The aim of this manuscript is to handle the nonlocal boundary value problem for a specific kind of nonlinear fractional differential equations involving a ξ-Hilfer derivative. The used fractional operator is generated by the kernel of the kind \(k(\vartheta,s)=\xi (\vartheta )-\xi (s)\) and the operator of differentiation \({ D}_{\xi } = ( \frac{1}{\xi ^{\prime }(\vartheta )}\frac{d}{d\vartheta } ) \). The existence and uniqueness of solutions are established for the considered system. Our perspective relies on the properties of the generalized Hilfer derivative and the implementation of Krasnoselskii’s fixed point approach and Banach’s contraction principle with respect to the Bielecki norm to obtain the uniqueness of solution on a bounded domain in a Banach space. Besides, we discuss the Ulam–Hyers stability criteria for the main fractional system. Finally, some examples are given to illustrate the viability of the main theories.

本手稿的目的是处理涉及ξ -Hilfer 导数的特定类型的非线性分数阶微分方程的非局部边值问题。使用的分数运算符由\(k(\vartheta,s)=\xi (\vartheta )-\xi (s)\)类型的核和微分运算符\({ D}_{\xi } = ( \frac{1}{\xi ^{\prime }(\vartheta )}\frac{d}{d\vartheta } ) \). 为所考虑的系统建立解决方案的存在性和唯一性。我们的观点依赖于广义 Hilfer 导数的性质以及 Krasnoselskii 不动点方法和 Banach 收缩原理相对于 Bielecki 范数的实现，以获得 Banach 空间中的有界域上解的唯一性。此外，我们讨论了主要分数系统的 Ulam-Hyers 稳定性标准。最后，给出了一些例子来说明主要理论的可行性。