Monotonicity and positivity analyses for two discrete fractional-order operator types with exponential and Mittag–Leffler kernels

Abstract

The discrete analysed fractional operator technique was employed to demonstrate positive findings concerning the Atangana-Baleanu and discrete Caputo-Fabrizo fractional operators. Our tests utilized discrete fractional operators with orders between 1 < phi < 2, as well as between 1 < phi < 3/2. We employed the initial values of Mittag-Leffler functions and applied the principle of mathematical induction to ensure the positivity of the discrete fractional operators at each time step. As a result, we observed a significant impact of the positivity of these operators on (del(Q)) (tau) within Np0+1 according to the Riemann- Liouville interpretation. Furthermore, we established a correlation between the discrete fractional operators based on the Liouville-Caputo and Riemann-Liouville definitions. In addition, we emphasized the positivity of (del(Q)) (tau) in the Liouville-Caputo sense by utilizing this relationship. Two examples are presented to validate the results.(c) 2023 The Author(s). Published by Elsevier B.V. on behalf of King Saud University. This is an open access article under the CC BY-NC-ND license (http://creativecommons.org/licenses/by-nc-nd/4.0/).