The asymptotic behavior of higher order delay nonlinear difference equations (Q1430939)

For the nonlinear difference equation with delay \[ x_{n+1} = \beta x_n^2/(1+x^2_{n-k})\;,\;\beta>0 \] there are considered the positive for \(n\geq 0\) solutions. An open problem stated by \textit{V. L. Kocic} and \textit{G. Ladas} [Global behavior of nonlinear difference equations of higher order with applications. (1993; Zbl 0787.39001); p. 159] is considered here, concerning global asymptotic stability of the equilibria of the equation. The cases \(\beta<2\), \(\beta=2\), \(\beta>2\) are tackled separately.