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  ).
In the Section 4, we consider the nonlinear almost periodic system of
where k is a positive integer, are almost periodic in n and satisfy
In the special case where are constant functions, system (1) is a mathematical model of gas dynamics and was treated by T. Carleman  and R. D. Jenks  . 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
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.  );
“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  , that is, satisfies
This result improves a stability criterion based on results of F. Nakajima  for differential equations.
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
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
uniformly on as , where K is a compact set in U. There are many properties of the discrete almost periodic functions  , which are corresponding properties of the continuous almost periodic functions [cf.   ]. We denote by the function space consisting of all translates of f, that is, , where
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
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   .
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;
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  ). 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  , 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
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
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 ,
Set . Then, satisfies
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
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,
Now the -th equation of system (9) takes the form of
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)
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
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,
is sufficient small number and .
When , we have
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
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
in Z for some as
Therefore, satisfies and clearly, for ,
Moreover we have , which implies by Lemma 1 that
From this and (15) it follows that
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
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
Set . Then, satisfies
We can assume that
in Z+ for some function as
in Z for some as .
Therefore is a solution of the system
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.    ).
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
where and is an matrix whose element is given by for . First of all, we can show that for each in , the system
has an exponential dichotomy on Z+ since (20) has at least one bounded solution, and as is well known (cf.  ), it is equivalent to show the system
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
where is almost periodic function of n with conditions
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
and its perturbed system
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
because . On the set Π, systems (23) and (24) are written as
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  . Then, we will omit the proof (cf.  ).
We consider the almost periodic nonlinear system
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
And Π is invariant for the above system. Considering the first approximation of system (29)
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
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
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,
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
because . Moreover,
We can assume that in Z for some function , as . Therefore are solutions of system (22), because in as , 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
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  , 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
where . Since is irreducible, the above system (34) has an almost periodic solution such that
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  ).
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;
where, denotes the determinant of matrix and
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  .
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
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
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
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.  ).
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
Since is non-increasing, we have
and there exists a subinterval of such that
Set . Then, we obtain
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
Here we note that because . Then
Then, we have
and there is a sequence such that
Moreover, since I is compact and is bounded on Z, we can assume that
Clearly B satisfies (36), (38) and . Taking a difference of both sides of (44) at and using relation (41), we find
where , since we have
By (45), we have
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
since for . Thus we have
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
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.
Before Example 1, we state the following lemma is a special case of Theorem 3 in  .
In the nonlinear system
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
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 .
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.
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