Local K-Convoluted C-Cosine Functions and Abstract Cauchy Problems

Abstract

Abstract. Let K be a locally integrable function from [0, T0) into  F(=R or C) , and C  a bounded linear operator on a Banach space X over F. In this paper, we shall deduce some basic properties of a nondegenerate local K-convoluted C-cosine function on X and some generation theorems of local K-convoluted C-cosine functions on X with or without the nondegeneracy, which can be applied to obtain some equivalence relations between the generation of a nondegenerate local K-convoluted C-cosine function on X with subgenerator A and the unique existence of solutions of the abstract Cauchy problem: u''(t) = Au(t) + f(t) for a.e. t in (0, T0),u(0) = x, u'(0)=y.