On asymptotic periodic solutions of fractional differential equations and applications
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- by Vu Trong Luong, Nguyen Duc Huy, Nguyen Van Minh and Nguyen Ngoc Vien;
- Proc. Amer. Math. Soc. 151 (2023), 5299-5312
- DOI: https://doi.org/10.1090/proc/16484
- Published electronically: September 25, 2023
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Abstract:
In this paper we study the asymptotic behavior of solutions of fractional differential equations of the form $D^{\alpha }_Cu(t)=Au(t)+f(t), u(0)=x, 0<\alpha \le 1, ( *)$ where $D^{\alpha }_Cu(t)$ is the derivative of the function $u$ in the Caputo’s sense, $A$ is a linear operator in a Banach space $\mathbb {X}$ that may be unbounded and $f$ satisfies the property that $\lim _{t\to \infty } (f(t+1)-f(t))=0$ which we will call asymptotic $1$-periodicity. By using the spectral theory of functions on the half line we derive analogs of Katznelson-Tzafriri and Massera Theorems. Namely, we give sufficient conditions in terms of spectral properties of the operator $A$ for all asymptotic mild solutions of Eq. (*) to be asymptotic $1$-periodic, or there exists an asymptotic mild solution that is asymptotic $1$-periodic.References
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Bibliographic Information
- Vu Trong Luong
- Affiliation: VNU University of Education, Vietnam National University at Hanoi, 144 Xuan Thuy, Cau Giay, Hanoi, Vietnam
- MR Author ID: 852217
- ORCID: 0000-0002-4640-4348
- Email: vutrongluong@gmail.com
- Nguyen Duc Huy
- Affiliation: VNU University of Education, Vietnam National University at Hanoi, 144 Xuan Thuy, Cau Giay, Hanoi, Vietnam
- MR Author ID: 789889
- Email: huynd@vnu.edu.vn
- Nguyen Van Minh
- Affiliation: Department of Mathematics and Statistics, University of Arkansas at Little Rock, 2801 S University Ave, Little Rock, Arkansas 72204
- MR Author ID: 249004
- ORCID: 0000-0002-2648-1610
- Email: mvnguyen1@ualr.edu
- Nguyen Ngoc Vien
- Affiliation: Faculty of Foundations, Hai Duong University, Hai Duong City, Vietnam
- ORCID: 0000-0003-3615-7563
- Email: uhdviennguyen.edu@gmail.com
- Received by editor(s): December 8, 2022
- Received by editor(s) in revised form: February 14, 2022, and March 2, 2023
- Published electronically: September 25, 2023
- Communicated by: Wenxian Shen
- © Copyright 2023 American Mathematical Society
- Journal: Proc. Amer. Math. Soc. 151 (2023), 5299-5312
- MSC (2020): Primary 34K37, 34G10; Secondary 34K30, 45J05
- DOI: https://doi.org/10.1090/proc/16484
- MathSciNet review: 4648926