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, ∞).

We investigate the global stability character of the equilibrium points and the period-two solutions of , 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 investigate the global asymptotic behavior of solutions of the system of difference equations xn+1 = (a+xn)/(b+yn), yn+1 = (d+yn)/(e+xn), n = 0,1,..., where the parameters a, b, d, and e are positive numbers and the initial conditions x0 and y0 are arbitrary nonnegative numbers. In certain range of parameters, we prove the existence of the global stable manifold of the unique positive equilibrium of this system which is the graph of an increasing curve. We show that the stable manifold of this system separates the positive quadrant of initial conditions into basins of attraction of two types of asymptotic behavior. In the case where a = d and b = e, we find an explicit equation for the stable manifold to be y = x.

We investigate the unbounded solutions of the second order difference equation where all parameters and C and initial conditions are nonnegative and such that for all n. We give a characterization of unbounded solutions for this equation showing that whenever an unbounded solution exists the subsequence of even indexed (resp. odd) terms tends to and the subsequence of odd indexed (resp. even) terms tends to a nonnegative number. We also show that two sets in the plane of initial conditions corresponding to the two cases are separated by the global stable manifold of the unique positive equilibrium. Our result answers two open problems posed by Kulenović and Ladas (2001, Dynamics of Second Order Rational Difference Equations with Open Problems and Conjectures, Boca Raton/London: Chapman and Hall/CRC).

We investigate global behavior of $x_{n+1} = T(x_{n}),\quad n=0,1,2,...$ (E) where $T:\mathcal{ R}\rightarrow \mathcal{ R}$ is a competitive (monotone with respect to the south-east ordering) map on a set $\mathcal{R}\subset \mathbb{R}^2$ with nonempty interior. We assume the existence of a unique fixed point $\overline{e}$ in the interior of $\mathcal{ R}$. We give very general conditions which are easily verifiable for (E) to exhibit either competitive-exclusion or competitive-coexistence. More specifically, we obtain sufficient conditions for the interior fixed point $\overline{ e}$ to be a global attractor when $\mathcal{ R}$ is a rectangular region. We also show that when $T$ is strongly monotone in $\mathcal{ R}^{\circ}$ (interior of $\mathcal{ R}$), $\mathcal{ R}$ is convex, the unique interior equilibrium $\overline{ e}$ is a saddle, and a technical condition is satisfied, the corresponding global stable and unstable manifolds are the graphs of monotonic functions, and the global stable manifold splits the domain into two connected regions, which under additional conditions on $\mathcal{R}$ and on $T$ are shown to be basins of attraction of fixed points on the boundary of $\mathcal{R}$. Applications of the main results to specific difference equations are given.

We investigate the global attractivity of the equilibrium of second-order difference equation where the parameters p, q, q < p and initial conditions x − 1, x 0 are nonnegative for all n. We prove that the unique equilibrium of this equation is global attractor which gives the affirmative answer to a conjecture of Kulenović and Ladas. The method of proof is innovative, and it has the potential to be used in the proof of global attractivity of equilibria of many similar equations.

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.

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.