Asymptotical stability of numerical methods with constant stepsize for pantograph equations

In this paper, the asymptotical stability of the analytic solution and the numerical methods with constant stepsize for pantograph equations is investigated using the Razumikhin technique. In particular, the linear pantograph equations with constant coefficients and variable coefficients are considered. The stability conditions of the analytic solutions of those equations and the numerical solutions of the theta-methods with constant stepsize are obtained. As a result Z. Jackiewicz's conjecture is partially proved. Finally, some experiments are given.