In this section, it will be shown that, under certain conditions, the system (1.4) has a bounded solution and an asymptotically periodic solution.
Theorem 3.1 Suppose that conditions (), () are satisfied. Let r be defined as
Then Eq. (1.4) has a bounded solution on such that for . Furthermore, if is any solution of Eq. (1.4), then as .
Proof If for , we replace and by and , respectively. We assume, henceforth, that and for all and fix a vector with . For each positive integer n with , we consider the following Cauchy problem:
We find that the conditions ()-() and ()-() in  can be satisfied by (), () in the present paper, then Corollary 5.1 in  can now be applied to guarantee the (c.p) has a unique solution on . We first prove that
In fact, otherwise there exists some such that , where is an arbitrary number such that . Let , by the continuity of , it follows easily that . Then implies , and by Lemma 2.1 and (), we have
For each , there exists an such that
for . Since , , . Thus, for ϵ with , there exists a sufficiently small such that . This contradicts the definition of τ. Then for all .
On the other hand, using the following differential inequality:
It thus follows that for all . Lemma 8.1 in  can now be applied to guarantee the existence on of a bounded solution of Eq. (1.4). By the continuity of , is a bounded solution on which also satisfies . If is any solution of Eq. (1.4) on , by Lemma 2.3, we have
By Lemma 2.4, we obtain , when , , and then as . This completes the proof. □
Theorem 3.2 Suppose that is asymptotically almost periodic in t uniformly for , where r is a positive number defined by Eq. (3.1) and , and is an asymptotically almost periodic function. Suppose, furthermore, that the condition () is satisfied. Then Eq. (1.4) has an asymptotically almost periodic solution on .
Proof First, we prove that is bounded. in , and is an almost periodic function in ℝ. For any , there is an , when , there is an , . For any , choose , then , and , so for any t, . While , we have a positive such that , then the condition () is satisfied. Conditions () and () are satisfied, let be a bounded solution of (1.4) on obtained in Theorem 3.1. Note that for all , where r is a number defined by Eq. (3.1).
Notice that is also an almost periodic function in t uniformly for . For each , there exist a positive number and a positive number such that any interval of length contains an ω,
By Lemma 2.1, () and Eq. (3.4), we have
for all . Solving this differential inequality, we have
where η is a positive number to be chosen later appropriately, and . We show that
is finite. In fact, this follows from the following estimates:
Let be a number such that
We will show that , where is a positive constant independent of ϵ and ω.
We must estimate for t large enough.
Since (when ), if , , , , then
When we choose and , that is, , then, for any , there exists a positive number such that any interval of length contains an ω, when , , ,
From Lemma 2.2, is an asymptotically almost periodic solution of Eq. (1.4). This completes the proof. □
Remark 3.1 In , employing the dissipative-type condition for , the authors gave some sufficient conditions to prove the existence of a bounded solution, a periodic or almost periodic solution of the equation . Extension of this result has been obtained in one direction: from periodic and almost periodic to asymptotically almost periodic forcing. The equation can be more widely used with asymptotically almost periodic functions.
Remark 3.2 The condition () implies the following hypothesis.
() Suppose that there exist and positive constants δ, , and such that
And for all ,
We know that () can also be used to prove the lemmas in Section 2 and the theorems in Section 3 leaving the conclusion unchanged. () as well as () yields the existence of a bounded solution, and the process of the proof is similar to the proof before, and we need not necessarily do it again.