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

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.