The Existence and Asymptotic Estimates of Solutions for a Third-Order Nonlinear Singularly Perturbed Boundary Value Problem

In this paper, we consider a class of third-order nonlinear differential equation with singular perturbation subject to three-point boundary value conditions, whose solution exhibits a boundary layer at one endpoint. By using the Schauder fixed point theorem, Green’s function and the method of upper–lower solutions, we first establish an existence result of corresponding boundary value problem without perturbation. Furthermore, by constructing an appropriate lower solution-upper solution pair, as well as analysis technique, the existence and asymptotic estimates of the solutions for the singularly perturbed boundary value problems are obtained.