The collocation method for the numerical approximation of the periodic solutions of functional differential equations

A collocation method with trigonometric trial functions is presented form-order non-linear functional differential equations with periodicity boundary conditions. In general, uniform approximation of an isolated solution and of its firstm?1 derivatives is achieved, while them-derivative is approximated in mean square. In some special cases we have also the uniform approximation of them-derivative. The solution of then-th non-linear collocation equation may be approximated by Newton's iteration with an arbitrary starting point belonging to a suitable neighbourhood of an isolated solution, for alln>n ₀ withn ₀ large enough.