Quantitative bounds for the recursive sequence yn+1=A+ynyn−k

This note provides new quantitative bounds for the recursive equation yn+1=A+ynyn−k,n=0,1,…, where y−k,y−k+1,…,y−1,y0,A∈(0,∞)y−k,y−k+1,…,y−1,y0,A∈(0,∞) and k∈{2,3,4,…}k∈{2,3,4,…}. Issues regarding exponential convergence of solutions are also considered. In particular, it is shown that exponential convergence holds for all (A,k)(A,k) for which global asymptotic stability was proven in [R.M. Abu-Saris, R. DeVault, Global stability of yn+1=A+ynyn−k, Appl. Math. Lett. 16 (2) (2003) 173–178].