**Journal of Prime Research in Mathematics**

Vol. 17 (2021), Issue 1, pp. 59 – 69

ISSN: 1817-3462E (Online) 1818-5495 (Print)

ISSN: 1817-3462E (Online) 1818-5495 (Print)

# On a third-order fuzzy difference equation

**Ibrahim Yalcinkaya\(^{a,*}\), Nur Atak\(^b\), Durhasan Turgut Tollu\(^a\)**

\(^a\)University, Faculty of Science, Department of Mathematics and Computer Sciences, Konya, Turkey.

\(^b\)Sarayonu Multi-Program Anatolian High School, Konya, Turkey.

Correspondence should be addressed to: Ibrahim Yalcinkaya at iyalcinkaya@erbakan.edu.tr

Copyright © 2021 Ibrahim Yalcinkaya, Nur Atak, Durhasan Turgut Tollu. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

**Published:**Received: 15 March 2020; Accepted: 5 December 2020; Published Online: 28 June 2021.

### Abstract

In this paper, we investigate the qualitative behavior of the fuzzy

difference equation

\begin{equation*}

z_{n+1}=\frac{z_{n-2}}{C+z_{n-2}z_{n-1}z_{n}}\

\end{equation*}

where \(n\in \mathbb{N}_{0}=\mathbb{N}\cup \left\{ 0\right\}\), \((z_{n})\) is a sequence of positive fuzzy numbers, \(C\) and initial conditions \(z_{-2},z_{-1},z_{0}\) are positive fuzzy numbers.

difference equation

\begin{equation*}

z_{n+1}=\frac{z_{n-2}}{C+z_{n-2}z_{n-1}z_{n}}\

\end{equation*}

where \(n\in \mathbb{N}_{0}=\mathbb{N}\cup \left\{ 0\right\}\), \((z_{n})\) is a sequence of positive fuzzy numbers, \(C\) and initial conditions \(z_{-2},z_{-1},z_{0}\) are positive fuzzy numbers.

#### Keywords:

Fuzzy difference equations, Existence of solutions, Boundedness, Convergence.