Peter M. Knopf

Global convergence properties of first-order homogeneous systems of rational difference equations

We establish the global convergence properties of the homogeneous system for all non-negative coefficients and C 2, and for all positive initial conditions. In order to prove these results, we reduce the problem to the study of a first-order rational difference equation of one variable that is quadratic in the numerator and the denominator. We also prove the convergence properties for a more general class of first-order equations, which we apply to our rational difference equation.

On the boundedness character of some rational difference equations

We prove the boundedness of all positive solutions to special cases of rational difference equations of the form where α, A, β i , and B i are nonnegative constants. If these results are applied to fourth-order equations, they confirm a conjecture in a paper by Camouzis et al. for 16 cases.

Boundedness properties of the difference equation x n+1=(α+βx n +δx n−2)/(x n−1)

Consider the third-order difference equation x n+1 = (α+βx n +δx n − 2)/(x n − 1) with α ∈ [0,∞) and β,δ ∈ (0,∞). It is shown that this difference equation has unbounded solutions if and only if δ>β.

Boundedness and some convergence properties of the difference equation

For the difference equation with positive parameters given in the title, we show that the solutions are bounded for all positive initial conditions whenever . Furthermore, if and for a certain class of initial conditions, then the solutions converge to period-two solutions.

On the boundedness and local stability of

We show that every positive solution of the difference equation in the title is bounded whenever and . This confirms a conjecture posed by Amleh et al.. We also establish local asymptotic stability properties of the difference equation.

On the boundedness of solutions of a class of third-order rational difference equations

ABSTRACT We consider the difference equation with and If we show that for all positive initial conditions the solutions of the difference equations are bounded. If then there exists positive initial conditions such that the solutions are unbounded.

Mappings with a single critical point and applications to rational difference equations

Convergence properties of first-order difference equations of the form are established for a general class of mappings f, where f has at most one critical point. Using these results, we find necessary and sufficient conditions for the convergence of the solutions for all difference equations of the formfor all possible choices of non-negative coefficients and positive initial values.

Contracting invariant intervals and convergence properties of difference equations

In this paper, we consider kth order difference equations of the form where every function is continuous and monotonic in each of its arguments. We generalize and improve upon some previous techniques to prove convergence using invariant intervals as the common theme. We apply our techniques to several different special cases of the equation above.

On the period-five trichotomy of the rational equation

We study the behavior of the third order difference equation for any p>0 and any positive initial conditions x − 2, x − 1, x 0. We show that the positive equilibrium , where , is globally asymptotically stable if and only if . This result, together with the known results of Camouzis et al., confirm the conjecture that the difference equation above has a period-five trichotomy.

Projection Theorems for Harmonic Measure in NTA Domains

AbstractSuppose D is an NTA domain, E