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Journal of the London Mathematical Society 1991 s2-44(2):310-338; doi:10.1112/jlms/s2-44.2.310
© 1991 by London Mathematical Society
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© Oxford University Press

Existence and Uniqueness Families for the Abstract Cauchy Problem

Ralph Delaubenfels

Department of Mathematics, Ohio University Athens, Ohio 45701, USA

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.


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