Extreme stability in terms of two measures for difference systems with finite delay (Q2778658)

This paper is concerned with the stability of the delay difference system NEWLINE\[NEWLINEx_{n+1}= f(n,x_n),\;n\in\mathbb{N}^+ =\{0,1,2,\dots\}, \tag{1}NEWLINE\]NEWLINE where \(f:\mathbb{N}^+ \times C\to\mathbb{R}^m\), \(C=C(\{-r,-r+1, \dots,0\} \to\mathbb{R}^m)\) and \(r\) is a positive integer.NEWLINENEWLINENEWLINEThe concept of extreme stability in terms of two measures are introduced. Roughly, let \(h_0\), \(h:\mathbb{N}^+ \times\mathbb{R}^m \times \mathbb{R}^m \to [0,\infty)\) which are continuous with respect to their second and third independent variables. For any \(\varphi,\psi\in C\), let NEWLINE\[NEWLINE\widetilde h_0 (n,\varphi, \psi)= \max_{s=-r,\dots,0} h_0\bigl(n+s, \varphi(s),\psi(s) \bigr), NEWLINE\]NEWLINE and NEWLINE\[NEWLINE\widetilde h(n,\varphi,\psi)=\max_{s=-r,\dots,0} h\bigl(n+s, \varphi (s), \psi(s)\bigr).NEWLINE\]NEWLINE The system (1) is said to be \((h_0,h)\)-extreme stable if for any \(\varepsilon >0\) and \(n_0\in\mathbb{N}^+\), there exists \(\delta= \delta( \varepsilon, n_0)>0\) such that when \(\widetilde h_0(n,\varphi, \psi)< \delta\), we have \(h(n,x(n), y(n))< \varepsilon\) for \(n\geq n_0\), where \(x(n)= x(n,n_0, \varphi)\) and \(y(n)= y(n,n_0,\psi)\) are solutions of (1).NEWLINENEWLINENEWLINEOther similar stability concepts such as extreme uniform stability are defined. Lyapunov like functions and stability theorems are established. Finally, an example stability theorem is established for the linear system NEWLINE\[NEWLINEx(n+1)= Ax(n)+ Bx(n-1).NEWLINE\]