Multisummability of formal solutions of singular perturbation problems
Linear perturbation problems of the form \((*)\) \(\varepsilon z^{r+1}x'=A(z,\varepsilon)x-f(z,\varepsilon)\) are considered where \(x'=\partial x/\partial z\), \(r\in \mathbb{N}\), \(A\) is a \(\nu \times \nu\)-matrix and \(f\) is a \(\nu\)-vector, both holomorphic for \(|z|\), \(|\varepsilon |<\rho\). The matrix \(A(0,0)\) is assumed to be invertible. The authors study (mainly in the case \(r\geq 1\)) formal power series solutions to \((*)\) (in one of the variables \((z,\varepsilon)\), with coefficients depending holomorphically on the other, or in both variables), the Gevrey orders of the formal solutions, their integral representations and (multi)summability properties (in sectorial domains, w.r.t. one of the variables \((z,\varepsilon)\), or w.r.t. both variables). In the proofs, the authors use the notion of Nagumo norm of order \(p\) of a function \(f\) holomorphic in a disc of radius \(R\) about the origin: \(\|f\|_p=\sup _{|z|<R}((R-|z|)^p\|f(z)\|)\).
scientific article; zbMATH DE number 1836805