On the Rational Recursive Sequence xn+1=(α−βxn)‎(γ−δxn−xn−k)‎

Abstract EN

We study the global stability, the periodic character, and the boundedness character of the positive solutions of the difference equation xn+1=(α−βxn)/(γ−δxn−xn−k), n=0,1,2,…, k∈{1,2,…}, in the two cases: (i) δ≥0, α>0, γ>β>0; (ii) δ≥0, α=0, γ,β>0, where the coefficients α, β, γ, and δ, and the initial conditions x−k,x−k+1,…,x−1,x0 are real numbers.

We show that the positive equilibrium of this equation is a global attractor with a basin that depends on certain conditions posed on the coefficients of this equation.