On the Spectrum of Almost Periodic Solution of Second-Order Neutral Delay Differential Equations with Piecewise Constant of Argument

Abstract

The spectrum containment of almost periodic solution of second-order neutral delay differential equations with piecewise constant of argument (EPCA, for short) of the form is considered. The main result obtained in this paper is different from that given by some authors for ordinary differential equations (ODE, for short) and clearly shows the differences between ODE and EPCA. Moreover, it is also different from that given for equation because of the difference between and .

1. Introduction and Some Preliminaries

Differential equations with piecewise constant argument, which were firstly considered by Cooke and Wiener [1] and Shah and Wiener [2], combine properties of both differential and difference equations and usually describe hybrid dynamical systems and have applications in certain biomedical models in the work of Busenberg and Cooke [3]. Over the years, more attention has been paid to the existence, uniqueness, and spectrum containment of almost periodic solutions of this type of equations (see, e.g., [4–12] and reference there in).

If and are almost periodic, then the module containment property can be characterized in several ways (see [13–16]). For periodic function this inclusion just means that the minimal period of is a multiple of the minimal period of . Some properties of basic frequencies (the base of spectrum) were discussed for almost periodic functions by Cartwright. In [17], Cartwright compared basic frequencies (the base of spectrum) of almost periodic differential equations (ODE) , , with those of its unique almost periodic solution. For scalar equation, , Cartwright's results in [17] implied that the number of basic frequencies of , is the same as that of basic frequencies of its unique solution.

The spectrum containment of almost periodic solution of equation was studied in [9, 10]. Up to now, there have been no papers concerning the spectrum containment of almost periodic solution of equation

(1.1)

where denotes the greatest integer function, , are nonzero real constants, , , and is almost periodic. In this paper, we investigate the existence, uniqueness, and spectrum containment of almost periodic solutions of (1.1). The main result obtained in this paper is different from that given in [17] for ordinary differential equations (ODE, for short). This clearly shows differences between ODE and EPCA. Moreover, it is also different from that given in [9, 10] for equation . This is due to the difference between and . As well known, both solutions of (1.1) and equation can be constructed by the solutions of corresponding difference equations. However, noticing the difference between and , the solution of difference equation corresponding to the latter can be obtained directly (see [4]), while the solution of difference equation corresponding to the former (i.e., (1.1) cannot be obtained directly. In fact, consists of two parts: and . We will first obtain by solving a difference equation and then obtain from . (Similar technology can be seen in [8].) A detailed account will be given in Section 2.

Now, We give some preliminary notions, definitions, and theorem. Throughout this paper , , and denote the sets of integers, real, and complex numbers, respectively. The following preliminaries can be found in the books, for example, [13–16].

Definition 1.1.

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

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

(1.2)

is relatively dense for each .

Definition 1.2.

Let be a bounded continuous function. If the limit

(1.3)

exists, then we call the limit mean of and denote it by .

If , then the limit

(1.4)

exists uniformly with respect to . Furthermore, the limit is independent of .

For any and since the function is in , the mean exists for this function. We write

(1.5)

then there exists at most a countable set of 's for which . The set

(1.6)

is called the frequency set (or spectrum) of . It is clear that if , then if , for some ; and if , for any . Thus, .

Members of are called the Fourier exponents of , and 's are called the Fourier coefficients of . Obviously, is countable. Let and . Thus can associate a Fourier series:

(1.7)

The Approximation Theorem

Let and . Then for any there exists a sequence of trigonometric polynomials

(1.8)

such that

(1.9)

where is the product of and certain positive number (depending on and ) and .

Definition 1.3.

() 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 .

() A bounded sequence is called an almost periodic sequence (abbreviated as ) if the -translation set of

(1.10)

is relatively dense for each .

For an almost periodic sequence , it follows from the lemma in [13] that

(1.11)

exists. The set

(1.12)

is called the Bohr spectrum of . Obviously, for almost periodic sequence , if , for some ; if , for any . So,

2. The Statement of Main Theorem

We begin this section with a definition of the solution of (1.1).

Definition 2.1.

A continuous function is called a solution of (1.1) if the following conditions are satisfied:

(i) satisfies (1.1) for , ;

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

In [8], the authors pointed out that if is a solution of (1.1), then are continuous at , which guarantees the uniqueness of solution of (1.1) and cannot be omitted.

To study the spectrum of almost periodic solution of (1.1), we firstly study the solution of (1.1). Let

(2.1)

Suppose that is a solution of (1.1), then exist and are continuous everywhere on . By a process of integrating (1.1) two times in or as in [7, 8, 18], we can easily get

(2.2)

These lead to the difference equations

(2.3)

(2.4)

Suppose that . First, multiply the two sides of (2.3) and (2.4) by and , respectively, then add the resulting equations to get

(2.5)

Similarly, one gets

(2.6)

Replacing by in (2.6) and comparing with (2.5), one gets

(2.7)

The corresponding homogeneous equation is

(2.8)

We can seek the particular solution as for this homogeneous difference equation. At this time, will satisfy the following equation:

(2.9)

From the analysis above one sees that if is a solution of (1.1) and , then one gets (2.3) and (2.4). In fact, a solution of (1.1) is constructed by the common solution of (2.3) and (2.4). Moreover, it is clear that consists of two parts: and . can be obtained by solving (2.7), and can be obtained by substituting into (2.5) or (2.6). Without loss of generality, we consider (2.5) only. These will be shown in Lemmas 2.5 and 2.6.

Lemma 2.2.

If , then , , .

Lemma 2.3.

Suppose that and , then the roots of polynomial are of moduli different from 1.

Lemma 2.4.

Suppose that is a Banach space, denotes the set of bounded linear operators from to , and , then is bounded invertible and

(2.10)

where , and is an identical operator.

The proofs of Lemmas 2.2, 2.3, and 2.4 are elementary, and we omit the details.

Lemma 2.5.

Suppose that and , then (2.7) has a unique solution .

Proof.

As the proof of Theorem in [8], define by , where is the Banach space consisting of all bounded sequences in with . It follows from Lemmas 2.2–2.4 that (2.7) has a unique solution .

Substituting into (2.5), we obtain . Easily, we can get . Consequently, the common solution of (2.3) and (2.4) can be obtained. Furthermore, we have that is unique.

Lemma 2.6.

Suppose that and , . Let be the common solution of (2.3) and (2.4). Then (1.1) has a unique solution such that . In this case the solution is given for by

(2.11)

where

(2.12)

for

The proof is easy, we omit the details. Since the almost periodic solution of (1.1) is constructed by the common almost periodic solution of (2.3) and (2.4), easily, we have that are continuous at . It must be pointed out that in many works only one of (2.3) and (2.4) is considered while seeking the unique almost periodic solution of (1.1), and it is not true for the continuity of on , consequently, it is not true for the uniqueness (see [8]).

The expressions of and are important in the process of studying the spectrum containment of the almost periodic solution of (1.1). Before giving the main theorem, we list the following assumptions which will be used later.

(H1), .

(H2) for all .

(H3)If , then , .

Our result can be formulated as follows.

Main Theorem

Let and () be satisfied. Then (1.1) has a unique almost periodic solution and . Additionally, if () and () are also satisfied, then , that is, the following spectrum relation holds, where the sum of sets and is defined as .

We postpone the proof of this theorem to the next section.

3. The Proof of Main Theorem

To show the Main Theorem, we need some more lemmas.

Lemma 3.1.

Let , then , . If () is satisfied, then , . Furthermore, if () and () are both satisfied, then .

Proof.

Since , by Lemma 2.2 we know that , . It follows from The Approximation Theorem that, for any , there exists such that where , and we can assume that and appear together in the trigonometric polynomial . Define

(3.1)

where

(3.2)

Obviously, , , for all . For any , , , thus, we have , .

Since and , for all . For all , we have

(3.3)

(3.4)

Thus, and imply and , respectively, .

If () is satisfied, then for any , we have

(3.5)

Easily, we have and , that is, , . By the arbitrariness of , we get and . So,

If () and () are both satisfied, suppose that there exists such that () implies . Moreover, since () holds, we have . leads to , which contradicts with . So, . Noticing that , we have . Similarly, we can get . The proof is completed.

Lemma 3.2.

Suppose that () is satisfied, then . If (), (), and () are all satisfied, then , where is the unique almost periodic sequence solution of (2.7).

Proof.

Since () holds, from Lemma 2.5 we know , where  for all . For any , it follows from Lemma 2.3 that . Noticing the expressions of and , we obtain

(3.6)

(3.7)

Those equalities and Lemma 3.1 imply that and , when () is satisfied. If (), (), and () are all satisfied, we only need to prove . Suppose that there exists , obviously, , such that From Lemma 3.1, Thus, that is, , which leads to . This contradicts with (). Thus, , that is, . Noticing that , so, . The proof is completed.

As mentioned above, the common almost periodic sequence solution of (2.3) and (2.4) consists of two parts: and , where is the unique solution of (2.7), and is obtained by substituting into (2.5). Obviously, . In the following, we give the spectrum containment of .

Lemma 3.3.

Suppose that () is satisfied, then . If (), (), and () are all satisfied, then .

Proof.

Since , . Noticing the expression of , for any , we have

(3.8)

where . If () is satisfied, it follows from Lemmas 3.1 and 3.2 that .

If (), (), and () are all satisfied, supposing there exists , obviously, , such that , that is, Noticing (3.3)–(3.7), this equality is equivalent to that is, . Considering equation , its roots are , , and , obviously, , . We claim that , , that is, this equation has no imaginary root. Otherwise, suppose that and , then by the relationship between roots and coefficient of three-order equation, we know , which leads to a contradiction. Thus ; this contradiction shows . Noticing that , thus, . The proof is completed.

Lemma 3.4.

Suppose that () is satisfied, then . If (), (), and () are all satisfied, then , where is defined in Lemma 2.6.

Proof.

From Lemma 2.6, we have , for all For any

(3.9)

Since () holds, it follows from Lemmas 3.1–3.3 that we have .

If (), (), and () are all satisfied, supposing there exists such that it follows from () that . Notice that (3.3)–(3.8), is equivalent to . This equality is equivalent to . Since , that is, this leads to We firstly claim that the equation has no imaginary root, that is, equations and both have no imaginary roots, where . If these two equations have imaginary roots, then , . Since , then or . If the first equation has imaginary roots, then , which contradicts with or . If the second equation has imaginary roots, then , which also contradicts with or . The claim follows. Thus and . Substituting into , we get . This is impossible. Thus, for any , we have that is, . Noticing that , we have . The proof has finished.

In Lemma 2.6, we have given the expression of the almost periodic solution of (1.1) explicitly by a known function . This brings more convenience to study the spectrum containment of almost periodic solution of (1.1). Now, we are in the position to show the Main Theorem.

The proof of Main Theorem

Since () is satisfied, by Lemma 2.6, (1.1) has a unique almost periodic solution satisfying . Thus, for any we have . Since () holds, then . We only need to prove when () is satisfied, and when ()–() are all satisfied.

When () is satisfied, we prove firstly. For any , it follows from Lemmas 3.1–3.4 that , and , that is, From the expression of given in Lemma 2.6, we know

(3.10)

As mentioned in Lemma 3.1, for any , there exists such that as . By simple calculation, we have

(3.11)

Therefore, , that is, , which implies .

Additionally, if () and () are also satisfied, to show the equality , we only need to show the inverse inclusion, that is, . For any , define , where and define

(3.12)

, then, , and as , , , in , in , which implies for any , as where , and are as in Lemma 3.1, . Similarly as above, for all , we can get

(3.13)

We claim: . Suppose that the claim is false, then there would exist such that . Noticing () and (), an elementary calculation leads to

(3.14)

From the above equality, we have , where, So, . Since , , we have , which is equivalent to . Since , that is, , this leads to . Noticing , it follows from ( that , that is, . From Lemma 3.4, we know that the equation has no imaginary root. Thus , which leads to a contradiction. The claim follows.

Now we are able to prove . For any let . Noticing ( and (, , and we have

(3.15)

The above equality is equivalent to . So, , which implies that , that is, . From the claim above, we get . This completes the proof.