We consider two systems of rational difference equations in the plane: and where the parameters are positive numbers and initial conditions and are positive numbers. For the first system we obtain the global dynamics, whereas for the second system we obtain substantial global results for all values of parameters. Global dynamics of two systems is substantially different to the contrary of their similarities.

We prove fixed point theorems for monotone mappings in partially ordered complete met- ric spaces which satisfy a weaker contraction condition for all points that are related by a given ordering. We also give a global attractivity result for all solutions of the difference equation

We prove fixed point theorems for mixed-monotone mappings in partially ordered complete metric spaces which satisfy a weaker contraction condition than the classical Banach contraction condition for all points that are related by given ordering. We also give a global attractivity result for all solutions of the difference equation , where satisfies mixed-monotone conditions with respect to the given ordering.

We investigate global dynamics of the following systems of difference equations , , , where the parameters , , , , and are positive numbers and initial conditions and are arbitrary nonnegative numbers such that . We show that this system has rich dynamics which depend on the part of parametric space. We show that the basins of attractions of different locally asymptotically stable equilibrium points are separated by the global stable manifolds of either saddle points or of nonhyperbolic equilibrium points.

We investigate the rate of convergence of solutions of some special cases of the equation , with positive parameters and nonnegative initial conditions. We give precise results about the rate of convergence of the solutions that converge to the equilibrium or period-two solution by using Poincaré's theorem and an improvement of Perron's theorem.

In this paper we prove that

We investigate the global character of solutions of the equation in the title with positive parameters and positive initial conditions. We obtain results about the global attractivity of the equilibrium, the existence and attractivity of the period-two solution and the semicycles.