The abstract Cauchy problem in locally convex spaces

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.