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Perturbation theorems for local integrated semigroups

100%

Kuo C.-C.

tom 197

nr 1

13-26

We apply the contraction mapping theorem to establish some bounded and unbounded perturbation theorems concerning nondegenerate local α-times integrated semigroups. Some unbounded perturbation results of Wang et al. [Studia Math. 170 (2005)] are also generalized. We also establish some growth properties of perturbations of local α-times integrated semigroups.

On α-times integrated C-semigroups and the abstract Cauchy problem

63%

Kuo C.-C. , Shaw S.-Y.

Studia Mathematica

2000

tom 142

nr 3

201-217

This paper is concerned with α-times integrated C-semigroups for α > 0 and the associated abstract Cauchy problem: $u'(t) = Au(t) + \frac{t^{α-1}}{Γ(α)}x$, t >0; u(0) = 0. We first investigate basic properties of an α-times integrated C-semigroup which may not be exponentially bounded. We then characterize the generator A of an exponentially bounded α-times integrated C-semigroup, either in terms of its Laplace transforms or in terms of existence of a unique solution of the above abstract Cauchy problem for every x in $(λ-A)^{-1}C(X)$.