Analytical study of $ \mathcal{ABC} $-fractional pantograph implicit differential equation with respect to another function

Sabri T. M. Thabet; Miguel Vivas-Cortez; Imed Kedim; Sabri T. M. Thabet; Miguel Vivas-Cortez; Imed Kedim

doi:10.3934/math.20231202

AIMS Mathematics

2023, Volume 8, Issue 10: 23635-23654. doi: 10.3934/math.20231202

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Analytical study of $ \mathcal{ABC} $-fractional pantograph implicit differential equation with respect to another function

1.
Department of Mathematics, Radfan University College, University of Lahej, Lahej, Yemen
2.
Faculty of Exact and Natural Sciences, School of Physical Sciences and Mathematics, Pontifical Catholic University of Ecuador, Av. 12 de octubre 1076 y Roca, Apartado Postal 17-01-2184, Sede Quito, Ecuador
3.
Department of Mathematics, College of Science and Humanities in Al-Kharj, Prince Sattam Bin Abdulaziz University, Al-Kharj 11942, Saudi Arabia

Received: 12 May 2023 Revised: 01 July 2023 Accepted: 10 July 2023 Published: 31 July 2023
MSC : 34A08, 34B15

This article aims to establish sufficient conditions for qualitative properties of the solutions for a new class of a pantograph implicit system in the framework of Atangana-Baleanu-Caputo ($ \mathcal{ABC} $) fractional derivatives with respect to another function under integral boundary conditions. The Schaefer and Banach fixed point theorems (FPTs) are utilized to investigate the existence and uniqueness results for this pantograph implicit system. Moreover, some stability types such as the Ulam-Hyers $ (\mathbb{UH}) $, generalized $ \mathbb{UH} $, Ulam-Hyers-Rassias $ (\mathbb{UHR}) $ and generalized $ \mathbb{UHR} $ are discussed. Finally, interpretation mathematical examples are given in order to guarantee the validity of the main findings. Moreover, the fractional operator used in this study is more generalized and supports our results to be more extensive and covers several new and existing problems in the literature.
Citation: Sabri T. M. Thabet, Miguel Vivas-Cortez, Imed Kedim. Analytical study of $ \mathcal{ABC} $-fractional pantograph implicit differential equation with respect to another function[J]. AIMS Mathematics, 2023, 8(10): 23635-23654. doi: 10.3934/math.20231202

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Abstract

This article aims to establish sufficient conditions for qualitative properties of the solutions for a new class of a pantograph implicit system in the framework of Atangana-Baleanu-Caputo ($ \mathcal{ABC} $) fractional derivatives with respect to another function under integral boundary conditions. The Schaefer and Banach fixed point theorems (FPTs) are utilized to investigate the existence and uniqueness results for this pantograph implicit system. Moreover, some stability types such as the Ulam-Hyers $ (\mathbb{UH}) $, generalized $ \mathbb{UH} $, Ulam-Hyers-Rassias $ (\mathbb{UHR}) $ and generalized $ \mathbb{UHR} $ are discussed. Finally, interpretation mathematical examples are given in order to guarantee the validity of the main findings. Moreover, the fractional operator used in this study is more generalized and supports our results to be more extensive and covers several new and existing problems in the literature.

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