### 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 L^{1}(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 L^{p} (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 ∈ L^{1}(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 language | English (US) |
---|---|

Pages (from-to) | 283-305 |

Number of pages | 23 |

Journal | Journal of Evolution Equations |

Volume | 8 |

Issue number | 2 |

DOIs | |

State | Published - 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)