Stability properties of nonlinear difference equations and conditions for boundedness
The paper deals with a difference equation of the form \[ x_{n+1}=f(x_n)g(x_{n-k}),\quad n = 0, 1,\dots \tag{1} \] with positive initial conditions where \(f, g \in C([0,\infty), [0, \infty))\), \(f\) is increasing, \(g\) is decreasing, and the equation \(x=f(x)g(x)\) has a unique positive solution. Sufficient conditions for the global asymptotic stability and boundedness of solutions of equation (1) are obtained. As an application, the rational recursive sequence \(x_{n+1}=(a+bx_n^2)/(c+x_{n-k}^2)\), \(n=0,1,\dots\), where \(a,b,c\in(0,\infty)\), \(k\in \{1,2,\dots\}\) is considered.
scientific article; zbMATH DE number 1391611