Abstract
An elementary inequality is discussed for autonomous Caputo fractional differential equation (FDE) of order \(\alpha \in (0,1)\) in \(\mathbb {R}^d\) for which the vector field satisfies a dissipativity condition. This inequality is fundamental for investigating qualitative and dynamical properties of such equations. Here its use and effectiveness are illustrated to show the global existence and uniqueness of solutions of such equations when the vector field is only locally Lipschitz. In addition, the existence of an absorbing set, which is positively invariant, is established. Finally, it is used to show that an equilibrium solution of a nonlinear Caputo FDE is locally asymptotically stable when the matrix of its linear part is negative definite.
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Acknowledgements
The author thanks the Vietnam Academy of Science and Technology in Hanoi for its hospitality. Financial support from a Simons Foundation grant to the Institute of Mathematics of the Vietnam Academy of Science and Technology is gratefully acknowledged. He also thanks one of the referees for drawing his attention to the result of Alikhanov [2].
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Kloeden, P.E. An elementary inequality for dissipative Caputo fractional differential equations. Fract Calc Appl Anal 26, 2166–2174 (2023). https://doi.org/10.1007/s13540-023-00192-x
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DOI: https://doi.org/10.1007/s13540-023-00192-x
Keywords
- Caputo fractional differential equations
- Dissipativity condition
- Absorbing sets
- Attracting sets
- Asymptotic stability