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Solution of Initial Value Problems in Classes of Generalized Analytic Functions (Paperback)
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Solution of Initial Value Problems in Classes of Generalized Analytic Functions (Paperback)
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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."
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