New results on a coupled system for second-order pantograph equations with $ \mathcal{ABC} $ fractional derivatives

Saeed M. Ali; Mohammed S. Abdo; Bhausaheb Sontakke; Kamal Shah; Thabet Abdeljawad; Saeed M. Ali; Mohammed S. Abdo; Bhausaheb Sontakke; Kamal Shah; Thabet Abdeljawad

doi:10.3934/math.20221071

AIMS Mathematics

2022, Volume 7, Issue 10: 19520-19538. doi: 10.3934/math.20221071

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New results on a coupled system for second-order pantograph equations with $ \mathcal{ABC} $ fractional derivatives

1.
Department of Basic Engineering Sciences, College of Engineering, Imam Abdulrahman Bin Faisal University, P.O. Box 1982, Dammam 34151, Saudi Arabia
2.
Department of Mathematics, Hodeidah University, Al-Hudaydah, Yemen
3.
Department of Mathematics, Pratishthan College, Dr. Babasaheb Ambedkar Marathwada University, Paithan. Dist. Aurangabd (M.S.), India
4.
Department of Mathematics and Sciences, Prince Sultan University, P.O. Box 11586, Riyadh, Saudi Arabia
5.
Department of Mathematics, University of Malakand, Chakdara Dir(L), Khyber Pakhtunkhwa, Pakistan
6.
Department of Medical Research, China Medical University, Taichung 40402, Taiwan

Received: 21 July 2022 Revised: 19 August 2022 Accepted: 19 August 2022 Published: 05 September 2022
MSC : 34A08, 34A12, 47H10

The aim of this paper is to demonstrate a coupled system of second-order fractional pantograph differential equations with coupled four-point boundary conditions. The proposed system involves Atangana-Baleanu-Caputo ($ \mathcal{ABC} $) fractional order derivatives. We prove the solution formula for the corresponding linear version of the given system and then convert the system to a fixed point system. The existence and uniqueness results are obtained by making use of nonlinear alternatives of Leray-Schauder fixed point theorem, and Banach's contraction mapping. In addition, the guarantee of solutions for the system at hand is shown by the stability of Ulam-Hyers. Pertinent examples are provided to illustrate the theoretical results.
- pantograph system,
- $ \mathcal{ABC} $ fractional derivative,
- existence,
- fixed point theorem
Citation: Saeed M. Ali, Mohammed S. Abdo, Bhausaheb Sontakke, Kamal Shah, Thabet Abdeljawad. New results on a coupled system for second-order pantograph equations with $ \mathcal{ABC} $ fractional derivatives[J]. AIMS Mathematics, 2022, 7(10): 19520-19538. doi: 10.3934/math.20221071

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Abstract

The aim of this paper is to demonstrate a coupled system of second-order fractional pantograph differential equations with coupled four-point boundary conditions. The proposed system involves Atangana-Baleanu-Caputo ($ \mathcal{ABC} $) fractional order derivatives. We prove the solution formula for the corresponding linear version of the given system and then convert the system to a fixed point system. The existence and uniqueness results are obtained by making use of nonlinear alternatives of Leray-Schauder fixed point theorem, and Banach's contraction mapping. In addition, the guarantee of solutions for the system at hand is shown by the stability of Ulam-Hyers. Pertinent examples are provided to illustrate the theoretical results.

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