Difference monotonicity analysis on discrete fractional operators with discrete generalized Mittag-Leffler kernels

Abstract In this paper, we present the monotonicity analysis for nabla fractional differences with discrete generalized Mittag-Leffler kernels $( {}^{ABR}_{a-1}{\nabla }^{\delta ,\gamma }y )(\eta )$ (a−1ABR∇δ,γy)(η) of order $0<\delta <0.5$ xmlns:mml="http://www.w3.org/1998/Math/MathML">0<δ<0.5 , $\beta =1$ xmlns:mml="http://www.w3.org/1998/Math/MathML">β=1 $0<\gamma \leq 1$ xmlns:mml="http://www.w3.org/1998/Math/MathML">0<γ≤1 starting at $a-1$ xmlns:mml="http://www.w3.org/1998/Math/MathML">a−1 . If $({}^{ABR}_{a-1}{\nabla ) ( \eta )\geq 0$ />a−1ABR∇δ,γy)(η)≥0 then deduce that $y(\eta xmlns:mml="http://www.w3.org/1998/Math/MathML">y(η) is $\delta ^{2}\gamma $ xmlns:mml="http://www.w3.org/1998/Math/MathML">δ2γ -increasing. That is, +1)\geq \delta ^{2} \gamma y(\eta xmlns:mml="http://www.w3.org/1998/Math/MathML">y(η+1)≥δ2γy(η) each $\eta \in \mathcal{N}_{a}:=\{a,a+1,\ldots\}$ xmlns:mml="http://www.w3.org/1998/Math/MathML">η∈Na:={a,a+1,…} Conversely, if increasing $y(a)\geq xmlns:mml="http://www.w3.org/1998/Math/MathML">y(a)≥0 \geq Furthermore, properties Caputo and right are concluded to. Finally, find a difference version mean value theorem as an application our results. One can see results cover some existing in literature.