Title

On The Summability Of Formal Solutions In Liouville-green Theory

Document Type

Article

Publication Date

1-1-2002

Publication Title

Journal Of Dynamical And Control Systems

Department

Mathematics and Computer Science

Abstract

We consider the second-order differential equation ∈2 y″ = (1+∈2ψ(x, ∈))y with a small parameter ∈, where ψ is even with respect to ∈. It is well known that it has two formal solutions y ±(x, ∈) = e ±x/∈ h±(x, ∈), where h ±(x, ∈) is a formal series in powers of ∈ whose coefficients are functions of x. It has been shown [4] that one resp. both of these solutions are 1-summable in certain directions if ψ satisfies certain conditions, in particular, concerning its x-domain. In the present article we give necessary (and sufficient) conditions for 1-summability of one or both of the above formal solutions in terms of ψ. The method of proof involves a certain inverse problem, i.e., the construction of a differential equation of the above form exhibiting a prescribed Stokes phenomenon with respect to ∈.

Volume

8

Issue

3

pp.

371-398

ISSN

0888-3203