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.
(A1) is an integer, and .
Without loss of generality we may (and do) assume the following.
Under these conditions, .
where we suppose that
In this case, .
where . Clearly, (1.1) with (1.6) (and similarly (1.4) with (1.6)) has a unique solution which exists for any . The solution is denoted by .
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.
(A4) is an integer, and .
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
Consider the general delay difference system (1.13) with the initial condition (1.6). This initial value problem has a unique solution which is denoted by .
for any .
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.
Assume . If
is finite with , then for every initial sequence the solution of (1.13) and (1.6) has a finite limit which obeys (3.3).
For the independence of conditions (3.4) and (3.5), see Section 4.1.
As a corollary, we get the next result.
- (a)If for an initial sequence the solution of (1.13) and (1.6) has a finite limit, then(3.7)
- (b)If either(3.8)
where for each .
Based on the above results we give a necessary and sufficient condition for the solutions of (3.11) to have a finite limit.
- (a)If for every initial sequence the solution of (3.11) and (1.6) has a finite limit, then(3.12)
- (b)Assume that for each . Then the next two statements are equivalent.
For every initial sequence the solution of (3.11) and (1.6) has 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) is satisfied, but condition (1.9) does not hold.
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.
for every matrix norm on .
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.
However, the implication discussed above may be lost if (4.15) is not satisfied, even if the matrices are nonnegative, as the following example shows.
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).
Assume . If either
for some matrix norm on , or
is finite with , then for every initial sequence the solution of (1.1) and (1.6) has a finite limit which obeys (3.3).
for each integer . So, in the constant delay case, from Theorem 4.6 we get the next result.
Assume and . If either
for some matrix norm on , or
is finite with , then for every initial sequence the solution of (1.4) and (1.6) has a finite limit which obeys (3.3).
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 ).
- (a)If either(4.36)for some matrix norm on , or(4.37)
then for every initial sequence the solution of (1.10) and (1.6) has a finite limit.
- (b)Assume that(4.38)
for each . Then the next two statements are equivalent.
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.
showing clearly that may be negative.
Of course, we have from Theorem 3.4(a) (using that , ) that if for every initial sequence the solution of (1.10) and (1.6) has a finite limit, then (1.12) holds.
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.
Fix an initial value .
so the series (5.8) is convergent.
and this gives the boundedness of the sequence (5.11).
The proof is complete.
- (a)By (3.6),(5.19)
- (b)Since conditions (3.8) and (3.9) imply that the matrix(5.21)
is invertible, we can apply Theorem 3.2 and (3.7).
- (a)Equations (3.7) in Corollary 3.3 and (3.1) imply that(5.22)
- (b)Suppose (i). We have proved that the matrix (5.23) is invertible. If(5.25)
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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