- Open Access
On the existence of asymptotically almost periodic solutions for nonlinear systems
Advances in Difference Equations volume 2013, Article number: 28 (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.
In recent years, almost periodic solutions and their various generalizations have attracted the attention of many researchers (see [1–11] and the references therein). The existence of a bounded solution and an asymptotically almost periodic solution are two important properties which have a close relation to the applications of neural networks, epidemiology, etc., so they have been widely studied. For example, Medvedev  gave a sufficient condition to guarantee the existence of a bounded solution of the following equation:
where , and , denotes an Euclidean n-space, for , is any convenient norm of x. Using this result, he also proved the existence of periodic and almost periodic solutions when and are periodic or almost periodic in t uniformly for x in a bounded subset of . Shigeo and Masato  extended the existence result in  by using a dissipative-type condition for . Thanh and Nguyen Truong  considered the following difference equation:
where ℕ is a natural number and is a bounded linear operator on a Banach space, the sequences are totally ergodic, is countable and the sequence is asymptotically almost periodic, then the sequence is asymptotically almost periodic. As an application, they studied a similar problem for an evolution equation of the form
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.
Motivated by the aforementioned discussion, in this paper, we consider the following system:
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;
() . Suppose that there exist positive constants δ, γ, such that
And for all ,
where is defined as follows (see Definition 2.5).
Definition 2.1 If for any , there exists a positive number such that any interval of length contains a τ for which
for all , then is said to be an almost periodic function.
Definition 2.2 If for any and any compact set S in , there exists a positive number such that any interval of length contains a τ for which
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.
Definition 2.5 Functional :
The following lemma on the functional is well known (see ).
Lemma 2.1 
Let x, y and z be in . Then the functional has the following properties:
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 2.2 is an asymptotically almost periodic function if and only if for any , there exist positive numbers and such that any interval of length contains an ω such that when ,
Lemma 2.3 
Suppose that () is satisfied. Let and be solutions of (1.1) on an interval . Then
for all .
In order to obtain our main results, we should prove the following lemma.
Lemma 2.4 Suppose that () is satisfied. Then
Proof It follows from () that there exists a such that
This means that
for each , we have
for all and = sup. Then
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.
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.
4 The example
In this section, we give a numerical example to illustrate the conditions required in our theorems. We construct the following differential equation:
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.
On the other hand, there exists such that
() is satisfied too.
Then, from Theorem 3.1 and Theorem 3.2, we get a bounded solution and an asymptotically almost periodic solution on of Eq. (4.1) as follows:
We show the semiflow in Figure 1.
Dads EA, Ezzinbi K, Arino O: Periodic and almost periodic solutions for some differential equations in Banach spaces. Nonlinear Anal., Theory Methods Appl. 1998, 31: 163–170. 10.1016/S0362-546X(96)00301-X
Gao H, Wang K, Wei F, Ding X: Massera-type theorem and asymptotically periodic logistic equations. Nonlinear Anal., Real World Appl. 2006, 7: 1268–1283. 10.1016/j.nonrwa.2005.11.008
He C: Almost Periodic Differential Equations. Higher Education Press, Beijing; 1992. (in Chinese)
Shigeo K, Masato I: On the existence of periodic solutions and almost periodic solutions for nonlinear systems. Nonlinear Anal., Theory Methods Appl. 1995, 24: 1183–1192. 10.1016/0362-546X(94)00189-O
Shigeo K: Some remarks on nonlinear ordinary differential equations in a Banach space. Nonlinear Anal. 1981, 5: 81–93. 10.1016/0362-546X(81)90073-0
Shigeo K: Some remarks on nonlinear differential equations in Banach space. Hokkaido Math. J. 1975, 4: 205–226.
Liu B, Huang L: Existence and stability of almost periodic solutions for shunting inhibitory cellular neural networks with time-varying delays. Chaos Solitons Fractals 2007, 31: 211–217. 10.1016/j.chaos.2005.09.052
Krasnosel’skii MA Translat. Math. Monoger. 19. The Operator of Translation Along the Trajectories of Differential Equations 1968.
Medvedev NV: Certain tests for the existence of bounded solutions of systems of differential equations. Differ. Uravn. (Minsk) 1968, 4: 1258–1264.
Thanh N: Asymptotically almost periodic solutions on the half-line. J. Differ. Equ. Appl. 2005, 11: 1231–1243. 10.1080/10236190500267897
Xia Y, Cao J, Lin M: New results on the existence and uniqueness of almost periodic solution for BAM neural networks with continuously distributed delays. Chaos Solitons Fractals 2007, 31: 928–936. 10.1016/j.chaos.2005.10.043
The authors would like to thank the two referees for their valuable suggestions and comments concerning improvement of the work.
The authors declare that they have no competing interests.
All authors completed the paper together. All authors read and approved the final manuscript.
About this article
Cite this article
Song, J., Cao, J. & Li, X. On the existence of asymptotically almost periodic solutions for nonlinear systems. Adv Differ Equ 2013, 28 (2013). https://doi.org/10.1186/1687-1847-2013-28
- Banach Space
- Positive Constant
- Linear Operator
- Periodic Solution
- Cauchy Problem