Research Article | Open Access

Stevo Stević, Kenneth S. Berenhaut, "The Behavior of Positive Solutions of a Nonlinear Second-Order Difference Equation", *Abstract and Applied Analysis*, vol. 2008, Article ID 653243, 8 pages, 2008. https://doi.org/10.1155/2008/653243

# The Behavior of Positive Solutions of a Nonlinear Second-Order Difference Equation

**Academic Editor:**Allan C. Peterson

#### Abstract

This paper studies the boundedness, global asymptotic stability, and periodicity of positive solutions of the equation , , where . It is shown that if and are nondecreasing, then for every solution of the equation the subsequences and are eventually monotone. For the case when and satisfies the conditions , is nondecreasing, and is increasing, we prove that every prime periodic solution of the equation has period equal to one or two. We also investigate the global periodicity of the equation, showing that if all solutions of the equation are periodic with period three, then and , for some positive and .

#### 1. Introduction

Recently there has been great interest in the study of nonlinear and rational difference equations (cf. [1–35] and the references therein).

In this paper, we study the boundedness, global asymptotic stability, and periodicity for positive solutions of the equation where

#### 2. Asymptotic Periodicity of (1.1)

In this section,we investigate asymptotic periodicity of (1.1). The asymptotic periodicity of some difference equations has been investigated, for example, in the papers [3–6, 10, 12, 15, 20, 22, 23, 25, 26, 27, 28, 29, 31, 32, 35]. Our first result is the following theorem regarding eventual monotonicity, which is a natural extension of [24, Theorem 2].

Theorem 2.1. *
Assume that and
are
nondecreasing functions which map the interval
into itself,
and assume that
is a solution
of (1.1). Then, the sequences and
are eventually
monotone.
*

*Proof. *Suppose that is a solution
to (1.1), and set
for . Note that it suffices to show that and are eventually
constant.

From (1.1), we have that
If and , then by (1.1) and the monotonicity of and we have that , and hence . Similarly, if and , then . Hence, if for some , then the sequences and are both
constant, as required. This confirms the statement in the theorem, in this
case. Otherwise, itself is a
constant sequence and the result again follows.

*Remark 2.2. *Note that Theorem 2.1 guarantees only the eventually
monotonicity of the sequences and Hence, for a
solution of (1.1), one of
these two subsequences can be infinite. See, for example, [27, Theorem 1],
where it was shown that for the case , (1.1) has unbounded solutions. The problem
was previously treated in the papers [1, 11] but the proofs appearing there
have a gap (for more details see [27]).

The first special case of the nonrational (1.1) was considered in the paper [22], where the second author considered the equation where the function satisfies the following conditions: (i) for (ii) is increasing on ,(iii) is increasing on

The following two conjectures, which were posed by the second author, have circulated among the experts in the field, since early 2001.

*Conjecture 2.3. *Assume that . Show that every solution of (2.3) is bounded.

*Conjecture 2.4. *Assume that . Show that every prime periodic solution of (2.3) has
period equal to one or two.

Conjecture 2.3 was confirmed in [28] where the following lemma was proved.

Lemma 2.5. *
Suppose
is a function
which satisfies the following conditions:
*(a)*,*(b)* is increasing
on *(c)* is
nondecreasing on .**
Then for given such that there exist and such that
*(1)* and ,*(2)

By Lemma 2.5, a result was proven concerning an extension of (2.3). We present the proof of the theorem for the case of (2.3), for the benefit of the reader, since the proof is instructive. For related results regarding boundedness, see, for example, [1, 6, 10, 22, 27, 31, 33, 34].

Theorem 2.6. *Assume that
is a function
which satisfies conditions (ii) and (iii) and that Consider (2.3) where
Then every
solution of (2.3)
is bounded and
persists.*

*Proof. *Choose and such that and By Lemma 2.5, we
may also assume that Now we may use
mathematical induction to prove the result. Assume the statement is true for that is,

Then

We claim that . But this is obvious since
Similarly, we have that
from which it follows that for completing the
proof of the theorem.

By Theorems 2.1 and 2.6 we confirm Conjecture 2.4. Indeed, by Theorem 2.6 we have that every solution of (2.3) is bounded. On the other hand, by Theorem 2.1, the sequences and are eventually monotone, thus convergent. Hence, if (2.3) has periodic solutions they have period one or two, as conjectured.

#### 3. Global Periodicity of (1.1)

*Definition 3.1. *Let be a function
defined on a subset of Say that the
difference equation
where is *periodic*, if every solution of (3.1) is
periodic.

Periodic equations have been investigated, for example, in [2, 7, 8, 12, 14, 16–19, 26] (see also the references therein).

In this section, we investigate periodic equations of type (1.1). In order to facilitate notation we will write (1.1) in the equivalent form

If every solution of (3.2) is periodic with period then it must hold that that is , which implies that and for some positive constant . Thus, (3.2) has the form Since every solution of (3.2) must be two periodic, it follows that Hence, the equation is a unique equation of type (3.2) for which all solutions are periodic with period two.

Further we consider those equations of type (3.2) for which all solutions are periodic with period three. For a mapping the sequence of iterates of is defined by ( is the identity function on ), and generally for any

Before we prove the result concerning the case, we need the following auxiliary result which is folklore.

Lemma 3.2. *Assume that is a continuous function such that
**
Then, or *

*Proof. *If , then from (3.4) it follows that
which implies that the function must be Since is a continuous
function we have that must be
strictly monotone.

First assume that is strictly
increasing. If there is a point such that , then by the monotonicity of we have
which is a contradiction.

If , then we have
arriving again at a contradiction.

From this it follows that for every .

Assume now that is strictly decreasing. Then the function is strictly increasing and
Similar to the first case, we obtain that , finishing the proof of the lemma.

*Remark 3.3. *Note that if then is decreasing,
maps interval , “,” and onto itself, and its graph is symmetric with
respect to the line since .

Theorem 3.4. *All solutions of (3.2)
are periodic with period three if and only if and
for some
positive constants and .*

*Proof. *Assume that and for some
positive constants and Then (3.2)
becomes
It is easy to see that every solution of the equation is periodic with period three (see, e.g., [12]).

Assume now that every solution of (3.2) is periodic with period three. Then, we have that
for every

Eliminating in (3.10) we
obtain that
Now, in each of the two equations (3.11), we choose that a variable is arbitrary and the other is
equal to and use the
changes
Then, we obtain
for every , where .

From (3.13) we have
which implies that

If we set in the second identity in (3.13) and in the third
identity, and then apply (3.14), we obtain

From (3.15) and (3.16) it follows that

Lemma 2.5 implies that or If then (3.13)
implies and by the
second identity in (3.13) we obtain which is a
contradiction.Hence,
Substituting (3.18) in (3.13) we obtain that
from which it follows that
as desired.

*Remark 3.5. *
It is expected that Theorem 3.4 can be generalized for the case when all solutions of (3.2) are periodic with period more than three.

In the case when all solutions of (3.2) are periodic with period four, the functions and must satisfy the following system of functional equations: From the system as in the proof of Theorem 3.4, it can be obtained that the functions and satisfy the following identities: where

An obvious solution of the system is and where We leave the
problem of finding all solutions of the system as a further direction for
investigation for
interested readers.
*Note*. An
early draft by the second author of the paper, containing only results in Section 3, has circulated among the experts since the end of 2005 and was the starting point for further important investigations in the research field, see, for example, [2, 30]. The paper in the present form is a slight modification of a version from March 2006. A minor publication mishap caused some wrong citations (see [30]), as well as a delay in publishing of it.

#### Acknowledgments

The authors are grateful to B. T. Lamb and A. C. Pecorella for useful discussions which led to the current proof of Theorem 2.1.

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#### Copyright

Copyright © 2008 Stevo Stević and Kenneth S. Berenhaut. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.