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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 \(f\).

47D06 One-parameter semigroups and linear evolution equations
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