We consider systems of delay differential equations (DDEs) of the form \documentclass[12pt]{minimal}
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$y'(t)=f(t,y(t),y(t-\tau(t)))$\end{document} with the initial condition \documentclass[12pt]{minimal}
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$y(t)=\phi(t), \ t\leq t_{0}$\end{document}. Recently, Torelli [10] introduced a concept of stability for numerical methods applied to dissipative nonlinear systems of DDEs (in some inner product norm), namely RN-stability, which is the straighforward generalization of the wellknown concept of BN-stability of numerical methods with respect to dissipative systems of ODEs. Dissipativity means that the solutions \documentclass[12pt]{minimal}
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$y(t)$\end{document} and \documentclass[12pt]{minimal}
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$z(t)$\end{document} corresponding to different initial functions \documentclass[12pt]{minimal}
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$\phi(t)$\end{document} and \documentclass[12pt]{minimal}
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$\psi(t)$\end{document}, respectively, satisfy the inequality \documentclass[12pt]{minimal}
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$\|y(t)-z(t)\|\leq \max_{x\leq t_0} \|\phi(x)-\psi(x)\|, \ t\geq t_{0}$\end{document}, and is guaranteed by suitable conditions on the Lipschitz constants of the right-hand side function \documentclass[12pt]{minimal}
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$f(t,y,x)$\end{document}. A numerical method is said to be RN-stable if it preserves this contractivity property. After showing that, under slightly more stringent hypotheses on the Lipschitz constants and on the delay function \documentclass[12pt]{minimal}
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$\tau(t)$\end{document}, the solutions \documentclass[12pt]{minimal}
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$y(t)$\end{document} and \documentclass[12pt]{minimal}
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$z(t)$\end{document} are such that \documentclass[12pt]{minimal}
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$\lim_{t \rightarrow +\infty}\|y(t)-z(t)\|=0$\end{document}, in this paper we prove that RN-stable continuous Runge-Kutta methods preserve also this asymptotic stability property.
