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

• Authors: Z. Bartoszewski
• Venue: Journal of Computational and Applied Mathematics
• Year: 2011

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 v"h defined by its values v"i=v"h(t"i) at grid points t"i, i=0,1,...,N, where N is a given positive integer and h=max"1"@?"i"@?"N(t"i-t"i"-"1). Under suitable assumptions, the existence of a fixed-point of A is equivalent to existence of so called @e(h)-approximate fixed-points of v"h=Av"h, which can be found by minimization of L"2^(^n^) norm of residuum v"h-Av"h with respect to the variables v"i. These @e(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.