Regularity of solutions to an abstract inhomogeneous linear differential equation

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 $t > 0$.