In this paper, we present the monotonicity analysis for the nabla fractional differences with discrete generalized Mittag-Leffler kernels \(( {}^{ABR}_{a-1}{\nabla }^{\delta ,\gamma }y )(\eta )\) of order \(0<\delta <0.5\), \(\beta =1\), \(0<\gamma \leq 1\) starting at \(a-1\). If \(({}^{ABR}_{a-1}{\nabla }^{\delta ,\gamma }y ) ( \eta )\geq 0\), then we deduce that \(y(\eta )\) is \(\delta ^{2}\gamma \)-increasing. That is, \(y(\eta +1)\geq \delta ^{2} \gamma y(\eta )\) for each \(\eta \in \mathcal{N}_{a}:=\{a,a+1,\ldots\}\). Conversely, if \(y(\eta )\) is increasing with \(y(a)\geq 0\), then we deduce that \(({}^{ABR}_{a-1}{\nabla }^{\delta ,\gamma }y )(\eta ) \geq 0\). 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.

在本文中，我们对离散的广义Mittag-Leffler核\（（{{} ^ {ABR} _ {a-1} {\ nabla} ^ {\ delta，\ gamma} y）的nabla分数差进行单调分析。（\ ETA）\）的顺序\（0 <\增量<0.5 \） ，\（\的β= 1 \） ，\（0 <\伽马\当量1 \）开始在\（A-1 \） 。如果\（（{{^^ {ABR} _ {a-1} {\ nabla} ^ {\ delta，\ gamma} y）（\ eta} \ geq 0 \），则我们推导出\（y（\ eta ）\）是\（\ delta ^ {2} \ gamma \）-增加。也就是说，\（Y（\ ETA +1）\ GEQ \增量^ {2} \伽马Y（\ ETA）\）为每个\（\ ETA \在\ mathcal {N} _ {A}：= \ { a，a + 1，\ ldots \} \）。相反，如果\（y（\ eta）\）随\（y（a）\ geq 0 \）增大，然后推导\（（{{^^ {ABR} _ {a-1} {\ nabla} ^ {\ delta，\ gamma} y）（\ eta）\ geq 0 \）。此外，总结了卡普托人的单调性和正确的分数差。最后，我们找到了平均值定理的分数差分形式，作为我们的结果的应用。可以看到我们的结果涵盖了文献中的一些现有结果。