Qiu, Lin; Mitsui, Taketomo; Kuang, Jiaoxun The numerical stability of the \(\theta\)-method for delay differential equations with many variable delays. (English) Zbl 0942.65087 J. Comput. Math. 17, No. 5, 523-532 (1999). The asymptotic stability of theoretical solutions and the numerical stability of the \(\theta\)-method are studied for the delay differential equations of the form \[ y'(t)=a y(t) + \sum_{i=1}^{m} b_{i} y(\lambda_{i} t),\quad t \geq 0,\quad y(0)=y_{0}, \] where \(a, b_{1}, b_{2}, \dots, b_{m}\) and \(y_{0}\) \(\in\mathbb{C}\), \(0<\lambda_{m} \leq \lambda_{m-1} \leq \dots \leq \lambda_{1} < 1\). In order the differential equations to be asymptotically stable, a sufficient condition is derived. Finally, it is proved that the linear \(\theta\)-method is \(\Lambda\text{GP}_{m}\)-stable if and only if \(\frac{1}{2} \leq \theta \leq 1\). Reviewer: T.E.Simos (Xanthi) Cited in 7 Documents MSC: 65L20 Stability and convergence of numerical methods for ordinary differential equations 65L05 Numerical methods for initial value problems involving ordinary differential equations 34K06 Linear functional-differential equations 65L07 Numerical investigation of stability of solutions to ordinary differential equations Keywords:delay differential equations; variable delays; stability; \(\theta\)-methods; asymptotic stability PDFBibTeX XMLCite \textit{L. Qiu} et al., J. Comput. Math. 17, No. 5, 523--532 (1999; Zbl 0942.65087)