We are now in a position to establish our main results.
Theorem 2.1 Assume that:
(H1) There exists a constant such that
(H2) The functions C, D satisfy
(H3) and are not impulsive points, , , and .
(H4) The functions P, Q satisfy
Then every solution of (1.1) tends to a constant as .
Proof Let be any solution of system (1.1). We will prove that the exists and is finite. Indeed, the system (1.1) can be written as
From (H2) and (H4), we choose constants sufficiently small such that and and sufficiently large, for ,
and, for ,
From (2.1), (2.10), we have
which lead to
In the following, for convenience, the expressions of functional equalities and inequalities will be written without its domain. This means that the relations hold for all sufficiently large t.
Let , where
Computing along the solution of (1.1) and using the inequality , we have
Calculating directly for , , , we have
Summing for , , we obtain
it follows that
Adding the above inequality with and using condition (2.11), we have
Applying (2.8), (2.9), and (2.10), it follows that
For , we have
It is easy to see that , , and .
From (2.12) and (2.13), we conclude that is decreasing. In view of the fact that , we have exist and .
By using (2.8), (2.9), (2.12), and (2.13), we have
Hence, for any and , we get
Thus, it follows from (2.4) and (2.5) that
Therefore, from the above estimations, we have , , and , respectively.
Thus, , that is,
Now, we will prove that the limit
exists and is finite. Setting
and using (1.1) and condition (H3), we have
In view of (2.14), it follows that
In addition, from (2.16) and (2.17), system (2.6)-(2.7) can be written as
If , then . If , then there exists a sufficiently large such that for any . Otherwise, there is a sequence with such that , and so as . This contradicts . Therefore, for any , and , we have or from the continuity of y on . Without loss of generality, we assume that on . It follows from (H3) that , and thus on . By using mathematical induction, we deduce that on . Therefore, from (2.14), we have
where and is finite. In view of (2.18), for sufficient large t, we have
Taking and using (H3), we have
which leads to
This implies that
Next, we shall prove that
Further, we first show that is bounded. Actually, if is unbounded, then there exists a sequence such that , , as and
where, if is not an impulsive point, then . Thus, we have
as , which contradicts (2.20). Therefore, is bounded.
If and , then , which implies that (2.21) holds. If and , then we deduce that and are eventually positive or eventually negative. Otherwise, there are two sequences and with and such that and . Therefore, and as . It is a contradiction to and .
Now, we will show that (2.21) holds. By condition (H2), we can find a sufficiently large such that for , . Set
Then we can choose two sequences and such that , as , and
For , we consider the following eight possible cases.
Case 1. When and for , we have
Thus, we obtain
Since and , it follows that . By (2.20), we obtain
which shows that (2.21) holds.
Case 2. When and for , we get
which leads to
Since and , we conclude that
which implies that (2.21) holds.
Case 3. , for . The method of proof is similar to the above two cases. Therefore, we omit it.
Case 4. , for . The method of proof is similar to the above two first cases. Therefore, we omit it.
Case 5. When and for , we have
Since and , we have . Thus
and so (2.21) holds.
Using similar arguments, we can prove that (2.21) also holds for the following cases:
Case 6. , .
Case 7. , .
Case 8. , .
Summarizing the above investigation, we conclude that (2.21) holds and so the proof is completed. □
Theorem 2.2 Let conditions (H1)-(H4) of Theorem 2.1 hold. Then every oscillatory solution of (1.1) tends to zero as .
Corollary 2.1 Assume that (H3) holds and
Then every solution of the equation
tends to a constant as .
Corollary 2.2 The conditions (2.23) and (2.24) imply that every solution of the equation
tends to a constant as .
Theorem 2.3 The conditions (H1)-(H4) of Theorem 2.1 together with
imply that every solution of (1.1) tends to zero as .
Proof From Theorem 2.2, we only have to prove that every nonoscillatory solution of (1.1) tends to zero as . Without loss of generality, we assume that is an eventually positive solution of (1.1). As in the proof of Theorem 2.1, (1.1) can be written as in the form (2.18). Integrating from to t both sides of the first equation of (2.18), one has
Applying (2.19) and (H3), we have
This, together with (2.27), implies that and . By Theorem 2.1, . This completes the proof. □
Corollary 2.3 Assume that (2.1), (2.2), (2.4), (2.5), and (2.27) hold. Then every solution of the equation
, tends to zero as .