Step forward on nonlinear differential equations with the Atangana–Baleanu derivative: Inequalities, existence, uniqueness and method

Abstract

In this study, we have found new results related to the fractional derivatives, integrals, and corresponding nonlinear ordinary differential and integral equations of Atangana–Baleanu. A comment on the initial condition problem when working with ordinary differential equations with the Atangana–Baleanu fractional derivative introduced the paper. We have developed a variety of new inequalities that are similar to the Gronwall and generalized-Gronwall inequalities using some criteria for the evaluation. The linear growth condition was utilized to obtain several new inequalities after the interval and initial condition dependency were established. We demonstrated a Picard–Tonelli predictor–corrector integration for the Atangana–Baleanu fractional nonlinear differential equation’s local solution. We established three distinct proofs of uniqueness using the Kransorsel’skii–Krein, Kooi’s, and Lipschitz conditions. The Nystrom midpoint approach has also been expanded to include nonlinear ordinary differential equations with the Atangana–Baleanu derivative; hence, the case of Caputo is straightforward. To assess the effectiveness of this method, some theoretical analyses and some practical cases were provided.