A qualitative investigation of a system of third-order difference equations with multiplicative reciprocal terms

Year 2024, Volume: 6 Issue: 2, 30 - 44

Durhasan Turgut Tollu , İbrahim Yalçınkaya

https://doi.org/10.54286/ikjm.1524180

Abstract

In this paper, we study the system of third-order difference equations
\begin{equation*}
x_{n+1}=a+\frac{a_{1}}{y_{n}}+\frac{a_{2}}{y_{n-1}}+\frac{a_{3}}{y_{n-2}}%
,\quad y_{n+1}=b+\frac{b_{1}}{x_{n}}+\frac{b_{2}}{x_{n-1}}+\frac{b_{3}}{%
x_{n-2}},\quad n\in \mathbb{N}_{0},
\end{equation*}%
where the parameters $a$, $a_{i}$, $b$, $b_{i}$, $i=1,2,3$, and the initial
values $x_{-j}$, $y_{-j}$, $j=0,1,2$, are positive real numbers. We first
prove a general convergence theorem. By applying this convergence theorem to
the system, we show that positive equilibrium is a global attractor. We also
study the local asymptotic stability of the equilibrium and show that it is
globally asymptotically stable. Finally, we study the invariant set of
solutions.

Keywords

Boundedness, equilibrium point, system of difference equations.

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