Boundary Layer Solutions to Problems with Infinite Dimensional Singular and Regular Perturbations

Lutz Recke , Oleh Omel'chenko

Preprint series: Institut für Mathematik, Humboldt-Universität zu Berlin (ISSN 0863-0976),

MSC 2000

34E15 Singular perturbations, general theory

34B15 Nonlinear boundary value problems

47J07 Abstract inverse mapping and implicit function theorems

58C15 Implicit function theorems; global Newton methods

Abstract
We prove existence, local uniqueness and asymptotic estimates for boundary layer solutions to singularly perturbed equations of the type $(\varepsilon(x)^2 u'(x))'=f(x,u(x))+ g(x,u(x),\varepsilon(x) u'(x)), 0< x <1$, with Dirichlet and Neumann boundary conditions. Here the functions~$\varepsilon$ and~$g$ are small and, hence, regarded as singular and regular functional perturbation parameters. The main tool of the proofs is a generalization (to Banach space bundles) of an Implicit Function Theorem of R. Magnus.