On the asymptotic stability of \(\theta\)-methods for delay differential equations (Q1094830)

Stability regions of \(\theta\)-methods for the linear delay differential test equations \(y'(t)=py(t)+qy(t-\tau),\) \(t>0\), \(y(t)=\phi (t)\), \(t\in [- \tau,0]\), where \(\tau\) is a positive constant, are presented. In the case that p and q are real constant coefficients, necessary and sufficient conditions on the stepsize for the stability of a \(\theta\)-method are obtained. Furthermore, when p and q are complex coefficients, sufficient conditions for the stability of the \(\theta\)-methods are also given.