\(C\)-semigroups and the Cauchy problem. (English) Zbl 0895.47029

deLaubenfels, Ralph

J. Funct. Anal. 111, No. 1, 44-61 (1993).

Summary: We extend the definition of generator to \(C\)-semigroups that may not be exponentially bounded, where the range of \(C\) may not be dense. We then characterize linear operators, \(A\), for which the associated abstract Cauchy problem has a unique solution, for every initial value in the domain of another operator, \(B\), without assuming that the domain of \(A\) is dense, or the solutions are exponentially bounded. We also give Hille-Yosida type characterizations of generators that may not be densely defined.

MSC:

47D06

One-parameter semigroups and linear evolution equations

Keywords:

generator; \(C\)-semigroups; exponentially bounded; abstract Cauchy problem; initial value; Hille-Yosida type characterizations of generators