Positivity analysis for mixed order sequential fractional difference operators

Pshtiwan Othman Mohammed; Dumitru Baleanu; Thabet Abdeljawad; Soubhagya Kumar Sahoo; Khadijah M. Abualnaja; Pshtiwan Othman Mohammed; Dumitru Baleanu; Thabet Abdeljawad; Soubhagya Kumar Sahoo; Khadijah M. Abualnaja

doi:10.3934/math.2023140

AIMS Mathematics

2023, Volume 8, Issue 2: 2673-2685. doi: 10.3934/math.2023140

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Positivity analysis for mixed order sequential fractional difference operators

1.
Department of Mathematics, College of Education, University of Sulaimani, Sulaimani 46001, Iraq
2.
Department of Mathematics, Cankaya University, Balgat 06530, Ankara, Turkey
3.
Institute of Space Sciences, Magurele-Bucharest R76900, 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, P.O. Box 66833, Riyadh 11586, Saudi Arabia
6.
Department of Medical Research, China Medical University, Taichung 40402, Taiwan
7.
Department of Mathematics, Institute of Technical Education and Research, Siksha 'O' Anusandhan University, Bhubaneswar 751030, Odisha, India
8.
Department of Mathematics and Statistics, College of Science, Taif University, P.O. Box 11099, Taif 21944, Saudi Arabia

Received: 30 August 2022 Revised: 03 October 2022 Accepted: 13 October 2022 Published: 08 November 2022
MSC : 26A48, 26A51, 33B10, 39A12, 39B62

We consider the positivity of the discrete sequential fractional operators $ \left(^{\rm RL}_{a_{0}+1}\nabla^{\nu_{1}}\, ^{\rm RL}_{a_{0}}\nabla^{\nu_{2}}{f}\right)(\tau) $ defined on the set $ \mathscr{D}_{1} $ (see (1.1) and Figure 1) and $ \left(^{\rm RL}_{a_{0}+2}\nabla^{\nu_{1}}\, ^{\rm RL}_{a_{0}}\nabla^{\nu_{2}}{f}\right)(\tau) $ of mixed order defined on the set $ \mathscr{D}_{2} $ (see (1.2) and Figure 2) for $ \tau\in\mathbb{N}_{a_{0}} $. By analysing the first sequential operator, we reach that $ \bigl(\nabla {f}\bigr)(\tau)\geqq 0, $ for each $ \tau\in{\mathbb{N}}_{a_{0}+1} $. Besides, we obtain $ \bigl(\nabla {f}\bigr)(3)\geqq 0 $ by analysing the second sequential operator. Furthermore, some conditions to obtain the proposed monotonicity results are summarized. Finally, two practical applications are provided to illustrate the efficiency of the main theorems.
- discrete delta Riemann-Liouville fractional difference,
- negative lower bound,
- convexity analysis,
- analytical and numerical results
Citation: Pshtiwan Othman Mohammed, Dumitru Baleanu, Thabet Abdeljawad, Soubhagya Kumar Sahoo, Khadijah M. Abualnaja. Positivity analysis for mixed order sequential fractional difference operators[J]. AIMS Mathematics, 2023, 8(2): 2673-2685. doi: 10.3934/math.2023140

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Abstract

We consider the positivity of the discrete sequential fractional operators $ \left(^{\rm RL}_{a_{0}+1}\nabla^{\nu_{1}}\, ^{\rm RL}_{a_{0}}\nabla^{\nu_{2}}{f}\right)(\tau) $ defined on the set $ \mathscr{D}_{1} $ (see (1.1) and Figure 1) and $ \left(^{\rm RL}_{a_{0}+2}\nabla^{\nu_{1}}\, ^{\rm RL}_{a_{0}}\nabla^{\nu_{2}}{f}\right)(\tau) $ of mixed order defined on the set $ \mathscr{D}_{2} $ (see (1.2) and Figure 2) for $ \tau\in\mathbb{N}_{a_{0}} $. By analysing the first sequential operator, we reach that $ \bigl(\nabla {f}\bigr)(\tau)\geqq 0, $ for each $ \tau\in{\mathbb{N}}_{a_{0}+1} $. Besides, we obtain $ \bigl(\nabla {f}\bigr)(3)\geqq 0 $ by analysing the second sequential operator. Furthermore, some conditions to obtain the proposed monotonicity results are summarized. Finally, two practical applications are provided to illustrate the efficiency of the main theorems.

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