LOCAL $K$-CONVOLUTED $C$-SEMIGROUPS AND ABSTRACT CAUCHY PROBLEMS

Abstract

Let $K :[0,T_0) \to \Bbb F$ be a locally integrable function, and $C :X\to X$ a bounded linear operator on a Banach space $X$ over the field $\Bbb F$(=$\Bbb R$ or $\Bbb C$). In this paper, we will deduce some basic properties of a nondegenerate local $K$-convoluted $C$-semigroup on $X$ and some generation theorems of local $K$-convoluted $C$-semigroups on $X$ with or without the nondegeneracy, which can be applied to obtain some equivalence relations between the generation of a nondegenerate local $K$-convoluted $C$-semigroup on $X$ with subgenerator $A$ and the unique existence of solutions of the abstract Cauchy problem:\[ \textrm{ACP}(A,f,x) \qquad \begin{cases} u'(t) = A u(t)+f(t) &\textrm{for a.e. $t \in (0,T_0)$},\\ u(0)=x \end{cases}\]when $K$ is a kernel on $[0,T_0)$, $C :X\to X$ an injection, and $A:\text{D}(A)\subset X\to X$ a closed linear operator in $X$ such that $CA\subset AC$. Here $0\lt T_0\leq\infty$, $x\in X$, and $f\in\text{L}_{loc}^{1}([0,T_0),X)$.