Projection and Iterated Projection Methods for Nonlinear Integral equations

Abstract

Consider the nonlinear operator equation $x = {\mathcal {K}}(x)$ with $\mathcal {K}$ a completely continuous mapping of a domain in the Banach space $\mathcal {X}$ into $\mathcal {X}$ and let $x^ * $ denote an isolated fixed point of $\mathcal {K}$. Let $\mathcal {K}_n $, $n \geqq 1$ denote a sequence of finite dimensional approximating subspaces, and let $P_n $, be a projection of $\mathcal {X}$ onto $\mathcal {X}_n $ The projection method for solving $x = {\mathcal {K}}(x)$ given by$x_n = P_n \mathcal {K}(x_n )$, and the iterated projection solution is defined as $\tilde{x}_n = \mathcal {K}(x_n )$. We analyze the convergence of $x_n $ and $\tilde x_n $ to $x^ * $, giving a general analysis that includes both the Galerkin and collocation methods. A more detailed analysis is then given for a large class of Urysohn integral operators in one variable, showing the superconvergence of $\tilde x_n $ to $x^ * $.