Positivity analysis for the discrete delta fractional differences of the Riemann-Liouville and Liouville-Caputo types

Pshtiwan Othman Mohammed; Hari Mohan Srivastava; Dumitru Baleanu; Ehab E. Elattar; Y. S. Hamed; Pshtiwan Othman Mohammed; Hari Mohan Srivastava; Dumitru Baleanu; Ehab E. Elattar; Y. S. Hamed

doi:10.3934/era.2022155

Electronic Research Archive

2022, Volume 30, Issue 8: 3058-3070. doi: 10.3934/era.2022155

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Positivity analysis for the discrete delta fractional differences of the Riemann-Liouville and Liouville-Caputo types

1.
Department of Mathematics, College of Education, University of Sulaimani, Sulaimani 46001, Kurdistan Region, Iraq
2.
Department of Mathematics and Statistics, University of Victoria, Victoria, British Columbia V8W 3R4, Canada
3.
Department of Medical Research, China Medical University Hospital, China Medical University, Taichung 40402, Taiwan
4.
Department of Mathematics and Informatics, Azerbaijan University, 71 Jeyhun Hajibeyli Street, AZ1007 Baku, Azerbaijan
5.
Section of Mathematics, International Telematic University Uninettuno, I-00186 Rome, Italy
6.
Department of Mathematics, Cankaya University, Balgat 06530, Ankara, Turkey
7.
Institute of Space Sciences, R76900 Magurele-Bucharest, Romania
8.
Lebanese American University, 11022801 Beirut, Lebanon
9.
Department of Electrical Engineering, College of Engineering, Taif University, P.O. Box 11099, Taif 21944, Saudi Arabia
10.
Department of Mathematics and Statistics, College of Science, Taif University, P. O. Box 11099, Taif 21944, Saudi Arabia

Academic Editor: Zuowei Cai

Received: 08 May 2022 Revised: 28 May 2022 Accepted: 01 June 2022 Published: 08 June 2022

In this article, we investigate some new positivity and negativity results for some families of discrete delta fractional difference operators. A basic result is an identity which will prove to be a useful tool for establishing the main results. Our first main result considers the positivity and negativity of the discrete delta fractional difference operator of the Riemann-Liouville type under two main conditions. Similar results are then obtained for the discrete delta fractional difference operator of the Liouville-Caputo type. Finally, we provide a specific example in which the chosen function becomes nonincreasing on a time set.
Citation: Pshtiwan Othman Mohammed, Hari Mohan Srivastava, Dumitru Baleanu, Ehab E. Elattar, Y. S. Hamed. Positivity analysis for the discrete delta fractional differences of the Riemann-Liouville and Liouville-Caputo types[J]. Electronic Research Archive, 2022, 30(8): 3058-3070. doi: 10.3934/era.2022155

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Abstract

In this article, we investigate some new positivity and negativity results for some families of discrete delta fractional difference operators. A basic result is an identity which will prove to be a useful tool for establishing the main results. Our first main result considers the positivity and negativity of the discrete delta fractional difference operator of the Riemann-Liouville type under two main conditions. Similar results are then obtained for the discrete delta fractional difference operator of the Liouville-Caputo type. Finally, we provide a specific example in which the chosen function becomes nonincreasing on a time set.

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