Asymptotic stability properties of Theta-methods for the pantograph equation

In this paper we consider asymptotic stability properties of Theta-methods for the following pantograph equation: [GRAPHICS] where a, b, c is an element of C. In recent years stability properties of numerical methods for this kind of equation have been studied by numerous authors who have considered meshes with fixed stepsize. In general the developed techniques give rise to non-ordinary recurrence relations. In this work, instead, we study constrained variable stepsize schemes, suggested by theoretical and computational reasons, which lead to a non-stationary difference equation. For a first insight, we focus our attention on the class of Theta-methods and show that asymptotic stability is obtained for Theta > 1/2. Finally, some preliminary considerations are devoted to the non-neutral and non-stationary pantograph equation. (C) 1997 Elsevier Science B.V.