On the stability of continuous quadrature rules for differential equations with several constant delays

The aim of the present paper is to study the stability properties of the numerical methods for pure delay differential equations. The methods we consider are based on a quadrature rule and an interpolant (NCE) to get an approximation of the retarded part (continuous quadrature rule). As a test equation we consider y'(t) = -SUM_(r=1)^R b(r)(t)y(t - r tau), t > 0; y(t) = phi(t), t less-than-or-equal-to 0 and we give sufficient conditions for the boundedness of the solutions. The same behaviour is preserved by the continuous quadrature rule under some restriction on the parameters. As a conclusion we give some examples.