Solving boundary value problems for delay differential equations by a fixed-point method

Abstract

A general linear boundary value problem for a nonlinear system of delay differential equations (DDE in short) is reduced to a fixed-point problem v=Av with a properly chosen (generally nonlinear) operator A. The unknown fixed-point v is approximated by piecewise linear function vh defined by its values vi=vh(ti) at grid points ti, i=0,1,…,N, where N is a given positive integer and h=max1≤i≤N(ti−ti−1). Under suitable assumptions, the existence of a fixed-point of A is equivalent to existence of so called ε(h)-approximate fixed-points of vh=Avh, which can be found by minimization of L2(n) norm of residuum vh−Avh with respect to the variables vi. These ε(h)-approximate fixed-points are used for obtaining approximate solutions of the original boundary value problem for a system of DDE. Numerical experiments with the boundary value problem for a system of delay differential equations of population dynamics as well as with two periodic boundary value problems: one for the periodic distributed delay Lotka–Volterra competition system and the second one modeling two coupled identical neurons with time-delayed connections show an efficiency of this kind of approach.