Abstract
An autonomous Caputo fractional differential equation of order α ∈ (0,1) in a finite dimensional space whose vector field satisfies a global Lipschitz condition is shown to generate a semi-dynamical system in the function space \(\mathfrak {C}\) of continuous functions with the topology uniform convergence on compact subsets. This contrasts with a recent result of Cong and Tuan (J. Integral Equ. Appl.: 29, 585–608, 2017), which showed that such equations do not, in general, generate a dynamical system on the state space.
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References
Agarwal, R., Hristova, S., O’Regan, D.: Lyapunov functions and stability of Caputo fractional differential equations with delays. Differ. Equ. Dyn. Syst., https://doi.org/10.1007/s12591-018-0434-6 (2018)
Area, I., Losada, J., Nieto, J.J.: On quasi-periodicity properties of fractional integrals and fractional derivatives of periodic functions. Integral Transforms Spec. Funct. 27, 1–16 (2016)
Cong, N.D., Tuan, H.T.: Generation of nonlocal fractional dynamical systems by fractional differential equations. J. Integral Equ. Appl. 29, 585–608 (2017)
Diethelm, K.: The Analysis of Fractional Differential Equations. Springer Lecture Notes in Mathematics, vol. 2004. Springer, Berlin (2010)
Doan, T.S., Phan, T.H., Kloeden, P.E., Hoang, T.T.: Asymptotic separation between solutions of Caputo fractional stochastic differential equations. Stoch. Anal. Appl. 39, 654–664 (2018)
Kaslik, E., Sivasundaram, S.: Non-existence of periodic solutions in fractional-order dynamical systems and a remarkable difference between integer and fractional-order derivatives of periodic functions. Nonlinear Anal. Real World Appl. 13, 1489–1497 (2012)
Kloeden, P.E., Rasmussen, M.: Nonautonomous Dynamical Systems. American Mathematical Society, Providence (2011)
Losada, J., Nieto, J.J., Pourhadi, E.: On the attractivity of solutions for a class of multi-term fractional functional differential equations. J. Comput. Appl. Math. 312, 2–12 (2017)
Miller, R.K.: Nonlinear Volterra Integral Equations. W.A Benjamin, Menlo Park (1971)
Miller, R.K., Sell, G.R.: Volterra Integral Equations and Topological Dynamics. Memoirs of the American Mathematical Society, vol. 102. American Mathematical Society, Providence (1970)
Sell, G.R.: Topological Dynamics and Ordinary Differential Equations. Van Nostrand Reinhold Mathematical Studies, London (1971)
Acknowledgements
The work of Thai Son Doan is funded by Vietnam National Foundation for Science and Technology Development (NAFOSTED) under grant number 101.03–2019.310. The authors would like to thank anonymous reviewers for several constructive comments that lead to an improvement of the paper.
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Doan, T.S., Kloeden, P.E. Semi-Dynamical Systems Generated by Autonomous Caputo Fractional Differential Equations. Vietnam J. Math. 49, 1305–1315 (2021). https://doi.org/10.1007/s10013-020-00464-6
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DOI: https://doi.org/10.1007/s10013-020-00464-6
Keywords
- Caputo fractional differential equation
- Existence and uniqueness solutions
- Continuous dependence on the initial condition
- Semi-dynamical systems
- Volterra integral equations