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

## TL;DR

This paper explores various forms of distributional chaos for sequences of linear operators in Fréchet spaces, including new notions and their properties, with applications to specific operators and partial differential equations.

## Contribution

It introduces and analyzes new notions of $m_{n}$-distributional chaos and reiterative chaos for linear operators in Fréchet spaces, extending existing chaos concepts.

## Key findings

- Characterization of $m_{n}$-distributional chaos for weighted shift operators
- Introduction of reiterative $m_{n}$-distributional chaos notions
- Applications to abstract partial differential equations

## Abstract

The main aim of this paper is to consider various notions of (dense) $m_{n}$-distributional chaos of type $s$ and (dense) reiterative $m_{n}$-distributional chaos of type $s$ for general sequences of linear not necessarily continuous operators in Fr\' echet spaces. Here, $(m_{n})$ is an increasing sequence in $[1,\infty)$ satisfying $\liminf_{n\rightarrow \infty}\frac{m_{n}}{n}>0$ and $s$ could be $0,1,2,2+,2\frac{1}{2},3,1+,2-,2_{Bd},2_{Bd}+.$ We investigate $m_{n}$-distributionally chaotic properties and reiteratively $m_{n}$-distributionally chaotic properties of some special classes of operators like weighted forward shift operators and weighted backward shift operators in Fr\' echet sequence spaces, considering also continuous analogues of introduced notions and some applications to abstract partial differential equations.