Study of Nonlinear Fractional Order Delay Problem Under Mittag-Leffler Power Law

Abstract

In the concerned article, the author's developed indispensable conditions for theresults of existence theory and Ulam type stability of the intended pantograph fractional differential equation. The desired results for existence of the solutions and stability analysis are investigated with a new class of fractional derivative introduced by Atangana-Baleanu-Caputo and so abbreviated as (ABC). The mentioned derivative contains Mittag-Leffler function as a nonsingular kernel instead of singular kernel. We use the tools of fixed point theory, i.e Banach's theorem and Krasnoselskii's type to develop the desired results. Furthermore, we investigate different types of stabilities results including generalized Ullam-Hyers and Rassias stabilities for the concerned problem. We also provide appropriate examples for illustrative purposes.