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

Abstract

In this paper, we focus on the analytical and numerical convexity analysis of discrete delta Riemann–Liouville fractional differences. In the analytical part of this paper, we give a new formula for the discrete delta Riemann-Liouville fractional difference as an alternative definition. We establish a formula for the $\Delta ^{2}$ Δ 2 , which will be useful to obtain the convexity results. We examine the correlation between the positivity of $({}^{\mathrm{RL}}_{w_{0}}\Delta ^{\alpha} \mathrm{f} )( \mathrm{t})$ ( w 0 RL Δ α f ) ( t ) and convexity of the function. In view of the basic lemmas, we define two decreasing subsets of $(2,3)$ ( 2 , 3 ) , $\mathscr{H}_{\mathrm{k},\epsilon}$ H k , ϵ and $\mathscr{M}_{\mathrm{k},\epsilon}$ M k , ϵ . The decrease of these sets allows us to obtain the relationship between the negative lower bound of $({}^{\mathrm{RL}}_{w_{0}}\Delta ^{\alpha} \mathrm{f} )( \mathrm{t})$ ( w 0 RL Δ α f ) ( t ) and convexity of the function on a finite time set $\mathrm{N}_{w_{0}}^{\mathrm{P}}:=\{w_{0}, w_{0}+1, w_{0}+2,\dots , \mathrm{P}\}$ N w 0 P : = { w 0 , w 0 + 1 , w 0 + 2 , … , P } for some $\mathrm{P}\in \mathrm{N}_{w_{0}}:=\{w_{0}, w_{0}+1, w_{0}+2,\dots \}$ P ∈ N w 0 : = { w 0 , w 0 + 1 , w 0 + 2 , … } . The numerical part of the paper is dedicated to examinin the validity of the sets $\mathscr{H}_{\mathrm{k},\epsilon}$ H k , ϵ and $\mathscr{M}_{\mathrm{k},\epsilon}$ M k , ϵ for different values of k and ϵ. For this reason, we illustrate the domain of solutions via several figures explaining the validity of the main theorem.

Some Properties of a Falling Function and Related Inequalities on Green’s Functions

“…Taking into account the previous considerations, an important topic in discrete fractional calculus is to achieve computations of boundary and initial value problems whose initial and boundary conditions are of the form of nabla or delta difference operators (cf. [7][8][9][10][11][12][13]). In recent years, boundary and initial value problem computations when considering the nabla fractional and the delta fractional with different types of discrete operators bases have been achieved (e.g., [14][15][16][17][18][19][20][21]).…”