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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 $f$.

47D06One-parameter semigroups and linear evolution equations