A Lyapunov type inequality for fractional operators with nonsingular Mittag-Leffler kernel

Abstract

Abstract In this article, we extend fractional operators with nonsingular Mittag-Leffler kernels, a study initiated recently by Atangana and Baleanu, from order α ∈ [ 0 , 1 ] $\alpha\in[0,1]$ to higher arbitrary order and we formulate their correspondent integral operators. We prove existence and uniqueness theorems for the Caputo ( A B C $ABC$ ) and Riemann ( A B R $ABR$ ) type initial value problems by using the Banach contraction theorem. Then we prove a Lyapunov type inequality for the Riemann type fractional boundary value problems of order 2 < α ≤ 3 $2<\alpha\leq3$ in the frame of Mittag-Leffler kernels. Illustrative examples are analyzed and an application as regards the Sturm-Liouville eigenvalue problem in the sense of this fractional calculus is given as well.

Keywords

• A B C $ABC$ fractional derivative
• A B R $ABR$ fractional derivative
• Lyapunov inequality
• boundary value problem
• higher order
• Mittag-Leffler kernel