Quantitative bounds for the recursive sequence yn + 1 = A + frac(yn, yn - k)

Abstract

This note provides new quantitative bounds for the recursive equation yn + 1 = A + frac(yn, yn - k), n = 0, 1, ..., where y- k, y- k + 1, ..., y- 1, y0, A ∈ (0, ∞) and 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) for which global asymptotic stability was proven in [R.M. Abu-Saris, R. DeVault, Global stability of yn + 1 = A + frac(yn, yn - k), Appl. Math. Lett. 16 (2) (2003) 173-178].