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

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.