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On positive solutions of a (\(k+1\))th order difference equation. (English) Zbl 1095.39010
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 M. R. S. Kulenović and 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)].

MSC:
39A11 Stability of difference equations (MSC2000)
39A20 Multiplicative and other generalized difference equations, e.g., of Lyness type
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