Eventually Periodic Solutions of a Max-Type Difference Equation

We study the following max-type difference equation x n = max⁡{A n/x n−r, x n−k}, n = 1,2,…, where {A n}n=1 +∞ is a periodic sequence with period p and k, r ∈ {1,2,…} with gcd(k, r) = 1 and k ≠ r, and the initial conditions x 1−d, x 2−d,…, x 0 are real numbers with d = max⁡{r, k}. We show that if p = 1 (or p ≥ 2 and k is odd), then every well-defined solution of this equation is eventually periodic with period k, which generalizes the results of (Elsayed and Stevic´ (2009), Iričanin and Elsayed (2010), Qin et al. (2012), and Xiao and Shi (2013)) to the general case. Besides, we construct an example with p ≥ 2 and k being even which has a well-defined solution that is not eventually periodic.


Introduction
The max operator arises naturally in certain models in automatic control theory (see [1]). In recent years, the discrete case involving difference equations with maximum has been receiving increasing attention, for some results in this area; see, for example, [2][3][4].
In this paper, we will generalize the results of [5][6][7][8] to the general case.

Main Results and Example
In this section, we are ready to state and prove the main results.
(1) If ≥ 2 and is odd, then every well-defined solution of (1) is eventually periodic with period .
Proof. Let { } +∞ =1− be a well-defined solution of (1). It follows from (1) that, for any ≥ 0 and any ≥ 0, 2 The Scientific World Journal Then, for every ≥ 0, { + } +∞ =0 is increasing, and + < 0 for all ≥ 0 or there exists > 0 such that We claim that { } +∞ =0 is a constant sequence eventually. Indeed, if { } +∞ =0 is not constant sequence eventually, then there exist From this we obtain that, for all ≥ 1, It follows that, for all ≥ 1, Therefore we have − < 0 eventually. By induction, we can show that − −( +1) < 0 eventually for all 1 ≤ ≤ − 1 and every If ≥ 2 and is odd, then we have eventually. This is a contradiction.
If = 1, then we write = for all ≥ 1 and choose which is a contradiction. This completes the proof of the claim.
By the above claim we may choose an Since is a periodic sequence, we can choose an 1 > such that 1 Then, for all ≥ 1 , Thus Proof. Let { } +∞ =1− be a well-defined solution of (1). Using arguments similar to the ones developed in the proof of Theorem 1, we know that, for every ≥ 0, { + } +∞ =0 is increasing, and + < 0 for all ≥ 0 or there exists > 0 such that + > 0 for all ≥ .
We may assume without loss of generality that 1 ≥ 0. We claim that { } +∞ =0 is a constant sequence eventually. Indeed, if { } +∞ =0 is not constant sequence eventually, then there exist < 1 < 2 < ⋅ ⋅ ⋅ such that > ( −1) with being a constant sequence for all ≥ 1. Thus we have The Scientific World Journal From this we obtain that, for all ≥ 1, Thus ≥ 0 and It follows that, for all ≥ 1, This is a contradiction.
Remark 5. Consider the max-type equation (1) If = 1 (or ≥ 2 and is odd), then it follows from Theorem 1 that, for every 1 ≤ ≤ , every well-defined solution of equation = max{ + / − , − } is eventually periodic with period . Thus every welldefined solution of (15) is eventually periodic with period .
(2) If ≥ 2 and, for every 1 ≤ ≤ , there exists some such that + ≥ 0, then it follows from Theorem 3 that for every 1 ≤ ≤ , every well-defined solution of equation = max{ + / − , − } is eventually periodic with period . Thus every welldefined solution of (15) is eventually periodic with period .
(3) If ≥ 2 and is even, then it follows from Example 4 that, for every 1 ≤ ≤ , we can construct an equation = max{ + / − , − } such that it has a welldefined solution which is not eventually periodic. Thus we can construct an equation such that it has a well-defined solution which is not eventually periodic.

Conflict of Interests
The authors declare that there is no conflict of interests regarding the publication of this paper.

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