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

• Published in: Journal of inequalities and applications Vol. 2017; no. 1; pp. 130 - 11
• Main Author: Abdeljawad, Thabet
• Format: Journal Article
• Language: English
• Published: Cham Springer International Publishing 2017; Springer Nature B.V; SpringerOpen
• Subjects: A B C $ABC$ fractional derivative; A B R $ABR$ fractional derivative; Analysis; Applications of Mathematics; boundary value problem; Boundary value problems; Existence theorems; Fractional calculus; higher order; Kernels; Lyapunov inequality; Mathematical analysis; Mathematics; Mathematics and Statistics; Mittag-Leffler kernel; Operators; Uniqueness theorems; Lyapunov inequality; fractional derivative; boundary value problem; Mittag-Leffler kernel; higher order; [Formula: see text] fractional derivative
• ISSN: 1029-242X; 1025-5834; 1029-242X
• DOI: 10.1186/s13660-017-1400-5

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 ] to higher arbitrary order and we formulate their correspondent integral operators. We prove existence and uniqueness theorems for the Caputo ( A B C ) and Riemann ( A B R ) 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 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.