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

Abstract

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 generalized implicit function theorem which is close to those of Fife and Greenlee (Uspechi Mat. Nauk 24 (1974), 103–130) and of Magnus (Proc. Royal Soc. Edinburgh 136A (2006), 559–583). Finally we show how to construct approximate solutions under certain natural conditions.