Strict bounds for the period of periodic orbits of difference equations (Q1824777)

This paper answers the following question: Suppose f is Lipschitz with constant L in a normed space and the autonomous difference equation \(x_{k+1}=x_ k+f(x_ k)\) admits an n-periodic solution. What is the minimum value for the Lipschitz constant L or, in other words, what is the maximal value for the period n? The answer is \(L\geq n/\lambda_ n\) where \(\lambda_ n\) is the largest eigenvalue of a computable (n- 1)\(\times (n-1)\) integer matrix. An example is given with equality, so the bounds are strict. If the period n is prime an explicit formula is found for \(\lambda_ n\). The proof of the main result mimics the proof of the analogous differential equation result by Busenberg, Fisher, Martelli [Am. Math. Monthly (to appear)].