Based on the lower and upper solutions method, we propose a new approach to the Hyers–Ulam and Hyers–Ulam–Rassias stability of first-order ordinary differential equations $$u'=f(t,u)$$ u ′ = f ( t , u ) , in the lack of Lipschitz continuity assumption. Apart from extending and improving the literature by dropping some assumptions, our result provides an estimate for the difference between the solutions of the exact and perturbed models better than from that one obtained by fixed point approach which is commonly used method in this topic. Some examples are also given to illustrate the improvement. Particularly, we examine our approach to the Hyers–Ulam stability problem of second-order elliptic differential equations $$-\Delta u =g(x,u)$$ - Δ u = g ( x , u ) with homogeneous Dirichlet boundary condition which arise in different applications such as population dynamics and population genetics. This investigation is not only of interest in its own right, but also it supports the usability of our approach to other types of boundary value problems such as p ( x )-Laplacian Dirichlet problems, Kirchhoff type problems, fractional differential equations and etc.

中文翻译：

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微分方程Hyers-Ulam-Rassias稳定性的一种新方法

基于上下解法，我们提出了一种求解一阶常微分方程的 Hyers-Ulam 和 Hyers-Ulam-Rassias 稳定性的新方法 $$u'=f(t,u)$$ u ' = f ( t , u ) ，缺乏 Lipschitz 连续性假设。除了通过删除一些假设来扩展和改进文献之外，我们的结果提供了对精确模型和扰动模型解之间差异的估计，而不是通过本主题中常用的定点方法获得的估计。还给出了一些例子来说明改进。特别地，我们检查了我们对二阶椭圆微分方程的 Hyers-Ulam 稳定性问题的方法 $$-\Delta u =g(x,u)$$ - Δ u = g ( x , u ) 具有齐次狄利克雷边界条件，这些边界条件出现在不同的应用中，例如种群动力学和种群遗传学。这项研究不仅本身很有趣，而且还支持我们的方法解决其他类型的边值问题的可用性，例如 p ( x )-Laplacian Dirichlet 问题、Kirchhoff 类型问题、分数阶微分方程等。