Abstract

We investigate asymptotic behavior and periodic nature of positive solutions of the difference equation , where and . We prove that every positive solution of this difference equation approaches or is eventually periodic with period 2.

1. Introduction

In the recent years, there has been a lot of interest in studying the global attractivity and the periodic nature of, so-called, max-type difference equations (see, e.g., [117] and references therein).

In [10], the following difference equation was proposed by Ladas: where are real numbers and initial conditions are nonzero real numbers.

In [17], asymptotic behavior of positive solutions of the difference equation was investigatedwhere and It was showed that every positive solution of this difference equation approaches or is eventually periodic with period 4.

In [14], it was proved that every positive solution of the difference equation, where , and converges to

In this paper, we investigate the difference equationwhere and initial conditions are positive real numbers. We prove that every positive solution of this difference equation approaches or is eventually periodic with period 2.

2. The Case

In this section, we consider the difference equationwhere Theorem 2.1. Let be a solution of (2.1). Then approaches Proof. Choose a number such that let for Then (2.1) implies the difference equation where and initial conditions are real numbers.
Let be a solution of (2.2). Then it suffices to prove . Observe that there exists a positive integer such thatBy computation, we get that and then So, This implies

3. The Case

In this section, we consider (1.4), where

Let Equation (1.4) implies the difference equationwhere initial conditions are real numbers.Lemma 3.1. Let be a solution of (3.1). Then for all Proof. From (3.1), we have the following statements:
if and then if and then if and then if and then
In general, we have
Theorem 3.2. if is a solution of (1.4), approaches Proof. Let be a solution of (3.1). To prove it suffices to prove
Choose a number such that Then from inequality (3.2), we get that
Let then and
From (3.4) and by induction, we get that
This implies

4. The Case

In this section, we consider (1.4). Let for Equation (1.4) implies the difference equation where and initial conditions are real numbers.Theorem 4.1. If is a solution of (1.4), then the following statements are true:
(a) approaches if there is an integer such that(b) is eventually periodic with period 2, if there is an integer such that
Proof. (a) Change of variables If and for then and Let be a solution of (4.1). So, to prove it suffices to prove From (4.1), there is at least an integer such that and for By computation from (4.1), we get that , and then
So, we havefor all This implies
(b) Change of variables , Let be a solution of (4.1).
If and for thenClearly, there is at least an integer such that for
Suppose that and
If then from (4.1), we have So, from (a) we get immediately that
If then we have and Then we get that from (a).
We assume that and . To prove the desired result, it suffices to show that is eventually periodic with period 2. By computation from (4.1), we get immediately for all This is the desired result.