Nonlinear singular perturbation problems: Proofs of correctness of a formal approximation based on a contraction principle in a Banach space

Abstract

For many singular perturbation problems formal approximations of the solution can be constructed. Here it will be shown that correctness of a formal approximation of sufficiently high order can be proven if (i) certain a priori estimates for the solution of the linearized problem can be obtained and (ii) the remainder term of linearization satisfies a certain smallness condition. The proof will be based on a contractive mapping in a Banach space. Applications are given for singularly perturbed nonlinear second-order elliptic problems with boundary conditions of Dirichlet type.