Strongly P-Positive Operators and Explicit Representations of the Solutions of Initial Value Problems for Second-Order Differential Equations in Banach Space

The initial value problems for two second-order differential equations with an unbounded operator coefficient A in a Banach space are considered. Using a linear fractional transform (the Cayley transform) of the operator A we give explicit formulas for the solution of these problems if the spectrum...

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Bibliographic Details
Published in:Journal of mathematical analysis and applications 1999-08, Vol.236 (2), p.327-349
Main Author: Gavrilyuk, Ivan P.
Format: Article
Language:English
Subjects:
Exact sciences and technology
Mathematical analysis
Mathematics
Operator theory
Ordinary differential equations
Sciences and techniques of general use
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Summary:The initial value problems for two second-order differential equations with an unbounded operator coefficient A in a Banach space are considered. Using a linear fractional transform (the Cayley transform) of the operator A we give explicit formulas for the solution of these problems if the spectrum of A is situated inside of a parabola. These formulas also provide the algorithmic representations of the operator cosine-function and of the operator Bessel-function with the generator A being a basis for approximate solutions for which error estimates are given. One of the important properties of our approach is the following: the accuracy of the approximate solutions depends automatically on the regularity of the initial data.
ISSN:0022-247X
1096-0813
DOI:10.1006/jmaa.1999.6430