The Cauchy problem for singularly perturbed weakly nonlinear second-order differential equations: An iterative method

For a singularly perturbed weakly nonlinear second-order differential equation, we construct a sequence converging to the Cauchy problem solution. This is an asymptotical sequence because the deviation (in the sense of the norm of the space of continuous functions) of its nth element from the solution to the problem is proportional to the (n + 1)th power of the perturbation parameter. Such a sequence can be used to justify the asymptotics obtained by using boundary functions.