TY - JOUR

T1 - Differentiability of convolutions, integrated semigroups of bounded semi-variation, and the inhomogeneous Cauchy problem

AU - Thieme, Horst

N1 - Funding Information:
I thank Pierre Magal for motivating discussions and him and Ralph deLaubenfels for helpful suggestions. This paper has been substantially improved through the constructive critique by an anonymous referee. This work was partially supported by NSF grant DMS-0314529.

PY - 2008/5

Y1 - 2008/5

N2 - 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.

AB - 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.

KW - (C, integrated) semigroup

KW - Bounded additive perturbation

KW - Commutative sum

KW - Convolution

KW - Hille-Yosida

KW - Inhomogeneous Cauchy problem

KW - Resolvent positive

KW - Semi-variation

KW - Stieltjes integral

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U2 - 10.1007/s00028-007-0355-2

DO - 10.1007/s00028-007-0355-2

M3 - Article

AN - SCOPUS:44549087021

VL - 8

SP - 283

EP - 305

JO - Journal of Evolution Equations

JF - Journal of Evolution Equations

SN - 1424-3199

IS - 2

ER -