## Abstract

In the present paper, we are aiming to study limiting behaviour of infinite dimensional Volterra operators. We introduce two classes V^{-}^{+} and V^{-−} of infinite dimensional Volterra operators. For operators taken from the introduced classes we study their omega limiting sets ω_{V} and ω_{V}^{(}w^{)} with respect to ℓ^{1}-norm and pointwise convergence, respectively. To investigate the relations between these limiting sets, we study linear Lyapunov functions for such kind of Volterra operators. It is proven that if Volterra operator belongs to V^{-}^{+}, then the sets ω_{V} (x) and ω_{V}^{(}w^{)}(x) coincide for every x ∈ S, and moreover, they are non empty. If Volterra operator belongs to V^{-−}, then ω_{V}(x) could be empty, and it implies the non-ergodicity (w.r.t. ℓ^{1}-norm) of V, while it is weak ergodic.

Original language | English |
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Pages (from-to) | 5875-5904 |

Number of pages | 30 |

Journal | Nonlinearity |

Volume | 33 |

Issue number | 11 |

DOIs | |

Publication status | Published - Nov 2020 |

## Keywords

- Ergodic
- Infinite dimensional
- Omega limiting sets
- Volterra operator

## ASJC Scopus subject areas

- Statistical and Nonlinear Physics
- Mathematical Physics
- General Physics and Astronomy
- Applied Mathematics