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

In this article we show the existence of solutions to the nonlinear difference equation
 \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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