Hypercyclic and Topologically Mixing Properties of Abstract Time-Fractional Equations With Discrete Shifts - Kostić, Marko (Sarajevo Journal of Mathematics, 2024)
The most valuable theoretical results about hypercyclic and topologically mixing properties of some special subclasses of the abstract time-fractional equations of the following form: \begin{align}\label{FDEab1} & {\mathbf D}_{t}^{\alpha_{n}}u(t)+ A_{n-1}{\mathbf D}_{t}^{\alpha_{n-1}}u(t)+\cdot \cdot \cdot + A_{1}{\mathbf D}_{t}^{\alpha_{1}}u(t)= A_{0}{\mathbf D}_{t}^{\alpha}u(t), \ t > 0, otag\\ & u^{(k)}(0)=u_k,\ k=0,\cdot \cdot \cdot, \lceil \alpha_{n} \rceil -1. \end{align} where $n\in {\mathbb N}\setminus \{1\},$ $A_{0},A_{1},\cdot \cdot \cdot ,A_{n-1}$ are closed linear operators acting on a separable infinite-dimensional complex Banach space $E,$ $0 \leq \alpha_{1}<\cdot \cdot \cdot<\alpha_{n},$ $0\leq \alpha<\alpha_{n},$ and ${\mathbf D}_{t}^{\alpha}$ denotes the Caputo fractional derivative of order $\alpha$ (\cite{bajlekova}), have been recently clarified in \cite{hf}-\cite{icdea}. In this paper, we continue the analysis contained in \cite{hf}-\cite{icdea} by assuming that, for every $j\in {\mathbb N}_{n-1},$ the operator $A_{j}$ is a certain function of unilateral backward shifts acting on weighted $l^{1}({\mathbb C})$-spaces. 2010 Mathematics Subject Classification. 47A16, 26D33, 47D06.