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 intendยญ 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 conยญ 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
CONTENT
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
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