Integrated semigroups and their applications to the abstract Cauchy problem

Integrated Semigroups and Their Applications to the Abstract Cauchy Problem

This paper is concerned with characterizations of those linear, closed, but not necessarily densely defined operators A on a Banach space E with nonempty resolvent set for which the abstract Cauchy problem u'(t) = Au(t), u(0) = x has unique, exponentially bounded solutions for every initial value x e D(A). Investigating these operators we are led to the class of "integrated semigroups". Among o...

Inhomogeneous Two-Parameter Abstract Cauchy Problem

inhomogeneous two-parameter abstract cauchy problem

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

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) = ∫ t 0 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 a...

A singular abstract cauchy problem.

In this note a singular abstract Cauchy problem is considered. A solution is obtained for this problem in terms of a regular abstract Cauchy problem. As an application we obtain a new solution of the initial value problem for a class of singular partial differential equations.