Reiterative $m_{n}$-distributional chaos of type $s$ in Fr\' echet spaces

Abstract

The main aim of this paper is to consider various notions of (dense) mn-distributional chaos of type s and (dense) reiterative mn-distributional chaos of type s for general sequences of linear not necessarily continuous operators in Fréchet spaces. Here, (mn) is an increasing sequence in [1, ∞) satisfying lim infn→∞ mn n > 0 and s could be 0, 1, 2, 2+, 2 1 2 , 3, 1+, 2−, 2 Bd , 2 Bd + . We investigate mn-distributionally chaotic properties and reiteratively mndistributionally chaotic properties of some special classes of operators like weighted forward shift operators and weighted backward shift operators in Fréchet sequence spaces, considering also continuous analogues of introduced notions and some applications to abstract partial differential equations.2010 Mathematics Subject Classification. 47A06, 47A16. Key words and phrases. mn-distributional chaos of type s, reiterative mn-distributional chaos of type s, λ-distributional chaos of type s, reiterative λ-distributional chaos of type s, Fréchet spaces.The author is partially supported by grant 174024 of Ministry of Science and Technological Development, Republic of Serbia.

Disjoint Li-Yorke Chaos in Frechet Spaces ´ M. Kostic

“…The final statement of theorem now follows similarly as in the proofs of [7,Theorem 15] and [30,Theorem 4.1]. Now we will state the following corollary of Theorem 1.1:…”

“…To the best knowledge of the author, this is the first paper, in both linear and non-linear setting, which considers the notion of disjoint Li-Yorke chaos. The genesis of paper is motivated by our recent results on the existence of special types of dense Li-Yorke irregular manifolds obtained in a joint research study with A. Bonilla [12], and later expanded by the author in [30].…”

Disjoint Li-Yorke chaos in Fréchet spaces

“…2n−1 n∈N and ω j,n n∈N := ( 2n 2n−1 ) ζj n∈N for all j ∈ N N ; see also [31,Theorem 3.5] and [30,Example 4.9]. Then for each j ∈ N N the corresponding operator T j is topologically mixing, absolutely Cesàro bounded and therefore not distributionally chaotic; albeit this basically follows from the argumentation used in the proof of [31,Theorem 3.5], we will include all relevant details for the sake of completeness.…”

“…Proof. The only non-trivial is to show that (i) implies (ii); see also the proof of [30,Proposition 2.16(i)]. So, let σ > 0, ǫ > 0 and (m n ) ∈ R be fixed.…”

“…The main result for applications is the following continuous counterpart of Theorem 1.1; the proof can be deduced similarly and therefore omitted (cf. [14] and [30] for more details): strongly continuous translation semigroup in the space L p ρ ([0, ∞)) of C 0,ρ ([0, ∞)), then the strongly continuous semigroups generated by the operators…”