MMN-3365

Solutions formulas for some general systems of difference equations

  • Y. Akrour, MAM laboratory, Department of Mathematics, University of Mohamed Seddik Ben Yahia, Jijel, and Departement des Sciences Exactes et d'Informatique, École Normale Supérieure, Constantine, Algeria., youssouf.akrour@gmail.com
  • M. Kara, Department of Mathematics, Kamil Ozdag Science Faculty, Karamanoglu Mehmetbey University, Karaman, Turkey, mervekara@kmu.edu.tr
  • N. Touafek, LMAM Laboratory, Department of Mathematics, University of Mohamed Seddik Ben Yahia, Jijel, Algeria, ntouafek@gmail.com
  • Y. Yazlik, Department of Mathematics, Faculty of Science and Art, Nevsehir Haci Bektas Veli University, Nevsehir, Turkey, yyazlik@nevsehir.edu.tr

Abstract

In this paper, we give explicit formulas of the solutions of the two general systems of difference equations \begin{equation*} x_{n+1}=f^{-1} \left( ag(y_n)+bf(x_{n-1})+cg(y_{n-2}) +df(x_{n-3}) \right), \end{equation*} \begin{equation*} y_{n+1}=g^{-1} \left( af(x_n)+bg(y_{n-1})+cf(x_{n-2})+dg(y_{n-3}) \right), \end{equation*} and \begin{equation*} x_{n+1}=f^{-1} \left( a+\frac{b}{g(y_n)}+ \frac{c}{g(y_n)f(x_{n-1})}+\frac{d}{g(y_n)f(x_{n-1})g(y_{n-2)}} \right), \end{equation*} \begin{equation*} y_{n+1}=g^{-1} \left( a+\frac{b}{f(x_n)}+ \frac{c}{f(x_n)g(y_{n-1})}+\frac{d}{f(x_n)g(y_{n-1})f(x_{n-2)}} \right), \end{equation*} where $n\in \mathbb{N}_0$, $f, g : D \longrightarrow \mathbb{R}$ are a $ ``1-1" $ continuous functions on $D\subseteq \mathbb{R}$, the initial values $x_{-i}$, $y_{-i}$, $i=0,1,2,3$ are arbitrary real numbers in $D$ and the parameters $a$, $b$, $c$ and $d$ are arbitrary real numbers. Our results considerably extend some existing results in the literature.


Vol. 22 (2021), No. 2, pp. 529-555
DOI: 10.18514/MMN.2021.3365


Download: MMN-3365