Discrete generalized fractional operators defined using h‐discrete Mittag‐Leffler kernels and applications to AB fractional difference systems

Abstract

This study investigates the h‐fractional difference operators with h‐discrete generalized Mittag‐Leffler kernels ( in the sense of Riemann type (namely, the ABR) and Caputo type (namely, the ABC). For which, we will discuss the region of convergent. Then, we study the h‐discrete Laplace transforms to formulate their corresponding AB‐fractional sums. Also, it is useful in obtaining the semi‐group properties. We will prove the action of fractional sums on the ABC type h‐fractional differences and then it can be used to solve the system of ABC h‐fractional difference. By using the h‐discrete Laplace transforms and the Picard successive approximation technique, we will solve the nonhomogeneous linear ABC h‐fractional difference equation with constant coefficient, and also we will remark the h‐discrete Laplace transform method for the continuous counterpart. Meanwhile, we will obtain a nontrivial solution for the homogeneous linear ABC h‐fractional difference initial value problem with constant coefficient for the case δ ≠ 1. We will formulate the relation between the ABC and ABR h‐fractional differences by using the h‐discrete Laplace transform. By iterating the fractional sums of order −(ϕ, δ, 1), we will generate the h‐fractional sum‐differences, and in view of this, a semigroup property will be proved. Due to these new powerful techniques, we can calculate the nabla h‐discrete transforms for the AB h‐fractional sums and the AB iterated h‐fractional sum‐differences. Furthermore, we will obtain some particular cases that can be found in examples and remarks. Finally, we will discuss the higher order case of the h‐discrete fractional differences and sums.