A Qualitative Study on Second-Order Nonlinear Fractional Differential Evolution Equations with Generalized ABC Operator

Abstract

This research paper is dedicated to an investigation of an evolution problem under a new operator ($\mathfrak{g}$-Atangana–Baleanu–Caputo type fractional derivative)(for short, $\mathfrak{g}$-ABC). For the proposed problem, we construct sufficient conditions for some properties of the solution like existence, uniqueness and stability analysis. Existence and uniqueness results are proved based on some fixed point theorems such that Banach and Krasnoselskii. Furthermore, through mathematical analysis techniques, we analyze different types of stability results. The symmetric properties aid in identifying the best strategy for getting the correct solution of fractional differential equations. An illustrative example is discussed for the control problem.