Abstract
We investigate the existence of almostperiodic weak solutions of secondorder neutral delaydifferential equations with piecewise constant argument of the form , where denotes the greatest integer function, is a real nonzero constant, and is almost periodic.
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1. Introduction and Preliminaries
Differential equations with piecewise constant argument, which were firstly considered by Cooke and Wiener [1], and Shah and Wiener [2], usually describe hybrid dynamical systems (a combination of continuous and discrete) and so combine properties of both differential and difference equations. Over the years, great attention has been paid to the study of the existence of almostperiodictype solutions of this type of equations. There are many remarkable works on this field (see [3–10] and references therein). Particularly, the secondorder neutral delaydifferential equations with piecewise constant argument of the form
have been intensively studied for by different methods, where denotes the greatest integer function, , are real nonzero constants, and is almost periodic. In [6], Li introduced the concepts of oddweak solution, evenweak solution, and weak solution of (1.1). Some theorems about the existence of almostperiodic weak solutions were obtained while putting restriction on the function . Papers [7, 8] concentrated on dealing with the existence and uniqueness of pseudoalmostperiodic solution by putting some restrictions on the roots of characteristic equation instead of on the function . If is replaced by a nonlinear function , some results about the existence and uniqueness of almostperiodic solution or pseudoalmostperiodic solution were obtained in [8–10].
Up to now, there have been no papers concerning the solutions or weak solutions of (1.1) when . In this paper, we study this case, namely, the equation
In constructing almostperiodictype solution or weak solution of (1.1) in [6–10], the condition is essential because it guarantees the convergence of the related series. To investigate such equation as (1.2), we have to give a quite different consideration.
Now we give some definitions. Throughout this paper, , , and denote the sets of integers, real, and complex numbers, respectively. The following definitions can be found in any book, say [11], on almostperiodic functions.
Definition 1.1.
(1). A subset of is said to be relatively dense in if there exists a number such that for all .
(2). A continuous function is called almost periodic (abbreviated as ) if the translation set of
is relatively dense for each .
Definition 1.2. (1) For a sequence , define and call it sequence interval with length . A subset of is said to be relatively dense in if there exists a positive integer such that for all .
(2) A bounded sequence (resp., ) is called an almostperiodic sequence (abbreviated as ) (resp., abbreviated as ) if the translation set of
is relatively dense for each .
As mentioned in [6], we have the following definitions.
Definition 1.3.
A continuous function is called an oddweak solution (resp., evenweak solution) of (1.2) if the following conditions are satisfied:

(i)
satisfies (1.2) for , ;

(ii)
the onesided derivatives exist at (resp., ), ;

(iii)
the onesided secondorder derivatives exist at (resp., ), .
Both oddweak solution (abbreviated as owsolution) and evenweak solution (abbreviated as ewsolution) of (1.2) are called weak solution (abbreviated as wsolution) of (1.2). It should be pointed out that if is an owsolution (resp., ewsolution) of (1.2), then are continuous at (resp., ), ; owsolution of (1.2) is not equivalent to ewsolution of (1.2); is a solution of (1.2) if it is an owsolution as well as an ewsolution of (1.2).
Let and , then the following hold.
Lemma 1.4.
Assume . The roots of polynomials are of modules different from 1, .
Proof.
It is clear that 1 and −1 are not the roots of because , . Denote the three roots of by , without loss of generality, let , , here is a real constant, thus we obtain , which is impossible. So, the modules of roots of polynomial are not 1.
If , it is obvious that the result holds for . If , denote the three roots of by , without loss of generality, let , , here is a real constant, then we have
This implies that and contradicts the hypothesis. The proof is complete.
The rest of this paper is organized as follows. Section 2 is devoted to the main theorems and their proofs. In Section 3, some examples are given to explain our results and illuminate the relationship among solution, owsolution, and ewsolution.
2. The Main Results
Let
To present the main results of this paper, we need the following assumption.
(H) is such that there exists such that , for all .
Remark 2.1. (1) is a translation invariant Banach space. For every , one has too. Set , then satisfies (H), and therefore there exist a great number of functions satisfying the assumption (H). (2) Reference [5] uses an assumption similar to (H) implicitly.
Let . We have the following lemma.
Lemma 2.2.
Under the assumption (H), one has .
Proof.
By (H), there exists such that , . Let and , it is easy to verify that ,, , ,, , , and , , for all that is, , . Set , similarly we can obtain , , and , for all that is, .
Lemma 2.3.
Suppose that is a Banach space, denotes the set of bounded linear operators from to , and , then is bounded invertible and
where , is an identical operator.
The proof of Lemma 2.3 can be found in any book of functional analysis. We remark that if is a linear operator and its inverse exists, then is also a linear operator.
To get wsolutions or solutions of (1.2), we start with its corresponding difference equations.
Suppose that is an owsolution of (1.2), then satisfies the three conditions in Definition 1.3. By a process of integrating (1.2) two times in as in [6–10], we can easily get
Similarly if is an ewsolution of (1.2), by the process of integrating (1.2) two times in , we get
These lead to the difference equations
From the analysis above, one sees that if is an owsolution (resp., ewsolution) of (1.2), then one gets (2.5) (resp., (2.6)); if is a solution of (1.2), then one gets both (2.5) and (2.6). Conversely, we will show that the wsolutions or solutions of (1.2) are obtained via the solutions of (2.5) and (2.6). In order to get the solutions of (2.5) and (2.6), we will consider the following difference equations:
Notice that for any sequences , and , one has . Especially, In virtue of studying (2.7) and (2.8), we have the following theorem.
Theorem 2.4.
Under the assumption (H), (2.7) (resp., (2.8)) has a unique solution (resp., ).
Proof.
As the proof of [7, Theorem 9], define by , where is the Banach space consisting of all bounded sequences in with . Notice Lemmas 1.4 and 2.3, we know that (2.7) has a unique solution . By the process of proving Lemma 2.2, we have that is, where (this follows in the same way as [7, Theorem 9]). Therefore, (2.7) has a unique solution .
Similarly, (2.8) has a unique solution and , that is, where . Therefore, (2.8) has a unique solution . This completes the proof.
Remark 2.5. (i) In Theorem 2.4, since and , we can easily get
It must be stressed that (2.9) and (2.10) are important, since they can guarantee the continuity of the wsolutions or solutions of (1.2) constructed in Theorems 2.6, 2.7, and 2.8.
(ii) Let with satisfying (2.9) , and with satisfying (2.10) . Notice the fact that the solution of (2.7) (resp., (2.8)) must be a solution of (2.5) (resp., (2.6)), it is false conversely. So, suppose the assumption (H) holds, it follows from Theorem 2.4 that (2.5) (resp., (2.6)) has solution (resp., ). Moreover, such solutions may not be unique. See Example 3.1 at the end of this paper.
In the following, we focus on seeking the almostperiodic wsolutions or solutions of (1.2) via the almostperiodic sequence solutions of (2.5) and (2.6). As mentioned above, it is due to that, to get almostperiodic wsolutions or solutions of (1.2), we have to use a way quite different from [6–10]. Our main idea is to construct solutions or wsolutions of (1.2) piecewise. It seems that this is a new technique.
Without loss of generality, suppose that (resp., ) is an arbitrary solution of (2.5) (resp., (2.6)). To prove the following theorems, we need to introduce some notations firstly:
where and . It can be easily verified that
For the existence of the almostperiodic owsolution of (1.2), we have the following.
Theorem 2.6.
Under the assumption (H), (1.2) has an owsolution such that .
Proof.
Under the assumption (H), define as
where
From (2.9), it follows that is continuous on and , . Moreover, for , , one has ; for , , one has . By simple calculation, for , , we have
Note that , this implies that the onesided derivatives exist at . Since , the secondorder derivatives are continuous at , . Therefore, is an owsolution of (1.2) such that , . Furthermore, it is easy to check that is almost periodic, we omit the details. The proof is complete.
For the existence of the almostperiodic ewsolution of (1.2), we have the following.
Theorem 2.7.
Under the assumption (H), (1.2) has an ewsolution such that .
Proof.
Under the assumption (H), define as
where
From (2.10), it follows that is continuous on and , . The rest of the proof is similar to that of Theorem 2.6, we omit the details.
For the existence of almostperiodic solution of (1.2), we have the following.
Theorem 2.8.
Under the assumption (H), if is the common solution of (2.5) and (2.6), then (1.2) has a solution such that , . If replaces , the conclusion is still true.
Proof.
Since and are solutions of (2.5) and (2.9) respectively, and they are also solutions of (2.6) and (2.10), respectively, it follows from Theorems 2.6 and 2.7 that, by simple calculation, the almostperiodic owsolution constructed as the proof of Theorem 2.6 with , , is the same as the almostperiodic ewsolution constructed as the proof of Theorem 2.7 with , . This implies is an almostperiodic solution of (1.2) such that , . If replaces , the proof is similar, we omit the details.
Remark 2.9.
As mentioned above, an owsolution of (1.2) is not equivalent to an ewsolution of (1.2), and a solution of (1.2) is an owsolution of (1.2) as well as an ewsolution of (1.2). See the examples in Section 3.
The following theorem is usually used for judging whether or not a wsolution of (1.2) is a solution of (1.2).
Theorem 2.10.
Suppose that is a solution of (1.2), then
Proof.
If is a solution of (1.2), then must be common solution of (2.5) and (2.6). Moreover, (2.6) is equivalent to
Substituting into both the above equation and (2.5), then add the resulting equations to get the result.
3. Some Examples
In this section, we first explain how to get almost periodic wsolutions and solutions of (1.2) specifically. And then, we present two examples: in Example 3.1, we aim mainly to obtain the almostperiodic solution, and in Example 3.2, we obtain the almostperiodic owsolution and ewsolution. Consequently, the relationship among owsolution, ewsolution, and solution is shown. Besides, Example 3.1 also illuminates that the solutions in (resp., ) of (2.5) (resp., (2.6)) may not be unique.
Under the assumption (H), it follows from the proof of Theorem 2.6 (resp., 2.7) that we can get almostperiodic owsolution (resp., ewsolution) of (1.2) by the following three steps.

(i)
Calculate , , , , , , , , and , .

(ii)
Seek the solution (resp., ) of (2.5) (resp., (2.6)). Calculate and (resp., and ).

(iii)
By the proof of Theorem 2.6 (resp., 2.7), we get the almostperiodic owsolution (resp., ewsolution ) such that , (resp., ), .
On the other hand, it follows from the proof of Theorem 2.8 that we can get the almostperiodic solution by the following steps.

(i)
Seek the solution in which is the common solution of (2.5) and (2.6). Calculate , , , and .

(ii)
Find the almost periodic owsolution such that , or ewsolution such that , by the above methods. From Theorem 2.8, we know they are the same, that is, it must be the almost periodic solution.
The following example shows that a solution of (1.2) is an owsolution of (1.2) as well as an ewsolution of (1.2), and the solutions in (resp., ) of (2.5) (resp., (2.6)) may not unique.
Example 3.1.
Let , , and , then , and , for all that is, satisfy the assumption (H). By simple calculation, we can obtain , , , , and , , for all .

(i)
We construct the almostperiodic solution of (1.2) as the proof of Theorem 2.8.
Let
then is the common solution of (2.5) and (2.6). Calculate , , , by the formulas mentioned above, we obtain , , Obviously, and . Define the owsolution and the ewsolution as the proofs of Theorems 2.6 and 2.7, respectively, it follows from the proof of Theorem 2.8 that they are the same, so it must be a solution, that is, an owsolution as well as an ewsolution. Specifically, it can be expressed as
where
It is easy to check that is an almostperiodic solution of (1.2) such that , .

(ii)
We show that (resp., ) is not unique solution of (2.5) (resp., (2.6)).
Let
Obviously, is another solution of (2.5).
Let and
where is an arbitrary constant, then it is clear that is another solution of (2.6).
The following example shows that owsolutions and ewsolutions of (1.2) are not equivalent.
Example 3.2.
As [6], let , , for all , then , setting , then and , for all that is, the assumption (H) holds. By simple calculation, we can obtain , , , , , , , .

(i)
We construct the almostperiodic owsolution of (1.2) as the proof of Theorem 2.6.
Let
then is the solution of (2.5). Calculate as the formulas mentioned above, we obtain , Obviously, , .
Define the owsolution as
where
It is easy to check that is an almostperiodic owsolution of (1.2). Since is not solution of (2.17), it follows from Theorem 2.10 that is not solution of (1.2) and consequently, is not an ewsolution of (1.2).

(ii)
Similarly to (i), by Theorem 2.7, we construct the almostperiodic ewsolution of (1.2).
Let and
where is an arbitrary constant, then is the solution of (2.6). Calculate as the formulas mentioned above, we obtain , Obviously, .
Define the ewsolution as
where
It is easy to verify that is an almostperiodic ewsolution of (1.2). Since is not solution of (2.17), it follows from Theorem 2.10 that is not solution of (1.2) and consequently, is not an owsolution of (1.2).
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Acknowledgment
The research is supported by the NSF of China no. 10671047.
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Wang, L., Zhang, C. AlmostPeriodic Weak Solutions of SecondOrder Neutral DelayDifferential Equations with Piecewise Constant Argument. Adv Differ Equ 2008, 816091 (2008). https://doi.org/10.1155/2008/816091
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DOI: https://doi.org/10.1155/2008/816091