Fractal Fract, Vol. 10, Pages 50: Stability for Caputo–Hadamard Fractional Uncertain Differential Equation

Fractal and Fractional doi: 10.3390/fractalfract10010050

Authors:
Shida Peng
Zhi Li
Jun Zhang
Yuncong Zhu
Liping Xu

This paper focuses on the Caputo-Hadamard fractional uncertain differential equations (CH-FUDEs) governed by Liu processes, which combine the Caputo–Hadamard fractional derivative with uncertain differential equations to describe dynamic systems involving memory characteristics and uncertain information. Within the framework of uncertain theory, this Liu process serves as the counterpart to Brownian motion. We establish some new Bihari type fractional inequalities that are easy to apply in practice and can be considered as a more general tool in some situations. As applications of those inequalities, we establish the well-posedness of a proposed class of equations under specified non-Lipschitz conditions. Building upon this result, we establish the notions of stability in distribution and stability in measure solutions to CH-FUDEs, deriving sufficient conditions to ensure these stability properties. Finally, the theoretical findings are verified through two numerical examples.

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