Forbidden Set of the Rational Difference Equation $x_{n+1}=x_n{x}_{n-k}/(a{x}_{n-k+1}+x_n{x}_{n-k+1}{x}_{n-k})$

Abstract

This short note aims to answer one of the open problems raised by F. Balibrea and A. Cascales in \cite{bc}. In particular, the forbidden set of the nonlinear difference equation $x_{n+1}=x_n{x}_{n-k}/(a{x}_{n-k+1}+x_n{x}_{n-k+1}{x}_{n-k})$, where $k$ is a positive integer and $a$ is a positive constant, is found by first computing the closed form solution of the given equation. Additional results regarding the limiting properties and periodicity of its solutions are also discussed. Numerical examples are also provided to illustrate the exhibited results. Lastly, a possible generalization of this present work is offered as an open problem.

arXiv commentBuletinul Academiei de Ştiinţe a Republicii Moldova. Matematica

References in corpus (2)

• On invariants and forbidden sets
• On the Closed-Form Solution of a Nonlinear Difference Equation and Another Proof to Sroysang's Conjecture