On a Third-Order System of Difference Equations with Variable Coefficients

Abstract

We show that the system of three difference equations $x_{n+1}=a_n^{\left(1\right)}{x}_{n-2}/\left(b_n^{\left(1\right)}{y}_n{z}_{n-1}{x}_{n-2}+c_n^{\left(1\right)}\right)$, $y_{n+1}=a_n^{\left(2\right)}{y}_{n-2}/\left(b_n^{\left(2\right)}{z}_n{x}_{n-1}{y}_{n-2}+c_n^{\left(2\right)}\right)$, and $z_{n+1}=a_n^{\left(3\right)}{z}_{n-2}/\left(b_n^{\left(3\right)}{x}_n{y}_{n-1}{z}_{n-2}+c_n^{\left(3\right)}\right)$, $n\in {\mathbb{N}}_0$, where all elements of the sequences $a_n^{\left(i\right)}$, $b_n^{\left(i\right)}$, $c_n^{\left(i\right)}$, $n\in {\mathbb{N}}_0$, $i\in \{\mathrm{1,2},3\}$, and initial values $x_{-j}$, $y_{-j}$, $z_{-j}$, $j\in \{\mathrm{0,1},2\}$, are real numbers, can be solved. Explicit formulae for solutions of the system are derived, and some consequences on asymptotic behavior of solutions for the case when coefficients are periodic with period three are deduced.