Singularly Perturbed Boundary Value Problems for Systems of Tichonov's Type in Case of Exchange of Stabilities

Abstract

We consider a system of ordinary differential equations consisting of a singularly perturbed scalar differential equation of second order and a scalar differential equation of first or second order and study a Neumann–Cauchy or a Neumann–Dirichlet problem. We assume that the degenerate equation has two intersecting solutions such that the standard theory for systems of Tichonov's type cannot be applied. We introduce the notation of a degenerate stable solution. By means of the technique of ordered lower and upper solutions we prove the existence of a solution of our problems near the degenerate stable solution for sufficiently small ε and determine its asymptotic behavior in ε.