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

Consider the boundary value problem ?y′′ = (y² ? t²)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,” $\text{t}\text{?}\text{; y(t, ?)}{}^2\text{?}\text{t}\text{?}{}^2\text{= 0}$. 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.