Global dynamics of perturbation of certain rational difference equation

Abstract

We investigate the global asymptotic stability of the difference equation of the form $x_{n+1;=;frac{Ax_n^2+F}{ax_a^2+ex_{n-1}}}$, n = 0; 1,......, with positive parameters and nonnegative initial conditions such that x0+x􀀀1 > 0. The map associated to this equationis always decreasing in the second variable and can be either increasing or decreasing in the first variable depending onthe parametric space. In some cases, we prove that local asymptotic stability of the unique equilibrium point impliesglobal asymptotic stability.