On Riemann—Liouville and Caputo Fractional Forward Difference Monotonicity Analysis

Abstract

Monotonicity analysis of delta fractional sums and differences of order υ∈(0,1] on the time scale hZ are presented in this study. For this analysis, two models of discrete fractional calculus, Riemann–Liouville and Caputo, are considered. There is a relationship between the delta Riemann–Liouville fractional h-difference and delta Caputo fractional h-differences, which we find in this study. Therefore, after we solve one, we can apply the same method to the other one due to their correlation. We show that y(z) is υ-increasing on Ma+υh,h, where the delta Riemann–Liouville fractional h-difference of order υ of a function y(z) starting at a+υh is greater or equal to zero, and then, we can show that y(z) is υ-increasing on Ma+υh,h, where the delta Caputo fractional h-difference of order υ of a function y(z) starting at a+υh is greater or equal to −1Γ(1−υ)(z−(a+υh))h(−υ)y(a+υh) for each z∈Ma+h,h. Conversely, if y(a+υh) is greater or equal to zero and y(z) is increasing on Ma+υh,h, we show that the delta Riemann–Liouville fractional h-difference of order υ of a function y(z) starting at a+υh is greater or equal to zero, and consequently, we can show that the delta Caputo fractional h-difference of order υ of a function y(z) starting at a+υh is greater or equal to −1Γ(1−υ)(z−(a+υh))h(−υ)y(a+υh) on Ma,h. Furthermore, we consider some related results for strictly increasing, decreasing, and strictly decreasing cases. Finally, the fractional forward difference initial value problems and their solutions are investigated to test the mean value theorem on the time scale hZ utilizing the monotonicity results.

On Discrete Delta Caputo–Fabrizio Fractional Operators and Monotonicity Analysis

“…Remark 5. As we have discussed in our recently published article [22], it is impossible to obtain fractional difference MVT for the CFC case, i.e., the following inequalities…”

On positivity and monotonicity analysis for discrete fractional operators with discrete Mittag–Leffler kernel

“…where A is some nonnegative number and Δ 𝛼 y is some type of fractional difference of y. Studies of this form include papers by Atici et al, 23 Du et al, 24 Goodrich, 25 Goodrich and Jonnalagadda, 26 Goodrich et al, 27 Jia et al, [28][29][30] Liu et al, 31 Mohammed et al, 32,33 Suwan et al, 34 and Suwan et al 35 ; we point out, off hand, that, somewhat curiously, there does not seem to be the same sort of activity in the continuous fractional frame, and, in fact, the only work of which we are aware in this direction is a paper by Diethelm. 36 Usually in these studies, the number A in (1.2) has been taken to be 0 since this is the natural and obvious analogue of the integer-order setting in which one has the result "(Δy)(𝜏) ≥ 0 ⇒ y is increasing."…”

“…On the one hand, there are papers that have studied a single fractional difference—i.e., theorems that take the following form: where A is some nonnegative number and Δ α y is some type of fractional difference of y. Studies of this form include papers by Atici et al, 23 Du et al, 24 Goodrich, 25 Goodrich and Jonnalagadda, 26 Goodrich et al, 27 Jia et al, 28–30 Liu et al, 31 Mohammed et al, 32,33 Suwan et al, 34 and Suwan et al 35 ; we point out, off hand, that, somewhat curiously, there does not seem to be the same sort of activity in the continuous fractional frame, and, in fact, the only work of which we are aware in this direction is a paper by Diethelm 36 . Usually in these studies, the number A in () has been taken to be 0 since this is the natural and obvious analogue of the integer‐order setting in which one has the result “ ” Nonetheless, the recent paper by Goodrich and Lizama 37 studied in extensive detail the case in which A > 0, and a subsequent paper by Goodrich and Muellner 38 also considered such cases.…”

On Convexity, Monotonicity and Positivity Analysis for Discrete Fractional Operators Defined Using Exponential Kernels

“…The discrete fractional calculus (DFC) is a field of mathematical analysis and it is a new branch of continuous fractional calculus that is responsible for studying the discrete operators of the sum and difference of fractional order on domains of discrete functions. The definitions of DFC models with singular and nonsingular kernels have been studied by researchers in recent years (see [1][2][3][4][5][6]). Currently, many definitions of nabla (or backward) fractional differences, ∇ υ , have been adopted in order to modify and generalize the concept of ordinary nabla difference, so that for υ = 1, the first-order nabla difference can be recovered, which is usually defined for any function h as ∇h(x) = h(x) − h(x − 1).…”