Stability on a boundary problem with RL-Fractional derivative in the sense of Atangana-Baleanu of variable-order

Yihui Xu; Benoumran Telli; Mohammed Said Souid; Sina Etemad; Jiafa Xu; Shahram Rezapour; Yihui Xu; Benoumran Telli; Mohammed Said Souid; Sina Etemad; Jiafa Xu; Shahram Rezapour

doi:10.3934/era.2024007

Electronic Research Archive

2024, Volume 32, Issue 1: 134-159. doi: 10.3934/era.2024007

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Stability on a boundary problem with RL-Fractional derivative in the sense of Atangana-Baleanu of variable-order

1.
School of Sciences and Arts, Suqian University, Suqian 223800, China
2.
Department of Mathematics, University of Tiaret, Tiaret 14035, Algeria
3.
Department of Economic Sciences, University of Tiaret, Tiaret 14035, Algeria
4.
Department of Mathematics, Azarbaijan Shahid Madani University, Tabriz, Iran
5.
School of Mathematical Sciences, Chongqing Normal University, Chongqing 401331, China
6.
Department of Medical Research, China Medical University Hospital, China Medical University, Taichung, Taiwan

Received: 22 August 2023 Revised: 05 December 2023 Accepted: 06 December 2023 Published: 18 December 2023

In this paper, we study the existence and stability of solutions in connection to a non-local multiterm boundary value problem (BVP) with differential equations equipped with the Riemann-Liouville (RL) fractional derivative in the sense of Atangana-Baleanu of variable-order. The results about the existence property are investigated and proved via Krasnoselskii's fixed point theorem. Note that all theorems in the present research are studied based on piece-wise constant functions defined on generalized intervals. We shall convert our main BVP with the RL-fractional derivative of the Atangana-Baleanu type of variable-order to an equivalent BVP of constant order of the RL-Atangana-Baleanu derivative. In the next step, we examine the Ulam-Hyers stability for the supposed variable-order RL-Atangana-Baleanu BVP. Finally, we provide some examples to validate that our results are applicable.
- Atangana-Baleanu derivative,
- variable-order,
- differential equation,
- fixed point,
- Ulam-Hyers stability
Citation: Yihui Xu, Benoumran Telli, Mohammed Said Souid, Sina Etemad, Jiafa Xu, Shahram Rezapour. Stability on a boundary problem with RL-Fractional derivative in the sense of Atangana-Baleanu of variable-order[J]. Electronic Research Archive, 2024, 32(1): 134-159. doi: 10.3934/era.2024007

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Abstract

In this paper, we study the existence and stability of solutions in connection to a non-local multiterm boundary value problem (BVP) with differential equations equipped with the Riemann-Liouville (RL) fractional derivative in the sense of Atangana-Baleanu of variable-order. The results about the existence property are investigated and proved via Krasnoselskii's fixed point theorem. Note that all theorems in the present research are studied based on piece-wise constant functions defined on generalized intervals. We shall convert our main BVP with the RL-fractional derivative of the Atangana-Baleanu type of variable-order to an equivalent BVP of constant order of the RL-Atangana-Baleanu derivative. In the next step, we examine the Ulam-Hyers stability for the supposed variable-order RL-Atangana-Baleanu BVP. Finally, we provide some examples to validate that our results are applicable.

References

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