Abstract This paper deals with Cauchy problems and nonlocal problems for non-linear Stieltjes differential equations corresponding to a certain function g. We establish existence and uniqueness results for nonlinear equations with initial value or nonlocal conditions in the space ℬ𝒞 g ([0, H], ℝ) using fixed point methods and g-topology theory. We introduce the concepts of Ulam-Hyers (UH) and generalized Ulam-Hyers-Rassias (UHR) stability and present Ulam type stability results for linear and nonlinear equations in the spaces 𝒜𝒞 g ([0, H], ℝ) ⊂ ℬ𝒞 g ([0, H], ℝ) and ℬ𝒞 g ([0, H], ℝ). Finally, numerical examples are given to illustrate our results.

中文翻译：

---

线性和非线性 Stieltjes 微分方程解的存在性和稳定性

摘要 本文研究了与某个函数g对应的非线性斯蒂尔杰斯微分方程的柯西问题和非局部问题。我们使用不动点方法和 g 拓扑理论在空间 ℬ𝒞 g ([0, H], ℝ) 中建立具有初始值或非局部条件的非线性方程的存在唯一性结果。我们介绍了 Ulam-Hyers (UH) 和广义 Ulam-Hyers-Rassias (UHR) 稳定性的概念，并给出了空间中线性和非线性方程的 Ulam 型稳定性结果 𝒜𝒞 g ([0, H], ℝ) ⊂ ℬ𝒞 g ([0, H], ℝ) 和 ℬ𝒞 g ([0, H], ℝ)。最后，给出了数值例子来说明我们的结果。