Monotonicity Results for Nabla Riemann–Liouville Fractional Differences

Abstract

Positivity analysis is used with some basic conditions to analyse monotonicity across all discrete fractional disciplines. This article addresses the monotonicity of the discrete nabla fractional differences of the Riemann–Liouville type by considering the positivity of ∇b0RLθg(z) combined with a condition on g(b0+2), g(b0+3) and g(b0+4), successively. The article ends with a relationship between the discrete nabla fractional and integer differences of the Riemann–Liouville type, which serves to show the monotonicity of the discrete fractional difference ∇b0RLθg(z).

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.…”