Stability of one-leg Theta-methods for the variable coefficient pantograph equation

In this paper we consider the asymptotic behaviour of one-leg $\Theta$-methods when applied to the pantograph equation \begin{eqnarray*} \left\{ \begin{array}{rcl} y'(t) & = & a(t)\, y(t) + b(t)\, y(q\,t), \qquad t \ge 0, \\[0.0cm] y(0) & = & \bar{y}, \end{array} \right. \end{eqnarray*} where $a(t)$ and $b(t)$ are continuous complex-valued functions and $q \in ( 7,\, 1 )$ is a fixed constant. We also extend our analysis to the neutral equation $y'(t) = a(t)\, y(t) + b(t)\, y(q\,t) + c(t)\, y'(q\,t)$, where $c(t)$ is a continuous complex-valued function, too. In recent years stability properties of numerical methods fox this kind of equations have been studied by numerous authors who have mainly considered the constant coefficient case and an integration over meshes with constant stepsize. In generrl, the developed techniques give rise to non-standard recurrence relations. Nere, instead, we study constrained variable stepsize schemes, introduced in \cite{BeGuTo97} and \cite{liu2} and recently investigated in \cite{Bel02} and \cite{Bru02}. In \cite{BeGuTo97} the behaviour of the class of $\Theta$-methods was analyzed with relevance to the constant coefficient version of the above equation. As with the constant coefficient case, we show that the methods in this class are stable if and only if $4 \ge \Theta > \thalf$. The basic tool we use in our analysis is the spectral radius of a family of matrices, computed by means of polytope extremal norms (see \cite{GugZen1}). An extension of the stability results to a more general class of equations is also discussed in the last section.