A complete analysis of a model nonlinear singular perturbation problem having a continuous locus of singular points

Abstract

Consider the boundary value problem ϵ y′′ = ( y 2 − t 2) y′, − 1 ≤ t ≤ 0, y(− 1) = A, y(0) = B. Depending on the choice of A and B, one can ensure the existence of “turning points,” t ̂ ; y(t, ϵ) 2 − t ̂ 2 = 0 . However, due to the nonlinear nature of the problem, one does not know the position or number of such turning points. In the case when A > f 0 = B Kedem, Parter and Steuerwalt gave a development of this problem based on an abstract bifurcation analysis which in turn was based on “degree theory.” In this paper we give a complete analysis of the problem based entirely on a priori estimates and the “shooting” method.