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$$N$$-times integrated $$C$$-semigroups and the abstract Cauchy problem. (English) Zbl 0892.47042
Summary: This paper is concerned with generation theorems for exponentially equicontinuous $$n$$-times integrated $$C$$-semigroups of linear operators on a sequentially complete locally convex space (SCLCS). The generator of a nondegenerate $$n$$-times integrated $$C$$-semigroup is characterized. The proofs will base on a SCLCS-version of the Widder-Arendt theorem about the Laplace transforms of Lipschitz continuous functions, and on some properties of a $$C$$-pseudo-resolvent. We also discuss the existence and uniqueness of solutions of the abstract Cauchy problem: $$u'= Au+f$$, $$u(0)= x$$, for $$x\in C(D(A^{n+ 1}))$$ and a suitable function $$f$$.

##### MSC:
 47D06 One-parameter semigroups and linear evolution equations
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