Historical Introduction

Turning point theory is a branch of the asymptotic theory of ordinary differential equations that depend in a singular manner on a parameter. “Turning points” are certain exceptional points in that theory. Their precise definition in a general framework is, in itself, a nontrivial matter which will be discussed later on. While turning points are exceptional, their analysis is essential to a full understanding of the asymptotic nature of the solutions of such differential equations. This is a general situation in Mathematics. In an analogous way, analytic functions can only be understood through a study of their singularities; the solutions of an ordinary differential equation depend decisively on the location and type of their critical points, etc. Also, the mathematical formulation of many problems of Physics and Engineering involves turning point problems, and this has been the principal motivation for most of the early investigations.