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

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