The novelty of the present article is to introduce the study of hybrid dynamical system of order \(\xi \in (\delta -1,\delta ]\) with finite time delay. These hybrid systems are more suitable to deal several dynamical process as particular cases. The importance of this manuscript is to discuss the concept of stability results including Ulam-Hyers stability (UHS), generalized Ulam-Hyers stability (GUHS), Ulam-Hyers Rassias stability (UHRS) and generalized Ulam-Hyers Rassias stability (GUHRS). Meanwhile, we investigate some sufficient conditions for existence result of the solution for proposed work by adopting the application of fixed point theorem of Banach algebra due to Dhage under mixed Lipschitz and Carathéodory conditions. Finally the paper is enriched by two interesting applications to demonstrate our results.

本文的新颖之处在于介绍了对\(\xi \in (\delta -1,\delta ]\)具有有限的时间延迟。这些混合系统更适合作为特殊情况处理多个动态过程。本手稿的重要性是讨论稳定性结果的概念，包括 Ulam-Hyers 稳定性 (UHS)、广义 Ulam-Hyers 稳定性 (GUHS)、Ulam-Hyers Rassias 稳定性 (UHRS) 和广义 Ulam-Hyers Rassias 稳定性 (GUHRS)。同时，我们通过在混合 Lipschitz 和 Carathéodory 条件下应用由 Dhage 引起的 Banach 代数不动点定理，研究了所提出的工作解的存在结果的一些充分条件。最后，通过两个有趣的应用程序丰富了本文以展示我们的结果。