- Research Article
- Open Access
- Published:
On the Spectrum of Almost Periodic Solution of Second-Order Neutral Delay Differential Equations with Piecewise Constant of Argument
Advances in Difference Equations volume 2009, Article number: 143175 (2009)
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

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

is relatively dense for each .
Definition 1.2.
Let be a bounded continuous function. If the limit

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

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

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

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:

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

such that

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

is relatively dense for each .
For an almost periodic sequence , it follows from the lemma in [13] that

exists. The set

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

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

These lead to the difference equations


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

Similarly, one gets

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

The corresponding homogeneous equation is

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

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

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

where

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

where

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


Thus, and
imply
and
, respectively,
.
If () is satisfied, then for any
, we have

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


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

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

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

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

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

, 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

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

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

The above equality is equivalent to . So,
, which implies that
, that is,
. From the claim above, we get
. This completes the proof.
References
Cooke KL, Wiener J: Retarded differential equations with piecewise constant delays. Journal of Mathematical Analysis and Applications 1984,99(1):265–297. 10.1016/0022-247X(84)90248-8
Shah SM, Wiener J: Advanced differential equations with piecewise constant argument deviations. International Journal of Mathematics and Mathematical Sciences 1983,6(4):671–703. 10.1155/S0161171283000599
Busenberg S, Cooke KL: Models of vertically transmitted diseases with sequential-continuous dynamics. In Nonlinear Phenomena in Mathematical Sciences. Edited by: Lakshmikantham V. Academic Press, New York, NY, USA; 1982:179–187.
Seifert G: Second-order neutral delay-differential equations with piecewise constant time dependence. Journal of Mathematical Analysis and Applications 2003,281(1):1–9.
Piao DX: Almost periodic solutions of neutral differential difference equations with piecewise constant arguments. Acta Mathematica Sinica 2002,18(2):263–276.
Li H-X: Almost periodic solutions of second-order neutral delay-differential equations with piecewise constant arguments. Journal of Mathematical Analysis and Applications 2004,298(2):693–709. 10.1016/j.jmaa.2004.05.034
Li H-X: Almost periodic weak solutions of neutral delay-differential equations with piecewise constant argument. Nonlinear Analysis: Theory, Methods & Applications 2006,64(3):530–545. 10.1016/j.na.2005.05.041
Dads EA, Lhachimi L: New approach for the existence of pseudo almost periodic solutions for some second order differential equation with piecewise constant argument. Nonlinear Analysis: Theory, Methods & Applications 2006,64(6):1307–1324. 10.1016/j.na.2005.06.037
Yuan R: On the spectrum of almost periodic solution of second order scalar functional differential equations with piecewise constant argument. Journal of Mathematical Analysis and Applications 2005,303(1):103–118. 10.1016/j.jmaa.2004.06.057
Wang L, Yuan R, Zhang C: Corrigendum to: "On the spectrum of almost periodic solution of second order scalar functional differential equations with piecewise constant argument". Journal of Mathematical Analysis and Applications 2009,349(1):299. 10.1016/j.jmaa.2008.08.008
Yang X, Yuan R: On the module containment of the almost periodic solution for a class of differential equations with piecewise constant delays. Journal of Mathematical Analysis and Applications 2006,322(2):540–555. 10.1016/j.jmaa.2005.09.036
Yuan R: On the spectrum of almost periodic solution of second-order differential equations with piecewise constant argument. Nonlinear Analysis: Theory, Methods & Applications 2004,59(8):1189–1205.
Corduneanu C: Almost Periodic Functions, Interscience Tracts in Pure and Applied Mathematics, no. 22. John Wiley & Sons, New York, NY, USA; 1968:x+237.
Fink AM: Almost Periodic Differential Equations, Lecture Notes in Mathematics. Volume 377. Springer, Berlin, Germany; 1974:viii+336.
Levitan BM, Zhikov VV: Almost Periodic Functions and Differential Equations. Cambridge University Press, Cambridge, UK; 1982:xi+211.
Zhang C: Almost Periodic Type Functions and Ergodicity. Science Press, Beijing, China; Kluwer Academic Publishers, Dordrecht, The Netherlands; 2003:xii+355.
Cartwright ML: Almost periodic differential equations and almost periodic flows. Journal of Differential Equations 1969,5(1):167–181. 10.1016/0022-0396(69)90110-7
Yuan R: Pseudo-almost periodic solutions of second-order neutral delay differential equations with piecewise constant argument. Nonlinear Analysis: Theory, Methods & Applications 2000,41(7–8):871–890. 10.1016/S0362-546X(98)00316-2
Author information
Authors and Affiliations
Corresponding author
Rights and permissions
Open Access This article is distributed under the terms of the Creative Commons Attribution 2.0 International License (https://creativecommons.org/licenses/by/2.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.
About this article
Cite this article
Wang, L., Zhang, C. On the Spectrum of Almost Periodic Solution of Second-Order Neutral Delay Differential Equations with Piecewise Constant of Argument. Adv Differ Equ 2009, 143175 (2009). https://doi.org/10.1155/2009/143175
Received:
Accepted:
Published:
DOI: https://doi.org/10.1155/2009/143175
Keywords
- Periodic Solution
- Difference Equation
- Trigonometric Polynomial
- Delay Differential Equation
- Approximation Theorem