Fractional difference operators with discrete generalized Mittag–Leffler kernels

Abstract

Abstract In this article we define fractional difference operators with discrete generalized Mittag–Leffler kernels of the form ( E θ , μ ¯ γ ( λ , t − ρ ( s ) ) for both Riemann type (ABR) and Caputo type (ABC) cases, where AB stands for Atangana–Baleanu. Then, we employ the discrete Laplace transforms to formulate their corresponding A B − fractional sums, and prove useful and applicable versions of their semi-group properties. The action of fractional sums on the ABC type fractional differences is proved and used to solve the A B C − fractional difference initial value problems. The nonhomogeneous linear ABC fractional difference equation with constant coefficient is solved by both the discrete Laplace transforms and the successive approximation, and the Laplace transform method is remarked for the continuous counterpart. In fact, for the case μ ≠ 1, we obtain a nontrivial solution for the homogeneous linear A B C − type initial value problem with constant coefficient. The relation between the ABC and ABR fractional differences are formulated by using the discrete Laplace transform. We iterate the fractional sums of order − ( θ , μ , 1 ) to generate fractional sum-differences for which a semigroup property is proved. The nabla discrete transforms for the A B − fractional sums and the A B − iterated fractional sum-differences are calculated. Examples and remarks are given to clarify and confirm the obtained results and some of their particular cases are highlighted. Finally, the discrete extension to the higher order case is discussed.