Asymptotic behavior of a system of linear fractional difference equations.

Kulenović, M.R.S.; Nurkanović, M.

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Asymptotic behavior of a competitive system of linear fractional difference equations.

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On the system of rational difference equations $x_{n + 1} = f (y_{n - q}, x_{n - s})$ , $y_{n + 1} = g (x_{n - t}, y_{n - p})$ .

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Global asymptotic behavior of $y_{n + 1} = (p y_{n} + y_{n - 1}) / (r + q y_{n} + y_{n - 1})$ .

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On the behavior of solutions of the system of rational difference equations: $x_{n + 1} = x_{n - 1} / (y_{n} x_{n - 1} - 1)$ , $y_{n + 1} = y_{n - 1} / (x_{n} y_{n - 1} - 1)$ , and $z_{n + 1} = z_{n - 1} / (y_{n} z_{n - 1} - 1)$ .

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Some boundedness results for systems of two rational difference equations

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We study k th order systems of two rational difference equations $x_{n} = \frac{α + \sum_{i = 1}^{k} β_{i} x_{n - 1} + \sum_{i = 1}^{k} γ_{i} y_{n - 1}}{A + \sum_{j = 1}^{k} B_{j} x_{n - j} + \sum_{j = 1}^{k} C_{j} y_{n - j}}, y_{n} = \frac{p + \sum_{i = 1}^{k} δ_{i} x_{n - i} + \sum_{i = 1}^{k} ε_{i} y_{n - i}}{q + \sum_{j = 1}^{k} D_{j} x_{n - j} + \sum_{j = 1}^{k} E_{j} y_{n - j}} n \in ℕ$ . 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.

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