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Global asymptotic stability of the higher order equation

In this paper, we investigate the local and global stability and the period two solutions of all nonnegative solutions of the difference equation, xn+1 = axn + bxn−k A + Bxn−k where a, b, A, B are all positive real numbers, k ≥ 1 is a positive integer, and the initial conditions x−k, x−k+1, ..., x0 are nonnegative real numbers. It is shown that the zero equilibrium point is globally asymptotically stable under the condition a+b ≤ A, and the unique positive solution is also globally asymptotically stable under the condition a − b ≤ A ≤ a + b. By the end, we study the global stability of such an equation through numerically solved examples.