# Some boundedness results for systems of two rational difference equations

Open Mathematics (2010)

• Volume: 8, Issue: 6, page 1058-1090
• ISSN: 2391-5455

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## Abstract

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We study k th order systems of two rational difference equations ${x}_{n}=\frac{\alpha +{\sum }_{i=1}^{k}{\beta }_{i}{x}_{n-1}+{\sum }_{i=1}^{k}{\gamma }_{i}{y}_{n-1}}{A+{\sum }_{j=1}^{k}{B}_{j}{x}_{n-j}+{\sum }_{j=1}^{k}{C}_{j}{y}_{n-j}},{y}_{n}=\frac{p+{\sum }_{i=1}^{k}{\delta }_{i}{x}_{n-i}+{\sum }_{i=1}^{k}{\epsilon }_{i}{y}_{n-i}}{q+{\sum }_{j=1}^{k}{D}_{j}{x}_{n-j}+{\sum }_{j=1}^{k}{E}_{j}{y}_{n-j}}n\in ℕ$ . In particular, we assume non-negative parameters and non-negative initial conditions, such that the denominators are nonzero. We develop several approaches which allow us to extend well known boundedness results on the k th order rational difference equation to the setting of systems in certain cases.

## How to cite

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Gabriel Lugo, and Frank Palladino. "Some boundedness results for systems of two rational difference equations." Open Mathematics 8.6 (2010): 1058-1090. <http://eudml.org/doc/269538>.

@article{GabrielLugo2010,
abstract = {We study k th order systems of two rational difference equations $x\_n = \frac\{\{\alpha + \sum \nolimits \_\{i = 1\}^k \{\beta \_i x\_\{n - 1\} + \} \sum \nolimits \_\{i = 1\}^k \{\gamma \_i y\_\{n - 1\} \} \}\}\{\{A + \sum \nolimits \_\{j = 1\}^k \{B\_j x\_\{n - j\} + \} \sum \nolimits \_\{j = 1\}^k \{C\_j y\_\{n - j\} \} \}\}, y\_n = \frac\{\{p + \sum \nolimits \_\{i = 1\}^k \{\delta \_i x\_\{n - i\} + \} \sum \nolimits \_\{i = 1\}^k \{\varepsilon \_i y\_\{n - i\} \} \}\}\{\{q + \sum \nolimits \_\{j = 1\}^k \{D\_j x\_\{n - j\} + \} \sum \nolimits \_\{j = 1\}^k \{E\_j y\_\{n - j\} \} \}\} n \in \mathbb \{N\}$ . In particular, we assume non-negative parameters and non-negative initial conditions, such that the denominators are nonzero. We develop several approaches which allow us to extend well known boundedness results on the k th order rational difference equation to the setting of systems in certain cases.},
author = {Gabriel Lugo, Frank Palladino},
journal = {Open Mathematics},
keywords = {Difference equation; Systems; Boundedness character; systems; boundedness; rational difference equations},
language = {eng},
number = {6},
pages = {1058-1090},
title = {Some boundedness results for systems of two rational difference equations},
url = {http://eudml.org/doc/269538},
volume = {8},
year = {2010},
}

TY - JOUR
AU - Gabriel Lugo
TI - Some boundedness results for systems of two rational difference equations
JO - Open Mathematics
PY - 2010
VL - 8
IS - 6
SP - 1058
EP - 1090
AB - We study k th order systems of two rational difference equations $x_n = \frac{{\alpha + \sum \nolimits _{i = 1}^k {\beta _i x_{n - 1} + } \sum \nolimits _{i = 1}^k {\gamma _i y_{n - 1} } }}{{A + \sum \nolimits _{j = 1}^k {B_j x_{n - j} + } \sum \nolimits _{j = 1}^k {C_j y_{n - j} } }}, y_n = \frac{{p + \sum \nolimits _{i = 1}^k {\delta _i x_{n - i} + } \sum \nolimits _{i = 1}^k {\varepsilon _i y_{n - i} } }}{{q + \sum \nolimits _{j = 1}^k {D_j x_{n - j} + } \sum \nolimits _{j = 1}^k {E_j y_{n - j} } }} n \in \mathbb {N}$ . In particular, we assume non-negative parameters and non-negative initial conditions, such that the denominators are nonzero. We develop several approaches which allow us to extend well known boundedness results on the k th order rational difference equation to the setting of systems in certain cases.
LA - eng
KW - Difference equation; Systems; Boundedness character; systems; boundedness; rational difference equations
UR - http://eudml.org/doc/269538
ER -

## References

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1. [1] Camouzis E., Kulenovic M.R.S., Ladas G., Merino O., Rational systems in the plane, J. Difference Equ. Appl., 2009, 15(3), 303–323 http://dx.doi.org/10.1080/10236190802125264 Zbl1169.39010
2. [2] Camouzis E., Ladas G., Global results on rational systems in the plane, part 1, J. Difference Equ. Appl., 2010, 16(8), 975–1013 http://dx.doi.org/10.1080/10236190802649727 Zbl1218.39001
3. [3] Camouzis E., Ladas G., Palladino F., Quinn E.P., On the boundedness character of rational equations, part 1, J. Difference Equ. Appl., 2006, 12(5), 503–523 http://dx.doi.org/10.1080/10236190500539311 Zbl1104.39003
4. [4] Knopf P.M., Huang Y.S., On the boundedness character of some rational difference equations, J. Difference Equ. Appl., 2008, 14(7), 769–777 http://dx.doi.org/10.1080/10236190701852695 Zbl1153.39016
5. [5] Lugo G., Palladino F.J., Unboundedness results for systems, Cent. Eur. J. Math., 2009, 7(4), 741–756 http://dx.doi.org/10.2478/s11533-009-0051-2 Zbl1185.39001
6. [6] Palladino F.J., Difference inequalities, comparison tests, and some consequences, Involve, 2008, 1(1), 91–100 Zbl1154.39012

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