Integrated semigroups and vibrating string problem

Abstract

Consider the abstract Cauchy problems

(1)

where A(∈) is a closed and non-densely defined operator on a Banach space X, and ∈ is a milti-parameter. Continuity with respect to parameter of integrated semigroup is a property that arises naturally when considering solutions of abstract Cauchy problem(1). Theorems are developed to determine continuity with respect to parameter ε of integrated semigroups generated by the operator A(∈). These obtained theorems are applied to a vibrating string problem, and the results of existence of an unique classical solution or integral solution of this equation and continuity in parameter εof such a solution are obtained.

Keywords:

• Integrated semigroup
• continuity in parameter
• vibrating string problem