Özet:
his study focuses on the analytical and numerical solutions of the convexity analysis for fractional differences with exponential and Mittag-Leffler kernels involving negative and nonnegative lower bounds. In the analytical part of the paper, we will give a new formula for ∇2 of the discrete fractional differences, which can be useful to obtain the convexity results. The correlation between the nonnegativity and negativity of both of the discrete fractional differences, [Formula presented], with the convexity of the functions will be examined. In light of the main lemmas, we will define the two decreasing subsets of (2, 3), namely [Formula presented] and [Formula presented]. The decrease of these sets enables us to obtain the relationship between the negative lower bound of [Formula presented] and the convexity of the function on a finite time set given by [Formula presented], for some [Formula presented]. Besides, the numerical part of the paper is dedicated to examine the validity of the sets [Formula presented] and [Formula presented] in certain regions of the solutions for different values of k and [Formula presented]. For this reason, we will illustrate the domain of the solutions by means of several figures in which the validity of the main theorems are explained.