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Nonlinear stability of Runge-Kutta methods applied to infinite-delay-differential equations. (English) Zbl 1068.65106

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.

MSC:

65L20 Stability and convergence of numerical methods for ordinary differential equations
65L06 Multistep, Runge-Kutta and extrapolation methods for ordinary differential equations
34K28 Numerical approximation of solutions of functional-differential equations (MSC2010)
65L05 Numerical methods for initial value problems involving ordinary differential equations

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References:

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