Asymptotic stability theorem and asymptotic stability region on difference systems (Q2749012)

This paper considers the recurrence systems NEWLINE\[NEWLINEx(n+1)=f \bigl(n,x(n) \bigr),\;n=0,1,\dots, \tag{1}NEWLINE\]NEWLINE and NEWLINE\[NEWLINEy(n+1)= g\bigl(y(n) \bigr),\;n=0,1,\dots, \tag{2}NEWLINE\]NEWLINE where \(f:\mathbb{N}^+ \times B_a\to\mathbb{R}^m\) is a function which satisfies \(f(n,0)=0\) and \(f(n,x)\) is continuous with respect to \(x\), \(g:\mathbb{R}^m \to \mathbb{R}^m\) is continuous and \(g(0)=0\), \(\mathbb{N}^+=\{0,1,\dots,\}\) and \(B_a=\{x \in \mathbb{R}^m \mid|x|< a\}\).NEWLINENEWLINENEWLINEAssuming the existence of Lyapunov type functions \(V\), the trivial solution of (1) is shown to be asymptotically stable. An example theorem is as follows: if increasing functions \(\varphi,\psi, \chi: [0,\infty)\to [0,\infty)\) exist such that \(\varphi(0)=\psi(0) =\chi(0)=0\) and if \(V:\mathbb{N}^+\times \mathbb{R}^m\to \mathbb{R}\) exists such that \(\varphi(|x|)\leq V(n,x)\leq \psi(|x|)\) and \(\Delta V(n,x(n)) \leq-\chi(|x(n)|)\) for all \(n\in\mathbb{N}^+\) and \(x\in\mathbb{R}^m\), and if there is \(\delta_0>0\) such that NEWLINE\[NEWLINE\bigl|\chi\circ \psi^{-1}(u_1)- \chi\circ \psi^{-1}(u_2)\bigr |\leq |u_1-u_2 |,\;u_1,u_2\in [0,\delta_0),NEWLINE\]NEWLINE then the trivial solution of (1) is asymptotically stable.NEWLINENEWLINENEWLINEBasins of attraction are also found for (2). For example, it is shown that if \(V,\varphi:\mathbb{R}^m \to\mathbb{R}^m\) are positive definite continuous functions such that \(\varphi(x)\) does not converge to 0 as \(|x|\to \infty\) and there exists \(b>0\) such that NEWLINE\[NEWLINEV\bigl( x(n+1)\bigr) \leq\biggl[1+ \varphi\bigl(x(n) \bigr)\biggr] V\bigl(x(n) \bigr)-b\varphi \bigl( x(n)\bigr),NEWLINE\]NEWLINE then for every \(x(0)\in \{x\in\mathbb{R}^m \mid V(x)<b\}\), the solution of (2) determined by \(x(0)\) will tend to 0.