Quantum Laplace Transforms for the Ulam–Hyers Stability of Certain q-Difference Equations of the Caputo-like Type

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We aim to investigate the stability property for the certain linear and nonlinear fractional q-difference equations in the Ulam–Hyers and Ulam–Hyers–Rassias sense. To achieve this goal, we prove that three types of the linear q-difference equations of the q-Caputo-like type are Ulam–Hyers stable by using the quantum Laplace transform and quantum Mittag–Leffler function. Moreover, after proving the existence property for a nonlinear Cauchy q-difference initial value problem, we use the same quantum Laplace transform and the q-Gronwall inequality to show that it is generalized Ulam–Hyers–Rassias stable.