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

Abstract

This article deals with analysing the positivity, monotonicity and convexity of the discrete nabla fractional operators with exponential kernels from the sense of Riemann and Caputo operators. These operators are called discrete nabla Caputo–Fabrizio–Riemann and Caputo–Fabrizio–Caputo fractional operators. Further, some of our results concern sequential nabla Caputo–Fabrizio–Riemann and Caputo–Fabrizio–Caputo fractional differences, such as ∇aCFRμ∇bCFCυh(x), for various values of start points a and b, and for orders υ and μ in different ranges. Three illustrative examples of the main lemmas in the case of Riemann–Liouville are given at the end of the article.

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

“…However, with discrete fractional calculus theory and discrete operators, this area has not received a lot of attention so far. Moreover, in terms of numerical simulations, some works can be found for monotonicity analysis of the discrete fractional calculus operators (see [27][28][29] and references therein) or convexity analysis of these operators (see [30][31][32] and references therein).…”

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.…”

Monotonicity Results for Nabla Riemann–Liouville Fractional Differences

“…In the context of discrete fractional calculus, the development of new positivity and monotonicity analyses is a source of interesting mathematical problems (see, for example, [16][17][18][19][20][21]). In recent years, several papers have been published devoted exclusively to the study of the problem of the monotonicity of discrete nabla/delta fractional operators with a certain kernel (and often under additional assumptions about the function).…”