Numerical solution of Urysohn integral equations using the iterated collocation method

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 Xn, n1, denote a sequence of finite-dimensional approximating subspaces, and let Pn be a projection of X onto Xn. The projection method for solving x = y+K(x) is given by xn = Pny+PnK(xn), and the iterated projection solution is defined as [image omitted]. We analyse the convergence of {xn} and {[image omitted]} 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 {[image omitted]} to x*.