Semnan University International Journal of Nonlinear Analysis and Applications 2008-6822 2026 06 19 On solution of a difference equation via generalized Fibonacci sequence 10740 10.22075/ijnaa.2025.39381.5561 EN Ömer Aktaş Instutite of Science, Karamanoğlu Mehmetbey University, Turkey 0000-0002-5763-0308 Merve Kara Department of Mathematics, Kamil Ozdag Science Faculty, Turkey 0000-0001-8081-0254 Journal Article 2025 10 14 We consider the following the difference equation<br />\begin{equation*}<br />w_n=\frac{w^{r+1}_{n-2}w_{n-3}}{w_{n-1}\left(\gamma_nw_{n-4}^r+\delta_nw_{n-2}w_{n-3}\right)},\quad n\in\mathbb{N}_0, <br />\end{equation*}<br />where $r\in\mathbb{N},$ the initial conditions $w_{-j}$, $j=\overline{1,4}$ are non zero real numbers and $\left(\gamma_n\right)_{n\in\mathbb{N}_0}$, $\left(\delta_n\right)_{n\in\mathbb{N}_0}$ are non zero real number sequences. In addition, the<br />solution of a more general difference equation defined by one to one continuous function is obtained. The solution of the mentioned equation is gained via a generalized Fibonacci sequence. Finally, we obtain the solution of the above difference equation when the sequences $\left(\gamma_n\right)_{n\in\mathbb{N}_0}$, $\left(\delta_n\right)_{n\in\mathbb{N}_0}$ are constant. https://ijnaa.semnan.ac.ir/article_10740_f75ce735a80f6ac091aea546866fb4bb.pdf