H. El-Metwally - Department of Mathematics, Faculty of Science, Mansoura University, Mansoura, 35516, Egypt. M. T. Alharthi - Department of Mathematics, Faculty of Science, Jeddah University, Jeddah, Saudi Arabia.
In this paper we study some qualitative properties of the solutions for the following difference equation \[ y_{n+1}=\frac{\alpha +\alpha _{0} y_{n}^{r}+\alpha _{1} y_{n-1}^{r}+\cdots+\alpha _{k} y_{n-k}^{r}}{\beta +\beta _{0} y_{n}^{r}+\beta _{1} y_{n-1}^{r}+\cdots+\beta _{k} y_{n-k}^{r}}% ,~~~~n\geq 0,\tag{I} \] where \(r, \alpha , \alpha _{0}, \alpha _{1},\ldots, \alpha _{k}, \beta , \beta _{0}, \beta _{1},\ldots, \beta _{k}\in (0,\infty )\) and \(k \) is a non-negative integer number. We find the equilibrium points for the considered equation. Then classify these points in terms of local stability or not. We investigate the boundedness and the global stability of the solutions for the considered equation. Also we study the existence of periodic solutions of Eq. (I).
H. El-Metwally, M. T. Alharthi, A qualitative investigation of some rational difference equations, Journal of Mathematics and Computer Science, 34 (2024), no. 1, 1--10
El-Metwally H., Alharthi M. T., A qualitative investigation of some rational difference equations. J Math Comput SCI-JM. (2024); 34(1):1--10
El-Metwally, H., Alharthi, M. T.. "A qualitative investigation of some rational difference equations." Journal of Mathematics and Computer Science, 34, no. 1 (2024): 1--10