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Global stability of a difference equation with maximum. (English) Zbl 1167.39007
The main results of the paper are the following two theorems.
Theorem 1. Every positive solution to the difference equation \[ x_n=\max \left \{\frac{1}{x_{n-1}^{\alpha_1}}, \frac{1}{x_{n-2}^{\alpha_2}}, \ldots, \frac{1}{x_{n-k}^{\alpha_k}}\right \}, \qquad n\in \mathbb{N}_0, \] where \(k\in \mathbb{N}\), \(\alpha_i>0\), \(i=1\), \(\ldots\), \(k\) and \(\alpha_1\alpha_k\in (0,1)\), converges to one.
Theorem 2. Assume that \(k\in \mathbb{N}\), \(\alpha_i\in (0,1)\), \(i=1\), \(\ldots\), \(k\) and \(A_i>0\), \(i=1\), \(\ldots\), \(k\). Then every positive solution to \[ x_n=\max\left \{\frac{A_1}{x_{n-1}^{\alpha_1}},\frac{A_2}{x_{n-2}^{\alpha_2}}, \ldots, \frac{A_k}{x_{n-k}^{\alpha_k}}\right \}, \qquad n\in \mathbb{N}_0 \] converges to \(\max\limits_{1\leq i\leq k}\left \{A_i^{\frac{1}{\alpha_i+1}}\right \}\).
Moreover, Theorem 2 solves the conjecture proposed by the author of this paper [Appl. Math. Comput. 210, No. 2, 525–529 (2009; Zbl 1167.39007)] and by F. Sun [Discrete Dyn. Nat. Soc. 2008, Article ID 243291, 6 p. (2008; Zbl 1155.39008)].

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