Analysis study on multi-order $ \varrho $-Hilfer fractional pantograph implicit differential equation on unbounded domains

Sabri T. M. Thabet; Sa'ud Al-Sa'di; Imed Kedim; Ava Sh. Rafeeq; Shahram Rezapour; Sabri T. M. Thabet; Sa'ud Al-Sa'di; Imed Kedim; Ava Sh. Rafeeq; Shahram Rezapour

doi:10.3934/math.2023938

AIMS Mathematics

2023, Volume 8, Issue 8: 18455-18473. doi: 10.3934/math.2023938

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Analysis study on multi-order $ \varrho $-Hilfer fractional pantograph implicit differential equation on unbounded domains

1.
Department of Mathematics, University of Lahej, Lahej, Yemen
2.
Department of Mathematics, University of Aden, Aden, Yemen
3.
Department of Mathematics, Faculty of Science, The Hashemite University, P.O. BOX 330127, Zarqa 13133, Jordan
4.
Department of Mathematics, College of Science and Humanities in Al-Kharj, Prince Sattam Bin Abdulaziz University, Al-Kharj 11942, Saudi Arabia
5.
Department of Mathematics, Faculty of Science, University of Zakho, Duhok-42001, Iraq
6.
Department of Mathematics, Azarbaijan Shahid Madani University, Tabriz, Iran
7.
Department of Mathematics, Kyung Hee University, 26 Kyungheedae-ro, Dongdaemun-gu, Seoul, Republic of Korea
8.
Department of Medical Research, China Medical University Hospital, China Medical University, Taichung, Taiwan

Received: 14 February 2023 Revised: 10 May 2023 Accepted: 17 May 2023 Published: 31 May 2023
MSC : 34A08, 34B10

In this paper, we investigate a multi-order $ \varrho $-Hilfer fractional pantograph implicit differential equation on unbounded domains $ (a, \infty), a\geq 0 $. The existence and uniqueness of solution are established for a such problem by utilizing the Banach fixed point theorem in an applicable Banach space. In addition, stability of the types Ulam-Hyers ($ \mathcal UH $), Ulam-Hyers-Rassias ($ \mathcal UHR $) and semi-Ulam-Hyers-Rassias (s-$ \mathcal UHR $) are discussed by using nonlinear analysis topics. Finally, a concrete example includes some particular cases is enhanced to illustrate rightful of our results.
- $ \varrho $-Hilfer fractional derivatives,
- multi-order,
- pantograph implicit differential equations,
- Ulam-Hyers-Rassias stability,
- fixed point theorems
Citation: Sabri T. M. Thabet, Sa'ud Al-Sa'di, Imed Kedim, Ava Sh. Rafeeq, Shahram Rezapour. Analysis study on multi-order $ \varrho $-Hilfer fractional pantograph implicit differential equation on unbounded domains[J]. AIMS Mathematics, 2023, 8(8): 18455-18473. doi: 10.3934/math.2023938

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Abstract

In this paper, we investigate a multi-order $ \varrho $-Hilfer fractional pantograph implicit differential equation on unbounded domains $ (a, \infty), a\geq 0 $. The existence and uniqueness of solution are established for a such problem by utilizing the Banach fixed point theorem in an applicable Banach space. In addition, stability of the types Ulam-Hyers ($ \mathcal UH $), Ulam-Hyers-Rassias ($ \mathcal UHR $) and semi-Ulam-Hyers-Rassias (s-$ \mathcal UHR $) are discussed by using nonlinear analysis topics. Finally, a concrete example includes some particular cases is enhanced to illustrate rightful of our results.

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