Quasilinearization and rate of convergence for higher-order nonlinear periodic boundary-value problems

We study the convergence of a sequence of approximate solutions for the following higher-order nonlinear periodic boundary-value problem: u((n))(t) =f(t, u(t)), t epsilon I=[0, T], u((i))(0) - u((i))(T)= c(i), i = 0,..., n - 1. Here, f epsilon C(I x R, R) is such that, for some k greater than or equal to 1, a(1)f/au(1) exists and is a continuous function for i = 0, 1,..., k. We prove that it is possible to construct two sequences of approximate solutions converging to the extremal solution with rate of convergence of order k.