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Existence and Uniqueness Families for the Abstract Cauchy Problem
Author(s) -
Delaubenfels Ralph
Publication year - 1991
Publication title -
journal of the london mathematical society
Language(s) - English
Resource type - Journals
SCImago Journal Rank - 1.441
H-Index - 62
eISSN - 1469-7750
pISSN - 0024-6107
DOI - 10.1112/jlms/s2-44.2.310
Subject(s) - uniqueness , mathematics , bounded function , banach space , pure mathematics , cauchy problem , finite rank operator , cauchy distribution , initial value problem , operator (biology) , bounded operator , complete metric space , mathematical analysis , biochemistry , chemistry , repressor , transcription factor , gene
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.