Electron. J. Diff. Equ., Vol. 2014 (2014), No. 264, pp. 1-14.

Solvability of nonlinear difference equations of fourth order

Stevo Stevic, Josef Diblik, Bratislav Iricanin, Zdenek Smarda

Abstract:
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.

Submitted September 21, 2014. Published December 22, 2014.
Math Subject Classifications: 39A10, 39A20.
Key Words: Solution to difference equation; long-term behavior of solutions; undefinable solutions.

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Stevo Stevic
Mathematical Institute of the Serbian Academy of Sciences
Knez Mihailova 36/III, 11000 Beograd, Serbia.
Department of Mathematics, King Abdulaziz University
P.O. Box 80203, Jeddah 21589, Saudi Arabia
email: sstevic@ptt.rs
Josef Diblik
Department of Mathematics and Descriptive Geometry
Faculty of Civil Engineering, 60200, Brno University of Technology
Brno, Czech Republic
email: diblik.j@fce.vutbr.cz, diblik@feec.vutbr.cz
Bratislav Iricanin
Faculty of Electrical Engineering
Belgrade University, Bulevar Kralja Aleksandra 73
11000 Beograd, Serbia
email: iricanin@etf.rs
Zdenek Smarda
Department of Mathematics, Faculty of Electrical Engineering and Communication
61600, Brno University of Technology
Brno, Czech Republic
email: smarda@feec.vutbr.cz

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