We present a global attractivity result for maps generated by systems of autonomous difference equations. It is assumed that the map of the system leaves invariant a box, is monotone in a coordinate-wise sense (but not necessarily monotone with respect to a standard cone), and satisfies certain algebraic condition. It is shown that there exists a unique equilibrium, and that it is a global attractor. As an application, it is shown that a discretized version of the Lotka-Volterra system of differential equations of order $k$ has a global attractor in the positive orthant for certain range of parameters.

Abstract We investigate the stability of solutions of the Gumowski–Mira equation with a period-two coefficient: y n + 1 = y n b n + y n 2 − y n − 1 , n = 0 , 1 , … , with b n = { α ⩾ 0 for  n = 2 k , β ⩾ 0 for  n = 2 k + 1 , k = 0 , 1 , … , and the initial values y −1 , y 0 are real numbers.

We investigate the global asymptotic behavior of solutions of the system of difference equations,,, where the parameters,,, and are positive numbers and the initial conditions and are arbitrary nonnegative numbers. We obtain some asymptotic results for the positive equilibrium of this system.

We investigate the global character of solutions of the system of difference equations with positive parameters and non-negative initial conditions.

Abstract Consider the nonlinear, nonautonomous differential equation y ″ ( t ) + p ( t ) f ( y ( t ) ) = 0 , t ≥ t 0 where p is a continuous, positive function and f is a continuous function such that u f ( u ) > 0 for u ≠ 0 , and ∫ 0 ± ∞ f ( u ) d u = ∞ . We obtain some results about the asymptotic behavior of the amplitudes of all oscillatory solutions of this equation. Combining this result with some known oscillation results, we obtain the global results for all solutions of our equation.

Our aim here is to present a summary of our recent work and a large number of open problems and conjectures on third order rational difference equations of the form with non-negative parameters and non-negative initial conditions.

The role of compensatory and overcompensatory dynamics in generating multiple attractors in density-dependent Leslie models with or without the Allee effects are studied. In the presence of the Allee effect the models support multiple attractors. However, in the absence of the Allee effect single attractors are supported when the dynamics are compensatory while multiple attractors are supported under overcompensatory dynamics. The existence of multiple attractors in density-dependent Leslie models implies that the qualitative population dynamics depend on initial conditions.

We investigate the periodic nature, the boundedness character and the global asymptotic stability of solutions of the difference equation where the parameter p n is a period-two sequence with positive values and the initial conditions are positive.

We investigate the global character of solutions of the equation in the title with positive parameters and non-negative initial conditions.

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 section, we present some open problems and conjectures about some interesting types of difference equations. Please submit your problems and conjectures with all relevant information to G. Ladas.

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.

Abstract Consider the system of neutral delay differential equations d d t ( x (t)+p x (t−τ))+Q(t) x (t−σ)= 0 , where p∈ R ,τ∈(0,∞),σ∈[0,∞] and Q is continuous matrix, and the system d d t ( x (t)+B x (t−τ))+Q(t) x (t−σ)= 0 , where B is a matrix. We obtain the sufficient condition for the existence of certain types of solutions of the above equation to be ∫ ∞ ||Q(s)|| d s for p≠−1 .

We investigate the global asymptotic behavior of solutions of the system of difference equations xn+1 = xn/ a + cyn, yn+1 = yn/ b + dxn, n =0,1,..., where the parameters a and b are in (0, 1), c and d are arbitrary positive numbers and the initial conditions x0 and y0 are arbitrary nonnegative numbers. We show that the stable manifold of this system separates the positive quadrant into basins of attraction of two types of asymptotic behavior. In the case where a = b we find an explicit equation for the stable manifold.