An iterative way for the approximate solution of perturbed equations

We propose an iterative approach to regularly perturbed differential equations. With this approach we construct a sequence that converges (with respect to the norm of the space of continuous functions) to the solution to the Cauchy problem for a perturbed by a small parameter first-order weakly nonlinear differential equation (weakly nonlinearity means the presence of a small factor in front of the nonlinear term). This sequence also converges to the solution to the problem in the asymptotic sense. The proof of convergence (both the ordinary and the asymptotic) of the sequence constructed and the estimate of the rate of convergence are based on the Banach fixed-point theorem for a contraction mapping of a complete metric space.