Stationary oscillation of non-autonomous periodic difference system (Q2721816)

The author considers the stationary oscillation of the non-autonomous discrete periodic system NEWLINE\[NEWLINEx(\tau+1) =f\bigl(\tau,x (\tau) \bigr), \quad\tau \in I=\{t_0+k \mid k\in Z\},\tag{1}NEWLINE\]NEWLINE where \(x\in\mathbb{R}^n\), \(f: \mathbb{Z}\times \mathbb{R}^n\to \mathbb{R}^n\) is continuous in \(x\) and for some positive integer \(N\) the relation \(f(\tau+N,x)= f(\tau,x)\), \(\tau\in I\), \(x\in \mathbb{R}^n\) is satisfied. A new concept of strongly extreme stability is introduced as follows:NEWLINENEWLINENEWLINEThe system (1) is called strongly and extremely stable, if there exists a nonnegative continuous function \(a(\tau)\) on \(I\) with \(\lim_{\tau \to+\infty} a(\tau)=0\) such that for any two solutions \(x(\tau,t_0,x_0)\), \(y( \tau, t_0,y_0)\) of system (1) we have NEWLINE\[NEWLINE\bigl\|x(\tau,t_0,x_0)-y(\tau, t_0, y_0) \bigr\|\leq a(\tau) \|x_0-y_0\|,\;\tau\in I.NEWLINE\]NEWLINE Using the fact that the strongly extreme stability of (1) implies the global asymptotic stability of the periodic solution of (1) and a fixed point theorem, the author establishes the next main result:NEWLINENEWLINENEWLINETheorem 1. If the system (1) is strongly and extremely stable, then system (1) possesses a stationary oscillation with period \(N\).NEWLINENEWLINENEWLINESome application consequence and an example are also indicated in details.