© Fang Li et al. 2011
Received: 21 October 2010
Accepted: 13 December 2010
Published: 20 December 2010
1. Introduction and Preliminaries
In 1987, Arendt  studied the -times integrated semigroups, which are more general than semigroups (there exist many operators that generate -times integrated semigroups but not semigroups).
In recent years, the -times integrated -regularized semigroups have received much attention because they can be used to deal with ill-posed abstract Cauchy problems and characterize the "weak" well-posedness of many important differential equations (cf., e.g., [2–18]).
Stimulated by the works in [2, 5–7, 9, 12–18], in this paper, we present a generation theorem of the -times integrated -regularized semigroups for the case that the domain of generator and the range of regularizing operator are not necessarily dense, and prove that the subgenerator of an exponentially bounded, differentiable -times integrated -regularized semigroup is also a subgenerator of a singular -times integrated -regularized semigroup.
Throughout this paper, is a Banach space; denotes the dual space of ; denotes the space of all linear and bounded operators from to , it will be abbreviated to ; denotes the dual space of . By we denote the space of all continuously differentiable -valued functions on . is the space of all continuous -valued functions on .
Lemma 1.2 (see [10, Lemma 3.2]).
2. The Main Results
Let , be constants, and let be a closed operator satisfying . Assume that is the nonnegative measurable function on . A necessary and sufficient condition for is the subgenerator of an -times integrated -semigroup satisfying
Assertion (i) is an immediate consequence of (2.11) and (A1).
If and , then is an integrated semigroup in the sense of Bobrowski .
and the proof is completed.
This completes the proof of (1).
The authors are grateful to the referees for their valuable suggestions. This work is supported by the NSF of Yunnan Province (2009ZC054M).
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