Advances in Difference Equations volume 2009, Article number: 380568 (2009)
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.
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.
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:
,
,
.
The space
is a Banach space.
We need the following property on 
.
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.
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.
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 
.
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.
If 
is an a.p. sequence, then there exists an almost periodic function 
such that 
for 
.
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]).
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).
is continuous in the second variable for any fixed 
.
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:
stable, if for any 
and any integer 
, there is 
such that 
implies that 
for all 
, where 
is any solution of (1.1);
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);
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);
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:
is 
, if it is uniformly stable, and there exists 
such that if 
and 
, then 
as 
, where 
is any solution of (1.1).
Here, we will assume what follows.
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).
Here, we will assume that
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.
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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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.
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