A study on a special case of the Sturm-Liouville equation using the Mittag-Leffler function and a new type of contraction

Zohreh Heydarpour; Maryam Naderi Parizi; Rahimeh Ghorbnian; Mehran Ghaderi; Shahram Rezapour; Amir Mosavi; Zohreh Heydarpour; Maryam Naderi Parizi; Rahimeh Ghorbnian; Mehran Ghaderi; Shahram Rezapour; Amir Mosavi

doi:10.3934/math.20221004

AIMS Mathematics

2022, Volume 7, Issue 10: 18253-18279. doi: 10.3934/math.20221004

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A study on a special case of the Sturm-Liouville equation using the Mittag-Leffler function and a new type of contraction

1.
Department of Mathematics, Payame Noor University (PNU), P.O. Box 19395-4697, Tehran, Iran
2.
Education District 1 Tabriz, Tabriz, Iran
3.
Department of Mathematics, Azarbaijan Shahid Madani University, Tabriz, Iran
4.
Department of Medical Research, China Medical University Hospital, China Medical University, Taichung, Taiwan
5.
Institute of Information Engineering, Automation and Mathematics, Slovak University of Technology in Bratislava, Bratislava, Slovakia
6.
John von Neumann Faculty of Informatics, Obuda University, 1034 Budapest, Hungary

Received: 10 June 2022 Revised: 14 July 2022 Accepted: 14 July 2022 Published: 11 August 2022
MSC : 34A40, 34C10

One of the most famous equations that are widely used in various branches of physics, mathematics, financial markets, etc. is the Langevin equation. In this work, we investigate the existence of the solution for two generalized fractional hybrid Langevin equations under different boundary conditions. For this purpose, the problem of the existence of a solution will become the problem of finding a fixed point for an operator defined in the Banach space. To achieve the result, one of the recent fixed point techniques, namely the $ \alpha $-$ \psi $-contraction technique, will be used. We provide sufficient conditions to use this type of contraction in our main theorems. In the calculations of the auxiliary lemmas that we present, the Mittag-Leffler function plays a fundamental role. The fractional derivative operators used are of the Caputo type. Two examples are provided to demonstrate the validity of the obtained theorems. Also, some figures and a table are presented to illustrate the results.
- $ \alpha $-$ \psi $-Contraction,
- fixed point,
- fractional differential equation,
- the Mittag-Leffler function,
- the Sturm-Liouville equation
Citation: Zohreh Heydarpour, Maryam Naderi Parizi, Rahimeh Ghorbnian, Mehran Ghaderi, Shahram Rezapour, Amir Mosavi. A study on a special case of the Sturm-Liouville equation using the Mittag-Leffler function and a new type of contraction[J]. AIMS Mathematics, 2022, 7(10): 18253-18279. doi: 10.3934/math.20221004

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Abstract

One of the most famous equations that are widely used in various branches of physics, mathematics, financial markets, etc. is the Langevin equation. In this work, we investigate the existence of the solution for two generalized fractional hybrid Langevin equations under different boundary conditions. For this purpose, the problem of the existence of a solution will become the problem of finding a fixed point for an operator defined in the Banach space. To achieve the result, one of the recent fixed point techniques, namely the $ \alpha $-$ \psi $-contraction technique, will be used. We provide sufficient conditions to use this type of contraction in our main theorems. In the calculations of the auxiliary lemmas that we present, the Mittag-Leffler function plays a fundamental role. The fractional derivative operators used are of the Caputo type. Two examples are provided to demonstrate the validity of the obtained theorems. Also, some figures and a table are presented to illustrate the results.

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