A Study on the Ulam Stability of Impulsive Dynamic Equations on Time Scales

This paper investigates the Hyers-Ulam and Hyers-Ulam-Rassias stability of first-order nonlinear impulsive dynamic equations defined on finite time scale intervals. Stability in the sense of Ulam addresses the behavior of approximate solutions and their closeness to exact ones, which is key to the qualitative examination of dynamic systems. The aim is to establish sufficient conditions ensuring such stability properties within the time scale framework that unifies discrete and continuous cases. To achieve this, we utilize tools from time scale calculus combined with an extended integral inequality technique to effectively handle impulsive effects. The analysis is carried out using a fixed-point approach based on the contraction mapping principle, which guarantees both existence and uniqueness of solutions. Explicit stability constants related to Hyers-Ulam and Hyers-Ulam-Rassias stability are derived. To validate the theoretical outcomes, an illustrative example is included. This study contributes to extending stability theory for nonlinear impulsive dynamic equations on time scales, offering a unified perspective for both continuous and discrete models.