On \(\alpha\)-times integrated \(C\)-semigroups and the abstract Cauchy problem. (English) Zbl 0979.47028

Kuo, Chung-Cheng; Shaw, Sen-Yen

Stud. Math. 142, No. 3, 201-217 (2000).

The authors consider \(\alpha\)-times integrated \(C\)-semigroups \((\alpha>0)\) and the associated Cauchy problem of the form \(u'(t)= Au(t)+{t^{\alpha- 1}\over \Gamma(\alpha)} x\), \(t>0\); \(u(0)= 0\). For exponentially bounded \(\alpha\)-times integrated \(C\)-semigroup they give the characterization of the generator \(A\) in terms of the Laplace transforms or via existence of a unique solution of the associated Cauchy problem for each \(x\in(\lambda- A)^{- 1}C(X)\). An example of a non-exponentially bounded \(\alpha\)-times integrated \(C\)-semigroup is also given.

Reviewer: Jan Cholewa (Katowice)

MSC:

47D60	\(C\)-semigroups, regularized semigroups
47D62	Integrated semigroups

Keywords:

\(\alpha\)-times integrated \(C\)-semigroup; generator; abstract Cauchy problem; Laplace transforms