Dynamical Systems of an Infinite-Dimensional Nonlinear Operator on the Banach Space $l_1$

U. R. Olimov^a, U. A. Rozikov<sup>bca

a</sup> V. I. Romanovskiy Institute of Mathematics, University st. 9, Tashkent, 100174 Uzbekistan
^b Karshi State University, Kuchabag st. 17, Karshi, 180119 Uzbekistan
^c National University of Uzbekistan, Universitet st. 4, Tashkent, 100174 Uzbekistan

Abstract: We investigate discrete-time dynamical systems generated by an infinite-dimensional nonlinear operator that maps the Banach space $l_1$ to itself. It is demonstrated that this operator possesses up to seven fixed points. By leveraging the specific form of our operator, we illustrate that analyzing the operator can be simplified to a two-dimensional approach. Subsequently, we provide a detailed description of all fixed points, invariant sets for the two-dimensional operator and determine the set of limit points for its trajectories. These results are then applied to find the set of limit points for trajectories generated by the infinite-dimensional operator.

Keywords: infinite-dimensional operator, trajectory, fixed point, limit point, partial order

MSC: 37L05, 37N05

Received: 24.02.2024
Accepted: 07.07.2024

Language: English

DOI: 10.20537/nd240804