in corresponding functional spaces for multi-functions with non-convex values, the classical Tychonov-Schauder theorem of a fixed point, and on the representation of the solution of a multi-valued equation as a convex compact set of its continuous selectors.
Alexander Tolstonogov
Chapter 3. Properties of Solutions
Abstract
In this Chapter we study interrelationships between a set of all, of some type or other, solutions of a differential inclusion with non-convex right hand side and a set of all, of the same type of solutions of a differential inclusions with convexified right hand side. It is shown that each, of some type or other, solution of the differential inclusion is a selector of the same solution of a multi-valued differential equation generated by a differential inclusion. We have established the density and co-density of a set of all, of some type or other, solutions of a differential inclusion with non-convex right hand side in a set of all Caratheodory type of solutions of a differential inclusion with convexified right hand side. In addition, in this Section properties of a set of all solutions of a differential inclusion such as the compactness, the connectedness and the dependence on initial conditions and parameters, are considered.
Alexander Tolstonogov
Chapter 4. Integral Funnel of the Differential Inclusion
Abstract
In this Chapter a study is made of the equation which is satisfied by an integral funnel (an attainable set) of a differential inclusion which is considered as a multi-function of time. Properties of solutions of this equation are revealed. It is shown that this equation is satisfied not only by the integral funnel of a differential inclusion but also by the integral funnel of an ordinary differential equation having a non-unique solution. An interconnection has been established between solution of the integral funnel equation and solutions of a multi-valued differential equation generated by a differential inclusion. Theorems are formulated for a continuous dependence of the integral funnel on initial conditions and parameters which are distinguished from traditional and known ones in a finite-dimensional space by the absence, in the assumptions, of conditions in explicit form which, with reference to ordinary differential equations, mean uniqueness of the solution.
Alexander Tolstonogov
Chapter 5. Inclusions with Non-Compact Right Hand Side
Abstract
In this Chapter differential inclusions with non-convex, non-compact right hand side are considered. Questions of the existence and properties of Caratheodory type of solution sets are studied.
Alexander Tolstonogov
Backmatter
Metadaten
Titel
Differential Inclusions in a Banach Space
verfasst von
Alexander Tolstonogov
Copyright-Jahr
2000
Verlag
Springer Netherlands
Electronic ISBN
978-94-015-9490-5
Print ISBN
978-90-481-5580-4
DOI
https://doi.org/10.1007/978-94-015-9490-5