Gronwall-Type Inequalities and Qualitative Studies onHigher-Variable Orders of Atangana-Baleanu Fractional Operators via Increasing Functions

This paper introduces a novel extension of Caputo-Atangana-Baleanu and Riemann-Atangana-Baleanu fractional derivatives from constant to increasing variable order. We generalize the fractional order from a fixed value in (0, 1] to a time-dependent function in (k, k + 1], where k ≥ 0. The corresponding Atangana-Baleanu fractional integral is also extended. Key properties ofthese new definitions are explored, including a generalized Gronwall inequality. We then delve into the analysis of higher-variable initial fractional differential equations using the Caputo-Atangana-Baleanu operator with an increasing function, establishing existence and uniqueness results via Picard’s iterative method. The findings presented in this work are expected to stimulate further research on inequalities and fractional differential equations related to Atangana-Baleanu fractional calculus with respect to increasing functions. Concrete examples are provided to illustrate the practical applications of our results.