Monotonicity and extremality analysis of difference operators in Riemann-Liouville family

Pshtiwan Othman Mohammed; Dumitru Baleanu; Thabet Abdeljawad; Eman Al-Sarairah; Y. S. Hamed; Pshtiwan Othman Mohammed; Dumitru Baleanu; Thabet Abdeljawad; Eman Al-Sarairah; Y. S. Hamed

doi:10.3934/math.2023266

AIMS Mathematics

2023, Volume 8, Issue 3: 5303-5317. doi: 10.3934/math.2023266

Previous Article Next Article

Research article Special Issues

Monotonicity and extremality analysis of difference operators in Riemann-Liouville family

1.
Department of Mathematics, College of Education, University of Sulaimani, Sulaimani 46001, Iraq
2.
Department of Mathematics, Cankaya University, 06530 Balgat, Ankara, Turkey
3.
Institute of Space Sciences, R76900 Magurele-Bucharest, Romania
4.
Department of Natural Sciences, School of Arts and Sciences, Lebanese American University, Beirut 11022801, Lebanon
5.
Department of Mathematics and Sciences, Prince Sultan University, Riyadh 11586, Saudi Arabia
6.
Department of Medical Research, China Medical University, Taichung 40402, Taiwan
7.
Department of Mathematics, Khalifa University, P.O. Box 127788, Abu Dhabi, United Arab Emirates
8.
Department of Mathematics, Al-Hussein Bin Talal University, P.O. Box 33011, Ma'an, Jordan
9.
Department of Mathematics and Statistics, College of Science, Taif University, P.O. Box 11099, Taif 21944, Saudi Arabia

Received: 27 October 2022 Revised: 08 December 2022 Accepted: 08 December 2022 Published: 14 December 2022
MSC : 39A12, 39B62, 33B10, 26A48, 26A51

In this paper, we will discuss the monotone decreasing and increasing of a discrete nonpositive and nonnegative function defined on $ \mathbb{N}_{r_{0}+1} $, respectively, which come from analysing the discrete Riemann-Liouville differences together with two necessary conditions (see Lemmas 2.1 and 2.3). Then, the relative minimum and relative maximum will be obtained in view of these results combined with another condition (see Theorems 2.1 and 2.2). We will modify and reform the main two lemmas by replacing the main condition with a new simpler and stronger condition. For these new lemmas, we will establish similar results related to the relative minimum and relative maximum again. Finally, some examples, figures and tables are reported to demonstrate the applicability of the main lemmas. Furthermore, we will clarify that the first condition in the main first two lemmas is solely not sufficient for the function to be monotone decreasing or increasing.
- Riemann-Liouville discrete operators,
- monotonicity analysis,
- extremality analysis
Citation: Pshtiwan Othman Mohammed, Dumitru Baleanu, Thabet Abdeljawad, Eman Al-Sarairah, Y. S. Hamed. Monotonicity and extremality analysis of difference operators in Riemann-Liouville family[J]. AIMS Mathematics, 2023, 8(3): 5303-5317. doi: 10.3934/math.2023266

Related Papers:

Abstract

In this paper, we will discuss the monotone decreasing and increasing of a discrete nonpositive and nonnegative function defined on $ \mathbb{N}_{r_{0}+1} $, respectively, which come from analysing the discrete Riemann-Liouville differences together with two necessary conditions (see Lemmas 2.1 and 2.3). Then, the relative minimum and relative maximum will be obtained in view of these results combined with another condition (see Theorems 2.1 and 2.2). We will modify and reform the main two lemmas by replacing the main condition with a new simpler and stronger condition. For these new lemmas, we will establish similar results related to the relative minimum and relative maximum again. Finally, some examples, figures and tables are reported to demonstrate the applicability of the main lemmas. Furthermore, we will clarify that the first condition in the main first two lemmas is solely not sufficient for the function to be monotone decreasing or increasing.

References

[1]	P. O. Mohammed, O. Almutairi, R. P. Agarwal, Y. S. Hamed, On convexity, monotonicity and positivity analysis for discrete fractional operators defined using exponential kernels, Fractal Fract., 6 (2022), 55. https://doi.org/10.3390/fractalfract6020055 doi: 10.3390/fractalfract6020055
[2]	F. M. Atici, P. W. Eloe, A transform method in discrete fractional calculus, Int. J. Differ. Equ., 2 (2007), 165–176.
[3]	F. M. Atici, P. W. Eloe, Discrete fractional calculus with the nabla operator, Electron. J. Qual. Theory Differ. Equ., 2009, 1–12.
[4]	F. M. Atici, P. W. Eloe, Initial value problems in discrete fractional calculus, Proc. Amer. Math. Soc., 137 (2009), 981–989. https://doi.org/10.1090/S0002-9939-08-09626-3 doi: 10.1090/S0002-9939-08-09626-3
[5]	T. Abdeljawad, Q. M. Al-Mdallal, M. A. Hajji, Arbitrary order fractional difference operators with discrete exponential kernels and applications, Discrete Dyn. Nat. Soc., 2017 (2017), 1–8. https://doi.org/10.1155/2017/4149320 doi: 10.1155/2017/4149320
[6]	T. Abdeljawad, F. M. Atici, On the definitions of nabla fractional operators, Abstr. Appl. Anal., 2012 (2012), 1–13. https://doi.org/10.1155/2012/406757 doi: 10.1155/2012/406757
[7]	T. Abdeljawad, D. Baleanu, Monotonicity analysis of a nabla discrete fractional operator with discrete Mittag-Leffler kernel, Chaos Solitons Fract., 102 (2017), 106–110. https://doi.org/10.1016/j.chaos.2017.04.006 doi: 10.1016/j.chaos.2017.04.006
[8]	P. O. Mohammed, H. M. Srivastava, D. Baleanu, K. M. Abualnaja, Modified fractional difference operators defined using Mittag-Leffler kernels, Symmetry, 14 (2022), 1519. https://doi.org/10.3390/sym14081519 doi: 10.3390/sym14081519
[9]	C. R. Chen, M. Bohner, B. G. Jia, Ulam-hyers stability of Caputo fractional difference equations, Math. Methods Appl. Sci., 42 (2019), 7461–7470. https://doi.org/10.1002/mma.5869 doi: 10.1002/mma.5869
[10]	R. A. C. Ferreira, A discrete fractional Gronwall inequality, Proc. Amer. Math. Soc., 140 (2012), 1605–1612. https://doi.org/10.1090/S0002-9939-2012-11533-3 doi: 10.1090/S0002-9939-2012-11533-3
[11]	M. Holm, Sum and difference compositions and applications in discrete fractional calculus, Cubo, 13 (2011), 153–184. http://dx.doi.org/10.4067/S0719-06462011000300009 doi: 10.4067/S0719-06462011000300009
[12]	G. C. Wu, D. Baleanu, Discrete chaos in fractional delayed logistic maps, Nonlinear Dyn., 80 (2015), 1697–1703. https://doi.org/10.1007/s11071-014-1250-3 doi: 10.1007/s11071-014-1250-3
[13]	C. Goodrich, A. C. Peterson, Discrete fractional calculus, Cham: Springer, 2015. https://doi.org/10.1007/978-3-319-25562-0
[14]	R. Dahal, C. S. Goodrich, A monotonicity result for discrete fractional difference operators, Arch. Math., 102 (2014), 293–299. https://doi.org/10.1007/s00013-014-0620-x doi: 10.1007/s00013-014-0620-x
[15]	F. M. Atici, M. Uyanik, Analysis of discrete fractional operators, Appl. Anal. Discrete Math., 9 (2015), 139–149. https://doi.org/10.2298/AADM150218007A doi: 10.2298/AADM150218007A
[16]	B. G. Jia, L. Erbe, A. Peterson, Some relations between the Caputo fractional difference operators and integer-order differences, Electron. J. Differ. Equ., 2015 (2015), 1–7.
[17]	J. Bravo, C. Lizama, S. Rueda, Second and third order forward difference operator: What is in between? RACSAM, 115 (2021), 1–20. https://doi.org/10.1007/s13398-021-01015-5 doi: 10.1007/s13398-021-01015-5
[18]	J. Bravo, C. Lizama, S. Rueda, Qualitative properties of nonlocal discrete operators, Math. Methods Appl. Sci., 45 (2022), 6346–6377. https://doi.org/10.1002/mma.8174 doi: 10.1002/mma.8174
[19]	R. Dahal, C. S. Goodrich, Mixed order monotonicity results for sequential fractional nabla differences, J. Differ. Equ. Appl., 25 (2019), 837–854. https://doi.org/10.1080/10236198.2018.1561883 doi: 10.1080/10236198.2018.1561883
[20]	K. Ahrendt, L. Castle, M. Holm, K. Yochman, Laplace transforms for the nabla-difference operator and a fractional variation of parameters formula, Commun. Appl. Anal., 16 (2012), 317–347.
[21]	L. Erbe, C. S. Goodrich, B. G. Jia, A. Peterson, Monotonicity results for delta fractional differences revisited, Math. Slovaca, 67 (2017), 895–906. https://doi.org/10.1515/ms-2017-0018 doi: 10.1515/ms-2017-0018
[22]	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. https://doi.org/10.7153/mia-19-57 doi: 10.7153/mia-19-57
[23]	C. S. Goodrich, A sharp convexity result for sequential fractional delta differences, J. Differ. Equ. Appl., 23 (2017), 1986–2003. https://doi.org/10.1080/10236198.2017.1380635 doi: 10.1080/10236198.2017.1380635
[24]	C. S. Goodrich, J. M. Jonnalagadda, B. Lyons, Convexity, monotonicity, and positivity results for sequential fractional nabla difference operators with discrete exponential kernels, Math. Methods Appl. Sci., 44 (2021), 7099–7120. https://doi.org/10.1002/mma.7247 doi: 10.1002/mma.7247
[25]	C. Goodrich, C. Lizama, A transference principle for nonlocal operators using a convolutional approach: fractional monotonicity and convexity, Israel J. Math., 236 (2020), 533–589. https://doi.org/10.1007/s11856-020-1991-2 doi: 10.1007/s11856-020-1991-2
[26]	C. Goodrich, C. Lizama, Positivity, monotonicity, and convexity for convolution operators, Discrete Contin. Dyn. Syst., 40 (2020), 4961–4983. https://doi.org/10.3934/dcds.2020207 doi: 10.3934/dcds.2020207
[27]	C. S. Goodrich, B. Lyons, M. T. Velcsov, Analytical and numerical monotonicity results for discrete fractional differences with negative lower bound, Commun. Pure Appl. Anal., 20 (2021), 339–358. https://doi.org/10.3934/cpaa.2020269 doi: 10.3934/cpaa.2020269
[28]	C. S. Goodrich, M. Muellner, An analysis of the sharpness of monotonicity results via homotopy for sequential fractional operators, Appl. Math. Lett., 98 (2019), 446–452. https://doi.org/10.1016/j.aml.2019.07.003 doi: 10.1016/j.aml.2019.07.003
[29]	P. O. Mohammed, T. Abdeljawad, F. K. Hamasalh, On discrete delta Caputo-Fabrizio fractional operators and monotonicity analysis, Fractal Fract., 5 (2021), 116. https://doi.org/10.3390/fractalfract5030116 doi: 10.3390/fractalfract5030116
[30]	P. O. Mohammed, H. M. Srivastava, D. Baleanu, E. E. Elattar, Y. S. Hamed, Positivity analysis for the discrete delta fractional differences of the Riemann-Liouville and Liouville-Caputo types, Electron. Res. Arch., 30 (2022), 3058–3070. https://doi.org/10.3934/era.2022155 doi: 10.3934/era.2022155
[31]	X. Liu, F. F. Du, D. Anderson, B. G. Jia, Monotonicity results for nabla fractional $h$-difference operators, Math. Methods Appl. Sci., 44 (2020), 1207–1218. https://doi.org/10.1002/mma.6823 doi: 10.1002/mma.6823
[32]	J. L. G. Guirao, P. O. Mohammed, H. M. Srivastava, D. Baleanu, M. S. Abualrub, Relationships between the discrete Riemann-Liouville and Liouville-Caputo fractional differences and their associated convexity results, AIMS Math., 7 (2022), 18127–18141. https://doi.org/10.3934/math.2022997 doi: 10.3934/math.2022997
[33]	F. M. Atici, S. Şengül, Modeling with fractional difference equations, J. Math. Anal. Appl., 369 (2010), 1–9. https://doi.org/10.1016/j.jmaa.2010.02.009 doi: 10.1016/j.jmaa.2010.02.009
[34]	F. M. Atici, M. Atici, N. Nguyen, T. Zhoroev, G. Koch, A study on discrete and discrete fractional pharmacokinetics-pharmacodynamics models for tumor growth and anti-cancer effects, Comput. Math. Biophys., 7 (2019), 10–24. https://doi.org/10.1515/cmb-2019-0002 doi: 10.1515/cmb-2019-0002
[35]	F. M. Atici, M. Atici, M. Belcher, D. Marshall, A new approach for modeling with discrete fractional equations, Fund. Inform., 151 (2017), 313–324. https://doi.org/10.3233/FI-2017-1494 doi: 10.3233/FI-2017-1494
[36]	T. Abdeljawad, Different type kernel $h$-fractional differences and their fractional $h$-sums, Chaos Solitons Fract., 116 (2018), 146–156. https://doi.org/10.1016/j.chaos.2018.09.022 doi: 10.1016/j.chaos.2018.09.022
[37]	T. Abdeljawad, On delta and nabla Caputo fractional differences and dual identities, Discrete Dyn. Nat. Soc., 2013 (2013), 1–12. https://doi.org/10.1155/2013/406910 doi: 10.1155/2013/406910

Reader Comments

Your name:*

Email:*
© 2023 the Author(s), licensee AIMS Press. This is an open access article distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/4.0)