Existence and Uniqueness Results for a Pantograph Boundary Value Problem Involving a Variable-Order Hadamard Fractional Derivative

Abstract

This paper discusses the problem of the existence and uniqueness of solutions to the boundary value problem for the nonlinear fractional-order pantograph equation, using the fractional derivative of variable order of Hadamard type. The main results are proved through the application of fractional calculus and Krasnoselskii’s fixed-point theorem. Moreover, the Ulam–Hyers–Rassias stability of the nonlinear fractional pantograph equation is analyzed. To conclude this paper, we provide an example illustrating our findings and approach.

A Comprehensive Study of the Langevin Boundary Value Problems with Variable Order Fractional Derivatives

“…Variable order fractional calculus is a key advancement in this discipline by offering the new concept that the order of differentiation or integration might be a variable rather than a fixed value. This breakthrough provides a strong tool for modeling complicated systems with various degrees of memory and non-local effects that results in a more realistic depiction of real-world processes and makes it a useful and adaptable tool (see [5][6][7][8][9] and the references therein).…”