Relationships between the discrete Riemann-Liouville and Liouville-Caputo fractional differences and their associated convexity results

Juan L. G. Guirao; Pshtiwan Othman Mohammed; Hari Mohan Srivastava; Dumitru Baleanu; Marwan S. Abualrub; Juan L. G. Guirao; Pshtiwan Othman Mohammed; Hari Mohan Srivastava; Dumitru Baleanu; Marwan S. Abualrub

doi:10.3934/math.2022997

AIMS Mathematics

2022, Volume 7, Issue 10: 18127-18141. doi: 10.3934/math.2022997

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Relationships between the discrete Riemann-Liouville and Liouville-Caputo fractional differences and their associated convexity results

1.
Departamento de Matemáca Aplicada y Estadística, Universidad Politécnica de Cartagena, ES-30202 Cartagena, Spain
2.
Nonlinear Analysis and Applied Mathematics (NAAM) - Research Group, Department of Mathematics, Faculty of Science, King Abdulaziz University, P.O. Box 80203, Jeddah 21589, Saudi Arabia
3.
Department of Mathematics, College of Education, University of Sulaimani, Sulaimani 46001, Iraq
4.
Department of Mathematics and Statistics, University of Victoria, Victoria, British Columbia V8W 3R4, Canada
5.
Department of Medical Research, China Medical University Hospital, China Medical University, Taichung 40402, Taiwan
6.
Department of Mathematics and Informatics, Azerbaijan University, 71 Jeyhun Hajibeyli Street, Baku AZ1007, Azerbaijan
7.
Section of Mathematics, International Telematic University Uninettuno, I-00186 Rome, Italy
8.
Department of Mathematics, Cankaya University, Balgat 06530, Ankara, Turkey
9.
Institute of Space Sciences, R76900 Magurele-Bucharest, Romania
10.
Department of Natural Sciences, School of Arts and Sciences, Lebanese American University, Beirut 11022801, Lebanon
11.
Mathematics Department (Prep. Program), Faculty of Science, Khalifa University, P.O. Box 127788, Abu Dhabi UAE

Received: 30 June 2022 Revised: 01 August 2022 Accepted: 01 August 2022 Published: 09 August 2022
MSC : 26A48, 26A51, 33B10, 39A12, 39B62

In this study, we have presented two new alternative definitions corresponding to the basic definitions of the discrete delta and nabla fractional difference operators. These definitions and concepts help us in establishing a relationship between Riemann-Liouville and Liouville-Caputo fractional differences of higher orders for both delta and nabla operators. We then propose and analyse some convexity results for the delta and nabla fractional differences of the Riemann-Liouville type. We also derive similar results for the delta and nabla fractional differences of Liouville-Caputo type by using the proposed relationships. Finally, we have presented two examples to confirm the main theorems.
- Riemann-Liouville fractional difference,
- Liouville-Caputo fractional difference,
- convexity analysis
Citation: Juan L. G. Guirao, Pshtiwan Othman Mohammed, Hari Mohan Srivastava, Dumitru Baleanu, Marwan S. Abualrub. Relationships between the discrete Riemann-Liouville and Liouville-Caputo fractional differences and their associated convexity results[J]. AIMS Mathematics, 2022, 7(10): 18127-18141. doi: 10.3934/math.2022997

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Abstract

In this study, we have presented two new alternative definitions corresponding to the basic definitions of the discrete delta and nabla fractional difference operators. These definitions and concepts help us in establishing a relationship between Riemann-Liouville and Liouville-Caputo fractional differences of higher orders for both delta and nabla operators. We then propose and analyse some convexity results for the delta and nabla fractional differences of the Riemann-Liouville type. We also derive similar results for the delta and nabla fractional differences of Liouville-Caputo type by using the proposed relationships. Finally, we have presented two examples to confirm the main theorems.

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