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Boundedness character of two classes of third-order difference equations. (English) Zbl 1182.39012

The author studies the boundedness character of positive solutions of the following difference equations \[ x_{n+1}=A+\frac{x_{n}^{p}}{x_{n-2}^{r}}\quad \text{and}\quad x_{n+1}=\max \left\{ A,\frac{x_{n}^{p}}{x_{n-2}^{r}}\right\} , \] \(n\in \mathbb{N}_{0}\), where the parameters \(A\) and \(r\) are positive and \(p\) is nonnegative real numbers. Two cases are considered: \(4p^{3}/27<r\) and \(4p^{3}/27\geq r\). The quantity \(r-4p^{3}/27\) arises as the value of the polynomial \(\lambda ^{3}-p\lambda ^{2}+r\) at its minimum \(\lambda =2p/3\). The polynomial is the characteristic polynomial of the equations by taking \( A=0\) and the logarithm of such obtained equation.

MSC:

39A22 Growth, boundedness, comparison of solutions to difference equations
39A20 Multiplicative and other generalized difference equations
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