## Taiwanese Journal of Mathematics

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

Chung-Cheng Kuo

#### 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 $f\in\text{L}_{loc}^{1}([0,T_0),X)$.

#### Article information

Source
Taiwanese J. Math., Volume 19, Number 4 (2015), 1227-1245.

Dates
First available in Project Euclid: 4 July 2017

https://projecteuclid.org/euclid.twjm/1499133698

Digital Object Identifier
doi:10.11650/tjm.19.2015.4737

Mathematical Reviews number (MathSciNet)
MR3384688

Zentralblatt MATH identifier
1357.34098

#### Citation

Kuo, Chung-Cheng. LOCAL $K$-CONVOLUTED $C$-SEMIGROUPS AND ABSTRACT CAUCHY PROBLEMS. Taiwanese J. Math. 19 (2015), no. 4, 1227--1245. doi:10.11650/tjm.19.2015.4737. https://projecteuclid.org/euclid.twjm/1499133698

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