The Discrete Galerkin method for nonlinear integral equations

Let K be a completely continuous nonlinear integral operator, and consider solving x = K(x) by Galerkin's method. This can be written as x n = PnK(x n ),Pn an orthogonal projection; the iterated Galerkin solution is defined by x n = K(x n ). We give a general framework and error analysis for the numerical method that results from replacing all integrals in Galerkin's method with numerical integrals. A special high order formula is given for integral equations arising from solving nonlinear two-point boundary value problems.