The periodic character of positive solutions of the difference equation χn+1 = (χn, χn−k)

In this note, we consider the nonlinear difference equation, xn+1=f(xn,xn−k),n=0,1,...xn+1=f(xn,xn−k),n=0,1,... where κ ∈ { 1,2,} and the initial values χk, χ k+1,, χ0 ∈ (0, +∞). We give sufficient conditions under which every positive solution of this equation converges to a (not necessarily prime) period two solution if k is odd and the unique positive equilibrium if k is even.