Atangana, AbdonAraz, Seda Igret2024-12-242024-12-2420230960-07791873-2887https://doi.org/10.1016/j.chaos.2023.113700https://hdl.handle.net/20.500.12604/6425In 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.eninfo:eu-repo/semantics/openAccessInequalitiesNonlinear differential equationsExistence and uniquenessAtangana-Baleanu derivativeNumerical schemeArticle173Q1WOS:001040309900001Q12-s2.0-8516340574510.1016/j.chaos.2023.113700