The Asymptotic Solution of a Class of Singularly Perturbed Nonlinear Boundary Value Problems via Differential Inequalities

Abstract

The singular perturbation problem $\varepsilon y'' = f(t,y,y')$, $0 < t < 1$, $y(0,\varepsilon )$, $y(1,\varepsilon )$ given, is studied under the principal assumption that $f_{y'y'} $ is never zero in the domain of interest. Solutions are shown to exhibit essentially two types of asymptotic behavior: (i) boundary layer behavior and/or (ii) smooth transition from one stable reduced root to another. In addition, an algorithm for the exact determination of conditions guaranteeing such behavior as well as several illustrative examples are discussed. The results are established by applying an extension of the classical Nagumo theory of differential inequalities.