On a class of Langevin equations in the frame of Caputo function-dependent-kernel fractional derivatives with antiperiodic boundary conditions

Abdelatif Boutiara; Mohammed S. Abdo; Manar A. Alqudah; Thabet Abdeljawad; Abdelatif Boutiara; Mohammed S. Abdo; Manar A. Alqudah; Thabet Abdeljawad

doi:10.3934/math.2021327

AIMS Mathematics

2021, Volume 6, Issue 6: 5518-5534. doi: 10.3934/math.2021327

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Research article

On a class of Langevin equations in the frame of Caputo function-dependent-kernel fractional derivatives with antiperiodic boundary conditions

1.
Laboratory of Mathematics and Applied Sciences University of Ghardaia, Algeria
2.
Department of Mathematics, Hodeidah University, Al-Hudaydah, Yemen
3.
Department of Mathematical Sciences, Faculty of Sciences, Princess Nourah bint Abdulrahman University, Riyadh 11671, Saudi Arabia
4.
Department of Mathematics and General Sciences, Prince Sultan University, Riyadh, Saudi Arabia
5.
Department of Medical Research, China Medical University, Taichung 40402, Taiwan
6.
Department of Computer Science and Information Engineering, Asia University, Taichung, Taiwan

Received: 16 December 2020 Accepted: 11 March 2021 Published: 18 March 2021
MSC : 26A33, 34A60

In this manuscript, we consider a class of nonlinear Langevin equations involving two different fractional orders in the frame of Caputo fractional derivative with respect to another monotonic function $ \vartheta $ with antiperiodic boundary conditions. The existence and uniqueness results are proved for the suggested problem. Our approach is relying on properties of $ \vartheta $-Caputo's derivative, and implementation of Krasnoselskii's and Banach's fixed point theorem. At last, we discuss the Ulam-Hyers stability criteria for a nonlinear fractional Langevin equation. Some examples justifying the results gained are provided. The results are novel and provide extensions to some of the findings known in the literature.
- $ \vartheta $-Caputo-type fractional Langevin equation,
- existence and U-H stability,
- fixed point theorem
Citation: Abdelatif Boutiara, Mohammed S. Abdo, Manar A. Alqudah, Thabet Abdeljawad. On a class of Langevin equations in the frame of Caputo function-dependent-kernel fractional derivatives with antiperiodic boundary conditions[J]. AIMS Mathematics, 2021, 6(6): 5518-5534. doi: 10.3934/math.2021327

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Abstract

In this manuscript, we consider a class of nonlinear Langevin equations involving two different fractional orders in the frame of Caputo fractional derivative with respect to another monotonic function $ \vartheta $ with antiperiodic boundary conditions. The existence and uniqueness results are proved for the suggested problem. Our approach is relying on properties of $ \vartheta $-Caputo's derivative, and implementation of Krasnoselskii's and Banach's fixed point theorem. At last, we discuss the Ulam-Hyers stability criteria for a nonlinear fractional Langevin equation. Some examples justifying the results gained are provided. The results are novel and provide extensions to some of the findings known in the literature.

References

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