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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 |
$ \big(\Delta_{1+a-\mu}^{\nu}\Delta_a^{\mu}f\big)(t) $ |
$ t\mapsto f(t) $ |
$ \begin{equation} \big(\Delta_{1+a-\mu}^{\nu}\Delta_a^{\mu}f\big)(t)\ge-\varepsilon f(a),\notag \end{equation} $ |
$ \varepsilon>0 $ |
$ f $ |
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. |
[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. |
[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. |
[4] |
F. M. Atici and P. W. Eloe,
A transform method in discrete fractional calculus, Int. J. Differ. Equ., 2 (2007), 165-176.
|
[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. |
[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. |
[7] |
F. M. Atici and M. Uyanik,
Analysis of discrete fractional operators, Appl. Anal. Discrete Math., 9 (2015), 139-149.
doi: 10.2298/AADM150218007A. |
[8] |
B. C. Carlson, Special Functions of Applied Mathematics, Academic Press, New York, 1977.
![]() |
[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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[24] |
C. Goodrich and A. C. Peterson, Discrete Fractional Calculus, Springer, New York, 2015.
doi: 10.1007/978-3-319-25562-0. |
[25] |
M. Holm,
Sum and difference compositions in discrete fractional calculus, Cubo, 13 (2011), 153-184.
doi: 10.4067/s0719-06462011000300009. |
[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. |
[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. |
[28] |
B. Jia, L. Erbe and A. Peterson,
Monotonicity and convexity for nabla fractional $q$-differences, Dynam. Systems Appl., 25 (2016), 47-60.
|
[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. |
[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. |
[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. |
[32] |
A. Kilbas, H. M. Srivastava and J. J. Trujillo, Theory and Applications of Fractional Differential Equations, North-Holland, New York, 2006. |
[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. |
[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. |
[35] |
R. Wong and R. Beals, Special Functions: A Graduate Text, Cambridge University Press, New York, 2010.
doi: 10.1017/CBO9780511762543.![]() ![]() |
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. |
[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. |
[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. |
[4] |
F. M. Atici and P. W. Eloe,
A transform method in discrete fractional calculus, Int. J. Differ. Equ., 2 (2007), 165-176.
|
[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. |
[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. |
[7] |
F. M. Atici and M. Uyanik,
Analysis of discrete fractional operators, Appl. Anal. Discrete Math., 9 (2015), 139-149.
doi: 10.2298/AADM150218007A. |
[8] |
B. C. Carlson, Special Functions of Applied Mathematics, Academic Press, New York, 1977.
![]() |
[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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[24] |
C. Goodrich and A. C. Peterson, Discrete Fractional Calculus, Springer, New York, 2015.
doi: 10.1007/978-3-319-25562-0. |
[25] |
M. Holm,
Sum and difference compositions in discrete fractional calculus, Cubo, 13 (2011), 153-184.
doi: 10.4067/s0719-06462011000300009. |
[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. |
[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. |
[28] |
B. Jia, L. Erbe and A. Peterson,
Monotonicity and convexity for nabla fractional $q$-differences, Dynam. Systems Appl., 25 (2016), 47-60.
|
[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. |
[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. |
[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. |
[32] |
A. Kilbas, H. M. Srivastava and J. J. Trujillo, Theory and Applications of Fractional Differential Equations, North-Holland, New York, 2006. |
[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. |
[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. |
[35] |
R. Wong and R. Beals, Special Functions: A Graduate Text, Cambridge University Press, New York, 2010.
doi: 10.1017/CBO9780511762543.![]() ![]() |


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