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

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.

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 almost-periodic-type solutions of this type of equations. There are many remarkable works on this field (see [3–10] and references therein). Particularly, the second-order neutral delay-differential equations with piecewise constant argument of the form

(1.1)

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].

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

(1.2)

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 .

(2). A continuous function is called almost periodic (abbreviated as ) if the -translation set of

(1.3)

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 almost-periodic sequence (abbreviated as ) (resp., abbreviated as ) if the -translation set of

(1.4)

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.

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

(1.5)

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

Let

(2.1)

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

(2.2)

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.

Suppose that is an ow-solution 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

(2.3)

Similarly if is an ew-solution of (1.2), by the process of integrating (1.2) two times in , we get

(2.4)

These lead to the difference equations

(2.5)

(2.6)

From the analysis above, one sees that if is an ow-solution (resp., ew-solution) 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 w-solutions 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:

(2.7)

(2.8)

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

(2.9)

(2.10)

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.

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:

(2.11)

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.

Under the assumption (H), define as

(2.12)

where

(2.13)

From (2.9), it follows that is continuous on and , . Moreover, for , , one has ; for , , one has . By simple calculation, for , , we have

(2.14)

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.

Under the assumption (H), define as

(2.15)

where

(2.16)

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.

Suppose that is a solution of (1.2), then

(2.17)

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

(2.18)

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.

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

Let

(3.1)

then is the common solution of (2.5) and (2.6). Calculate , , , by the formulas mentioned above, we obtain , , Obviously, and . Define the ow-solution and the ew-solution 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 ow-solution as well as an ew-solution. Specifically, it can be expressed as

(3.2)

where

(3.3)

It is easy to check that is an almost-periodic solution of (1.2) such that , .

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

Let

(3.4)

Obviously, is another solution of (2.5).

Let and

(3.5)

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.

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

Let

(3.6)

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

Define the ow-solution as

(3.7)

where

(3.8)

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

• (ii) Similarly to (i), by Theorem 2.7, we construct the almost-periodic ew-solution of (1.2).

Let and

(3.9)

where is an arbitrary constant, then is the solution of (2.6). Calculate as the formulas mentioned above, we obtain , Obviously, .

Define the ew-solution as

(3.10)

where

(3.11)

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).