Formal power series solutions with coefficients defined on shrinking discs for some partial differential equations

TL;DR

This paper investigates conditions for the invertibility of integro-differential operators and explores the existence of unique formal power series solutions with Gevrey order coefficients on shrinking domains for certain PDEs.

Contribution

It provides new criteria for operator invertibility and establishes the existence of formal power series solutions with coefficients defined on shrinking discs.

Findings

Identifies conditions under which an integro-differential operator is a linear automorphism.

Proves the existence of unique formal power series solutions with Gevrey order coefficients.

Analyzes solutions on domains that shrink progressively.

Abstract

In this paper conditions, under which an integro-differential operator is a linear automorphism, are provided. Alternatively, the problem can be considered in terms of existence of a unique formal power series solution for a linear Cauchy problem, where the coefficients of the solution are of a certain Gevrey order on progressively shrinking domains.