N-TIMES INTEGRATED C-SEMIGROUPS AND THE ABSTRACT CAUCHY PROBLEM

Abstract

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 $C$-pseudoresolvent. 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 suitable function $f$.