Hypercyclic and topologically mixing properties of abstract timefractional equations with discrete shifts

Abstract

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: Dn t u(t) +An−1D αn−1 t u(t) + · · ·+A1D α1 t u(t) = A0D α t u(t), t > 0, u(0) = uk, k = 0, · · ·, dαne − 1. (1) where n ∈ N {1}, A0, A1, · · ·, An−1 are closed linear operators acting on a separable infinite-dimensional complex Banach space E, 0 ≤ α1 < · · · < αn, 0 ≤ α < αn, and Dt denotes the Caputo fractional derivative of order α ([1]), have been recently clarified in [12]-[13]. In this paper, we continue the analysis contained in [12]-[13] by assuming that, for every j ∈ Nn−1, the operator Aj is a certain function of unilateral backward shifts acting on weighted l(C)-spaces.