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
If for some , then
which implies that and
Thus for any . In similar fashion, we can show for any . Let and , then
which implies .
Case 2.
for any . Since is decreasing in , it follows that for any
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
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
It follows from Lemma 2.1 that , which implies
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.