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September 10, 2025Open Access

Stability of hybrid differential equations in the sense of Hyers-Ulam using Gronwall lemma

Key Points

The research demonstrates a new framework for understanding hybrid differential equations.
Using Gronwall's lemma, the study establishes both Hyers-Ulam and Hyers-Ulam-Rassias stability.
Two examples are introduced to validate the proposed method and illustrate its effectiveness.
Graphical representations underscore the results, providing visual clarity on complex concepts.

Abstract

In this paper, for the first time, we use Gronwall?s lemma to show the Hyers-Ulam and Hyers-Ulam-Rassias stability of hybrid differential equations, offering a fresh and rigorous framework for studying these complex structures. Two examples are provided to illustrate the correctness of our method, and graphical illustrations are used to graphically represent the results.

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