Boundary layer solutions to singularly perturbed problems via the implicit function theorem

We prove existence, local uniqueness and asymptotic estimates for boundary layer solutions to singularly perturbed problems of the type ε2u′′ = f(x, u, εu′, ε), 0 < x < 1, with Dirichlet and Neumann boundary conditions. For that we assume that there is given a family of approximate solutions which satisfy the differential equation and the boundary conditions with certain low accuracy. Moreover, we show that, if this accuracy is high, then the closeness of the approximate solution to the exact solution is correspondingly high. The main tool of the proofs is a modification of an Implicit Function Theorem of R. Magnus. Finally we show how to construct approximate solutions under certain natural conditions.