Global attractivity of a higher-order nonlinear difference equation

Abstract

The main goal of this paper is to investigate the locally asymptotically stable, period-two solutions, invariant intervals and global attractivity of all negative solutions of the nonlinear difference equation x n + 1 = 1 - x n A + x n - k , n = 0 , 1 , … , where A ∈ ( - ∞ , - 1 ) , k is a positive integer and initial conditions x - k , … , x 0 ∈ ( - ∞ , 0 ] . It is shown that the unique negative equilibrium of the equation is a global attractor with a basin that depends on certain conditions of the coefficient.