On a third order difference equation. (English) Zbl 1424.39020
Summary: In this paper, the authors solve the difference equation
\[
x_{n+1} =\frac{x_nx_{n-2}}{-ax_n + bx_{n-2}}, \quad n = 0, 1, \dots,
\]
where \(a\) and \(b\) are positive real numbers and the initial values \(x_{-2}, x_{-1}\) and \(x_0\) are real numbers. The authors find invariant sets and discuss the global behavior of the solutions of that equation. It is shown that when \(a > \frac{4}{27}b^3\), under certain conditions there exist solutions, either periodic or converge to periodic solutions.
MSC:
39A20 | Multiplicative and other generalized difference equations |
39A22 | Growth, boundedness, comparison of solutions to difference equations |
39A23 | Periodic solutions of difference equations |