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Published in:
2020 | OriginalPaper | Chapter
On the Boundedness Character of a Rational System of Difference Equations with Non-constant Coefficients
Authors : Yevgeniy Kostrov, Zachary Kudlak, Patrick Vernon
Published in: Difference Equations and Discrete Dynamical Systems with Applications
Publisher: Springer International Publishing
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Abstract
We investigate the boundedness character of nonnegative solutions of the nonautonomous rational system where the coefficient sequences \(\{\alpha _n\}\), \(\{\beta _n\}\) and \(\{\gamma _n\}\) are bounded both above and below by positive numbers, the initial conditions \((x_0, y_0), (x_{-1},y_{-1}),\ldots , (x_{-k+1},y_{-k+1})\) are positive, and g takes on only positive values for positive values of \(x_n,\ldots , x_{n-k+1}, y_n, y_{n-k+1}\) and nonnegative integers n. Special cases of this system, such as with periodic coefficients, are also investigated.
$$ \left\{ \begin{array}{l} x_{n+1}=\displaystyle \frac{\alpha _n + \gamma _n x_n}{\beta _n x_n + y_n}\\ \\ y_{n+1}= g(x_n,\ldots ,x_{n-k+1},y_n,\ldots , y_{n-k+1}, n) \end{array}\right. \text { for } n=0,1,\ldots $$