论分数阶微分方程的渐近周期解及其应用
On asymptotic periodic solutions of fractional differential equations and applications
AI总结：
本文研究Caputo型分数阶微分方程解的渐近行为，借助半直线函数谱理论得到经典定理的类似结果，给出算子谱条件以判定渐近温和解的渐近1-周期性。
摘要：
本文研究形式为$D^α_Cu(t)=Au(t)+f(t), u(0)=x, 0<α\le1, ( *)$的分数阶微分方程解的渐近行为，其中$D^α_Cu(t)$是函数$u$的Caputo意义导数，$A$是Banach空间$\X$中可能无界的线性算子，$f$满足$\lim_{t\to \infty} (f(t+1)-f(t))=0$的性质，我们称之为渐近1-周期性。通过运用半直线上的函数谱理论，我们推导得到Katznelson-Tzafriri定理和Massera定理的类似结论。具体而言，我们根据算子$A$的谱性质给出了充分条件，使得方程(*)的所有渐近温和解均为渐近1-周期的，或者存在一个渐近1-周期的渐近温和解。
英文摘要：
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.
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