On the dynamics of the nonlinear rational difference equation xn+1=Axn+Bxn−k+Cxn−l+bxn−k/dxn−k−exn−l

Abstract

Abstract In this article, we study the periodicity, the boundedness and the global stability of the positive solutions of the following nonlinear difference equation x n + 1 = Ax n + Bx n - k + Cx n - l + bx n - k dx n - k - ex n - l , n = 0 , 1 , 2 , … where the coefficients A , B , C , b , d , e ∈ ( 0 , ∞ ) , while k and l are positive integers. The initial conditions x - l , … , x - k , … , x - 1 , x 0 are arbitrary positive real numbers such that k l . Some numerical examples will be given to illustrate our results.