On the existence of asymptotically almost periodic solutions for nonlinear systems
© Song et al.; licensee Springer 2013
Received: 13 October 2012
Accepted: 22 January 2013
Published: 7 February 2013
In this paper, a nonlinear differential equation is considered. Some new sufficient conditions for the existence of a bounded solution and an asymptotically almost periodic solution, which generalize and improve the previously known results, are established by using a dissipative-type condition for . Finally, an example is presented to illustrate the feasibility and effectiveness of the new results.
where and is a linear operator on a Banach space, which is periodic, and is asymptotically almost periodic. They showed a bounded mild solution x is asymptotically almost periodic.
where and . By employing the dissipative-type condition for , when and are asymptotically almost periodic functions, we present some new criteria ensuring the existence of a bounded solution and an asymptotically almost periodic solution of Eq. (1.4). The remaining part is organized as follows. In the next section, we introduce some definitions and lemmas. In Section 3, we obtain two theories, which guarantee the existence of a bounded solution and an asymptotically almost solution of Eq. (1.4). In Section 4, a numerical simulation is carried out to illustrate the main results.
Firstly, to establish our main results, it is necessary to make the following assumptions:
where N is a positive constant;
where is defined as follows (see Definition 2.5).
for all , then is said to be an almost periodic function.
for all and all , then is said to be almost periodic in t uniformly for .
Definition 2.3 If and in , is an almost periodic function in ℝ and is continuous in , , then is called an asymptotically almost periodic function on .
Definition 2.4 If and in , and is an almost periodic function in t uniformly on and is continuous in , uniformly on , where Ω is an open set on and H is a compact set, then is said to be an asymptotically almost periodic function in t.
The following lemma on the functional is well known (see ).
Lemma 2.1 
- (4)Let u be a function from a real interval J into such that exists for an interior point of J. Then exists and
where denotes the right derivative of at .
Lemma 2.3 
for all .
In order to obtain our main results, we should prove the following lemma.
This completes the proof. □
3 Existence of bounded solutions and asymptotically periodic solutions
In this section, it will be shown that, under certain conditions, the system (1.4) has a bounded solution and an asymptotically periodic solution.
Then Eq. (1.4) has a bounded solution on such that for . Furthermore, if is any solution of Eq. (1.4), then as .
for . Since , , . Thus, for ϵ with , there exists a sufficiently small such that . This contradicts the definition of τ. Then for all .
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).
We will show that , where is a positive constant independent of ϵ and ω.
We must estimate for t large enough.
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.
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.
4 The example
where is an asymptotically almost periodic function in uniformly on x which belongs to a compact set and is an asymptotically almost periodic function on .
we know that () is satisfied.
() is satisfied too.
The authors would like to thank the two referees for their valuable suggestions and comments concerning improvement of the work.
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