Analysis of Nonlocal Fractional Hilfer Pantograph Langevin Equation and Inclusion

Abstract

This work aims to explore the existence and uniqueness of results for pantograph Langevin equations and their inclusion in the weighted space involving the Hilfer fractional derivative, along with nonlocal conditions. We will do this by using common fixed-point theorem tools for both single-valued and multi-valued functions. We employ Krasnosel skii’s fixed-point theorem to derive the existence results for the single-valued problem and Banach’s fixed-point theorem to get the uniqueness result. On the other hand, using the nonlinear alternative for contractive maps, the existence results of the inclusion problem are obtained when the right-hand side is convex. We provide examples to support the validity of our results.