Fractional variable order differential equations with impulses: A study on the stability and existence properties

Amar Benkerrouche; Sina Etemad; Mohammed Said Souid; Shahram Rezapour; Hijaz Ahmad; Thongchai Botmart; Amar Benkerrouche; Sina Etemad; Mohammed Said Souid; Shahram Rezapour; Hijaz Ahmad; Thongchai Botmart

doi:10.3934/math.2023038

AIMS Mathematics

2023, Volume 8, Issue 1: 775-791. doi: 10.3934/math.2023038

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Research article

Fractional variable order differential equations with impulses: A study on the stability and existence properties

1.
Department of Mathématiques, Ziane Achour University of Djelfa, Algeria
2.
Department of Mathematics, Azarbaijan Shahid Madani University, Tabriz, Iran
3.
Department of Economic Sciences, University of Tiaret, Tiaret, Algeria
4.
Department of Medical Research, China Medical University Hospital, China Medical University, Taichung, Taiwan
5.
Section of Mathematics, International Telematic University Uninettuno, Corso Vittorio Emanuele Ⅱ, 00186 Rome, Italy
6.
Department of Mathematics, Faculty of Science, Khon Kaen University, Khon Kaen, 40002, Thailand

Received: 21 August 2022 Revised: 26 September 2022 Accepted: 06 October 2022 Published: 12 October 2022
MSC : 34A08, 34A37, 34B15

In this paper, for the first time, we study the existence and uniqueness of solutions of a Caputo variable-order initial value problem (IVP) in the impulsive settings. Our existence results are proved by using two fixed point theorems. The Ulam-Hyers stability of solutions is established for the variable order impulsive initial value problem. Finally, we provide an example to show the correctness of the results.
- impulsive condition,
- differential equations,
- fractional variable order,
- Ulam-Hyers stability,
- existence of solutions,
- fixed point theorem
Citation: Amar Benkerrouche, Sina Etemad, Mohammed Said Souid, Shahram Rezapour, Hijaz Ahmad, Thongchai Botmart. Fractional variable order differential equations with impulses: A study on the stability and existence properties[J]. AIMS Mathematics, 2023, 8(1): 775-791. doi: 10.3934/math.2023038

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Abstract

In this paper, for the first time, we study the existence and uniqueness of solutions of a Caputo variable-order initial value problem (IVP) in the impulsive settings. Our existence results are proved by using two fixed point theorems. The Ulam-Hyers stability of solutions is established for the variable order impulsive initial value problem. Finally, we provide an example to show the correctness of the results.

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