New classifications of monotonicity investigation for discrete operators with Mittag-Leffler kernel

Pshtiwan Othman Mohammed; Christopher S. Goodrich; Aram Bahroz Brzo; Dumitru Baleanu; Yasser S. Hamed; Pshtiwan Othman Mohammed; Christopher S. Goodrich; Aram Bahroz Brzo; Dumitru Baleanu; Yasser S. Hamed

doi:10.3934/mbe.2022186

Mathematical Biosciences and Engineering

2022, Volume 19, Issue 4: 4062-4074. doi: 10.3934/mbe.2022186

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New classifications of monotonicity investigation for discrete operators with Mittag-Leffler kernel

1.
Department of Mathematics, College of Education, University of Sulaimani, Sulaimani 46001, Kurdistan Region, Iraq
2.
School of Mathematics and Statistics, UNSW Sydney, Sydney, NSW 2052, Australia
3.
Department of Physics, College of Education, University of Sulaimani, Sulaimani 46001, Kurdistan Region, Iraq
4.
Department of Mathematics, Cankaya University, Balgat 06530, Ankara, Turkey
5.
Institute of Space Sciences, Magurele-Bucharest R76900, Romania
6.
Department of Mathematics, King Abdul Aziz University, Jeddah 21577, Saudi Arabia
7.
Department of Mathematics and Statistics, College of Science, Taif University, P. O. Box 11099, Taif 21944, Saudi Arabia

Academic Editor:Maria Luz Gandarias

Received: 07 December 2021 Revised: 26 January 2022 Accepted: 10 February 2022 Published: 15 February 2022

This paper deals with studying monotonicity analysis for discrete fractional operators with Mittag-Leffler in kernel. The $ \nu- $monotonicity definitions, namely $ \nu- $(strictly) increasing and $ \nu- $(strictly) decreasing, are presented as well. By examining the basic properties of the proposed discrete fractional operators together with $ \nu- $monotonicity definitions, we find that the investigated discrete fractional operators will be $ \nu^2- $(strictly) increasing or $ \nu^2- $(strictly) decreasing in certain domains of the time scale $ \mathbb{N}_a: = \{a, a+1, \dots\} $. Finally, the correctness of developed theories is verified by deriving mean value theorem in discrete fractional calculus.
- discrete fractional calculus,
- nabla AB-fractional operators,
- monotonicity investigation
Citation: Pshtiwan Othman Mohammed, Christopher S. Goodrich, Aram Bahroz Brzo, Dumitru Baleanu, Yasser S. Hamed. New classifications of monotonicity investigation for discrete operators with Mittag-Leffler kernel[J]. Mathematical Biosciences and Engineering, 2022, 19(4): 4062-4074. doi: 10.3934/mbe.2022186

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Abstract

This paper deals with studying monotonicity analysis for discrete fractional operators with Mittag-Leffler in kernel. The $ \nu- $monotonicity definitions, namely $ \nu- $(strictly) increasing and $ \nu- $(strictly) decreasing, are presented as well. By examining the basic properties of the proposed discrete fractional operators together with $ \nu- $monotonicity definitions, we find that the investigated discrete fractional operators will be $ \nu^2- $(strictly) increasing or $ \nu^2- $(strictly) decreasing in certain domains of the time scale $ \mathbb{N}_a: = \{a, a+1, \dots\} $. Finally, the correctness of developed theories is verified by deriving mean value theorem in discrete fractional calculus.

References

[1]	C. Goodrich, A. C. Peterson, Discrete Fractional Calculus, Springer, New York, 2015.
[2]	T. Abdeljawad, Different type kernel $h$–fractional differences and their fractional $h$–sums, Chaos Solitons Fractals, 116 (2018), 146–156. https://doi.org/10.1016/j.chaos.2018.09.022 doi: 10.1016/j.chaos.2018.09.022
[3]	T. Abdeljawad, F. Jarad, A. Atangana, P. O. Mohammed, On a new type of fractional difference operators on h-step isolated time scales, J. Fractional Calculus Nonlinear Syst., 1 (2021), 46–74. https://doi.org/10.48185/jfcns.v1i1.148 doi: 10.48185/jfcns.v1i1.148
[4]	P. O. Mohammed, T. Abdeljawad, Discrete generalized fractional operators defined using h-discrete Mittag-Leffler kernels and applications to AB fractional difference systems, Math. Meth. Appl. Sci., 2020 (2020), 1–26, https://doi.org/10.1002/mma.7083 doi: 10.1002/mma.7083
[5]	T. Abdeljawad, D. Baleanu, Discrete fractional differences with nonsingular discrete Mittag-Leffler kernels, Adv. Differ. Equation, 2016 (2016), 232. https://doi.org/10.1186/s13662-016-0949-5 doi: 10.1186/s13662-016-0949-5
[6]	T. Abdeljawad, F. Madjidi, Lyapunov-type inequalities for fractional difference operators with discrete Mittag-Leffler kernel of order $2 < \alpha < 5/2$, Eur. Phys. J. Spec. Top., 226 (2017), 3355–3368. https://doi.org/10.1140/epjst/e2018-00004-2 doi: 10.1140/epjst/e2018-00004-2
[7]	T. Abdeljawad, Q. M. Al-Mdallal, Q. M. Hajji, Arbitrary order fractional difference operators with discrete exponential kernels and applications, Discrete Dyn. Nat. Soc., 2017 (2017). https://doi.org/10.1155/2017/4149320 doi: 10.1155/2017/4149320
[8]	M. Yavuz, Characterizations of two different fractional operators without singular kernel, Math. Model. Nat. Phenom., 14 (2019), 302. https://doi.org/10.1051/mmnp/2018070 doi: 10.1051/mmnp/2018070
[9]	A. Keten, M. Yavuz, D. Baleanu, Nonlocal cauchy problem via a fractional operator involving power kernel in Banach spaces, Fractal Fractional, 3 (2019), 27. https://doi.org/10.3390/fractalfract3020027 doi: 10.3390/fractalfract3020027
[10]	F. M. Atici, M. Atici, M. Belcher, D. Marshall, A new approach for modeling with discrete fractional equations, Fundam. Inf., 151 (2017), 313–324. https://doi.org/10.3233/FI-2017-1494 doi: 10.3233/FI-2017-1494
[11]	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
[12]	F. M. Atici, S. Sengul, 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
[13]	Z. Wang, B. Shiri, D. Baleanu, Discrete fractional watermark technique, Front. Inf. Technol. Electron. Eng., 21 (2020), 880–883. https://doi.org/10.1631/FITEE.2000133 doi: 10.1631/FITEE.2000133
[14]	G. Wu, D. Baleanu, Y. Bai, Discrete fractional masks and their applications to image enhancement, Handb. Fractional Calculus Appl., 8 (2019), 261–270. https://doi.org/10.1515/9783110571929 doi: 10.1515/9783110571929
[15]	T. Abdeljawad, Q. M. Al-Mdallal, Discrete Mittag-Leffler kernel type fractional difference initial value problems and Gronwall's inequality, J. Comput. Appl. Math., 339 (2018), 218–230. https://doi.org/10.1016/j.cam.2017.10.021 doi: 10.1016/j.cam.2017.10.021
[16]	A. Khan, H. M. Alshehri, T. Abdeljawad, Q. M. Al-Mdallal, H. Khan, Stability analysis of fractional nabla difference COVID-19 model, Results Phys., 22 (2021), 103888. https://doi.org/10.1016/j.rinp.2021.103888 doi: 10.1016/j.rinp.2021.103888
[17]	A. Shaikh, K. S. Nisar, V. Jadhav, S. K.Elagan, M. Zakarya, Dynamical behaviour of HIV/AIDS model using fractional derivative with Mittag-Leffler kernel, Alexandria Eng. J., 61 (2022), 2601–2610. https://doi.org/10.1016/j.aej.2021.08.030 doi: 10.1016/j.aej.2021.08.030
[18]	C. Ravichandran, K. Logeswari, S. K. Panda, K. S. Nisar, On new approach of fractional derivative by Mittag-Leffler kernel to neutral integro-differential systems with impulsive conditions, Chaos Solitons Fractals, 139 (2020), 110012. https://doi.org/10.1016/j.chaos.2020.110012 doi: 10.1016/j.chaos.2020.110012
[19]	H. Dong, Y.Gao, Existence and uniqueness of bounded stable solutions to the Peierls-Nabarro model for curved dislocations, Calculus Variations Partial Differ. Equation, 60 (2021), 62. https://doi.org/10.1007/s00526-021-01939-1 doi: 10.1007/s00526-021-01939-1
[20]	Y. Gao, J. G. Liu, Z. Liu, Existence and rigidity of the vectorial Peierls-Nabarro model for dislocations in high dimensions, Nonlinearity, 34 (2021), 7778.
[21]	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 Fractional, 6 (2022), 55. https://doi.org/10.3390/fractalfract6020055 doi: 10.3390/fractalfract6020055
[22]	R. Dahal, C. S. Goodrich, A monotonicity result for discrete fractional difference operators, Arch. Math. (Basel), 102 (2014), 293–299. https://doi.org/10.1007/s00013-014-0620-x doi: 10.1007/s00013-014-0620-x
[23]	T. Abdeljawad, D. Baleanu, Monotonicity analysis of a nabla discrete fractional operator with discrete Mittag-Leffler kernel, Chaos Solitons Fractals, 116 (2017), 1–5. https://doi.org/10.1016/j.chaos.2017.04.006 doi: 10.1016/j.chaos.2017.04.006
[24]	I. Suwan, T. Abdeljawad, F. Jarad, Monotonicity analysis for nabla $h$-discrete fractional Atangana-Baleanu differences, Chaos Solitons Fractals, 117 (2018), 50–59. https://doi.org/10.1016/j.chaos.2018.10.010 doi: 10.1016/j.chaos.2018.10.010
[25]	T. Abdeljawad, B. Abdallaa, Monotonicity results for delta and nabla Caputo and Riemann fractional differences via dual identities, preprint, arXiv: 1601.05510.
[26]	C. S. Goodrich, J. M. Jonnalagadda, An analysis of polynomial sequences and their application to discrete fractional operators, J. Differ. Equations Appl., 27 (2021), 1081–1102. https://doi.org/10.1080/10236198.2021.1965132 doi: 10.1080/10236198.2021.1965132
[27]	P. O. Mohammed, T. Abdeljawad, F. K. Hamasalh, On Riemann-Liouville and Caputo fractional forward difference monotonicity analysis, Mathematics, 9(2021), 1303. https://doi.org/10.3390/math9111303 doi: 10.3390/math9111303
[28]	P. O. Mohammed, F. K. Hamasalh, T. Abdeljawad, Difference monotonicity analysis on discrete fractional operators with discrete generalized Mittag-Leffler kernels, Adv. Differ. Equation, 2021, 2021, 213. https://doi.org/10.1186/s13662-021-03372-2 doi: 10.1186/s13662-021-03372-2
[29]	J. Bravo, C. Lizama, S. Rueda, Second and third order forward difference operator: what is in between?, Rev. R. Acad. Cienc. Exactas Fís. Nat. Ser. A Mat. RACSAM, 115 (2021), 1–20. https://doi.org/10.1007/s13398-021-01015-5 doi: 10.1007/s13398-021-01015-5
[30]	C. S. 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
[31]	C. S. 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
[32]	C. S. Goodrich, B. Lyons, Positivity and monotonicity results for triple sequential fractional differences via convolution, Analysis, 40 (2020), 89–103. https://doi.org/10.1515/anly-2019-0050 doi: 10.1515/anly-2019-0050
[33]	C. S. Goodrich, B. Lyons, M. T. Velcsov, Analytical and numerical monotonicity results for discrete fractional sequential 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
[34]	C. S. Goodrich, J. M. Jonnalagadda, B. Lyons, Convexity, monotonicity, and positivity results for sequential fractional nabla difference operators with discrete exponential kernels, Math. Meth. Appl. Sci., 44 (2021), https://doi.org/10.1002/mma.7247 doi: 10.1002/mma.7247
[35]	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
[36]	F. M. Atici, M. Uyanik, Analysis of discrete fractional operators, Appl. Anal. Discrete Math., 9 2015,139–149. https://doi.org/10.2298/AADM150218007A

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