On positive solutions of a (\(k+1\))th order difference equation (Q2488728)
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| Language | Label | Description | Also known as |
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| English | On positive solutions of a (\(k+1\))th order difference equation |
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On positive solutions of a (\(k+1\))th order difference equation (English)
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11 May 2006
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The author proves that the following difference equation \[ x_{n+1}=\frac{x_{n-k}}{1+x_{n}+\cdots + x_{n-k+1}},\quad n=0, 1, \cdots, \] where \(k\in {\mathbb N}\), has a positive solution which converges to zero. This result solves Open Problem 11.4.10 (a) of \textit{M. R. S. Kulenović} and \textit{G. Ladas} [Dynamics of second order rational difference equations. With open problems and conjectures. (Boca Raton, FL: Chapman and Hall/CRC) (2002; Zbl 0981.39011)].
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Rational difference equations
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positive solutions
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convergence to zero
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