A Unified Framework for Impulsive Fractional Dynamic Equations with Delay on Time Scales: Existence, Uniqueness, and Stability via Multiple Fixed-Point Theorems

This work investigates a broad class of impulsive fractional dynamic equations with delay and nonlocal initial conditions on arbitrary time scales. Our framework combines several important aspects that are often studied separately, including Caputo fractional derivatives with a general Ψ-kernel, fixed delays, instantaneous and non-instantaneous impulses, nonlocal initial conditions, and nonlinear terms satisfying sublinear growth assumptions. By establishing an appropriate integral formulation of the problem, we analyze the existence, uniqueness, and stability of solutions within a unified setting. Our results not only unify several earlier contributions but also considerably broaden the existing theory of impulsive fractional dynamic equations on time scales. We illustrate the applicability of our theoretical developments in models arising in population dynamics, pharmacokinetics, and neural networks.