- Research Article
- Open Access

# Almost-Periodic Weak Solutions of Second-Order Neutral Delay-Differential Equations with Piecewise Constant Argument

- Li Wang
^{1}Email author and - Chuanyi Zhang
^{1}

**2008**:816091

https://doi.org/10.1155/2008/816091

© L. Wang and C. Zhang. 2008

**Received:**10 December 2007**Accepted:**4 June 2008**Published:**10 June 2008

## Abstract

We investigate the existence of almost-periodic weak solutions of second-order neutral delay-differential equations with piecewise constant argument of the form , where denotes the greatest integer function, is a real nonzero constant, and is almost periodic.

## Keywords

- Banach Space
- Weak Solution
- Difference Equation
- Simple Calculation
- Bounded Linear Operator

## 1. Introduction and Preliminaries

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 odd-weak solution, even-weak solution, and weak solution of (1.1). Some theorems about the existence of almost-periodic weak solutions were obtained while putting restriction on the function . Papers [7, 8] concentrated on dealing with the existence and uniqueness of pseudo-almost-periodic 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 almost-periodic solution or pseudo-almost-periodic solution were obtained in [8–10].

In constructing almost-periodic-type 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 almost-periodic functions.

Definition 1.1.

(1). A subset of is said to be relatively dense in if there exists a number such that for all .

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 .

is relatively dense for each .

As mentioned in [6], we have the following definitions.

Definition 1.3.

A continuous function is called an odd-weak solution (resp., even-weak solution) of (1.2) if the following conditions are satisfied:

- (i)
satisfies (1.2) for , ;

- (ii)
the one-sided derivatives exist at (resp., ), ;

- (iii)
the one-sided second-order derivatives exist at (resp., ), .

Both odd-weak solution (abbreviated as ow-solution) and even-weak solution (abbreviated as ew-solution) of (1.2) are called weak solution (abbreviated as w-solution) of (1.2). It should be pointed out that if is an ow-solution (resp., ew-solution) of (1.2), then are continuous at (resp., ), ; ow-solution of (1.2) is not equivalent to ew-solution of (1.2); is a solution of (1.2) if it is an ow-solution as well as an ew-solution 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.

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, ow-solution, and ew-solution.

## 2. The Main Results

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.

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 w-solutions or solutions of (1.2), we start with its corresponding 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.

It must be stressed that (2.9) and (2.10) are important, since they can guarantee the continuity of the w-solutions 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 almost-periodic w-solutions or solutions of (1.2) via the almost-periodic sequence solutions of (2.5) and (2.6). As mentioned above, it is due to that, to get almost-periodic w-solutions or solutions of (1.2), we have to use a way quite different from [6–10]. Our main idea is to construct solutions or w-solutions of (1.2) piecewise. It seems that this is a new technique.

where and . It can be easily verified that

For the existence of the almost-periodic ow-solution of (1.2), we have the following.

Theorem 2.6.

Under the assumption (H), (1.2) has an ow-solution such that .

Proof.

Note that , this implies that the one-sided derivatives exist at . Since , the second-order derivatives are continuous at , . Therefore, is an ow-solution 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 almost-periodic ew-solution of (1.2), we have the following.

Theorem 2.7.

Under the assumption (H), (1.2) has an ew-solution such that .

Proof.

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 almost-periodic 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 almost-periodic ow-solution constructed as the proof of Theorem 2.6 with , , is the same as the almost-periodic ew-solution constructed as the proof of Theorem 2.7 with , . This implies is an almost-periodic solution of (1.2) such that , . If replaces , the proof is similar, we omit the details.

Remark 2.9.

As mentioned above, an ow-solution of (1.2) is not equivalent to an ew-solution of (1.2), and a solution of (1.2) is an ow-solution of (1.2) as well as an ew-solution of (1.2). See the examples in Section 3.

The following theorem is usually used for judging whether or not a w-solution of (1.2) is a solution of (1.2).

Theorem 2.10.

Proof.

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 w-solutions and solutions of (1.2) specifically. And then, we present two examples: in Example 3.1, we aim mainly to obtain the almost-periodic solution, and in Example 3.2, we obtain the almost-periodic ow-solution and ew-solution. Consequently, the relationship among ow-solution, ew-solution, 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 almost-periodic ow-solution (resp., ew-solution) 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 almost-periodic ow-solution (resp., ew-solution ) such that , (resp., ), .

On the other hand, it follows from the proof of Theorem 2.8 that we can get the almost-periodic 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 ow-solution such that , or ew-solution 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 ow-solution of (1.2) as well as an ew-solution of (1.2), and the solutions in (resp., ) of (2.5) (resp., (2.6)) may not unique.

Example 3.1.

- (i)
We construct the almost-periodic solution of (1.2) as the proof of Theorem 2.8.

- (ii)
We show that (resp., ) is not unique solution of (2.5) (resp., (2.6)).

Obviously, is another solution of (2.5).

where is an arbitrary constant, then it is clear that is another solution of (2.6).

The following example shows that ow-solutions and ew-solutions of (1.2) are not equivalent.

Example 3.2.

- (i)
We construct the almost-periodic ow-solution of (1.2) as the proof of Theorem 2.6.

then is the solution of (2.5). Calculate as the formulas mentioned above, we obtain , Obviously, , .

- (ii)
Similarly to (i), by Theorem 2.7, we construct the almost-periodic ew-solution 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, .

It is easy to verify that is an almost-periodic ew-solution 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 ow-solution of (1.2).

## Declarations

### Acknowledgment

The research is supported by the NSF of China no. 10671047.

## Authors’ Affiliations

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