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Asymptotic Constancy in Linear Difference Equations: Limit Formulae and Sharp Conditions
Advances in Difference Equations volume 2010, Article number: 789302 (2010)
Abstract
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.
1. Introduction
Consider the nonautonomous linear delay difference system
where the following are considered.
(A_{1}) is an integer, and .
(A_{2}) and are sequences of nonnegative integers such that
is finite.
Without loss of generality we may (and do) assume the following.
(A_{3})For each and ,
Under these conditions, .
Whenever the delays are constants, we get the system
where we suppose that
(A_{4}) are nonnegative integers and
In this case, .
Together with the above equations we assume initial conditions of the form
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 .
Driver et al. [1] have shown that if
for some matrix norm on , then every solution of (1.4) tends to a finite limit at infinity which will be denoted by
In the paper of [2] the same statement has been proved under the condition
As we will show in Section 4.1 (see Example 4.1), conditions (1.7) and (1.9) are independent if the coefficients are time dependent. In the special case of (1.4) with constant coefficients (each is independent of )
conditions (1.7) and (1.9) coincide and each reduces to
Moreover, considering (1.10) under the condition (), the existence of the finite limit of each solution (for whatever reason) implies that
(See [1].)
In the nonautonomous case with constant delays, it has been proved by Pituk [2] 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.
In this paper we prove similar results for the general delay difference system
where
(A_{4}) 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 timedependent 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.
2. Preliminaries
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 .
In our first theorem we give a new limit formula in terms of the initial values. To this end, we introduce the linear mapping which is defined by
for any .
Theorem 3.1.
Assume . For an initial sequence , the solution of (1.13) and (1.6) has a finite limit if and only if
is finite, and in this case
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.
Theorem 3.2.
Assume . If
either
for some matrix norm on ,or
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.
Corollary 3.3.
Assume , and for each let the limit
be finite. Then the following are considered.

(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)
for some matrix norm on , or
then for every initial sequence the solution of (1.13) and (1.6) has a finite limit which obeys
Now consider the equation
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.
Theorem 3.4.
Consider (3.11).

(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.

(i)
For every initial sequence the solution of (3.11) and (1.6) has a finite limit.

(ii)
And

(i)
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.
Example 4.1.
Let ,,, and . Elementary considerations show the following.

(a)
If
(4.1)then condition (1.7) is satisfied, but condition (1.9) does not hold.

(b)
If
(4.2)
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.
Example 4.2.
Let the matrices and be defined by
Since
yield that
for every matrix norm on , hence
for every matrix norm on .
On the other hand
We can see that there are situations in which (3.5) is satisfied but (3.4) is not.
Example 4.3.
Let the matrices and be defined by
Observe that
and therefore there exists a matrix norm on such that
From
and from (4.10), it follows that there is an integer such that
This together with (4.10) gives that
Finally,
We can see that (3.4) does not imply (3.5) in general.
Suppose that and the limit
is finite for each . In this case condition (3.4) guarantees that condition (3.5) also holds. Really,
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.
Example 4.4.
Let the matrix be defined by
Using the norm on , we have
while
4.3. Application to Delay Difference Equations
For every and let the function be defined on the set of integers by
Lemma 4.5.
Assume . Then the delay difference (1.1) is equivalent to (1.13) if for every is defined by
Proof.
It is easy to see that
By using (4.20) we get
Thus (1.1) can be written in the form
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).
Theorem 4.6.
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).
Now consider the constant delay equation (1.4). For every , let the function be defined on the set of integers by
In (1.4) and for every and ; thus the function defined in (4.20) satisfies
for each integer . So, in the constant delay case, from Theorem 4.6 we get the next result.
Theorem 4.7.
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).
Remark 4.8.
Our condition (4.29) is weaker than condition (1.9). In fact
Therefore
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.
Example 4.9.
Consider
An elementary calculation shows that
while
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 ).
Theorem 4.10.
Assume and with for each and . Then the following are considered.

(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.
Remark 4.11.
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.
Example 4.12.
Consider the scalar difference equation
which is a special case of (1.10). Then
Consequently, the conditions in (4.38) have the form
showing clearly that may be negative.
Remark 4.13.
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.
Since
it comes from (1.13) that
Now a simple calculation confirms that
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)].
Lemma 5.1.
Consider the initial value problem
where is a given integer, , and . The unique solution of (5.4) is denoted by . Let be a norm on . If
is satisfied for some matrix norm on , then there are numbers and such that
Proof of Theorem 3.2.
Fix an initial value .
Since
it is enough to show that the series
is convergent.
Suppose (3.4). Let be a norm on . According to Lemma 5.1,
so the series (5.8) is convergent.
Now suppose (3.5). Obviously, the convergence of the series (5.8) comes from the convergence of the series
Moreover, the members of the previous series are nonnegative, so it suffices to prove that the sequence
is bounded.
Let be the matrix in , where for each pair . By the definition of the matrix , for each positive number there exists a nonnegative integer such that
The property insures that we can choose a positive number such that . We set
Equation (1.13) implies that
Introducing the notation
and using (5.12) and (5.13), we calculate
Hence
Because the matrix was chosen to satisfy and is nonnegative, is invertible and is nonnegative too. Therefore, (5.16) yields that
and this gives the boundedness of the sequence (5.11).
The proof is complete.
Proof of Corollary 3.3.

(a)
By (3.6),
(5.19)
From (3.6) it also follows that
Equations (5.19) and (5.20) together with Theorem 3.1 give the result.

(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).
Proof of Theorem 3.4.

(a)
Equations (3.7) in Corollary 3.3 and (3.1) imply that
(5.22)
Our goal is to prove that the matrix
is invertible. To this end, we choose initial sequences of the form . Then (5.22) shows that the linear mapping
is surjective, whence it is an isomorphism. Consequently, (5.23) is invertible. Now the result follows from (5.22).

(b)
Suppose (i). We have proved that the matrix (5.23) is invertible. If
(5.25)
is also satisfied, then we have (ii) (see [4]). To prove this, choose initial sequences of the form . Then, by (5.22)
Therefore, we have only to observe that implies that . It is enough to show that yields
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.
References
Driver RD, Ladas G, Vlahos PN: Asymptotic behavior of a linear delay difference equation. Proceedings of the American Mathematical Society 1992,115(1):105112. 10.1090/S00029939199211112170
Pituk M: The limits of the solutions of a nonautonomous linear delay difference equation. Computers & Mathematics with Applications 2001,42(3–5):543550. 10.1016/S08981221(01)001766
Győri I, Horváth L:A new view of the theory of system of higher order difference equations. Computers and Mathematics with Applications 2010,59(8):29182932. 10.1016/j.camwa.2010.02.010
Schaefer HH: Banach Lattices and Positive Operators, Die Grundlehren der Mathematischen Wissenschaften. Volume 215. Springer, New York, NY, USA; 1974:xi+376.
Acknowledgment
This paper is supported by Hungarian National Foundations for Scientific Research Grant no. K73274.
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Győri, I., Horváth, L. Asymptotic Constancy in Linear Difference Equations: Limit Formulae and Sharp Conditions. Adv Differ Equ 2010, 789302 (2010). https://doi.org/10.1155/2010/789302
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DOI: https://doi.org/10.1155/2010/789302