On a novel impulsive boundary value pantograph problem under Caputo proportional fractional derivative operator with respect to another function

Songkran Pleumpreedaporn; Chanidaporn Pleumpreedaporn; Weerawat Sudsutad; Jutarat Kongson; Chatthai Thaiprayoon; Jehad Alzabut; Songkran Pleumpreedaporn; Chanidaporn Pleumpreedaporn; Weerawat Sudsutad; Jutarat Kongson; Chatthai Thaiprayoon; Jehad Alzabut

doi:10.3934/math.2022438

AIMS Mathematics

2022, Volume 7, Issue 5: 7817-7846. doi: 10.3934/math.2022438

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Research article

On a novel impulsive boundary value pantograph problem under Caputo proportional fractional derivative operator with respect to another function

1.
Department of Mathematics, Faculty of Science and Technology, Rambhai Barni Rajabhat University, Chanthaburi 22000, Thailand
2.
Department of Statistics, Faculty of Science, Ramkhamhaeng University, Bangkok 10240, Thailand
3.
Department of Mathematics, Faculty of Science, Burapha University, Chonburi 20131, Thailand
4.
Center of Excellence in Mathematics, CHE, Sri Ayutthaya Rd., Bangkok 10400, Thailand
5.
Department of Mathematics and General Sciences, Prince Sultan University, 11586 Riyadh, Saudi Arabia
6.
Department of Industrial Engineering, OSTİM Technical University, Ankara 06374, Turkey

Received: 14 October 2021 Revised: 27 January 2022 Accepted: 06 February 2022 Published: 17 February 2022
MSC : 26A33, 34A37, 34A08, 34D20, 38B82

In this manuscript, we study the existence and Ulam's stability results for impulsive multi-order Caputo proportional fractional pantograph differential equations equipped with boundary and integral conditions with respect to another function. The uniqueness result is proved via Banach's fixed point theorem, and the existence results are based on Schaefer's fixed point theorem. In addition, the Ulam-Hyers stability and Ulam-Hyers-Rassias stability of the proposed problem are obtained by applying the nonlinear functional analysis technique. Finally, numerical examples are provided to supplement the applicability of the acquired theoretical results.
- existence and uniqueness,
- fractional differential equations,
- fixed point theorems,
- impulsive conditions,
- Ulam-Hyers stability
Citation: Songkran Pleumpreedaporn, Chanidaporn Pleumpreedaporn, Weerawat Sudsutad, Jutarat Kongson, Chatthai Thaiprayoon, Jehad Alzabut. On a novel impulsive boundary value pantograph problem under Caputo proportional fractional derivative operator with respect to another function[J]. AIMS Mathematics, 2022, 7(5): 7817-7846. doi: 10.3934/math.2022438

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Abstract

In this manuscript, we study the existence and Ulam's stability results for impulsive multi-order Caputo proportional fractional pantograph differential equations equipped with boundary and integral conditions with respect to another function. The uniqueness result is proved via Banach's fixed point theorem, and the existence results are based on Schaefer's fixed point theorem. In addition, the Ulam-Hyers stability and Ulam-Hyers-Rassias stability of the proposed problem are obtained by applying the nonlinear functional analysis technique. Finally, numerical examples are provided to supplement the applicability of the acquired theoretical results.

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