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Analytical and numerical monotonicity results for discrete fractional sequential differences with negative lower bound

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  • We investigate the relationship between the sign of the discrete fractional sequential difference $ \big(\Delta_{1+a-\mu}^{\nu}\Delta_a^{\mu}f\big)(t) $ and the monotonicity of the function $ t\mapsto f(t) $. More precisely, we consider the special case in which this fractional difference can be negative and satisfies the lower bound

    $ \begin{equation} \big(\Delta_{1+a-\mu}^{\nu}\Delta_a^{\mu}f\big)(t)\ge-\varepsilon f(a),\notag \end{equation} $

    for some $ \varepsilon>0 $. We prove that even though the fractional difference can be negative, the monotonicity of the function $ f $, nonetheless, is still implied by the above inequality. This demonstrates a significant dissimilarity between the fractional and non-fractional cases. Because of the challenges of a purely analytical approach, our analysis includes numerical simulation.

    Mathematics Subject Classification: Primary: 26A48, 33F05, 39A12, 65D15, 65Q20; Secondary: 39A99, 39B62.


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  • Figure 1.  Graphical representation of the set $ \mathscr{E}_{k, 0.1} $ for $ k \le 72 $

    Figure 2.  Graphical representation of the set $ \mathscr{E}_{k, 0.01} $ for $ k \le 72 $

    Figure 3.  Graphical representation of the set $ \mathscr{E}_{k, 0.001} $ for $ k \le 72 $

    Figure 4.  Graphical representation of the set $ \mathscr{E}_{k, 0.0001} $ for $ k \le 72 $

    Figure 5.  Graphical representation of the set $ \mathscr{E}_{k, 0.00001} $ for $ k \le 72 $

    Figure 6.  Heat maps for the cardinality of the set $ \{ k\ : \ (\mu, \nu ) \in \mathscr{E}_{k,\varepsilon}\} $ for $ \varepsilon = 0.01, 0.001, 0.0001, 0.00001, 0.000001, 0.0000001 $. The cardinality increases from small (dark blue) to large (dark red) and the actual cardinalities are shown along the sidebar of each subplot

    Figure 7.  Heat maps for the cardinality of the set $ \{ k\ : \ (\mu, \nu ) \in \mathscr{E}_{k,\varepsilon}\} $ for $ \varepsilon = 1/100, 1/150, 1/400, 1/650, 1/900, 1/1000 $. These correspond to the interval of $ \varepsilon $ reflected in the top two subplots of Figure 6. Notice the change of cardinality values as $ \varepsilon $ decreases

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