On asymptotic periodic solutions of fractional differential equations and applications
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| Format: | Preprint |
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2023
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| _version_ | 1866914020603199488 |
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| author | Luong, Vu Trong Huy, Nguyen Duc Van Minh, Nguyen Vien, Nguyen Ngoc |
| author_facet | Luong, Vu Trong Huy, Nguyen Duc Van Minh, Nguyen Vien, Nguyen Ngoc |
| contents | In this paper we study the asymptotic behavior of solutions of fractional differential equations of the form $
D^α_Cu(t)=Au(t)+f(t), u(0)=x, 0<α\le1, ( *) $
where $D^α_Cu(t)$ is the derivative of the function $u$ in the Caputo's sense, $A$ is a linear operator in a Banach space $\X$ 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. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2304_04850 |
| institution | arXiv |
| publishDate | 2023 |
| record_format | arxiv |
| spellingShingle | On asymptotic periodic solutions of fractional differential equations and applications Luong, Vu Trong Huy, Nguyen Duc Van Minh, Nguyen Vien, Nguyen Ngoc Classical Analysis and ODEs Analysis of PDEs Dynamical Systems 34K37, 34G10, 34K30, 45J05 In this paper we study the asymptotic behavior of solutions of fractional differential equations of the form $ D^α_Cu(t)=Au(t)+f(t), u(0)=x, 0<α\le1, ( *) $ where $D^α_Cu(t)$ is the derivative of the function $u$ in the Caputo's sense, $A$ is a linear operator in a Banach space $\X$ 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. |
| title | On asymptotic periodic solutions of fractional differential equations and applications |
| topic | Classical Analysis and ODEs Analysis of PDEs Dynamical Systems 34K37, 34G10, 34K30, 45J05 |
| url | https://arxiv.org/abs/2304.04850 |