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On a third-order system of difference equations with variable coefficients. (English) Zbl 1242.39011
Summary: We show that the system of three difference equations \(x_{n+1} = a^{(1)}_n x_{n-2}/(b^{(1)}_n y_n z_{n-1} x_{n-2} + c^{(1)}_n), y_{n+1} = a^{(2)}_n y_{n-2}/(b^{(2)}_n z_n x_{n-1} y_{n-2} + c^{(2)}_n)\), and \(z_{n+1} = a^{(3)}_n z_{n-2}/(b^{(3)}_n x_n y_{n-1} z_{n-2} + c^{(3)}_n), n \in \mathbb N_0\), where all elements of the sequences \(a^{(i)}_n, b^{(i)}_n, c^{(i)}_n, n \in \mathbb N_0, i \in \{1, 2, 3\}\), and initial values \(x_{-j}, y_{-j}, z_{-j}, j \in \{0, 1, 2\}\), 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.

MSC:
39A12 Discrete version of topics in analysis
39A22 Growth, boundedness, comparison of solutions to difference equations
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