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

##### MSC:
 47D60 $$C$$-semigroups, regularized semigroups 47D62 Integrated semigroups
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