A Nonlinear Singular Perturbation Problem for Second Order Systems

Abstract

The existence and asymptotic behavior as $\varepsilon \to 0^ + $ of solutions of nonlinear boundary value problems for second order systems are studied using differential inequality techniques. Conditions are given under which two point problems for $\varepsilon x'' = f(t,x,x',\varepsilon )$ have unique solutions which converge uniformly as $\varepsilon \to 0^ + $ outside boundary layers at each endpoint of width $\sqrt \varepsilon $ to a solution of the reduced equation $0 = f(t,x,x',0)$.