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

Authors

  • Hasanen A. Hammad Sohag University
  • Manuel De la Sen Univesity of the Basque Country, Spain

DOI:

https://doi.org/10.29020/nybg.ejpam.v17i4.5592

Keywords:

Fractional derivatives; variable order derivatives; fixed point techniques; existence results

Abstract

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 of
these 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.

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Published

2024-10-31

Issue

Section

Nonlinear Analysis

How to Cite

Gronwall-Type Inequalities and Qualitative Studies onHigher-Variable Orders of Atangana-Baleanu Fractional Operators via Increasing Functions. (2024). European Journal of Pure and Applied Mathematics, 17(4), 3687-3707. https://doi.org/10.29020/nybg.ejpam.v17i4.5592