Research Article Arbitrary Order Fractional Difference Operators with Discrete Exponential Kernels and Applications Thabet Abdeljawad, 1 Qasem M. Al-Mdallal, 2 and Mohamed A. Hajji 2 1 Department of Mathematics and Physical Sciences, Prince Sultan University, P.O. Box 66833, Riyadh 11586, Saudi Arabia 2 Department of Mathematical Sciences, United Arab Emirates University, P.O. Box 17551, Al Ain, Abu Dhabi, UAE Correspondence should be addressed to Tabet Abdeljawad; tabdeljawad@psu.edu.sa Received 3 April 2017; Accepted 25 May 2017; Published 21 June 2017 Academic Editor: Garyfalos Papashinopoulos Copyright © 2017 Tabet Abdeljawad et al. Tis is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. Recently, Abdeljawad and Baleanu have formulated and studied the discrete versions of the fractional operators of order 0<≤1 with exponential kernels initiated by Caputo-Fabrizio. In this paper, we extend the order of such fractional diference operators to arbitrary positive order. Te extension is given to both lef and right fractional diferences and sums. Ten, existence and uniqueness theorems for the Caputo (CFC) and Riemann (CFR) type initial diference value problems by using Banach contraction theorem are proved. Finally, a Lyapunov type inequality for the Riemann type fractional diference boundary value problems of order 2<≤3 is proved and the ordinary diference Lyapunov inequality then follows as  tends to 2 from right. Illustrative examples are discussed and an application about Sturm-Liouville eigenvalue problem in the sense of this new fractional diference calculus is given. 1. Introduction In the last few decades, the continuous and discrete fractional diferential equations have received considerable interest due to their importance in many scientifc felds; see, by way of example not exhaustive enumeration, [1–7]. In [8], the authors introduced a fractional derivative with an exponential kernel which tends to the ordinary derivative as  tends to 1. More properties of this fractional derivative have been studied in [9], where the correspondent fractional integral operator was formulated. Ten, the authors in [7] defned the lef and right fractional derivatives with exponential kernel in the Riemann sense and formulated the right fractional derivatives in the sense of Caputo-Fabrizio with complete investigation to the correspondent fractional integrals and all the discrete versions with integration and summation by parts applied in the fractional and discrete fractional variational calculus. Ten, very recently, the same authors proved an interesting monotonicity result in the sense of this new fractional diference calculus in [10]. In the same direction, for the purpose of providing more fractional derivatives with diferent nonsingular ker- nels, the authors in [11] defned a fractional operator with Mittag-Lefer kernel and in [12, 13] the complete details and discrete versions have been studied. Te exponential kernel fractional derivatives and hence their discrete counterparts are quite diferent from the Mittag-Lefer kernel fractional operators. For example, the integral operator corresponding to exponential kernel fractional derivatives consists of a mul- tiple of the function  added to a multiple of the integration of , whereas the Mittag-Lefer kernel correspondent integral operator consists of a multiple of  and a Riemann-Liouville fractional integral of the same order. Also, the monotonicity coefcient of the CFR fractional diference operator of order 0<≤1 is  as shown in [10], whereas for the discrete Mittag-Lefer CFR operator is  2 as proven in [14]. Motivated, by what we mentioned above, we extend the order of fractional diference type operators with dis- crete exponential kernels to arbitrary positive order, prove existence and uniqueness theorems for the fractional initial value diference problems, and fnally prove a Lyapunov type inequality for the CFR fractional diference operators of order 2<≤3. Te ordinary discrete Lyapunov inequality is then confrmed as  tends to 2 from the right not as in the case of the classical fractional diference as  tends to 2 from the lef [15]. For various fractional Lyapunov extensions we refer, for Hindawi Discrete Dynamics in Nature and Society Volume 2017, Article ID 4149320, 8 pages https://doi.org/10.1155/2017/4149320