In this paper we investigate the existence, uniqueness and approximation of solutions of delay differential equations (DDEs) with the right-hand side functions $f=f(t,x,z)$ that are Lipschitz continuous with respect to $x$ but only H\"older continuous with respect to $(t,z)$. We give a construction of the randomized two-stage Runge-Kutta scheme for DDEs and investigate its upper error bound in the $L^p(\Omega)$-norm for $p\in [2,+\infty)$. Finally, we report on results of numerical experiments.

翻译：本文研究了右端函数$f=f(t,x,z)$关于$x$满足Lipschitz连续、仅关于$(t,z)$满足Hölder连续的延迟微分方程（DDEs）解的存在性、唯一性及逼近问题。我们构造了面向DDEs的随机两阶段龙格-库塔格式，并研究了该格式在$p\in [2,+\infty)$情形下$L^p(\Omega)$范数的误差上界。最后，我们报告了数值实验的结果。