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.

Keywords

• fractional operators of variable order
• Krasnoselskii’s fixed-point result
• piecewise continuous functions