THEORETICAL AND NUMERICAL COMPUTATIONS OF CONVEXITY ANALYSIS FOR FRACTIONAL DIFFERENCES USING LOWER BOUNDEDNESS

This 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.