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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 =max1 x n-1 α 1 ,1 x n-2 α 2 ,...,1 x n-k α k ,n 0 ,

where k, α i >0, i=1, ..., k and α 1 α k (0,1), converges to one.

Theorem 2. Assume that k, α i (0,1), i=1, ..., k and A i >0, i=1, ..., k. Then every positive solution to

x n =maxA 1 x n-1 α 1 ,A 2 x n-2 α 2 ,...,A k x n-k α k ,n 0

converges to max 1ik A i 1 α i +1 .

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:
39A11Stability of difference equations (MSC2000)
39A20Generalized difference equations
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