A self-invertibility condition for global periodicity of difference equations

Given a non-degenerate interval of real numbers DD, and a continuous function f:Dk→Df:Dk→D with k≥2k≥2, we consider a kkth-order difference equation of the form yn+1=f(yn,…,yn−k+1);n=0,1,2,…. We develop an easy-to-apply necessary condition so that all solutions of the above-mentioned equation are periodic of the same period.