On the periodic nature of some max-type difference equation

We study some qualitative behavior of solutions of some max-type difference equations with periodic coefficients. Some new results of the periodicity character of solutions of that type of difference equations will be established.


Introduction
Recently there has been a lot of interest in studying the global attractivity, the boundedness character, and the periodicity nature of nonlinear difference equations.In [5,6,8] some global convergence results were established which can be applied to nonlinear difference equations in proving that every solution of these equations converges to a periodic solution (which need not necessarily be stable).The periodic nature of nonlinear difference equations of the max type has been investigated by many authors.See for example [1,2,3,4].
Our main objective in this paper is to extend the study of boundedness and periodicity to solutions of some max-type difference equations.We deal with the following difference equation: where {A n } ∞ n=0 = {...,α,β,α,β,...} is a periodic sequence of positive numbers of period two with β > α > 1.The case where {A n } ∞ n=0 is a periodic sequence of positive numbers of period three and A n ∈ (0,1] was investigated in [4].

Invariant interval and boundedness
In this section, we show that every solution of (1.1) is bounded and persists.
The following lemmas are quite important results in their own; however these lemmas will be used in the subsequent discussion.

Proof. Let {x n } ∞
n=−1 be a solution of (1.1).It follows from (1.1) for an integer number N ≥ 0 that That is, there exists a positive real number m such that Thus from (1.1), we see that (2.5)

Hence
x n ≤ M ∀n ≥ N. (2.6) Thus from inequalities (2.4) and (2.6) we get Therefore every solution of (1.1) is bounded and persists.
Lemma 2.3.Every solution of (1.1) which is bounded below by n=−1 be a positive solution of (1.1) and there exists N ≥ 0 such that (2.17) It follows from (1.1) that Similarly, we see that The rest of the proof follows by Lemma 2.2.

The main result
In this section, we study the periodicity character of solutions of (1.1).
In the following we study the existence of periodic solutions of (1.1) with period four.
Theorem 3.1.Assume that {x n } ∞ n=−1 is a positive solution of (1.1) with n=−1 be a positive solution of (1.1).Suppose there exists N ≥ 0 such that Assume that Observe from (1.1) that We consider the following two cases.

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Then clearly the solution becomes in the form In this case we see that where and so the solution becomes in the form ..., p, q, α p , β q , p, q, α p , β q ,... .(3.9) The proof is complete.
Theorem 3.2.Every positive solution of (1.1) which is bounded from below by 1/ √ α is eventually periodic with period four.
Proof.Let {x n } ∞ n=−1 be a positive solution of (1.1).By Lemma 2.3, we assume From (1.1), we see that We consider the following two cases.(A 1 ) x N+1 = 1/x N .In this case 1/x N > α/x N−1 , and we see that We consider the following two cases.
(A 11 ) x N+2 = x N .In this case x N > β/x N , and we see that We see that the solution is in the form Therefore {x n } ∞ n=−1 is a periodic solution with period four.(A 12 ) x N+2 = β/x N .In this case β/x N > x N , and we see that E. M. Elabbasy et al. 2233 where (3.17) Therefore {x n } ∞ n=−1 is a periodic solution with period four as follows: In this case α/x N−1 > 1/x N , and we see that

.19)
We consider the following two cases.(A 21 ) x N+2 = x N−1 /α.In this case x N−1 /α > β/x N , and we see that where (3.21) We consider the following two cases.(A 211 ) x N+5 = x N−1 /αβ.In this case x N−1 /αβ > α/x N−1 , and we see that where (3.24) Therefore the solution can be written as Then {x n } ∞ n=−1 is a periodic solution with period four.We consider the following two cases.(A 212 ) x N+5 = α/x N−1 .In this case α/x N−1 > x N−1 /αβ, and we see that (3.26) E. M. Elabbasy et al. 2235 It is also easy to see that the solution takes the form which is periodic with period four.(A 22 ) x N+2 = β/x N .In this case β/x N > x N−1 /α, and we see that (3.28) We consider the following two cases.(A 221 ) x N+3 = x N−1 .In this case x N−1 > x N /β, and we see that (3.29) One can easily see that the solution will be in the form and so the solution is periodic with period four.
(A 222 ) x N+3 = x N /β.In this case x N /β > x N−1 , and we see that where where (3.33) Then the solution can be written in the form and so the solution is periodic with period four.This completes the proof.The proof of Theorem 3.2 is thus completed. where (3.43) We consider the following two cases.
(B 1 ) x m+3 = x m /α.In this case we see that x m+3 = β/x m+1 .This can be treated similarly to the case x m+3 = x m /α and the solution is either periodic with period four or lim The proof is complete.Remark 3.4.Observe by assumption that x m , x m+1 < 1/ √ α is not possible as can be seen from (1.1).Now, we can state the main result in this section.Theorem 3.5.Every solution of (1.1) is periodic with period four.
Proof.The proof of this theorem follows from Theorem 3.2 and Lemma 3.3.
We consider the following two cases.(B11 ) x m+10 = x m .In this case the solution eventually will be periodic with period four as