#### 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., [1β17] 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:

βif and then βif and then βif and then βif 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.