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

Journal of Inequalities and Applications

Thabet Abdeljawad¹ 

[1] Department of Mathematics and General Sciences, Prince Sultan University;

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

DOI  :  10.1186/s13660-017-1400-5

来源: DOAJ

【 摘 要 】

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.

【 授权许可】

Unknown