Abstract
In this paper, we analyse the iterated collocation method for the nonlinear operator equation x = y+K(x) with K a smooth kernel. The paper expands the study begun by H. Kaneko and Y. Xu concerning the superconvergence of the iterated Galerkin method for Hammerstein equations. Let x* denote an isolated fixed point of K. Let X n , n≥1, denote a sequence of finite-dimensional approximating subspaces, and let P n be a projection of X onto X n . The projection method for solving x = y+K(x) is given by x n = P n y+P n K(x n ), and the iterated projection solution is defined as . We analyse the convergence of {x n } and {} to x*, giving a general analysis that includes the collocation method. A detailed analysis is then given for a large class of Urysohn integral operators in one variable, showing the superconvergence of {} to x*.