TY - JOUR

T1 - On omega limiting sets of infinite dimensional Volterra operators

AU - Mukhamedov, Farrukh

AU - Khakimov, Otabek

AU - Embong, Ahmad Fadillah

N1 - Funding Information:
The present work is supported by the UAEU UPAR Grant No. 31S391. The third named author (AFE) acknowledges the Ministry of Higher Education (MOHE) and Research Management Centre-UTM, Universiti Teknologi Malaysia (UTM) for the financial support through the research grant (vote number 17J93).
Publisher Copyright:
© 2020 IOP Publishing Ltd & London Mathematical Society

PY - 2020/11

Y1 - 2020/11

N2 - 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 ωV and ωV(w) with respect to ℓ1-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 ωV (x) and ωV(w)(x) coincide for every x ∈ S, and moreover, they are non empty. If Volterra operator belongs to V-−, then ωV(x) could be empty, and it implies the non-ergodicity (w.r.t. ℓ1-norm) of V, while it is weak ergodic.

AB - 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 ωV and ωV(w) with respect to ℓ1-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 ωV (x) and ωV(w)(x) coincide for every x ∈ S, and moreover, they are non empty. If Volterra operator belongs to V-−, then ωV(x) could be empty, and it implies the non-ergodicity (w.r.t. ℓ1-norm) of V, while it is weak ergodic.

KW - Ergodic

KW - Infinite dimensional

KW - Omega limiting sets

KW - Volterra operator

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U2 - 10.1088/1361-6544/ab9a1c

DO - 10.1088/1361-6544/ab9a1c

M3 - Article

AN - SCOPUS:85092570440

VL - 33

SP - 5875

EP - 5904

JO - Nonlinearity

JF - Nonlinearity

SN - 0951-7715

IS - 11

ER -