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 investigate the global dynamics of solutions of four distinct competitive rational systems of difference equations in the plane. 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 nonhyperbolic equilibrium points. Our results give complete answer to Open Problem 2 posed recently by Camouzis et al. (2009).

Let T be a competitive map on a rectangular region R ⊂ R 2 , and assume T is C 1 in a neighborhood of a fixed point x ∈ R. The main results of this paper give conditions on T that guarantee the existence of an invariant curve emanating from x when both eigenvalues of the Jacobian of T at x are nonzero and at least one of them has absolute value less than one, and establish that C is an increasing curve that separates R into invariant regions. The results apply to many hyperbolic and nonhyperbolic cases, and can be effectively used to determine basins of attraction of fixed points of competitive maps, or equivalently, of equilibria of competitive systems of difference equations. Several applications to planar systems of difference equations with non-hyperbolic equilibria are given.

A global bifurcation result is obtained for families of competitive systems of difference equations $x_{n+1} = f_\alpha(x_n,y_n) $ $y_{n+1} = g_\alpha(x_n,y_n)$ where $\alpha$ is a parameter, $f_\alpha$ and $g_\alpha$ are continuous real valued functions on a rectangular domain $\mathcal{R}_\alpha \subset \mathbb{R}^2$ such that $f_\alpha(x,y)$ is non-decreasing in $x$ and non-increasing in $y$, and $g_\alpha(x, y)$ is non-increasing in $x$ and non-decreasing in $y$. A unique interior fixed point is assumed for all values of the parameter $\alpha$. As an application of the main result for competitive systems a global period-doubling bifurcation result is obtained for families of second order difference equations of the type $x_{n+1} = F_\alpha(x_n, x_{n-1}), \quad n=0,1, \ldots $ where $\alpha$ is a parameter, $F_\alpha:\mathcal{I_\alpha}\times \mathcal{I_\alpha} \rightarrow \mathcal{I_\alpha}$ is a decreasing function in the first variable and increasing in the second variable, and $\mathcal{I_\alpha}$ is a interval in $\mathbb{R}$, and there is a unique interior equilibrium point. Examples of application of the main results are also given.

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.

Professor Gerry Ladas has had a long and distinguished career in mathematics. He received a B.S. degree in Mathematics from the University of Athens, Greece, in 1961. He received a M.S. degree in Mathematics from New York University’s Courant Institute in 1966 and a Ph.D. in Mathematics from the Courant Institute in 1968. He was an Assistant Professor of Mathematics at Fairfield University during the academic year 1968–1969. In the fall of 1969, he joined the faculty of the Mathematics Department of the University of Rhode Island, where he remains today as a Professor of Mathematics. During his stay at the University of Rhode Island, he has been the major professor of 21 Ph.D. students and is currently the major professor of 2 Ph.D. candidates. Professor Ladas is one of the very few recipients in the history of the University of Rhode Island of both the university wide Excellence in Teaching Award (1987) and the Excellence in Research Award (1996) at the University of Rhode Island. Professor Ladas is a Editor-in-Chief (with Saber Elaydi) of this journal and an Associate Editor of 13 other mathematics journals. He is a referee for numerous journals, as well as a referee of mathematics research proposals for different research granting agencies. So far, he has edited the proceedings of eight different conferences.

Sharkovsky’s Theorem and the ‘Period three implies chaos’ result of Li and Yorke, are profound and deep results showing the rich periodic character of first-order, non-linear difference equations. The (3x þ 1)-conjecture of L. Collatz is one of the most challenging problems of contemporary mathematics. In the authors’ words, ‘During the last ten years, we have been fascinated discovering non-linear difference equations of order greater than one which for certain values of their parameters have one of the following characteristics:

We investigate the global stability character of the equilibrium points and the period-two solutions of yn+1 = (pyn + yn−1)/(r + qyn + yn−1), n = 0,1, . . . , with positive parameters and nonnegative initial conditions. We show that every solution of the equation in the title converges to either the zero equilibrium, the positive equilibrium, or the period-two solution, for all values of parameters outside of a specific set defined in the paper. In the case when the equilibrium points and period-two solution coexist, we give a precise description of the basins of attraction of all points. Our results give an affirmative answer to Conjecture 9.5.6 and the complete answer to Open Problem 9.5.7 of Kulenović and Ladas, 2002.

We study global attractivity of the period-two coefficient version of the delay logistic difference equation, also known as Pielou's equation, where We prove that for , zero is the unique equilibrium point. If , then zero is globally asymptotically stable, with basin of attraction given by the nonnegative quadrant of initial conditions. If , then zero is unstable, and a sequence converges to zero if and only if . If , then the sequence converges to the unique period-two solution where and are uniquely determined by the equations

We first investigate the Lyapunov stability of the period-three solution of Todd's equation with a period-three coefficient: where α,β, and γ positive. Then for k = 2,3,… we extend our stability result to the k-order equation, where pn is a periodic coefficient of period k with positive real values and x-k+1,…,x-1, x0 ∈ (0, ∞).