Stability investigation of invariant sets of difference equations by means of Lyapunov functions (Q2782059)

The author considers a system of difference equations NEWLINE\[NEWLINE x_{n+1} =f(x_n),\tag{1} NEWLINE\]NEWLINE where \( x\in{\mathbb R}^n \) and \( n\in {\mathbb Z}\). It is assumed that the right-hand part of system (1) is definite in some domain \( D\subset {\mathbb R}^n \) and for each \( x_0\in D \) there exists a unique solution \(x_n(x_0)\) of system (1) such that \( x_0(x_0) = x_0 \) up to the boundary of the domain \(D\). The Lyapunov function and its first difference are used to prove a theorem on a asymptotic stability of the unilateral invariant set of system (1) determined by \( V(x)=0\), \( x\in D_1\), where \( D_1\subset D \) is a bounded domain.