Solvability of nonlinear difference equations of fourth order

In this article we show the existence of solutions to the nonlinear difference equation $$ x_n=\frac{x_{n-3}x_{n-4}}{x_{n-1}(a_n+b_nx_{n-2}x_{n-3}x_{n-4})}, \quad n\in\mathbb{N}_0, $$ where the sequences $(a_n)_{n\in\mathbb{N}_0}$ and $(b_n)_{n\in\mathbb{N}_0}$, and initial the values $x_{-j}$, $j=\...

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Bibliographic Details
Published in:Electronic journal of differential equations 2014-12, Vol.2014 (264), p.1-14
Main Authors: Stevo Stevic, Josef Diblik, Bratislav Iricanin, Zdenek Smarda
Format: Article
Language:English
Subjects:
long-term behavior of solutions
Solution to difference equation
undefinable solutions
Online Access:Get full text
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Summary:In this article we show the existence of solutions to the nonlinear difference equation $$ x_n=\frac{x_{n-3}x_{n-4}}{x_{n-1}(a_n+b_nx_{n-2}x_{n-3}x_{n-4})}, \quad n\in\mathbb{N}_0, $$ where the sequences $(a_n)_{n\in\mathbb{N}_0}$ and $(b_n)_{n\in\mathbb{N}_0}$, and initial the values $x_{-j}$, $j=\overline{1,4}$, are real numbers. Also we find the set of initial values for which solutions are undefinable when $a_n\ne 0$ and $b_n\neq 0$ for every $n\in\mathbb{N}_0$. When these two sequences are constant, we describe the long-term behavior of the solutions in detail.
ISSN:1072-6691