A method of the study of singularly perturbed differential equations

We propose an iterative approach to singularly perturbed differential equations. With this approach we construct a sequence that converges (both in the asymptotic and in the usual sense) to the solution to the Cauchy problem for a singularly perturbed first-order weakly nonlinear differential equation with two small parameters. The proof of convergence of the sequence constructed is based on the Banach fixed-point theorem for a contraction mapping, and the estimate of the contraction coefficient allows one to estimate the rate of convergence of the sequence. This sequence can be used for justification of the asymptotics obtained by other asymptotic methods.