On One Method for Studying the Cauchy Problem for a Singularly Perturbed Nonlinear First-Order Differential Operator

A sequence that converges to the solution of the Cauchy problem for a singularly perturbed nonlinear first-order differential operator has been constructed. The sequence is asymptotic in the sense that any deviation (in the norm of the space of continuous functions) of its nth element from the problem solution is proportional to the (n + 1)th power of the perturbation parameter. The possibility has been shown for applying the sequence to validating an asymptotics obtained with the method of boundary functions.