Li, Y.-C.; Shaw, S.-Y. \(N\)-times integrated \(C\)-semigroups and the abstract Cauchy problem. (English) Zbl 0892.47042 Taiwanese J. Math. 1, No. 1, 75-102 (1997). 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 \(f\). Cited in 22 Documents MSC: 47D06 One-parameter semigroups and linear evolution equations Keywords:exponentially equicontinuous \(n\)-times integrated \(C\)-semigroups; sequentially complete locally convex space; generator; Widder-Arendt theorem; Laplace transforms of Lipschitz continuous functions; \(C\)-pseudo-resolvent; abstract Cauchy problem PDFBibTeX XMLCite \textit{Y. C. Li} and \textit{S. Y. Shaw}, Taiwanese J. Math. 1, No. 1, 75--102 (1997; Zbl 0892.47042) Full Text: DOI