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

Christopher S. Goodrich; Benjamin Lyons; Mihaela T. Velcsov

doi:10.3934/cpaa.2020269

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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

Christopher S. Goodrich ^1,, , Benjamin Lyons ^2, and Mihaela T. Velcsov ^3,

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.

Keywords: Monotonicity, discrete fractional calculus, Gamma function, sequential difference, numerical approximation.

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

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
[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. Du, B. Jia, L. 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. Jia, L. Erbe, C. 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. Jia, L. 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. Jia, L. Erbe and A. Peterson, Monotonicity and convexity for nabla fractional $q$-differences, Dynam. Systems Appl., 25 (2016), 47-60. Google Scholar
[29]	B. Jia, L. 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. Du, B. Jia, L. 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. Jia, L. Erbe, C. 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. Jia, L. 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. Jia, L. Erbe and A. Peterson, Monotonicity and convexity for nabla fractional $q$-differences, Dynam. Systems Appl., 25 (2016), 47-60. Google Scholar
[29]	B. Jia, L. 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 $

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

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

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

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

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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

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

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