Eur. Phys. J. Special Topics 226, 3355–3368 (2017)
https://doi.org/10.1140/epjst/e2018-00004-2

Regular Article

Lyapunov-type inequalities for fractional difference operators with discrete Mittag-Leffler kernel of order 2 < α < 5/2

¹ Department of Mathematics and General Sciences, Prince Sultan University, Riyadh, Saudi Arabia
² Department of Mathematics, University of Mohamed Boudiaf, M’sila, Algeria
³ Laboratoire des Mathématiques Appliquées, Université de Bejaia, Bejaia 06000, Algeria

^a e-mail: tabdeljawad@psu.edu.sa

Received: 4 April 2017
Received in final form: 9 June 2017
Published online: 25 July 2018

Abstract

Fractional difference operators with discrete-Mittag-Leffler kernels of order α > 1 are defined and their corresponding fractional sum operators are confirmed. We prove existence and uniqueness theorems for the discrete fractional initial value problems in the frame of discrete Caputo (ABC) and Riemann (ABR) operators by using Banach contraction theorem. Then, we prove Lyapunov type inequality for a Riemann type fractional difference boundary value problem of order 2 < α < 5∕2 within discrete Mittag-Leffler kernels, where the limiting case α → 2⁺ results in the ordinary difference Lyapunov inequality. Examples are given to clarify the applicability of our results and an application about the discrete fractional Sturm-Liouville eigenvalue problem is analyzed.