Asymptotic stability in the numerical solution of linear pure delay differential equations as abstract Cauchy problems

In Bellen and Maset (Preprint) an approach is presented for solving the delay differential equation (DDE) (∗)y′(t)=Ly(t)+My(t−τ), t⩾0,y(t)=ϕ(t), −τ⩽t⩽0,where τ>0, L,M∈Cm×m and ϕ∈C([−τ,0],Cm), passing through a reformulation of (∗) as an abstract Cauchy problem and its discretization in a system of ordinary differential equations (ODEs). In this paper we investigate the asymptotic stability of this approach on the class of DDEs (∗) with L=0.