Ergodic approach to the study of boundedness of solutions of one type of the first-order semilinear difference equations

Abstract

We investigate the sufficient conditions for boundedness of one type of difference equations of the form $x(n+1)=ax(n)+f(x(n)) + y(n), \ n\geq 1$ in critical case $|a|=1$. For this equation the following assumptions are introduced: 1) The function $f: \mathbb{C} \to \mathbb{C}$ and the input sequence $\{y(n)\}_{n\geq1}$ are assumed to be bounded. 2) $\text{Re}\left(\overline{f(\rho \, e^{2\pi i \theta})} \; \cdot ae^{2\pi i \theta}\right)$ converges uniformly on $[0,1) \ni \theta$ to some real-valued function $\Phi(\theta)$ as $\rho \to +\infty$ . Combining the celebrated results of the probability and ergodic theory together with the geometric consideration of the problem, we show that under fairly general conditions this type of semilinear difference equations has all the solutions bounded. Subsequently we formulate the quantitative version of our theorem and give the example of its application. In addition, in the last section we discuss the conditions of our main result and provide the constructions, which highlight the importance of these conditions.