Existence and Stability Property of Almost Periodic Solutions in Discrete Almost Periodic Systems ()
1. Introduction
System of almost periodic difference equations has been studied to describe phenomena of oscillations in the natural and social sciences. The investigation of almost periodic systems has been developed quite widely during the twentieth century, since relationships with the stability theory have been found. A main interest of the subject is the existence theorem for almost periodic solutions. Obviously an almost periodic solution is a bounded solution, but the existence of bounded solutions does not necessarily imply the existence of almost periodic solutions. Therefore, in order to prove the existence of almost periodic solutions, we need some additional conditions to the existence of bounded solutions. A main subject of the investigation has been to find such additional conditions, and up to now, many conditions have been considered (for example, in the linear system, J. Favard’s separation condition [1] ).
In the Section 4, we consider the nonlinear almost periodic system of
(1)
where k is a positive integer,
are almost periodic in n and satisfy
i)
ii)
In the special case where
are constant functions, system (1) is a mathematical model of gas dynamics and was treated by T. Carleman [2] and R. D. Jenks [3] . In the main theorem, we show that if the
matrix
is irreducible, then there exists a positive almost periodic solution which is unique and some stability. Moreover, we can see that this result gives R. D. Jenks’ result in the case where
are constant functions. In the Section 5, we consider the linear almost periodic system with variable coefficients
(2)
where
. Even in nonlinear problems, system (2) plays an important role, as their variational equations and moreover, it is requested to determine the uniformly asymptotic stability of the zero solution from the condition about
. When
is a constant matrix, it is well known that the stability is equivalent to the following condition (cf. [4] );
“Absolute values of all eigenvalues of
are less than one.”
However, it is not true in the case of variable coefficients, and hence we need additional conditions to (2). In the main theorem, we show that one of the such conditions is the diagonal dominance matrix condition on
[5] , that is,
satisfies
This result improves a stability criterion based on results of F. Nakajima [6] for differential equations.
2. Preliminaries
We denote by Rm the real Euclidean m-space. Let
and
. Z is the set of integers, Z+ is the set of nonnegative integers. For
, let
be the Euclidean norm of x and
be the i-th component. Let
and
We introduce an almost periodic function
, where U is an open set in Rm.
Definition 1.
is said to be almost periodic in n uniformly for
, if for any
and any compact set K in U there exists a positive integer
such that any interval of length
contains an integer τ for which
for all
and all
. Such a number τ in above inequality is called an ò-translation number of
.
In order to formulate a property of almost periodic functions, which is equivalent to the above definition, we discuss the concept of the normality of almost periodic functions. Namely, let
be almost periodic in n uniformly for
. Then, for any sequence
, there exist a subsequence
of
and a function
such that
(3)
uniformly on
as
, where K is a compact set in U. There are many properties of the discrete almost periodic functions [7] , which are corresponding properties of the continuous almost periodic functions
[cf. [8] [9] ]. We denote by
the function space consisting of all translates of f, that is,
, where
(4)
Let
denote the uniform closure of
in the sense of (4).
is called the hull of f. In particular, we denote by
the set of all limit functions
such that for some sequence
,
as
and
uniformly on
for any compact subset S in Rm. Specially, for a function
on Z with values in Rm,
denotes the set of all function
such that for some sequence
,
where the symbol “?” stands for the uniformly convergence on any compact set in Z (in short, “in Z”). Clearly,
.
By (3), if
is almost periodic in n uniformly for
, so is a function in
.
We define the irreducible matrix to need after.
Definition 2. An
matrix
is said to be irreducible if for any two nonempty disjoint subsets I and J of the set of m integers
with
, there exists an i in I and a j in J such that
. In the case where
is scalar,
is said to be irreducible if
. Otherwise,
is said to be reducible, and we can assume that
takes the form of
where * is
square matrix, *' is
matrix,
is
zero or a square irreducible matrix.
3. Linear Systems
We consider the system of linear difference equation
(5)
where
and the
matrix
is bounded on Z and almost periodic function in n. We state discretization of Jenks and Nakajima' results for differential equations [3] [10] .
Now we define stability properties with respect to the subset K in Rm. Here, we denote by
the solution of system (5) with initial condition
.
Definition 3. The bounded solution
of system (5) defined on Z is said to be;
i) uniformly stable (in short, U.S.) in K on Z+ if for any
there exists a
such that
for all
whenever
and
at some
in Z+.
ii) uniformly asymptotically stable (in short, U.A.S.) in K on Z+ if it is U.S. in K on Z+ and if there exists a
and, if for any
there exists a
such that
for all
whenever
and
at some
in Z+.
iii) uniformly asymptotically stable (in short, U.A.S.) in the whole K on Z+ if it is U.S. in K on Z+ and if for any
and
there exists a
such that
for all
whenever
and
, at some
in Z+.
When Z+ in the definitions (i), (ii) and (iii) is replaced by Z, we say that
is U.S. in K on Z, U.A.S. in K on Z and U.A.S. in the whole K on Z, respectively. Clearly Definition 3 agrees with the definitions of the usual stability properties in the case where
.
Throughout this paper, we suppose the following conditions;
i)
ii)
and
iii) each element in
is irreducible.
First of all, we prove the following lemmas.
Lemma 1. Consider the m-equations
,
, where
is continuous on second variable x in Rm, and assume that the initial value problem has a unique solution.
a) If
, then the set Π is invariant.
b) If
for
and
, then the set D is positively invariant, and in addition,
if
, then the set Ω is positively invariant.
In the case of differential equations, the proof of the similar lemma is obvious (for instance, see [11] ). We modify it to prove this lemma, but we omit it.
Lemma 2. If conditions (i) and (ii) are satisfied, then the trivial solution of system (5) is U.S. in P on Z and also it is U.S. on Z.
By modifying theorem in [5] , we can easily prove Lemma 2 at same technique.
Lemma 3. If each element in
is irreducible, then the each element in
, we say
and
, has the property that for any two nonempty disjoint subsets I and J of the set of m integers
with
, there exists an
and
such that
Proof. Suppose not, Then there exists a
in
and two nonempty disjoint subsets I and J of
with
such that
Since
is bounded on Z, there exists a subsequence
as
, such that
where
. Clearly,
This show the reducibility of
, which is a contradiction. This proves Lemma 3.
For system (5), we consider the system in
of
(6)
where
.
Lemma 4. Assume that conditions (ii) and (iii) are satisfied for system (5), and let
be a nontrivial solution of system (6) such that
on Z. Then there exists a constant
such that
Proof. Let
be a solution of system (6) such that
on Z. First of all, we show that if
at some
, then
Since
satisfies the equation
where
. Moreover, since
, we have
(7)
which implies
Thus, we obtain
Because
and
on Z. Now suppose that Lemma 4 is not true. Then for some B in
, the corresponding system (6) has a nontrivial solution
,
on Z, such that for some sequence
,
(8)
Set
. Then,
satisfies
and
Since the sequence
is bounded,
is uniformly bounded on any finite interval in Z, and hence there is a convergent subsequence of
, which is again denoted by
, such that
in Z for some function
as
.
We can also assume that
where
and
. Therefore,
is the solution of the system
(9)
on Z and
. Moreover (8) implies that
. Thus, as was proved above, we have
For this
, we define two subsets I and J of
by
for
, where
depends on
and
. Then
,
and
since
. By Lemma 3,
(10)
Now the
-th equation of system (9) takes the form of
and hence
(11)
because of the definition of the set I. Since each term in the left hand side of (11) is nonnegative, all of them are equal to zero. Therefore
which implies, by (10),
This contradicts the definition of the set of J. The proof is completed.
The following proposition is an immediate result of Lemma 4.
Proposition 1. Under conditions (ii) and (iii), system (6) has no nontrivial solution
such that
where
for some
.
We next consider a non-homogeneous system corresponding to system (5)
(12)
and assume that
satisfies conditions (i), (ii) and (iii).
Lemma 5. If
is bounded on Z with values in Rm and
is bounded on Z+, then all solutions of system (12) are bounded on Z+.
Proof. It is sufficient to show that (12) has at least one bounded solution on Z+, because the trivial solution of (5) is U.S. by Lemma 2. We consider the system with real parameter ò
(13)
and show that for a sufficiently small ò, system (13) has a bounded solution on Z+, which implies the existence of a bounded solution on Z+ for system (12) by replacing x in (13) with
. For a
and for the m-vector e each of whose components is 1, let
be a convex cone defined by
where
denotes the inner product and
. Clearly,
. Every solution
of (13) satisfies
because of condition (i). By replacing n with n-1 of the above both sides,
where
is sufficient small number and
.
When
, we have
and hence,
(14)
Therefore, in order to show the boundedness of
with
in Ω, it is sufficient to prove that
on Z+ if ò is sufficiently small. Now suppose that for each solution
of (13) with
in Ω, there exists an
such that
We can assume that
and
where
and
denote the boundary and the closure of the set K, respectively. If we set
,
is a solution of the system
such that
at
and
for
. Thus, by (14),
The same argument in the proof of Lemma 4 enables us to assume that
in Z for some function
as
and
in Z for some
as
Therefore,
satisfies
and clearly, for
,
(15)
and
(16)
Moreover we have
, which implies by Lemma 1 that
From this and (15) it follows that
(17)
Now we show that
. In fact, if
, we have
Thus
because
, and hence
which contradicts (16). Therefore (16) and (17) hold for
. Moreover this enables us to assume that
and
Because
is compact in the sense of the convergence. This contradicts the conclusion in Proposition 1. This proves that
on Z+ if ò is sufficiently small. The proof is completed.
Lemma 6. Under the assumptions (i) and (ii), if for each B in
, the trivial solution of the system
is U.S. on Z and U.A.S. on Z+, then the trivial solution of system (5) is U.A.S. on Z.
Proof. Let
be the solution of (5). Since the trivial solution of (5) is U.S. on Z by Lemma 2, as is seen from (ii) in Definition 3, it is sufficient to show that for any
there exists a
such that
whenever
and
, where
is the number in (i) of Definition 3.
Now suppose that there exists an
and sequences
in Z and
in Rm such that
and
Since
,
Set
. Then,
satisfies
and
We can assume that
in Z+ for some function
as
and
in Z for some
as
.
Therefore
is a solution of the system
(18)
and
On the other hand, we have
because the trivial solution of (18) is U.A.S. on Z+. Therefore there arises a contradiction. Thus the proof is completed.
We show the following theorem, before we will mention a definition of the exponential dichotomy of a linear system;
System (5) is said to possess an exponential dichotomy if there exists a projection matrix P and positive constants
and
such that
where, I is a identical matrix and F is a fundamental matrix solution of system (5) (cf. [4] [5] [8] ).
Theorem 1. Assume that system (5) satisfies conditions (i), (ii) and (iii), Then the trivial solution of system (5) is U.A.S. in P on Z.
Proof. On the set Π which is invariant for system (5), the system is written as the
-system
(19)
where
and
is an
matrix whose
element is given by
for
. First of all, we can show that for each
in
, the system
(20)
has an exponential dichotomy on Z+ since (20) has at least one bounded solution, and as is well known (cf. [12] ), it is equivalent to show the system
(21)
possesses at least one bounded solution on Z+ for any bounded function
on Z+. For each
in
there corresponds some
in
such that the
element of
is equal to
for
. For
, let
be defined by
Obviously
and
are bounded on Z+. Applying Lemma 5 to the m-system
we obtain the bounded solution
on Z+ with
, and
which yields
Hence we can verify that
is a bounded solution on Z+ of system (21). The exponential dichotomy of (20) implies further that the trivial solution is U.A.S. on Z+, because the trivial solution is U.S. on Z by Lemma 2. Therefore it follows from Lemma 6 that the trivial solution of (19) is U.A.S. on Z, i.e., the trivial solution of (5) is U.A.S. in P on Z. The proof is completed.
4. Nonlinear Systems
We consider the nonlinear almost periodic system of
(22)
where
is almost periodic function of n with conditions
(iv)
and
(v)
In addition, assume that
are continuously differentiable for
,
and
for real number
, where
is the derivative of
at u.
We first consider the linear system
(23)
and its perturbed system
(24)
where
is an
matrix function, almost periodic function in n,
is continuous with respect to its second argument and
uniformly for
. Assume that the set Π is invariant for both system (23) and (24).
First of all, we can prove the following lemmas.
Lemma 7. If the trivial solution of system (23) is U.A.S. in Π on Z, then the trivial solution of system (24) has also the same stability property.
Proof. Let
for
. Then there are positive constants
and
such that
(25)
because
. On the set Π, systems (23) and (24) are written as
(26)
and
(27)
respectively, where the
element of
, is given by
for
and
uniformly for
. Inequality (25) shows that the trivial solution of (23) is U.A.S. in P if and only if the trivial solution of (26) is U.A.S., and we have also the same equivalence between (24) and (27). As is well known, if the trivial solution of (26) is U.A.S., then the trivial solution of (27) has also the same stability property. Thus our assertion is proved.
The following lemma is obtained by the slight modification of the difference equation to Seifert’s result [13] . Then, we will omit the proof (cf. [9] ).
We consider the almost periodic nonlinear system
(28)
where
is almost periodic in n uniformly for
and for a constant
,
for
and
.
Lemma 8. Assume that the set Ω is positively invariant for system (28) and all solutions in Ω on Z are U.A.S. in Ω on Z. Then the set of such solutions is finite and consists of only almost periodic solutions
which satisfy
on Z for
and some constant
.
Now we can show the following theorem. Since the last statements of the following theorem are alternative, under each assumption of these statements we can prove the existence of almost periodic solutions in Ω and the module containment.
Theorem 2. Under the assumptions (iv) and (v), system (22) has a nontrivial almost periodic solution in Ω whose module is contained in the module of
. In addition to the above assumptions, if
is irreducible, then the almost periodic solution of (22) is unique in Ω, which remains in
on Z, and it is U.A.S. in the whole Ω on Z, where
. Moreover, if
is reducible, then at least one of the above almost periodic solutions
satisfies that
on Z, where
.
Proof. First of all, we consider the case where
is irreducible. Since system (22) satisfies the conditions of Lemma 1, the set Ω is positively invariant, namely,
on Z+ for a solution
of (22) with
, and furthermore we can assume that
because of the almost periodicity of
. We can show that this
is U.A.S. in Ω on Z. If we set
in system (22), then
for x in Ω and
(29)
And Π is invariant for the above system. Considering the first approximation of system (29)
(30)
where
is defined by
, condition (iv) implies that Π is also invariant for (30). Then, by Lemma 6, if the trivial solution of (30) is shown to be U.A.S. in Π on Z, then the trivial solution of (29) has the same stability, and consequently
is U.A.S. in Ω on Z. Therefore it is sufficient to show that the trivial solution of (30) is U.A.S. in Π on Z. Clearly
is bounded and we have
and
(31)
because of conditions (iv) and (v), respectively. Thus
satisfies conditions (i) and (ii). Condition (iii) will be verified in the following way. Applying the same argument as in the proof of Lemma 4 to system (22), we can see that there exists a constant
such that
(32)
and hence there is a constant
such that
Therefore, (31) implies
which guarantees that each element of
is irreducible, because
is irreducible and almost periodic. Thus it follows from Theorem 1 that the trivial solution of (30) is U.A.S. in Π on Z, i.e., all solutions of system (22) in Ω on Z are U.A.S. in Ω on Z. Therefore Lemma 8 concludes that system (22) possesses an almost periodic solution in Ω which remains in
by (32), and the set of solutions in Ω on Z is finite and consists of only almost periodic solutions
which satisfy
on Z for
and some constant
.
Next we can show that there exists a
such that each solution
of (22) with
satisfies that for some
and the constant
of (ii) in Definition 3,
which implies
(33)
because
is U.A.S. in Ω. Suppose that this is not true. Then there exists a small constant
less than β and sequences
in Z and
in Ω such that
Since
is almost periodic in n uniformly for
, we can choose a sequence
, such that
If we set
for
and
, these
functions
satisfy
and
because
. Moreover,
We can assume that
in Z for some function
, as
. Therefore
are solutions of system (22), because
in
as
, and
and
which shows that system (22) has
distinct solutions in Ω on Z. This is contradiction. Therefore,
is U.A.S. in the whole Ω on Z, if the uniqueness of
is shown.
Now we will prove the uniqueness of
. Suppose
for
and set
and
Then
and
are open sets in Ω, and moreover these sets are nonempty and disjoint, because
on Z for
. On the other hand , (33) shows that
, which contradicts the connectedness of Ω. Thus the uniqueness of an almost periodic solution is proved, and moreover, as is seen from [14] , this uniqueness guarantees the module containment of the almost periodic solution.
Now consider the case where
is reducible. We can assume
takes the form of
where
is zero or a square irreducible matrix of order
. If
is zero, system (22) obviously has the constant solution
in
such that
for
and
. In the latter case, if we set in system (22)
then system (22) is reduced to the lower dimensional system
(34)
where
. Since
is irreducible, the above system (34) has an almost periodic solution
such that
(35)
and furthermore the module of
is contained in the module of
, i.e., of the module of
. Thus, system (22) has an almost periodic solution
in
on Z such that
for
and
for
. The proof is completed.
Remark 1. As will be seen from the module containment, the above almost periodic solution is a critical point in the case where
is a constant. Hence Theorem 2 is a discretization of Nakajimas’ result (Theorem 2 in [10] ).
5. A Stability Criteria of Linear Systems
We consider a stability criterion for solutions of a linear system with coefficient matrix of diagonal dominance type.
We again consider a linear system (5).
Let
be an
matrix of functions for
. We assume the following conditions;
(36)
(37)
where,
denotes the determinant of matrix
and
(38)
At first, we need the following lemmas for main theorem.
Lemma 9. If a square matrix A is irreducible and satisfies (38) and if for at least one j,
then A is nonsingular.
For the proof, see [15] .
Lemma 10. If a nonsingular
matrix
satisfies (38), then all principal minors of A are nonsingular, namely,
Proof. Let
be an
principal minor of A. Then, for a permutation matrix Q,
where
has
rows and l columns and
denotes the transposed matrix of Q. Moreover, from the definition of irreducibility, we can choose an
permutation matrix
such that
(381)
where
is an
irreducible matrix,
, and
has
row and
columns for
. In particular, in the case where
is irreducible,
must be
itself, and the matrices
are not present. Setting
for
unit matrix I, where
is the direct sum of
and I, we have
where
and
has
rows and
columns. Since the diagonal dominance condition (38) is invariant under the permutation of indexes, B also satisfies (38). Hence, letting
for a fixed
, we have
(39)
where the summations on j are taken along columns and
for convenience. If
or
, then
and hence for this k,
by (39). Therefore it follows from Lemma 9 that
(40)
since
is irreducible. If
and
, then we have the form of
which also implies (40), because
. In any case, we have
. Since these are true for all
, it follows from (381) that
this proves Lemma 10.
Lemma 11. If system (5) satisfies conditions (36) and (38), then the norm of solution
such that
, is non-increasing, and consequently the zero solution is U.S..
For the proof, we can see (cf. [5] ).
In the following theorem, we can prove that the zero solution is U.A.S., if
is bounded on Z and if condition (36), (37) and (38) are satisfied.
Theorem 3. In system (5), let
be bounded on Z. Assume that conditions (36) and (38) are satisfied for all
and that there is a constant
such that
Then the zero solution is U.A.S..
Proof. As is stated in Lemma 11, the zero solution is U.S., and hence it is sufficient to show that for any
there exists a
such that
whenever
. Suppose that this is not true. Then there exists a constant
, a sequence of solution
of (5) and a sequence
such that
and
Since
is non-increasing, we have
and there exists a subinterval
of
such that
Set
. Then, we obtain
(41)
(42)
and
(43)
Since
is bounded, it follows from (41) and (42) that
is uniformly bounded on any finite interval of Z, and thus, taking a subsequence,
can be assumed to converge uniformly on any finite interval of Z. Defining
by
it follows from (42) and (43) that there is a constant
, such that
Since
is defined on Z, we can choose an interval
(for some
) such that
and
Here we note that
because
. Then
Let
(44)
Then, we have
and there is a sequence
such that
(45)
Moreover, since I is compact and
is bounded on Z, we can assume that
and
Clearly B satisfies (36), (38) and
. Taking a difference of both sides of (44) at
and using relation (41), we find
and
where
, since we have
and
By (45), we have
(46)
Since B satisfies (36) and (38),
Therefore each term of right hand side of (46) is non-positive, because we have
Then it follows from (46) that
which implies
since
for
. Thus we have
(461)
On the other hand, B satisfies (38) and
, and thus it follows from Lemma 10 that all principal minors of B are nonsingular, which contradicts (461). This proves that the zero solution of system (5) is U.A.S..
Corollary 1. If system (5) is defined only for
and all assumptions of Theorem 3 are satisfied for
, then the zero solution is U.A.S. for
.
Proof. We construct the system defined on Z by
(47)
where
Since system (47) satisfies all assumptions of Theorem 3 on Z, the zero solution is U.A.S. on Z, and furthermore, since system (5) coincides with system (47) for
, this prove our conclusion.
6. Application
Before Example 1, we state the following lemma is a special case of Theorem 3 in [13] .
In the nonlinear system
(48)
let
be almost periodic in n uniformly for
and for any
, let there exists a constant
such that
Lemma 12. If
is a bounded solution of (48) on
and if for any solution
of (48),
is monotone decreasing to zero as
, then
is a unique almost periodic solution and its module is contained in the module of
.
Example 1. Consider the variational linear difference equation
(49)
where
We now assume that
is at least one bounded solution of (49) on
and
is any solution of (49) on
, and
,
are some bounded functions on
such that
for some positive constants
and
such that
. We can verify that
satisfies all assumptions in Corollary 1. First of all,
is bounded in the future for
, because
and
are bounded function on
. It is clear that the diagonal elements of
are negative and
The diagonal dominance condition (38) for
requires that
which is equivalent to
and this is satisfied by
Therefore, by Theorem 3, the zero solution of (49) is U.A.S. and
where the convergence is monotone decreasing by Lemma 11. Thus, applying Lemma 12 to system (49), we find that there exists a unique almost periodic solution with the module contained in the module of
.
7. Conclusion
In this paper, we obtain the existence and stability property of almost periodic solutions in discrete almost periodic systems. First, in Section 1, the research background is introduced. In Section 2, the fundamental concepts of the almost periodic solutions in discrete almost periodic systems is given. In Section 3, we are introduced to the several lemmas and have uniformly asymptotically stability theory of the linear system, and moreover, in Section 4, we consider the generalized gas almost periodic system, and if linear part is irreducible matrix, then we obtain the existence of almost periodic solutions of this system. Finally, in Sections 5 and 6, we consider and obtain an uniformly asymptotically stability criterion for solutions of a linear system with coefficient matrix of diagonal dominance conditions, and this result applies to meaningful example of a linear discrete system.