The asymptotic behavior of monotone difference equations of higher order (Q1600410)

The difference equation \[ u(t+n)= f\bigl(u(t), u(t+1),\dots, u(t+n-1) \bigr),\;n\geq 1,\;t\in\mathbb{N}_0, \] is studied for a positively homogeneous \(f\) satisfying a strict monotonicity property. It is proved that \(u(t)\lambda^{-1}\) converges for all solutions as \(t\to\infty\) where \(\lambda\) is the unique positive zero of \(\lambda^n= f(1,\lambda, \dots, \lambda^{n-1})\).