Nonlinear stability of Runge-Kutta methods applied to infinite-delay-differential equations (Q1765016)

The infinite delay-differential equation \[ \begin{gathered} y^1(t)= f(t, y(t), y(pt)),\qquad t> 0,\\ y(0)= \mu\end{gathered} \] is studied in \(\mathbb{C}^d\), where \(p\in (0,1)\), \(f(t,y,z)\) is contractive with constants \(\alpha\) and \(\beta\) toward \(y\) and \(z\), respectively, and \(\beta\leq p\alpha\). The authors divide \([0,\infty]\) into a sum of subintervals \(D_q= (T_q, T_{q+1}]\) and apply a Runge-Kutta \((k, 1)\)-algebraically stable method to the resolution of the problem. The global stability is a concept of long-time stability and the authors prove that the Runge-Kutta-method for a nonnegative matrix \(D\) is globally stable and asymptotically stable. In the proof of this result new conditions on the sizes \(\alpha\), \(\beta\), \(\gamma\), \(p\) and the choice of the step \(h\) are imposed.