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
SN - 0951-7715
VL - 33
SP - 5875
EP - 5904
JO - Nonlinearity
JF - Nonlinearity
IS - 11
ER -