Borzdyko’s uniqueness theorems for fractal-fractional ordinary differential equations with power-law kernels and hysteresis

Abstract

Fractal–fractional differential equations have emerged as a powerful mathematical framework for modeling complex systems exhibiting memory effects, non-locality, and hysteresis phenomena. This study investigates a class of fractal-fractional ordinary differential equations characterized by a power-law memory kernel and influenced by hysteresis behavior. The continuity of the function g (t, w(t)) over closed subsets of $ \mathbb{R}$, is used to establish the foundational results. A supporting lemma is introduced to facilitate the development of a uniqueness theorem. Drawing upon Borzdyko’s framework, we derive existence results pertinent to the targeted family of equations.

Keywords

Power law / Fractal-fractional operators / Uniqueness / Borzdyko’s conditions

References

[1] Chen W, Sun H, Zhang X, Korošak D. Anomalous diffusion modeling by fractal and fractional derivatives. Comput Math Appl. 2010; 59(5):1754-1758. https://doi.org/10.1016/j.camwa.2009.08.019

[2] Chen W. Time-space fabric underlying anomalous diffusion. Chaos Solitons Fractals. 2006; 28(4):923-929. https://doi.org/10.1016/j.chaos.2005.08.199

[3] Sun HG, Meerschaert MM, Zhang Y, Zhu J, Chen W. A fractal Richards’ equation to capture the non-Boltzmann scaling of water transport in unsaturated media. Adv Water Resour. 2013; 52:292-295. https://doi.org/10.1016/j.advwatres.2012.11.005

[4] Atangana A. Fractal-fractional differentiation and integration: Connecting fractal calculus and fractional calculus to predict complex systems. Chaos Solitons Fractals. 2017; 102:396-406. https://doi.org/10.1016/j.chaos.2017.04.027

[5] Atangana A, Jain S. A new numerical approximation of the fractal ordinary differential equation. Eur Phys J Plus. 2018; 133:37. https://doi.org/10.1140/epjp/i2018-11895-1

[6] Jain S, El-Khatib Y. Modelling chaotic dynamical attractors with fractal-fractional differential operators. AIMS Math. 2021; 6(12):13689-13725. https://doi:10.3934/math.2021795

[7] Jain S. Numerical analysis for the fractional diffusion and fractional Buckmaster equation by the two-step Laplace Adam-Bashforth method. Eur Phys J Plus. 2018; 133(1):19. https://doi.org/10.1140/EPJP/I2018-11854-X

[8] Agarwal RP, Lakshmikantham V. Uniqueness and Nonuniqueness Criteria for Ordinary Differential Equations. Singapore: World Scientific; 1993. https://doi.org/10.1142/1988

[9] Julia G. Mémoire sur l’itération des fonctions rationnelles. J Math Pures Appl. 1918; 8:47-245. http://eudml.org/doc/234994

[10] Tan L. Similarity between the Mandelbrot set and Julia sets. Commun Math Phys. 1990; 134:587-617. https://doi.org/10.1007/BF02098448