Stability analysis of linear multistep methods for delay differential equations (Q1091085)

This paper deals with the stability analysis of some linear multistep methods based on the test equation \(y'(t)=ay(\lambda t)+by(t)\), \(t\geq 0\), \(y(0)=y_ 0\), where a, b, and \(\lambda\) are real and \(0<\lambda <1\). Assuming that \(y_ h(t_ j)\) for \(j\leq i\) are bounded by a constant M and \(\| y_ h\|_{<t_ 0,t_ i>}\) is bounded by \(\gamma\) M for some \(\gamma\geq 1\), the authors are looking for conditions such that \(y_ h(t_{i+1})\) and \(\| y_ h\|_{<t_ i,t_{i+1}>}\) are bounded by the same constants M and \(\gamma\) M, respectively. Some sufficient stability conditions for linear multistep methods with Lagrange interpolation and for low order parametrized Adams-Bashforth and Adams-Moulton methods are given.