Consider the linear difference system
where , and is a matrix sequence. In what follows, we denote by any convenient norm either of a vector or of a matrix.
Definition 2 The difference system (3.1) is said to possess an exponential dichotomy on if there exists a projection P, that is, an matrix P such that , and constants , such that
where is the fundamental solution matrix of (3.1) and .
Consider the following almost periodic difference system:
where is an almost periodic matrix sequence and is an almost periodic vector sequence.
Theorem 5 If the linear system (3.1) admits an exponential dichotomy, then system (3.2) has a bounded solution in the form
where is the fundamental solution matrix of (3.1).
Proof By direct substitution, we obtain
It follows that
Moreover, we have
where . By Theorem 1, is a bounded solution of system (3.2). The proof is complete. □
Theorem 6 ()
be an almost periodic sequence on
then the linear system
admits an exponential dichotomy on .
If we define the norm =, for any , then one can easily deduce that is a Banach space.
We assume that
Theorem 7 Let A.1 and A.2 hold. Then, there exists a unique positive almost periodic solution of (1.3) in .
Proof For any , we consider an auxiliary equation
Since , it follows from Theorem 6 that the linear system
admits an exponential dichotomy on . By Theorem 5 and Theorem 6, we deduce that system (3.6) has a bounded solution of the form
In virtue of Theorem 2, Theorem 3, Theorem 4, and using the almost periodicity of and the fact that the uniform limit of almost periodic sequences is also almost periodic, we deduce that is also almost periodic.
Define a mapping by setting
It is easy to see that is a closed subset of . For any , we have
By the fact that , we obtain
Using that , we end up with
On the other hand, we have
By virtue of the fact that , we obtain
This tells that the mapping T is a self-mapping from to .
Let . Then
In virtue of the fact that , we observe that
By A.1, we get
Therefore, by (1.3), (2.1) and (3.12), we have
Therefore, we end up with
which implies by A.2 that the mapping T is contractive on . Therefore, the mapping T possesses a unique fixed point such that . Thus, is an almost periodic solution of (1.3) in the . The proof is complete. □
We assume that
Theorem 8 Let A.1 and A.3 hold. Further, assume is a positive almost periodic solution of (1.3) in the set . Then, the solution of (1.3) with converges exponentially to as .
Proof Set and , where . Then
The result of Lemma 1 tells that is positive and bounded on and
Define a function by setting
It is clear that Φ is continuous on . Then, by A.3 we have
which implies that there exist two constants and such that
We consider the discrete Lyapunov functional
Calculating the difference of along the solution of (3.13), we have
for all .
Then, we claim that
Assume, on the contrary, that there exists such that
which implies that
In virtue of (3.12), (3.18) and (3.20), we obtain
which contradicts (3.16). Hence (3.19) holds. It follows that for all . The proof is complete. □