## Differential and Integral Equations

### Regularized semigroups, existence families and the abstract Cauchy problem

#### Abstract

For an arbitrary bounded linear operator $C$, on a Banach space, and a closable linear operator $A$, we introduce a $C$-regularized semigroup for $A$. We present equivalences between $A$ having a $C$-regularized semigroup, the corresponding abstract Cauchy problem, well-posedness on a continuously embedded subspace, and (for exponentially bounded $C$-regularized semigroups) the Laplace transform.

#### Article information

Source
Differential Integral Equations, Volume 8, Number 6 (1995), 1477-1496.

Dates
First available in Project Euclid: 15 May 2013

Permanent link to this document
https://projecteuclid.org/euclid.die/1368638176

Mathematical Reviews number (MathSciNet)
MR1329851

Zentralblatt MATH identifier
0823.47033

#### Citation

deLaubenfels, Ralph; Sun, Guo Zheng; Wang, Sheng Wang. Regularized semigroups, existence families and the abstract Cauchy problem. Differential Integral Equations 8 (1995), no. 6, 1477--1496. https://projecteuclid.org/euclid.die/1368638176