The construction of asymptotic solutions of certain problems involving parameters

In this article we present a theory for the construction of approximate solutions of certain differential equations involving parameters. The basic idea in this theory consists in removing the nonregular dependence of the solution on the parameter, by going to a space of higher dimension. We construct the asymptotic expansion of the solution to the problem regularized in this way, and establish a definite algorithm that allows us to construct the corresponding approximations for the original problem. The theory is presented by means of certain singularly perturbed nonlinear systems, and is applied to a nonhomogeneous problem with a “turning point.”