Solution of Initial Value Problems in Classes of Generalized Analytic Functions

Wolfgang Tutschke

ISBN 10: 3540502165 / ISBN 13: 9783540502166

Editore: Springer, 1989

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illustrated edition. 192 pages. 7.90x5.30x0.50 inches. In Stock. Codice articolo x-3540502165

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Titolo: Solution of Initial Value Problems in ...

Casa editrice: Springer

Data di pubblicazione: 1989

Legatura: Paperback

Condizione: Brand New

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Riassunto:

The purpose of the present book is to solve initial value problems in classes of generalized analytic functions as well as to explain the functional-analytic background material in detail. From the point of view of the theory of partial differential equations the book is intend­ ed to generalize the classicalCauchy-Kovalevskayatheorem, whereas the functional-analytic background connected with the method of successive approximations and the contraction-mapping principle leads to the con­ cept of so-called scales of Banach spaces: 1. The method of successive approximations allows to solve the initial value problem du CTf = f(t,u), (0. 1) u(O) = u , (0. 2) 0 where u = u(t) ist real o. r vector-valued. It is well-known that this method is also applicable if the function u belongs to a Banach space. A completely new situation arises if the right-hand side f(t,u) of the differential equation (0. 1) depends on a certain derivative Du of the sought function, i. e. , the differential equation (0,1) is replaced by the more general differential equation du dt = f(t,u,Du), (0. 3) There are diff. erential equations of type (0. 3) with smooth right-hand sides not possessing any solution to say nothing about the solvability of the initial value problem (0,3), (0,2), Assume, for instance, that the unknown function denoted by w is complex-valued and depends not only on the real variable t that can be interpreted as time but also on spacelike variables x and y, Then the differential equation (0.

Contenuti:

0. Introduction.- 1. Initial Value Problems in Banach Spaces.- 2. Scales of Banach Spaces.- 3. Solution of Initial Value Problems in Scales of Banach Spaces.- 4. The Classical Cauchy-Kovalevskaya Theorem.- 5. The Holmgren Theorem.- 6. Basic Properties of Generalized Analytic Functions.- 7. Initial Value Problems with Generalized Analytic Initial Functions.- 8. Contraction-Mapping Principles in Scales of Banach Spaces.- 9. Further Existence Theorems for Initial Value Problems in Scales of Banach Spaces.- 10. Further Uniqueness Theorems.- References.