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.
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ISBN-13
978-94-015-9490-5 (9789401594905)
DOI
10.1007/978-94-015-9490-5
Schweitzer Klassifikation
1. Multi-Valued Differential Equation Generated by A Differential Inclusion.- 1. Background of the theory of measurable multi-valued mappings and differential inequalities.- 2. Existence of local solutions of a multi-valued differential equation with conditions of compactness type.- 3. Existence of local solutions of a multi-valued differential equation. Comparison theorems.- 4. Global solutions of a multi-valued differential equation.- 5. Existence theorems of solutions of the multi-valued operator equation.- 6. Notes and Remarks.- 2. Differential Inclusions. Existence of Solutions.- 1. Types of solutions of a differential inclusion.- 2. Semicontinuous multi-functions.- 3. Strongly measurable selectors of multi-functions.- 4. Continuous selectors of multi-functions with decomposable values.- 5. Continuous selectors of a multi-valued mappings generated by differential inclusions with non-convex right hand side.- 6. Multi-valued selectors of a multi-valued mapping generated by a differential inclusion.- 7. Existence of solutions of a differential inclusion with nonconvex right hand side continuous in x.- 8. Existence of the solution of a differential inclusion with right hand side semicontinuous in x.- 9. Notes and Remarks.- 3. Properties of Solutions.- 1. Auxiliary results.- 2. The density theorems.- 3. The co-density theorems.- 4. Compactness of the set of solutions.- 5. Dependence of the set of solutions on initial conditions and parameters.- 6. Connectedness of the set of solutions.- 7. Notes and remarks.- 4. Integral Funnel of the Differential Inclusion.- 1. Auxiliary lemmas.- 2. The equation of the integral funnel.- 3. Solutions of the equation of the integral funnel.- 4. Properties of solutions of the equation of the integral funnel.- 5. Properties of the integralfunnel.- 6. Extreme points of the set of solutions of a linear differential inclusion.- 7. Notes and Remarks.- 5. Inclusions with Non-Compact Right Hand Side.- 1. Continuous selectors of fixed point sets of multi-functions with decomposable values.- 2. Properties of the multi-valued Nemytskii operator.- 3. Continuous selectors of fixed point sets of the multi-valued Nemytskii operator.- 4. Existence and properties of solution sets of differential inclusion.- 5. Notes and Remarks.- Appendices.- A-.- 1. Non-convex problem of calculus of variations.- 2. Existence of optimal control without assumptions of convexity.- 3. Extension in continuity of multi-valued mappings.- 4. Equivalence of differential inclusions and control systems.- References.- Symbols.
