Existence and Uniqueness Families for the Abstract Cauchy Problem

Suppose that C is an arbitrary bounded operator on a Banach space. We define a pair of families of operators, one of which yields uniqueness and one of which yields existence, of solutions of the abstract Cauchy problem, for all initial data in the image of C. For exponentially bounded solutions, Hi...

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Bibliographic Details
Published in:Journal of the London Mathematical Society 1991-10, Vol.s2-44 (2), p.310-338
Main Author: Delaubenfels, Ralph
Format: Article
Language:English
Subjects:
Exact sciences and technology
Mathematical analysis
Mathematics
Operator theory
Sciences and techniques of general use
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Summary:Suppose that C is an arbitrary bounded operator on a Banach space. We define a pair of families of operators, one of which yields uniqueness and one of which yields existence, of solutions of the abstract Cauchy problem, for all initial data in the image of C. For exponentially bounded solutions, Hille-Yosida type sufficient conditions are given. We also give a perturbation theory. We apply our results to matrices of operators, acting on the product of (possibly different) Banach spaces.
ISSN:0024-6107
1469-7750
DOI:10.1112/jlms/s2-44.2.310