Browsing by Subject "difference equation"

Now showing 1 - 3 of 3

Access status: Open Access ,
Further properties of the rational recursive sequence xn + 1 = axn - 1 / (b + cxnxn - 1)
(2006) Andruch-Sobiło, Anna; Migda, Małgorzata
In this paper we consider the difference equation $x_{n+1}=\frac{ax_{n-1}}{b+cx_{n}x_{n-1}}, \quad n=0,1,...(E)$ with positive parameters $a$ and $c$, negative parameter $b$ and nonnegative initial conditions. We investigate the asymptotic behavior of solutions of equation $\text{(E)}$.
Access status: Open Access ,
Periodic, nonperiodic, and chaotic solutions for a class of difference equations with negative feedback
(Wydawnictwa AGH, 2023) Kennedy, Benjamin B.
We study the scalar difference equation $x(k+1) = x(k) + \frac{f(x(k-N))}{N},$ where $f$ is nonincreasing with negative feedback. This equation is a discretization of the well-studied differential delay equation $x'(t) = f(x(t-1)).$ We examine explicit families of such equations for which we can find, for infinitely many values of $ and appropriate parameter values, various dynamical behaviors including periodic solutions with large numbers of sign changes per minimal period, solutions that do not converge to periodic solutions, and chaos. We contrast these behaviors with the dynamics of the limiting differential equation. Our primary tool is the analysis of return maps for the difference equations that are conjugate to continuous self-maps of the circle.
Access status: Open Access ,
Right focal boundary value problems for difference equations
(2010) Henderson, Johnny; Liu, Xueyan; Lyons, Jeffrey W.; Neugebauer, Jeffrey T.
An application is made of a new Avery et al. fixed point theorem of compression and expansion functional type in the spirit of the original fixed point work of Leggett and Williams, to obtain positive solutions of the second order right focal discrete boundary value problem. In the application of the fixed point theorem, neither the entire lower nor entire upper boundary is required to be mapped inward or outward. A nontrivial example is also provided.