#### 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

#### Provider Link

#### Citation

Bodine, Sigrun, and R. Schäfke. 2002. "On the summability of formal solutions in Liouville-Green theory." Journal Of Dynamical And Control Systems 8(3): 371-398.