Asymptotic Constancy in Linear Difference Equations: Limit Formulae and Sharp Conditions
© I. Győri and L. Horváth. 2010
Received: 20 January 2010
Accepted: 23 March 2010
Published: 6 April 2010
It is found that every solution of a system of linear delay difference equations has finite limit at infinity, if some conditions are satisfied. These are much weaker than the known sufficient conditions for asymptotic constancy of the solutions. When we impose some positivity assumptions on the coefficient matrices, our conditions are also necessary. The novelty of our results is illustrated by examples.
where the following are considered.
Without loss of generality we may (and do) assume the following.
where we suppose that
In the nonautonomous case with constant delays, it has been proved by Pituk  that the value of the limit can be characterized in an implicit formula by using a special solution of the adjoint equation to (1.4) and the initial values.
The main novelty of our paper is that we prove the existence of the limit of the solutions of the above equations under much weaker conditions than (1.9). Moreover, utilizing our new limit formula, we show that some of our sufficient conditions are also necessary.
After recalling some preliminary facts on matrices in the next section, we state our main results on the asymptotic constancy of the solutions of (1.13), and derive a generalization of the limit formula (1.12) to the time-dependent case (Section 3). Section 4 is divided into three parts. In Section 4.1 we illustrate the independence of conditions (1.7) and (1.9). The relation between our new conditions is studied in Section 4.2. In the third part of Section 4 we specialize to (1.1), (1.4), and (1.10). The proofs of the main results are included in Section 5.
If is an integer, the space of all matrices is denoted by , the zero matrix by , and the identity matrix by . is a lattice under the canonical ordering defined by what follows: means that for every , , where and . Of course, the absolute value of is given by . The spectral radius of a matrix is denoted by . It is well known that for any norm on we have . We use that , , , , and imply that .
3. The Main Results
In the next theorem we prove the convergence of the solutions of (1.13) under a condition much weaker than (1.9), as it is illustrated in Section 4.3.
For the independence of conditions (3.4) and (3.5), see Section 4.1.
As a corollary, we get the next result.
Based on the above results we give a necessary and sufficient condition for the solutions of (3.11) to have a finite limit.
4. Discussion and Applications
4.1. Comparison of Conditions (1.7) and (1.9)
The independence of conditions (1.7) and (1.9) is illustrated by the next example.
then condition (1.7) does not hold, but condition (1.9) is satisfied.
4.2. Independence of Conditions (3.4) and (3.5)
It is illustrated by the following two examples that condition (3.4) does not generally imply condition (3.5) and conversely.
We can see that there are situations in which (3.5) is satisfied but (3.4) is not.
We can see that (3.4) does not imply (3.5) in general.
4.3. Application to Delay Difference Equations
The proof is complete.
The following result is an immediate consequence of Theorem 3.2 and Lemma 4.5, and it gives sufficient conditions for the convergence of the solutions of (1.1).
assuming that (1.9) holds.
In the next example our condition (4.29) holds, but neither condition (1.9) nor condition (1.7) can be applied.
By applying Theorem 4.7 and Theorem 3.4(b), we give sufficient and also necessary conditions for the solutions of (1.4) to be asymptotically constant, if in addition each matrix is constant (independent of ).
Condition (4.38) does not require the positivity of the coefficient matrices . To illustrate this, see the following example. To the best of our knowledge, no similar result has been proved for (1.10) with both positive and negative coefficients.
5. Proofs of the Main Results
Proof of Theorem 3.1.
From (5.3) the assertion and the desired relation (3.3) follow, bringing the proof to an end.
In order to prove Theorem 3.2, we need the following Lemma from [3, Corollary (b)].
Proof of Theorem 3.2.
so the series (5.8) is convergent.
and this gives the boundedness of the sequence (5.11).
The proof is complete.
is invertible, we can apply Theorem 3.2 and (3.7).
but this follows from (3.11) by an easy induction argument.
Now, suppose (ii). Then (i) comes from Corollary 3.3(b) (see the second condition).
The proof is complete.
This paper is supported by Hungarian National Foundations for Scientific Research Grant no. K73274.
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