Fractional operators with generalized Mittag-Leffler kernels and their iterated differintegrals.

Depending on a previous work about fractional operators of Riemann type (ABR) and Caputo type (ABC) with kernels of Mittag-Leffler in three parameters [Eα,μ γ(λ,t-s)], we derive the corresponding fractional integrals with arbitrary order by using the infinite binomial theorem, and study their semi-group properties and their action on the ABC type fractional derivatives to prove the existence and uniqueness theorem for the ABC-fractional initial value problems. In fact, as advantages to the obtained extension, we find that for μ≠1, we obtain a nontrivial solution for the linear ABC-type initial value problem with constant coefficient and prove a certain semigroup property in the parameters μ and γ simultaneously. Iterated type fractional differ-integrals are constructed by iterating fractional integrals of order (α,μ,1) to add a fourth parameter, and a semigroup property is derived under the existence of the fourth parameter. The Laplace transforms for the Atangana-Baleanu (AB) fractional integrals and the AB iterated fractional differ-integrals are calculated. An alternative representation of the ABR-derivatives is given and is compared, in the case γ=1, with the iterated AB differ-integrals with negative order (α,μ,1),-1. An example and several remarks are given to illustrate part of the proven results and to point out some particular cases. The obtained results generalized and improved some recent results.