A new approach to numerical solution of fixed-point problems and its application to delay differential equations

In this paper we consider a certain approximation of fixed-points of a continuous operator A mapping the metric space into itself by means of finite dimensional epsilon(h)-fixed-points of A. These finite dimensional functions are obtained from functions defined on discrete space grid points (related to a parameter h -> 0) by applying suitably chosen extension operators p(h). A theorem specifying necessary and sufficient conditions for existence of fixed-points of A in terms of epsilon(h)-fixed-points of A is given. A corollary which follows the theorem yields an approximate method for a fixed-point problem and determines conditions for its convergence. An example of application of the obtained general results to numerical solving of boundary value problems for delay differential equations is provided. Numerical experiments carried out on three examples of boundary value problems for second order delay differential equations show that the proposed approach produces much more accurate results than many other numerical methods when applied to the same examples. (C) 2010 Elsevier Inc. All rights reserved.