On Local $\alpha$-Times Integrated $C$-Semigroups

Abstract

This paper presents several characterizations of a local $\alpha$-times integrated $C$-semigroup $\{T(t); 0 \leq t \lt \tau\}$ by means of functional equation, subgenerator, and well-posedness of an associated abstract Cauchy problem. We also discuss properties concerning the nondegeneracy of $T(\cdot )$, the injectivity of $C$, the closability of subgenerators, the commutativity of $T(\cdot )$, and extension of solutions of the associated abstract Cauchy problem.