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Proceedings of the American Mathematical Society

Published by the American Mathematical Society since 1950, Proceedings of the American Mathematical Society is devoted to shorter research articles in all areas of pure and applied mathematics.

ISSN 1088-6826 (online) ISSN 0002-9939 (print)

The 2024 MCQ for Proceedings of the American Mathematical Society is 0.85.

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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

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