On a class of piece-wise fractional order derivative delay differential equation with integral type condition

Volume 34, Issue 4, pp 350--360 https://dx.doi.org/10.22436/jmcs.034.04.03

Publication Date: April 05, 2024 Submission Date: October 27, 2023 Revision Date: January 19, 2024 Accteptance Date: February 19, 2024

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Authors

K. Shah - Department of Mathematics and Sciences, Prince Sultan University, Riyadh 11586, Saudi Arabia. - Department of Mathematics, University of Malakand, Dir(L), Khyber Pakhtunkhwa, Pakistan. M. Sher - Department of Mathematics, University of Malakand, Dir(L), Khyber Pakhtunkhwa, Pakistan. M. Sarwar - Department of Mathematics and Sciences, Prince Sultan University, Riyadh 11586, Saudi Arabia. - Department of Mathematics, University of Malakand, Dir(L), Khyber Pakhtunkhwa, Pakistan. Th. Abdeljawad - Department of Mathematics and Sciences, Prince Sultan University, Riyadh 11586, Saudi Arabia.

Abstract

In this paper, we present a detailed study of a class of fractional-order delay differential equations, highlighting that many real-world problems exhibit multifaceted behaviors in their dynamical interpretations. To capture the aforementioned behavior in a more realistic way, the use of piecewise derivatives of fractional orders has increasingly been applied. Given the significant role of delay differential equations in modeling various real-world scenarios, this work specifically addresses a type of delay differential equation with a proportional delay term. Employing piecewise fractional derivatives and Ulam-Hyers (U-H) type stability analysis, we explore the qualitative theory of the analyzed problem. Utilizing fixed-point theory and techniques from functional analysis, we aim to achieve the desired outcomes. To demonstrate our findings, several illustrative examples are provided.

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Keywords

• Piecewise operator
• theoretical results
• Ulam type stability

MSC

•  26A33
•  34A08
•  35A09

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