We investigate error of the Euler scheme in the case when the right-hand side function of the underlying ODE satisfies nonstandard assumptions such as local one-sided Lipschitz condition and local H\"older continuity. Moreover, we assume two cases in regards to information availability: exact and noisy with respect to the right-hand side function. Optimality analysis of the Euler scheme is also provided. Finally, we present the results of some numerical experiments.

翻译：本文研究了当原常微分方程右端函数满足非标准假设（如局部单侧利普希茨条件和局部赫尔德连续性）时，欧拉法的误差。此外，我们针对信息可用性考虑了两种情形：右端函数信息精确与含噪声。同时给出了欧拉法的最优性分析。最后，我们展示了若干数值实验的结果。