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January  2021, 20(1): 339-358. doi: 10.3934/cpaa.2020269

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

1. 

School of Mathematics and Statistics, UNSW Sydney, NSW 2052 Australia

2. 

Department of Mathematics, Creighton Preparatory School, Omaha, NE 68114 USA

3. 

Department of Mathematics, University of Nebraska-Omaha, Omaha, NE 68182-0243 USA

* Corresponding author

Received  April 2020 Revised  August 2020 Published  January 2021 Early access  November 2020

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.
Citation: Christopher S. Goodrich, Benjamin Lyons, Mihaela T. Velcsov. Analytical and numerical monotonicity results for discrete fractional sequential differences with negative lower bound. Communications on Pure & Applied Analysis, 2021, 20 (1) : 339-358. doi: 10.3934/cpaa.2020269
References:
[1]

T. Abdeljawad and B. Abdalla, Monotonicity results for delta and nabla Caputo and Riemann fractional differences via dual identities, Filomat, 31 (2017), 3671-3683.  doi: 10.2298/fil1712671a.  Google Scholar

[2]

T. Abdeljawad and D. Baleanu, Monotonicity results for fractional difference operators with discrete exponential kernels, Adv. Differ. Equ., (2017), 9 pp. doi: 10.1186/s13662-017-1126-1.  Google Scholar

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F. M. Atici and P. W. Eloe, A transform method in discrete fractional calculus, Int. J. Differ. Equ., 2 (2007), 165-176.   Google Scholar

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F. M. Atici and P. W. Eloe, Initial value problems in discrete fractional calculus, P. Am. Math. Soc., 137 (2009), 981-989.  doi: 10.1090/S0002-9939-08-09626-3.  Google Scholar

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F. M. Atici and P. W. Eloe, Two-point boundary value problems for finite fractional difference equations, J. Differ. Equ. Appl., 17 (2011), 445-456.  doi: 10.1080/10236190903029241.  Google Scholar

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F. M. Atici and M. Uyanik, Analysis of discrete fractional operators, Appl. Anal. Discrete Math., 9 (2015), 139-149.  doi: 10.2298/AADM150218007A.  Google Scholar

[8] B. C. Carlson, Special Functions of Applied Mathematics, Academic Press, New York, 1977.   Google Scholar
[9]

R. Dahal and C. S. Goodrich, A monotonicity result for discrete fractional difference operators, Arch. Math., 102 (2014), 293-299.  doi: 10.1007/s00013-014-0620-x.  Google Scholar

[10]

R. Dahal and C. S. Goodrich, An almost sharp monotonicity result for discrete sequential fractional delta differences, J. Differ. Equ. Appl., 23 (2017), 1190-1203.  doi: 10.1080/10236198.2017.1307351.  Google Scholar

[11]

R. Dahal and C. S. Goodrich, Mixed order monotonicity results for sequential fractional nabla differences, J. Differ. Equ. Appl., 25 (2019), 837-854.  doi: 10.1080/10236198.2018.1561883.  Google Scholar

[12]

F. DuB. JiaL. Erbe and A. Peterson, Monotonicity and convexity for nabla fractional $(q, h)$-differences, J. Differ. Equ. Appl., 22 (2016), 1224-1243.  doi: 10.1080/10236198.2016.1188089.  Google Scholar

[13]

L. Erbe, C. S. Goodrich, B. Jia and A. Peterson, Survey of the qualitative properties of fractional difference operators: monotonicity, convexity, and asymptotic behavior of solutions, Adv. Differ Equ., (2016), 31 pp. doi: 10.1186/s13662-016-0760-3.  Google Scholar

[14]

R. A. C. Ferreira, A discrete fractional Gronwall inequality, P. Am. Math. Soc., 140 (2012), 1605-1612.  doi: 10.1090/S0002-9939-2012-11533-3.  Google Scholar

[15]

C. S. Goodrich, On discrete sequential fractional boundary value problems, J. Math. Anal. Appl., 385 (2012), 111-124.  doi: 10.1016/j.jmaa.2011.06.022.  Google Scholar

[16]

C. S. Goodrich, A note on convexity, concavity, and growth conditions in discrete fractional calculus with delta difference, Math. Inequal. Appl., 19 (2016), 769-779.  doi: 10.7153/mia-19-57.  Google Scholar

[17]

C. S. Goodrich, A sharp convexity result for sequential fractional delta differences, J. Differ. Equ. Appl., 23 (2017), 1986-2003.  doi: 10.1080/10236198.2017.1380635.  Google Scholar

[18]

C. S. Goodrich, A uniformly sharp monotonicity result for discrete fractional sequential differences, Arch. Math., 110 (2018), 145-154.  doi: 10.1007/s00013-017-1106-4.  Google Scholar

[19]

C. S. Goodrich, Sharp monotonicity results for fractional nabla sequential differences, J. Differ. Equ. Appl., 25 (2019), 801-814.  doi: 10.1080/10236198.2018.1542431.  Google Scholar

[20]

C. S. Goodrich and C. Lizama, A transference principle for nonlocal operators using a convolutional approach: fractional monotonicity and convexity, Israel J. Math., 236 (2020), 533-589.  doi: 10.1007/s11856-020-1991-2.  Google Scholar

[21]

C. S. Goodrich and C. Lizama, Positivity, monotonicity, and convexity for convolution operators, Discrete Contin. Dyn. Syst., 40 (2020), 4961-4983.  doi: 10.3934/dcds.2020207.  Google Scholar

[22]

C. S. Goodrich and B. Lyons, Positivity and monotonicity results for triple sequential fractional differences via convolution, Analysis (Berlin), 40 (2020), 89-103.  doi: 10.1515/anly-2019-0050.  Google Scholar

[23]

C. S. Goodrich and M. Muellner, An analysis of the sharpness of monotonicity results via homotopy for sequential fractional operators, Appl. Math. Lett., 98 (2019), 446-452.  doi: 10.1016/j.aml.2019.07.003.  Google Scholar

[24]

C. Goodrich and A. C. Peterson, Discrete Fractional Calculus, Springer, New York, 2015. doi: 10.1007/978-3-319-25562-0.  Google Scholar

[25]

M. Holm, Sum and difference compositions in discrete fractional calculus, Cubo, 13 (2011), 153-184.  doi: 10.4067/s0719-06462011000300009.  Google Scholar

[26]

B. JiaL. ErbeC. S. Goodrich and A. Peterson, Monotonicity results for delta fractional differences revisited, Math. Slovaca, 67 (2017), 895-906.  doi: 10.1515/ms-2017-0018.  Google Scholar

[27]

B. JiaL. Erbe and A. Peterson, Two monotonicity results for nabla and delta fractional differences, Arch. Math., 104 (2015), 589-597.  doi: 10.1007/s00013-015-0765-2.  Google Scholar

[28]

B. JiaL. Erbe and A. Peterson, Monotonicity and convexity for nabla fractional $q$-differences, Dynam. Systems Appl., 25 (2016), 47-60.   Google Scholar

[29]

B. JiaL. Erbe and A. Peterson, Convexity for nabla and delta fractional differences, J. Differ. Equ. Appl., 21 (2015), 360-373.  doi: 10.1080/10236198.2015.1011630.  Google Scholar

[30]

B. Jia, L. Erbe and A. Peterson, Some relations between the Caputo fractional difference operators and integer-order differences, Electron. J. Differ. Equ., (2015), 7 pp.  Google Scholar

[31]

J. M. Jonnalagadda, An ordering on Green's function and a Lyapunov-type inequality for a family of nabla fractional boundary value problems, Fract. Differ. Calc., 9 (2019), 109-124.  doi: 10.7153/fdc-2019-09-08.  Google Scholar

[32]

A. Kilbas, H. M. Srivastava and J. J. Trujillo, Theory and Applications of Fractional Differential Equations, North-Holland, New York, 2006.  Google Scholar

[33]

C. Lizama, The Poisson distribution, abstract fractional difference equations, and stability, P. Am. Math. Soc., 145 (2017), 3809-3827.  doi: 10.1090/proc/12895.  Google Scholar

[34]

C. Lizama and M. Murillo-Arcila, Well posedness for semidiscrete fractional Cauchy problems with finite delay, J. Comput. Appl. Math., 339 (2018), 356-366.  doi: 10.1016/j.cam.2017.07.027.  Google Scholar

[35] R. Wong and R. Beals, Special Functions: A Graduate Text, Cambridge University Press, New York, 2010.  doi: 10.1017/CBO9780511762543.  Google Scholar

show all references

References:
[1]

T. Abdeljawad and B. Abdalla, Monotonicity results for delta and nabla Caputo and Riemann fractional differences via dual identities, Filomat, 31 (2017), 3671-3683.  doi: 10.2298/fil1712671a.  Google Scholar

[2]

T. Abdeljawad and D. Baleanu, Monotonicity results for fractional difference operators with discrete exponential kernels, Adv. Differ. Equ., (2017), 9 pp. doi: 10.1186/s13662-017-1126-1.  Google Scholar

[3]

G. A. Anastassiou, Nabla discrete fractional calculus and nabla inequalities, Math. Comput. Model., 51 (2010), 562-571.  doi: 10.1016/j.mcm.2009.11.006.  Google Scholar

[4]

F. M. Atici and P. W. Eloe, A transform method in discrete fractional calculus, Int. J. Differ. Equ., 2 (2007), 165-176.   Google Scholar

[5]

F. M. Atici and P. W. Eloe, Initial value problems in discrete fractional calculus, P. Am. Math. Soc., 137 (2009), 981-989.  doi: 10.1090/S0002-9939-08-09626-3.  Google Scholar

[6]

F. M. Atici and P. W. Eloe, Two-point boundary value problems for finite fractional difference equations, J. Differ. Equ. Appl., 17 (2011), 445-456.  doi: 10.1080/10236190903029241.  Google Scholar

[7]

F. M. Atici and M. Uyanik, Analysis of discrete fractional operators, Appl. Anal. Discrete Math., 9 (2015), 139-149.  doi: 10.2298/AADM150218007A.  Google Scholar

[8] B. C. Carlson, Special Functions of Applied Mathematics, Academic Press, New York, 1977.   Google Scholar
[9]

R. Dahal and C. S. Goodrich, A monotonicity result for discrete fractional difference operators, Arch. Math., 102 (2014), 293-299.  doi: 10.1007/s00013-014-0620-x.  Google Scholar

[10]

R. Dahal and C. S. Goodrich, An almost sharp monotonicity result for discrete sequential fractional delta differences, J. Differ. Equ. Appl., 23 (2017), 1190-1203.  doi: 10.1080/10236198.2017.1307351.  Google Scholar

[11]

R. Dahal and C. S. Goodrich, Mixed order monotonicity results for sequential fractional nabla differences, J. Differ. Equ. Appl., 25 (2019), 837-854.  doi: 10.1080/10236198.2018.1561883.  Google Scholar

[12]

F. DuB. JiaL. Erbe and A. Peterson, Monotonicity and convexity for nabla fractional $(q, h)$-differences, J. Differ. Equ. Appl., 22 (2016), 1224-1243.  doi: 10.1080/10236198.2016.1188089.  Google Scholar

[13]

L. Erbe, C. S. Goodrich, B. Jia and A. Peterson, Survey of the qualitative properties of fractional difference operators: monotonicity, convexity, and asymptotic behavior of solutions, Adv. Differ Equ., (2016), 31 pp. doi: 10.1186/s13662-016-0760-3.  Google Scholar

[14]

R. A. C. Ferreira, A discrete fractional Gronwall inequality, P. Am. Math. Soc., 140 (2012), 1605-1612.  doi: 10.1090/S0002-9939-2012-11533-3.  Google Scholar

[15]

C. S. Goodrich, On discrete sequential fractional boundary value problems, J. Math. Anal. Appl., 385 (2012), 111-124.  doi: 10.1016/j.jmaa.2011.06.022.  Google Scholar

[16]

C. S. Goodrich, A note on convexity, concavity, and growth conditions in discrete fractional calculus with delta difference, Math. Inequal. Appl., 19 (2016), 769-779.  doi: 10.7153/mia-19-57.  Google Scholar

[17]

C. S. Goodrich, A sharp convexity result for sequential fractional delta differences, J. Differ. Equ. Appl., 23 (2017), 1986-2003.  doi: 10.1080/10236198.2017.1380635.  Google Scholar

[18]

C. S. Goodrich, A uniformly sharp monotonicity result for discrete fractional sequential differences, Arch. Math., 110 (2018), 145-154.  doi: 10.1007/s00013-017-1106-4.  Google Scholar

[19]

C. S. Goodrich, Sharp monotonicity results for fractional nabla sequential differences, J. Differ. Equ. Appl., 25 (2019), 801-814.  doi: 10.1080/10236198.2018.1542431.  Google Scholar

[20]

C. S. Goodrich and C. Lizama, A transference principle for nonlocal operators using a convolutional approach: fractional monotonicity and convexity, Israel J. Math., 236 (2020), 533-589.  doi: 10.1007/s11856-020-1991-2.  Google Scholar

[21]

C. S. Goodrich and C. Lizama, Positivity, monotonicity, and convexity for convolution operators, Discrete Contin. Dyn. Syst., 40 (2020), 4961-4983.  doi: 10.3934/dcds.2020207.  Google Scholar

[22]

C. S. Goodrich and B. Lyons, Positivity and monotonicity results for triple sequential fractional differences via convolution, Analysis (Berlin), 40 (2020), 89-103.  doi: 10.1515/anly-2019-0050.  Google Scholar

[23]

C. S. Goodrich and M. Muellner, An analysis of the sharpness of monotonicity results via homotopy for sequential fractional operators, Appl. Math. Lett., 98 (2019), 446-452.  doi: 10.1016/j.aml.2019.07.003.  Google Scholar

[24]

C. Goodrich and A. C. Peterson, Discrete Fractional Calculus, Springer, New York, 2015. doi: 10.1007/978-3-319-25562-0.  Google Scholar

[25]

M. Holm, Sum and difference compositions in discrete fractional calculus, Cubo, 13 (2011), 153-184.  doi: 10.4067/s0719-06462011000300009.  Google Scholar

[26]

B. JiaL. ErbeC. S. Goodrich and A. Peterson, Monotonicity results for delta fractional differences revisited, Math. Slovaca, 67 (2017), 895-906.  doi: 10.1515/ms-2017-0018.  Google Scholar

[27]

B. JiaL. Erbe and A. Peterson, Two monotonicity results for nabla and delta fractional differences, Arch. Math., 104 (2015), 589-597.  doi: 10.1007/s00013-015-0765-2.  Google Scholar

[28]

B. JiaL. Erbe and A. Peterson, Monotonicity and convexity for nabla fractional $q$-differences, Dynam. Systems Appl., 25 (2016), 47-60.   Google Scholar

[29]

B. JiaL. Erbe and A. Peterson, Convexity for nabla and delta fractional differences, J. Differ. Equ. Appl., 21 (2015), 360-373.  doi: 10.1080/10236198.2015.1011630.  Google Scholar

[30]

B. Jia, L. Erbe and A. Peterson, Some relations between the Caputo fractional difference operators and integer-order differences, Electron. J. Differ. Equ., (2015), 7 pp.  Google Scholar

[31]

J. M. Jonnalagadda, An ordering on Green's function and a Lyapunov-type inequality for a family of nabla fractional boundary value problems, Fract. Differ. Calc., 9 (2019), 109-124.  doi: 10.7153/fdc-2019-09-08.  Google Scholar

[32]

A. Kilbas, H. M. Srivastava and J. J. Trujillo, Theory and Applications of Fractional Differential Equations, North-Holland, New York, 2006.  Google Scholar

[33]

C. Lizama, The Poisson distribution, abstract fractional difference equations, and stability, P. Am. Math. Soc., 145 (2017), 3809-3827.  doi: 10.1090/proc/12895.  Google Scholar

[34]

C. Lizama and M. Murillo-Arcila, Well posedness for semidiscrete fractional Cauchy problems with finite delay, J. Comput. Appl. Math., 339 (2018), 356-366.  doi: 10.1016/j.cam.2017.07.027.  Google Scholar

[35] R. Wong and R. Beals, Special Functions: A Graduate Text, Cambridge University Press, New York, 2010.  doi: 10.1017/CBO9780511762543.  Google Scholar
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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