Differentiability of convolutions, integrated semigroups of bounded semi-variation, and the inhomogeneous Cauchy problem

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Abstract

If T = {T (t); t ≥ 0} is a strongly continuous family of bounded linear operators between two Banach spaces X and Y and f L1(0, b, X), the convolution of T with f is defined by (T*f )(t) = ∫0 t T(s)f(t - s)ds. It is shown that T*f is continuously differentiable for all f C(0, b, X) if and only if T is of bounded semi-variation on [0, b]. Further T*f is continuously differentiable for all f Lp (0, b, X) (1 ≤ p < ∞) if and only if T is of bounded semi-p-variation on [0, b] and T(0) = 0. If T is an integrated semigroup with generator A, these respective conditions are necessary and sufficient for the Cauchy problem u′ = Au + f, u(0) = 0, to have integral (or mild) solutions for all f in the respective function vector spaces. A converse is proved to a well-known result by Da Prato and Sinestrari: the generator A of an integrated semigroup is a Hille-Yosida operator if, for some b > 0, the Cauchy problem has integral solutions for all f ∈ L1(0, b, X). Integrated semigroups of bounded semi-p-variation are preserved under bounded additive perturbations of their generators and under commutative sums of generators if one of them generates a C 0-semigroup.

Original languageEnglish (US)
Pages (from-to)283-305
Number of pages23
JournalJournal of Evolution Equations
Volume8
Issue number2
DOIs
StatePublished - May 1 2008

Keywords

  • (C, integrated) semigroup
  • Bounded additive perturbation
  • Commutative sum
  • Convolution
  • Hille-Yosida
  • Inhomogeneous Cauchy problem
  • Resolvent positive
  • Semi-variation
  • Stieltjes integral

ASJC Scopus subject areas

  • Mathematics (miscellaneous)

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