Short proofs and a counterexample for analytical and numerical stability of delay equations with infinite memory

In this paper we study a class of linear systems of delay differential equations with variable coefficients and variable delay with infinite memory. This kind of problem includes the well-known class of equations with proportional delay (the pantograph equations). The aims of this paper are those of investigating the asymptotic behaviour of both the analytical and the numerical solutions, which are obtained when suitable discrete methods are applied. Relevant to the constant-coefficient equations with proportional delay, we first give an alternative short proof of an important result due to Iserles concerning sufficient conditions for the asymptotic stability of the solutions. Then we establish a new stability result for the more general case, which improves the conditions known in the literature. Afterwards, we focus our attention on the behaviour of one-leg -methods implemented on special integration meshes and prove a numerical stability result under suitable assumptions on the coefficient matrices. Doing this, we note that the numerical stability result applies both to the constant-coefficient and to the general variable-coefficient case under formally analogous conditions. Nevertheless, we prove that, in the general variable-coefficient framework, such conditions do not imply asymptotic stability of the true solutions. This is proved by constructing an explicit counterexample.