Monotonicity Analysis of Fractional Proportional Differences

Abstract

In this work, the nabla discrete new Riemann–Liouville and Caputo fractional proportional differences of order 0<ε<1 on the time scale ℤ are formulated. The differences and summations of discrete fractional proportional are detected on ℤ, and the fractional proportional sums associated to ∇cRχε,ρz with order 0<ε<1 are defined. The relation between nabla Riemann–Liouville and Caputo fractional proportional differences is derived. The monotonicity results for the nabla Caputo fractional proportional difference are proved; specifically, if ∇c−1Rχε,ρz>0 then χz is ερ −increasing, and if χz is strictly increasing on ℕc and χc>0, then ∇c−1Rχε,ρz>0. As an application of our findings, a new version of the fractional proportional difference of the mean value theorem (MVT) on ℤ is proved.

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.…”

An unexpected property of fractional difference operators: Finite and eventual monotonicity

“…The first investigation of the relationship between the monotonicity of $$$ u $$$ at $$$ n $$$ and the associated sign of $$$ \left({\Delta}^{\alpha }u\right)(n) $$$ was conducted by Dahal and Goodrich [14], which was then subsequently followed by an analogous study of convexity by Goodrich [15]. Numerous related results and improvements then quickly followed, and these include contributions by Abdeljawad and Abdalla [16], Abdeljawad and Baleanu [17, 18], Atici and Uyanik [19], Bravo, Lizama, and Rueda [20], Du, Jia, Erbe, and Peterson [21], Goodrich and Jonnalagadda [22], Jia, Erbe, and Peterson [23–26], Liu, Du, Anderson, and Jia [27], Mohammed, Abdeljawad, and Hamasalh [28, 29], Suwan, Abdeljawad, and Jarad [30], Suwan, Owies, and Abdeljawad [31, 32], and Suwan, Owies, Abussa, and Abdeljawad [33]. We note that each of these papers investigated non‐sequential fractional operators , by which we mean a single fractional difference.…”