On the dynamics of a certain four-order fractional difference equations

This paper is concerned with the following rational recursive sequences x(n+1) = x(n-1)x(n-2)/A+By(n-3), y(n+1) = y(n-1)y(n-2)/C+Dx(n-3), n = 0,1,.., where the parameters A, B, C, D are positive constants. The initial condition x(-3), x(-2), x(-1), x(0), and y(-3), y(-2), y(-1), y(0) are arbitrary nonnegative real numbers. We give sufficient conditions under which the equilibrium (0,0) of the system is globally asymptotically stable, which extends and includes corresponding results obtained in the cited references [12-17]. Moreover, the asymptotic behavior of others equilibrium points is also studied. Our approach to the problem is based on new variational iteration method for the more general nonlinear difference equations and inequality skills as well as the linearization techniques.