文章基本信息

标题：Normal periodic solutions for the fractional abstract Cauchy problem
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作者：Jennifer Bravo ; Carlos Lizama
期刊名称：Boundary Value Problems
印刷版ISSN：1687-2762
电子版ISSN：1687-2770
出版年度：2021
卷号：2021
期号：1
页码：1
DOI：10.1186/s13661-021-01529-2
出版社：Hindawi Publishing Corporation
摘要：We show that if A is a closed linear operator defined in a Banach space X and there exist $t_{0} \geq 0$ and $M>0$ such that $\{(im)^{\alpha }\}_{ m > t_{0}} \subset \rho (A)$ , the resolvent set of A, and $$ \bigl\Vert (im)^{\alpha }\bigl(A (im)^{\alpha }I \bigr)^{-1} \bigr\Vert \leq M \quad \text{ for all } \vert m \vert > t_{0}, m \in \mathbb{Z}, $$ then, for each $\frac{1}{p}<\alpha \leq \frac{2}{p}$ and $1< p < 2$ , the abstract Cauchy problem with periodic boundary conditions $$ \textstyle\begin{cases} _{GL}D^{\alpha }_{t} u(t) Au(t) = f(t), & t \in (0,2\pi ); \\ u(0)=u(2\pi ), \end{cases} $$ where $_{GL}D^{\alpha }$ denotes the Grünwald–Letnikov derivative, admits a normal 2π-periodic solution for each $f\in L^{p}_{2\pi }(\mathbb{R}, X)$ that satisfies appropriate conditions. In particular, this happens if A is a sectorial operator with spectral angle $\phi _{A} \in (0, \alpha \pi /2)$ and $\int _{0}^{2\pi } f(t)\,dt \in \operatorname{Ran}(A)$ .
关键词：35R11 ; 35B10 ; 43A50