Global stability of a higher order rational recursive sequence

In this paper, we study the qualitative behavior of solutions of the class of delay difference equationxn+1=βxn-k+1+γxn-2k+1A+Bxn-k+1,n=0,1,2,…,where the initial conditions x−2k+1, … , x−1, x0 are positive, k ∈ {1, 2, …}, and the parameters β, γ, A, B are positive. Our concentration is on invariant intervals and the global stability of the above mentioned equation. We obtain sufficient conditions for the global attractivity of all positive solutions about the zero and positive equilibrium points with basins that depend on specific conditions posed on the coefficients. Furthermore, the oscillation and the character of semicycles about the positive equilibrium are thoroughly discussed. It is also illustrated that for some special case of parameters, the solution will be either a period-2 solution, or will converge to the equilibrium point, or will have unbounded solutions.