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.