On \(\alpha\)-times integrated \(C\)-semigroups and the abstract Cauchy problem (Q2717535)

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.