The abstract Cauchy problem in locally convex spaces
Karsten Kruse

TL;DR
This paper establishes new criteria for the existence and uniqueness of solutions to the abstract Cauchy problem in locally convex spaces, extending previous results beyond Fréchet spaces using an asymptotic Laplace transform.
Contribution
It introduces a novel approach based on an asymptotic Laplace transform to analyze the Cauchy problem in broader locally convex spaces, surpassing earlier limitations.
Findings
Derived sufficient conditions for solution existence and uniqueness
Extended classical results to a wider class of locally convex spaces
Utilized asymptotic Laplace transform as a key analytical tool
Abstract
We derive sufficient criteria for the uniqueness and existence of solutions of the abstract Cauchy problem in locally convex Hausdorff spaces. Our approach is based on a suitable notion of an asymptotic Laplace transform and extends results of Langenbruch beyond the class of Fr\'echet spaces.
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