Discrete Mittag-Leffler kernel type fractional difference initial value problems and Gronwall's inequality

In this article, we studied the Caputo and Riemann–Liouville type discrete fractional difference initial value problems with discrete Mittag-Leffler kernels. The existence and uniqueness of the solution is proved by using Banach contraction principle. The linear type equations are used to prove new discrete fractional versions of the Gronwall's inequality. The nabla discrete Laplace transform is used to obtain solution representations. The proven Gronwall's inequality under a new defined α-Lipschitzian is used to prove that small changes in the initial conditions yield small changes in solutions. Numerical examples are discussed to demonstrate the reliability of the theoretical results.