Positivity analysis for the discrete delta fractional differences of the Riemann-Liouville and Liouville-Caputo types

Abstract

<abstract><p>In this article, we investigate some new positivity and negativity results for some families of discrete delta fractional difference operators. A basic result is an identity which will prove to be a useful tool for establishing the main results. Our first main result considers the positivity and negativity of the discrete delta fractional difference operator of the Riemann-Liouville type under two main conditions. Similar results are then obtained for the discrete delta fractional difference operator of the Liouville-Caputo type. Finally, we provide a specific example in which the chosen function becomes nonincreasing on a time set.</p></abstract>

A Study of Monotonicity Analysis for the Delta and Nabla Discrete Fractional Operators of the Liouville–Caputo Family

“…This mathematically rich behavior was first documented in a monotonicity study by Dahal and Goodrich [16] in 2014. Since their initial study, numerous studies have been published, including those by Atici and Uyanik [17]; Baoguo, Erbe and Peterson [18]; Bravo, Lizama and Rueda [19,20]; Dahal and Goodrich [21]; Du, Jia, Erbe and Peterson [22]; Wang, Jia, Du and Liu [23]; Du and Lu [24]; Goodrich [25]; Baoguo, Erbe and Peterson [26]; Goodrich and Lizama [27]; Baoguo, Erbe and Peterson [28]; Abdeljawad and Abdalla [29]; Chen, Bohner and Jia [30]; Mohammed, Abdeljawad and Hamasalh [31,32]; Liu, Du, Anderson, and Jia [33]; Mohammed, Almutairi, Agarwal and Hamad [34]; and Mohammed, Srivastava, Baleanu, Jan and Abualnaja [35]. These papers investigate a variety of questions surrounding the qualitative properties inferred from the sign of a fractional difference acting on a function.…”

On convexity analysis for discrete delta Riemann–Liouville fractional differences analytically and numerically

“…Furthermore, the positivity, monotonicity, and convexity analysis is important in understanding the nature of the discrete fractional problems from the perspective of continuous fractional problems. Many authors have developed many interesting results on optimality and duality in the setting of Riemann-Liouville and Liouville-Caputo fractional differences; see, for instance, [5,[23][24][25][26] and the references therein.…”