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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

${x}_{n}=max\left\{\frac{1}{{x}_{n-1}^{{\alpha }_{1}}},\frac{1}{{x}_{n-2}^{{\alpha }_{2}}},...,\frac{1}{{x}_{n-k}^{{\alpha }_{k}}}\right\},\phantom{\rule{2.em}{0ex}}n\in {ℕ}_{0},$

where $k\in ℕ$, ${\alpha }_{i}>0$, $i=1$, $...$, $k$ and ${\alpha }_{1}{\alpha }_{k}\in \left(0,1\right)$, converges to one.

Theorem 2. Assume that $k\in ℕ$, ${\alpha }_{i}\in \left(0,1\right)$, $i=1$, $...$, $k$ and ${A}_{i}>0$, $i=1$, $...$, $k$. Then every positive solution to

${x}_{n}=max\left\{\frac{{A}_{1}}{{x}_{n-1}^{{\alpha }_{1}}},\frac{{A}_{2}}{{x}_{n-2}^{{\alpha }_{2}}},...,\frac{{A}_{k}}{{x}_{n-k}^{{\alpha }_{k}}}\right\},\phantom{\rule{2.em}{0ex}}n\in {ℕ}_{0}$

converges to $\underset{1\le i\le k}{max}\left\{{A}_{i}^{\frac{1}{{\alpha }_{i}+1}}\right\}$.

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 Generalized difference equations
##### References:
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