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Existence of Periodic and Almost Periodic Solutions of Abstract Retarded Functional Difference Equations in Phase Spaces
Advances in Difference Equations volume 2009, Article number: 380568 (2009)
Abstract
The existence of periodic, almost periodic, and asymptotically almost periodic of periodic and almost periodic of abstract retarded functional difference equations in phase spaces is obtained by using stability properties of a bounded solution.
1. Introduction
In this paper, we study the existence of periodic, almost periodic, and asymptotic almost periodic solutions of the following functional difference equations with infinite delay:

assuming that this system possesses a bounded solution with some property of stability. In (1.1) , and
denotes an abstract phase space which we will define later.
The abstract space was introduced by Hale and Kato [1] to study qualitative theory of functional differential equations with unbounded delay. There exists a lot of literature devoted to this subject; we refer the reader to Corduneanu and Lakshmikantham [2], Hino et al. [3]. The theory of abstract retarded functional difference equations in phase space has attracted the attention of several authors in recent years. We only mention here Murakami [4, 5], Elaydi et al. [6], Cuevas and Pinto [7, 8], Cuevas and Vidal [9], and Cuevas and Del Campo [10].
As usual, we denote by ,
, and
the set of all integers, the set of all nonnegative integers, and the set of all nonpositive integers, respectively. Let
be the
-dimensional complex Euclidean space with norm
.
the set
.
If is a function, we define for
, the function
by
,
. Furthermore
is the function given for
, with
.
The abstract phase space , which is a subfamily of all functions from
into
denoted by
, is a normed space (with norm denoted by
) and satisfies the following axioms.
-
(A)
There is a positive constant
and nonnegative functions
and
on
with the property that
is a function, such that
, then for all
, the following conditions hold:
-
(i)
,
-
(ii)
,
-
(iii)
.
-
(i)
-
(B)
The space
is a Banach space.
We need the following property on
.
-
(C)
The inclusion map
is continuous, that is, there is a constant
, such that
, for all
, where
represents the bounded functions from
into
.
Axiom (C) says that any element of the Banach space of the bounded functions equipped with the supremum norm is on
.
Remark 1.1.
Using analogous ideas to the ones of [3], it is not difficult to prove that Axiom (C) is equivalente to the following.
-
(C')
If a uniformly bounded sequence
in
converges to a function
compactly on
(i.e., converges on any compact discrete interval in
) in the compact-open topology, then
belong to
and
as
.
Remark 1.2.
We will denote by (
, and
) or simply by
, the solution of (1.1) passing through
, that is,
, and the functional equation (1.1) is satisfied.
During this paper we will assume that the sequences and
are bounded. The paper is organized as follows. In Section 2 we see some important implications of the fading memory spaces. Section 3 is devoted to recall definitions and some important basic results about almost periodic sequences, asymptotically almost periodic sequences, and uniformly asymptotically almost periodic functions. In Section 4 we analyze separately the cases where
is periodic and when it is almost periodic. Thus, in Section 4.1 assuming that the system (1.1) is periodic and the existence of a bounded solution (particular solution) which is uniformly stable and the phase space satisfies only the axioms (A)–(C), we prove the existence of an almost periodic solution and an asymptotically almost periodic solution. If additionally the particular solution is uniformly asymptotically stable, we prove the existence of a periodic solution. Similarly, in Section 4.2 considering that system (1.1) is almost periodic and the existence of a bounded solution and whenever the phase space satisfies the axioms (A)–(C), but here it is also necessary that
verifies the fading memory property. If the particular solution is asymptotically almost periodic, then system (1.1) has an almost periodic solution. While, if the particular solution is uniformly asymptotically stable, we prove the existence of an asymptotically almost periodic solution.
In [11, 12] the problem of existence of almost periodic solutions for functional difference equations is considered in the first case for the discrete Volterra equation and in the second reference for the functional difference equations with finite delay; in both cases the authors assume the existence of a bounded solution with a property of stability that gives information about the existence of an almost periodic solution. In an analogous way in [13] the problem of the existence of almost periodic solutions for functional difference equations with infinite delay is considered. These results can be applied to several kinds of discrete equations. However, our approach differs from Hamaya's because, firstly, in our work we consider both cases, namely, when is periodic and when it is almost periodic in the first variable. And secondly, we analyze very carefully the implications of the existence of a bounded solution of (1.1) with each property: uniformly stable, uniformly asymptotically stable, and globally uniformly stable.
Furthermore, we cite the articles [14–16] which are devoted to study almost periodic solutions of difference equations, but a little is known about almost periodic solutions, and in particular, for periodic solutions of nonlinear functional difference equations in phase space via uniform stability, uniformly asymptotically stability, and globally uniformly stability properties of a bounded solution.
2. Fading Memory Spaces and Implications
Following the terminology given in [3], we introduce the family of operators on ,
, as

with . They constitute a family of linear operators on
having the semigroup property
for
. Immediately, the following result holds from Axiom (A):

Now, given any function such that
, we have the following decomposition:

where

Then, we have the following decomposition of ,
for
, where

and for all
. Note that

Let

be a subset of , and let
be the restriction of
to
. Clearly, the family
,
, is also a strongly continuous semigroup of bounded linear operators on
. It is given explicitly by

for .
Definition 2.1.
A phase space that satisfies axioms (A)-(B) and (
) or (
) and such that the semigroup
is strongly stable is called a fading memory space.
Remark 2.2.
Remember that a strongly continuous semigroup is strongly stable if for all ,
as
.
Thus, we have the following result.
Lemma 2.3.
Let , with
, where
is a fading memory space. If
as
, then
as
.
Proof.
Firstly, we note that as before, , where
, for
and

Then, by definition as
because
. On the other hand, by hypothesis,
as
, so it follows from Axiom (C') that
. Therefore, we conclude that
as
.
3. Notations and Preliminary Results
In this section, we review the definitions of (uniformly) almost periodic, asymptotically almost periodic sequence, which have been discussed by several authors and present some related properties.
For our purpose, we introduce the following definitions and results about almost periodic discrete processes which are given in [3, 17, 18] for the continuous case. For the discrete case we mention [11, 12].
Definition 3.1.
A sequence is called an almost periodic sequence if the
-translation set of
,

is a relatively dense set in for all
; that is, for any given
, there exists an integer
such that each discrete interval of length
contains
such that

is called the
-translation number of
. We will denote by
the set of all such sequences. We will write that
is a.p. if
.
Definition 3.2.
A sequence is called an asymptotically almost periodic sequence if

where is an almost periodic sequence, and
as
. We will denote by
the set of all such sequences. We will write that
is a.a.p. if
.
In general, we will consider a Banach space.
Definition 3.3.
A function or sequence is said to be almost periodic (abbreviated a.p.) in
if for every
there is
such that among
consecutive integers there is one; call it
, such that

Denote by all such sequences, and
is said to be an almost periodic (a.p.) in
.
Definition 3.4.
A sequence , (or
),
, equivalently, a function
(or,
) is called asymptotically almost periodic if
, where
and
(or,
) satisfying
as
(or,
). Denote by
(or
all such sequences, and
is said to be an asymptotically almost periodic on
(or on
) (a.a.p.) in
.
Remark 3.5.
Almost periodic sequences can be also defined for any sequence (
) or
by requiring that
consecutive integers are in
.
Definition 3.6.
Let .
is said to be almost periodic in
uniformly for
, if for any
and every compact
, there exists a positive integer
such that any interval of length
(i.e., among
consecutive integers) contains an integer (or equivalently, there is one); call it
, for which

is called the
-translation number of
. We will denote by
the set of all such sequences. In brief we will write that
is u.a.p. if
.
Definition 3.7.
The hull of , denoted by
, is defined by

for some sequence , where
is any compact set in
.
For our purpose, we introduce the following definitions and results about almost periodic discrete processes which are given in [3, 17, 18] for the continuous case. For the discrete case we mention [11, 12]. With the objective to make this manuscript self contained we decided to include the majority of the proofs.
Lemma 3.8.
-
(a)
If
is an a.p. sequence, then there exists an almost periodic function
such that
for
.
-
(b)
If
is an a.p. function, then
is an a.p. sequence.
Lemma 3.9. (a) If is an a.p. sequence, then
is bounded.
(b) is an a.p. sequence if and only if for any sequence
there exists a subsequence
such that
converges uniformly on
as
. Furthermore, the limits sequence is also an almost periodic sequence.
(c) is an a.p. sequence if and only if for any sequence of integers
,
there exist subsequences
,
such that

where for
.
(d),
(or,
) is an a.a.p. sequence if and only if for any sequence
(or,
) such that
and
as
(or,
as
), there exists a subsequence
such that
converges uniformly on
(or
) as
.
Lemma 3.10.
Let be an a.a.p. periodic sequence. Then its decomposition,

where is an a.p. sequence while
as
, is unique.
Lemma 3.11.
Let be almost periodic in
uniformly for
and continuous in
. Then
is bounded and uniformly continuous on
for any compact set
in
.
Lemma 3.12.
Let be the same as in the previous lemma. Then, for any sequence
, there exist a subsequence
of
and a function
continuous in
such that
uniformly on
as
, where
is any compact set in
. Moreover,
is also almost periodic in
uniformly for
.
Lemma 3.13.
Let be the same as in the previous lemma. Then, there exists a sequence
,
as
such that
uniformly on
as
, where
is any compact set in
.
Lemma 3.14.
Let be almost periodic in
uniformly for
and continuous in
, and let
be an almost periodic sequence in
such that
for all
, where
is a compact set in
. Then
is almost periodic in
.
Lemma 3.15.
Let be almost periodic in
uniformly for
and continuous in
, and let
be an almost periodic sequence in
such that
for all
, where
is a compact set in
and
for
. Then
is almost periodic in
.
Remark 3.16.
If is a.a.p., then the decomposition
, in the definition of an a.a.p. function, is unique (see [18]).
4. Existence of Almost Periodic Solutions
From now on we will assume that the system (1.1) has a unique solution for a given initial condition on and without loss of generality
, thus
.
We will make the following assumptions on (1.1).
-
(H1)
is continuous in the second variable for any fixed
.
-
(H2)
System (1.1) has a bounded solution
, passing through
,
, that is,
.
For this bounded solution , there is an
such that
for all
. So, we will have to assume that
for all
, and
. Next, we will point out the definitions of stability for functional difference equations adapting it from the continuous case according to Hino et al. in [3].
Definition 4.1.
A bounded solution of (1.1) is said to be:
-
(i)
stable, if for any
and any integer
, there is
such that
implies that
for all
, where
is any solution of (1.1);
-
(ii)
uniformly stable, abbreviated as "
'', if for any
and any integer
, there is
(
does not depend on
) such that
implies that
for all
, where
is any solution of (1.1);
-
(iii)
uniformly asymptotically stable, abbreviated as "
'', if it is uniformly stable and there is
such that for any
, there is a positive integer
such that if
and
, then
for all
, where
is any solution of (1.1);
-
(iv)
globally uniformly asymptotically stable, abbreviated as "
'', if it is uniformly stable and
as
, whenever
is any solution of (1.1).
Remark 4.2.
It is easy to see that an equivalent definition for , being
, is the following:
-
(iii)
is
, if it is uniformly stable, and there exists
such that if
and
, then
as
, where
is any solution of (1.1).
4.1. The Periodic Case
Here, we will assume what follows.
-
(H3)
The function
in (1.1) is periodic in
, that is, there exists a positive integer
such that
for all
.
Moreover, we will assume what follows.
-
(
) The sequences
and
in Axiom (A)(iii) are bounded by
and
, respectively and
.
Lemma 4.3.
Suppose that condition () holds. If
is a bounded solution of (1.1) such that
, then
is also bounded in
.
Proof.
Let us say that for all
. Then by Axiom (A)(iii) and hypothesis (
) we have

Lemma 4.4.
Suppose that condition () holds. Let
be a sequence in
such that
for all
. Assume that
as
for every
and
, then
in
as
for each
. In particular, if
as
uniformly in
, then
in
as
uniformly in
.
Proof.
By Axiom (A)(iii) and hypotheses we have that

In the particular case we obtain

and so as
. On the other hand, since
is fixed, it follows that

for each . Therefore, we have concluded the proof.
Theorem 4.5.
Suppose that condition () and (H1)–(H3) hold. If the bounded solution
of (1.1) is
, then
is an a.a.p. sequence in
, equivalently, (1.1) has an a.a.p. solution.
Proof.
By Lemma 4.3 there exists such that
for all
, and a bounded (or compact) set
such that
for all
. Let
be any integer sequence such that
and
as
. For each
, there exists a nonnegative integer
such that
. Set
. Then
for all
. Since
is a bounded set, we can assume that, taking a subsequence if necessary,
for all
, where
. Now, set
. Thus,

which implies that is a solution of the system,

through . It is clear that if
is
, then
is also
with the same pair
as the one for
.
Since is bounded for all
and
, we can use the diagonal method to get a subsequence
of
such that
converges for each
as
. Thus, we can assume that the sequence
converges for each
as
. Since
, by Lemma 4.4 it follows that
is also convergent for each
. In particular, for any
there exists a positive integer
such that if
(
is the constant given in Axiom A(ii), then

where is the number given by the uniform stability of
. Since
, it follows from Definition 4.1 and (4.7) that

and by Axiom A(ii) it follows that

This implies that for any positive integer sequence ,
as
, there is a subsequence
of
for which
converges uniformly on
as
. Thus, the conclusion of the theorem follows from Lemma 3.9(d).
Before proving our following result we remark that if is a.a.p. then there are unique sequences
such that
, with
a.p. and
as
as
. By Lemma 3.9(a) it follows that
is bounded and thus
. Hence, by Axiom (C) we must have that
for all
. In particular,
for all
.
Theorem 4.6.
Suppose that and (H1)–(H3) hold and the bounded solution
of (1.1) is
, then system (1.1) has an a.p. solution, which is also
.
Proof.
It follows from Theorem 4.5 that is an a.a.p. Set
(
), where
is a.p. sequence and
as
. For the positive integer sequence
, by Lemma 3.9(b)–(d) and arguments of the previous theorem, we can choice a subsequence
of
such that
converges uniformly in
and
uniformly on
as
and
is also a.p. Then,
uniformly in
, and thus by Lemma 4.4
uniformly in
on
as
and
. Since

as , we have
for
, that is, the system (1.1) has an almost periodic solution, and so we have proved the first statement of the theorem.
In order to prove the second affirmation, notice that since
. For any
, let
be a solution of (1.1) such that
and
. Again, by Lemma 4.4
as
for each
, so there is a positive integer
such that if
, then

Thus, for , we have

Then,

Therefore, there is such that if
, then

and hence, for all
, where
is a pair for the uniform stability of
. This shows that if
, then

for all , which implies that
for all
if
because
is arbitrary. This proves that
is
.
In the case when we have an asymptotically stable solution of (1.1) we obtain the following result.
Theorem 4.7.
Suppose that and (H1)–(H3) hold and the bounded solution
of (1.1) is
, then the system (1.1) has a periodic solution of period
for some positive integer
, which is also
.
Proof.
Set ,
. By the proof of Theorem 4.5, there is a subsequence
which converges to a solution
of (4.6) for each
and hence by Lemma 4.4,
as
. Thus, there is a positive integer
such that
(
), where
is obtained from the uniformly asymptotic stability of
. Let
, and notice that
is a solution of (1.1). Since
for
, that is,
, we have

and hence,

because is
(see also Remark 4.2). On the other hand,
is a.a.p. by Theorem 4.5, then

where is a.p. and
as
. It follows from (4.17) and (4.18) that

which implies that for all
because
is a.p.
For the integer sequence ,
, we have
. Then
uniformly for all
as
, and again by Lemma 4.4,
uniformly in
as
. Since
, we have
for
, which implies that (1.1) has a periodic solution
of period
.
Now, we will proceed to prove that by the use of definition
in Remark 4.2. Notice that since
then
is a
solution of (1.1) with the same
as the one for
. Let
be any solution of (1.1) such that
. Set
. Again, for sufficient large
, we have the similar relations (4.12) and (4.14) with
and
. Thus,

as if
, because
,
, and
satisfy (1.1). This completes the proof.
Finally, if the particular solution is , we will prove that system (1.1) has a periodic solution.
Theorem 4.8.
Suppose that and (H1)–(H3) hold and that the bounded solution
of (1.1) is
, then the system (1.1) has a periodic solution of period
.
Proof.
By Theorem 4.5, is a.a.p. Then
), where
(
) is an a.p. sequence and
as
. Notice that
is also a solution of (1.1) satisfying
. Since
is
, we have that
as
, which implies that
for all
. Using same technique as in the proof of Theorem 4.7, we can show that
is a
-periodic solution of (1.1).
4.2. The Almost Periodic Case
Here, we will assume that
-
(H4)
the function
in (1.1) is almost periodic in
uniformly in the second variable.
By we denote the uniform closure of
, that is,
. Note that
by Lemma 3.12 and
by Lemma 3.13.
Lemma 4.9.
Suppose that Axiom (C) is true, and that is an a.p. sequence with
, then
is a.p.
Proof.
We know that, given , there exists an integer
such that each discrete interval of length
contains a
such that

By Axiom (C) we have

Lemma 4.10.
Suppose that is a fading memory space and
is a.a.p. with
, then
is a.a.p.
Proof.
Since is a.a.p. there are unique sequences
and
such that
is a.p. and
as
. Then by Lemma 4.9 it follows that
is a.p., and by Lemma 2.3 it follows that
as
. Therefore,
is a.a.p.
Theorem 4.11.
Suppose that conditions , (H1)-(H2), and (H4) hold and that
is a fading memory space. If the bounded solution
of (1.1) is an a.a.p. sequence, then the system (1.1) has an a.p. solution.
Proof.
Since the solution is a.a.p., it follows from Lemma 3.10 that
has a unique decomposition
, where
is a.p. and
as
. Notice that
is bounded. By Lemma 4.3 there is a compact set
in
such that
for all
. By Lemma 3.13, there is an integer sequence
,
, such that
as
and
uniformly on
as
. Taking a subsequence if necessary, we can also assume that
uniformly on
, and by Lemma 3.9(b) we have that
is also an a.p. sequence. For any
, there is a positive integer
such that if
, then
. In this case, we see that
uniformly for all
as
, and hence by Lemma 4.4
in
in
as
. Since

and from the previous considerations the first term of the right-hand side of (4.23) tends to zero as and since
as
, we have that
for all
, which implies that (1.1) has an a.p. solution
passing through
, where
for
.
We are now in a position to prove the following result.
Theorem 4.12.
Suppose that the assumptions , (H1), (H2), and (H4) hold, and that
is a fading memory space. If the bounded solution
of (1.1) is
, then
is a.a.p. Consequently, (1.1) has an a.p. solution which is
.
Proof.
Let the bounded solution of (1.1) be
with the triple
. Let
be any positive integer such that
as
. Set
. As previously
is a solution of

and is
with the same triple
. By Lemma A.2, for the set
and any
there exists
such that
and
for some
implies that
for all
, where
is a bounded solution of

passing through and
for
. Since
is uniformly bounded for all
and
, taking a subsequence if necessary, we can assume that
is convergent for each
and
uniformly on
, for some a.p. function
. In this case, by Lemma 4.4 there is a positive integer
such that if
, then

On the other hand, for
is a solution of (4.25) with
, that is,

where is defined by the relation

To apply Lemma A.2 to (4.24) and its associated equation (4.27), we will point out some properties of the sequence . Since
uniformly on
, for the above
, there is a positive integer
such that if
, then

which implies that for all
. Applying Lemma A.2 to (4.24) and its associated equation (4.27) with the above arguments and condition (4.26), we conclude that for any positive integer sequence
,
as
, and
, there is a positive integer
such that

and hence by Axiom A(ii) for all
if
. This implies that the bounded solution
of (1.1) is a.a.p. by Lemma 3.9(d). Furthermore, (1.1) has an a.p. solution, which is
by Theorem 4.11. This ends the proof.
Appendix
The proof of the following lemmas used ideas developed by Hino et al. in [3] for the functional differential equations with infinite delay and by Song [12] for functional difference equations with finite delay.Lemma A.1. Suppose that , (H1), (H2), and (H4) hold and that
is a fading memory space. Let
be the bounded solution of (1.1). Let
be a positive integer sequence such that
,
, and
uniformly on
as
, where
is any compact subset in
and
. If the bounded solution
is
, then the solution
of

through , is
. In addition, if
is
, then
is also
.Proof. Set
. It is easy to see that
is a solution of

passing though and
for all
. Since
is
, then
is also
with the same pair
as the one for
. Taking a subsequence if necessary, we can assume that
converges to a vector
for each
as
. From (4.23) with
, we can see that
is the unique solution of (A.1), satisfying
because
.To show that the solution
of (A.1) is
, we need to prove that for any
and any integer
, there exists
such that
implies that
for all
, where
is a solution of (A.1) with
.We know from Lemma 4.4 that
as
for each
; thus, for any given
, if
is sufficiently large; say
, we have

where comes from the uniform stability of
. Let
be such that

and let be the solution of (1.1) such that
. Then
is a solution of (A.2) with
. Since
is
and
for
, we have

It follows from (A.5) that

Then there exists a number such that
for all
and
, which implies that there is a subsequence of
for each
, denoted by
again, such that
for each
, and hence by Lemma 4.4
for all
as
. Clearly,
, and the set
is compact set
. Since
is almost periodic in
uniformly for
, we can assume that, taking a subsequence if necessary,
uniformly on
as
. Taking
in
, we have
, namely,
is the unique solution of (A.1), passing through
with
. On the other hand, for any integer
, there exists
such that if
, then

From (A.5) and (A.7), we obtain

Since is arbitrary, we have
for all
if
and
, which implies that the solution
of (A.1) is
.Now, we consider the case where
is
. Then the solution
of (A.2) is also
with the same pair
as the one for
. Let
be the pair for uniform stability of
.For any given
, if
is sufficiently large; say
, we have

where is the one for uniformly asymptotic stability of
. Let
such that
, and let
, for each fixed
, be the solution of (1.1) such that
. Then
is a solution of (A.2) with
. Since
is
and
for each fixed
, we have

By the same argument as above, there is a subsequence of , which we will continue calling
, such that
converges to the solution
of (A.1) through
and
uniformly on
as
, where
is a compact set in
with
for all
and
. Then
is the unique solution of (A.1), passing through
with
. On the other hand, by Lemma 4.4 for any integer
there exists
such that if
, then

and hence for
. Since
is arbitrary, we have

if and
; thus,
and the proof is complete.Now, we need to prove the following important lemma. Lemma A.2. Suppose that the assumptions
, (H1), (H2), and (H4) hold, that
is a fading memory space, that the bounded solution
of (1.1) is
, and that for each
, the solution of (A.1) is unique for any given initial data. Let
be a given compact set in
. Then for any
, there exists
such that if
,
, and
is a sequence with
for
, one has
for all
, where
is any bounded solution of the system

passing through and such that
for all
.Proof. Suppose that the bounded solution
of (1.1) is
with the triple
. The proof will be by contradiction, we assume that Lemma A.2 is not true. Then for some compact set
, there exist
,
, sequences
,
, mapping sequences
,
, and

for sufficiently large , where
is a solution of

passing through such that
for all
and
. Since
is a bounded subset of
, it follows that
and
are uniformly bounded for all
and
. We first consider the case where
contains an unbounded subsequence. Set
. Taking a subsequence if necessary, we may assume from Lemmas 3.12 and 3.9(b) that there is
such that
uniformly on
,
, and
for
as
, where
are some bounded functions. Since

passing to the limit as , by the similar arguments in the proof of Theorem 4.11, we conclude that
is the solution of the following equation:

Similarly, is also a solution of (A.17). By Lemma 4.4
and
in
as
; it follows from (A.14) that
. Notice that
is a solution of (A.17), passing through
, and is
by Lemma A.1. We have
. On the other hand, since

as for each
, it follows from (A.14) that

This is a contradiction. Thus, the sequence must be bounded. Taking a subsequence if necessary, we can assume that
. Moreover, we may assume that
and
for each
, and
uniformly on
, for some functions
,
on
, and
. Since
and
in
as
, we have
by (A.14), and hence
, that is,
for all
. Moreover,
and
satisfy the same relation:

The uniqueness of the solutions for the initial value problems implies that for
, and hence
. On the other hand, and again from Lemma 4.4,
and
in
as
, then from (A.14) we have

This is a contradiction, that proves Lemma A.2.
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Vidal, C. Existence of Periodic and Almost Periodic Solutions of Abstract Retarded Functional Difference Equations in Phase Spaces. Adv Differ Equ 2009, 380568 (2009). https://doi.org/10.1155/2009/380568
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DOI: https://doi.org/10.1155/2009/380568
Keywords
- Positive Integer
- Phase Space
- Periodic Solution
- Functional Differential Equation
- Bounded Solution