On Boundedness of Solutions of the Difference Equation for

Abstract

We study the boundedness of the difference equation , where and the initial values . We show that the solution of this equation converges to if or for all ; otherwise is unbounded. Besides, we obtain the set of all initial values such that the positive solutions of this equation are bounded, which answers the open problem 6.10.12 proposed by Kulenović and Ladas (2002).

1. Introduction

In this paper, we study the following difference equation:

(1.1)

where with and the initial values .

The global behavior of (1.1) for the case is certainly folklore. It can be found, for example, in [1] (see also a precise result in [2,]).

The global stability of (1.1) for the case follows from the main result in [3] (see also Lemma 1 in Stević's paper [4]). Some generalizations of Copson's result can be found, for example, in papers [5–8]. Some more sophisticated results, such as finding the asymptotic behavior of solutions of (1.1) for the case (even when ) can be found, for example, in papers [4](see also [8–11]). Some other properties of (1.1) have been also treated in [4].

The case was treated for the first time by Stević's in paper [12]. The main trick from [12] has been later used with a success for many times; see, for example, [13–15].

Some existing results for (1.1) are summarized as follows[16].

Theorem 1 A.

If , then the zero equilibrium of (1.1) is globally asymptotically stable.

If , then the equilibrium of (1.1) is globally asymptotically stable.

If , then every positive solution of (1.1) converges to the positive equilibrium .

If , then every positive solution of (1.1) converges to a period-two solution.

If , then (1.1) has unbounded solutions.

In [16], Kulenović and Ladas proposed the following open problem.

Open problem B (see Open problem 6.10.12of [16])

Assume that .

• (a) Find the set of all initial conditions such that the solutions of (1.1) are bounded.

• (b) Let . Investigate the asymptotic behavior of .

In this paper, we will obtain the following results: let with , and let be a positive solution of (1.1) with the initial values . If for all (or for all ), then converges to . Otherwise is unbounded.

For closely related results see [17–34].

2. Some Definitions and Lemmas

In this section, let and be the positive equilibrium of (1.1). Write and define by, for all ,

(2.1)

It is easy to see that if is a solution of (1.1), then for any . Let

(2.2)

Then . The proof of Lemma 2.1 is quite similar to that of Lemma 1 in [35] and hence is omitted.

Lemma 2.1.

The following statements are true.

• (1) is a homeomorphism.

• (2) and .

• (3) and .

• (4) and .

• (5) and .

Lemma 2.2.

Let , and let be a positive solution of (1.1).

• (1) If and , then .

• (2) If and , then .

Proof.

We show only (1) because the proof of (2) follows from (1) by using the change and the fact that (1) is autonomous. Since and , by (1.1) we have

(2.3)

Also it follows from (1.1) that

(2.4)

from which we have and . This completes the proof.

Lemma 2.3.

Let , and let be a positive solution of (1.1) with the initial values . If there exists some such that , then .

Proof.

Since , it follows from Lemma 2.1 that for any . Without loss of generality we may assume that , that is, . Now we show Suppose for the sake of contradiction that , then

(2.5)

(2.6)

By (2.5) we have

(2.7)

and by (2.6) we get

(2.8)

Claim 1.

If , then

(2.9)

Proof of Claim 1

Let , then we have

(2.10)

Since , it follows

(2.11)

This completes the proof of Claim 1.

By (2.8), we have

(2.12)

or

(2.13)

Claim 2.

We have

(2.14)

(2.15)

Proof of Claim 2

Since

(2.16)

we have

(2.17)

The proof of (2.14) is completed.

Now we show (2.15). Let

(2.18)

Note that ; it follows that if , then

(2.19)

which implies that is decreasing for . Since and

(2.20)

it follows that

(2.21)

Thus

(2.22)

This implies that

(2.23)

Finally we have

(2.24)

The proof of (2.15) is completed.

Note that since . By (2.12), (2.13), (2.14), and (2.15), we see which contradicts to (2.7). The proof of Lemma 2.3 is completed.

3. Main Results

In this section, we investigate the boundedness of solutions of (1.1). Let , and let be a positive solution of (1.1) with the initial values , then we see that for some or for all or for all .

Theorem 3.1.

Let , and let be a positive solution of (1.1) such that for all or for all , then converges to .

Proof.

Case 1.

for any . If for some , then

(3.1)

If for some , then

(3.2)

which implies that and

(3.3)

Thus for any . In similar fashion, we can show for any . Let and , then

(3.4)

which implies .

Case 2.

for any . Since is decreasing in , it follows that for any

(3.5)

In similar fashion, we can show that . This completes the proof.

Lemma 3.2 (see [20, Theorem 5]).

Let be a set, and let be a function which decreases in and increases in , then for every positive solution of equation , and do exactly one of the following.

• (1) They are both monotonically increasing.

• (2) They are both monotonically decreasing.

• (3) Eventually, one of them is monotonically increasing, and the other is monotonically decreasing.

Remark 3.3.

Using arguments similar to ones in the proof of Lemma 3.2, Stevi proved Theorem 2 in [25]. Beside this, this trick have been used by Stević in [18, 28, 29].

Theorem 3.4.

Let , and let be a positive solution of (1.1) such that for some , then is unbounded.

Proof.

We may assume without loss of generality that and (the proof for is similar). From Lemma 2.1 we see for all .If is eventually increasing, then it follows from Lemma 2.3 that is eventually increasing. Thus and , it follows from Lemma 2.2 that .

If is not eventually increasing, then there exists some such that

(3.6)

from which we obtain , since and .

Since is increasing in and is decreasing in , we have that for any . It follows from Lemma 3.2 that is eventually decreasing. Thus and . It follows from Lemma 2.2 that . This completes the proof.

By Theorems 3.1 and 3.4 we have the following.

Corollary 3.5.

Let , and let be a positive bounded solution of (1.1), then for all or for all .

Now one can find out the set of all initial values such that the positive solutions of (1.1) are bounded. Let For any let

(3.7)

It follows from Lemma 2.1 that , which implies

(3.8)

for any .

Let be the set of all initial values such that the positive solutions of (1.1) are bounded. Then we have the following theorem.

Theorem 3.6.

.

Proof.

Let be a positive solution of (1.1) with the initial values .

If , then for any , which implies for any . It follows from Theorem 3.1 that .

If , then , which implies for any . It follows from Theorem 3.1 that .

Now assume that is a positive solution of (1.1) with the initial values .

If , then it follows from Lemma 2.1 that , which along with Theorem 3.4 implies that is unbounded.

If , then there exists such that . Thus . By Lemma 2.1, we obtain and , which along with Theorem 3.4 implies that is unbounded.

If , then there exists such that and . Again by Lemma 2.1 and Theorem 3.4, we have that is unbounded. This completes the proof.