On the rate of convergence of solutions for a class of difference equations (Q2726138)

Consider the initial-value problem NEWLINE\[NEWLINE\begin{cases} x(n)-x(n-1)=p(n)G \biggl(x(n), x\bigl(n-k(n) \bigr)\biggr),\\ x(n_0-j)=a_j,\quad j\in\{0,1, \dots, k\}. \end{cases}\tag{*}NEWLINE\]NEWLINE Here \(G\) is continuous function such that \(G(x,.)\) is nondecreasing, \(G(.,y)\) is nonincreasing and \(G(x,x)\equiv 0\). Under some additional assumptions every solution of (*) tends to a constant as \(n\to\infty\) and the rate of convergence can be estimated.