In this paper, we extend to the non-Hille–Yosida case a variation of constants formula for a nonautonomous and nonhomogeneous Cauchy problems first obtained by Gühring and Räbiger. By using this variation of constants formula, we derive a necessary and sufficient condition for the existence of an exponential dichotomy for the evolution family generated by the associated nonautonomous homogeneous problem. We also prove a persistence result of the exponential dichotomy for small perturbations. Finally, we illustrate our results by considering two examples. The first example is a parabolic equation with nonlocal and nonautonomous boundary conditions, and the second example is an age-structured model that is a hyperbolic equation.

We characterize algebra homomorphisms from the Lebesgue algebra $L^1_\omega(\mathbb{R})$ into a Banach algebra $\mathcal{A}$. As a consequence of this result, every bounded algebra homomorphism $\varPhi:L^1_\omega(\mathbb{R})\to\mathcal{A}$ is approached through a uniformly bounded family of fractional homomorphisms, and the Hille–Yosida theorem for $C_0$-groups is proved.

We introduce the concept of a distributional resolvent linear system and solve the linear quadratic optimal control problem for this class of systems. The class of distributional resolvent linear systems includes all linear time-invariant systems that have been studied in the control literature.

We show connections with integrated semigroups and distribution semigroups and we consider some examples of systems described by partial differential equations that could not be handled by the existing theory.