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Proceedings of the American Mathematical Society

ISSN 1088-6826(online) ISSN 0002-9939(print)

 

 

Regularity of solutions to an abstract inhomogeneous linear differential equation


Author: G. F. Webb
Journal: Proc. Amer. Math. Soc. 62 (1977), 271-277
MSC: Primary 34G05; Secondary 47D05
MathSciNet review: 0432996
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Abstract: Let $ T(t),t \geqslant 0$, be a strongly continuous semigroup of linear operators on a Banach space X with infinitesimal generator A satisfying $ T(t)X \subset D(A)$ for all $ t > 0$. Let f be a function from $ [0,\infty )$ to X of strong bounded variation. It is proved that $ u(t){ = ^{{\text{def}}}}T(t)x + {\smallint ^{t0}}T(t - s)f(s)ds,x \in X$, is strongly differentiable and satisfies $ du(t)/dt = Au(t) + f(t)$ for all but a countable number of $ t > 0$.


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DOI: https://doi.org/10.1090/S0002-9939-1977-0432996-3
Keywords: Strongly continuous semigroup, infinitesimal generator, inhomogeneous equation, strong bounded variation
Article copyright: © Copyright 1977 American Mathematical Society