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

Abstract

AbstractIn 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.