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Approximations of Laplace Transforms and Integrated Semigroups

✍ Scribed by Ti-Jun Xiao; Jin Liang

Publisher: Elsevier Science
Year: 2000
Tongue: English
Weight: 156 KB
Volume: 172
Category: Article
ISSN: 0022-1236
DOI: 10.1006/jfan.1999.3545

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✦ Synopsis

Let [ f m ; m # N] be a sequence of functions from [0, ) to a Banach space E. We give a new and essential condition on f m , which is weaker than the usual ``local Lipschitz continuity'' condition, ensuring that the convergence of f m is equivalent to the convergence of their Laplace transforms. This enables us to establish new approximation theorems for r-times integrated semigroups on E, for all r 0. As a consequence, an open problem for the convergence of integrated semigroups on the whole space E, is solved in essence. Moreover, we present an application to nonhomogeneous Cauchy problems.

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