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

Tom 142 / 2000

Chung-Cheng Kuo, Sen-Yen Shaw Studia Mathematica 142 (2000), 201-217 DOI: 10.4064/sm-142-3-201-217


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 $(λ-A)^{-1}C(X)$.


  • Chung-Cheng Kuo
  • Sen-Yen Shaw

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