# Solution of Initial Value Problems in Classes of Generalized Analytic Functions

• Wolfgang Tutschke Book

1. Front Matter
Pages 1-6
2. Wolfgang Tutschke
Pages 7-9
3. Wolfgang Tutschke
Pages 9-18
4. Wolfgang Tutschke
Pages 19-25
5. Wolfgang Tutschke
Pages 26-50
6. Wolfgang Tutschke
Pages 51-68
7. Wolfgang Tutschke
Pages 69-84
8. Wolfgang Tutschke
Pages 84-111
9. Wolfgang Tutschke
Pages 112-128
10. Wolfgang Tutschke
Pages 128-158
11. Wolfgang Tutschke
Pages 158-177
12. Wolfgang Tutschke
Pages 177-180
13. Back Matter
Pages 181-188

### Introduction

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.

### Keywords

Banach Space analytic function banach spaces derivative differential equation partial differential equation

#### Authors and affiliations

• Wolfgang Tutschke
• 1
1. 1.Sektion MathematikMartin-Luther-UniversitätHalleGDR

### Bibliographic information

• DOI https://doi.org/10.1007/978-3-662-09943-8
• Copyright Information Springer-Verlag Berlin Heidelberg 1989
• Publisher Name Springer, Berlin, Heidelberg
• eBook Packages
• Print ISBN 978-3-540-50216-6
• Online ISBN 978-3-662-09943-8
• Buy this book on publisher's site