# Periodicities which preserve and periodicities which destroy boundedness

Original Research

First Online:

- 46 Downloads
- 1 Citations

## Abstract

It is known that every positive solution of the difference equation with positive parameter is bounded. In this note we study the difference equation . We show that every positive solution of this equation is bounded when {

*β*> 0$$
x_{n + 1} = \beta + \frac{{x_{n - 2} }}
{{x_n }}, n = 0,1, \ldots
$$

$$
x_{n + 1} = \beta _n + \frac{{x_{n - 2} }}
{{x_n }}, n = 0,1, \ldots
$$

*β*_{ n }}_{n=0}^{∞}is a period-2 sequence of positive real numbers, that is, “Period-2 Preserves Boundedness.” We also show that there exist prime period-3*m*, sequences {*β*_{ n }}_{n=0}^{∞}of positive real numbers such that the equation has unbounded solutions. That is,“Period-3*m*Destroys Boundedness.”## Keywords

Existence of unbounded solutions Periodic coefficients Rational difference equations## Mathematics Subject Classification (2000)

39A10 39A11## Preview

Unable to display preview. Download preview PDF.

## References

- 1.Camouzis E., On the boundedness of some rational difference equations,
*J. Difference Equ. Appl.*,**12**, 69–94, (2006)zbMATHCrossRefMathSciNetGoogle Scholar - 2.Camouzis E. and Ladas G., Dynamics of Third-Order Rational Difference Equations; With Open Problems and Conjectures, Chapman & Hall/CRC Press, November (2007)Google Scholar

## Copyright information

© Foundation for Scientific Research and Technological Innovation 2010