Advances in Difference Equations

, 2008:932831 | Cite as

Asymptotic Representation of the Solutions of Linear Volterra Difference Equations

Open Access
Research Article

Abstract

This article analyses the asymptotic behaviour of solutions of linear Volterra difference equations. Some sufficient conditions are presented under which the solutions to a general linear equation converge to limits, which are given by a limit formula. This result is then used to obtain the exact asymptotic representation of the solutions of a class of convolution scalar difference equations, which have real characteristic roots. We give examples showing the accuracy of our results.

Keywords

Difference Equation Asymptotic Representation Nonnegative Matrix Finite Limit Unique Positive Solution 

1. Introduction

The literature on the asymptotic theory of the solutions of Volterra difference equations is extensive, and application of this theory is rapidly increasing to various fields. For the basic theory of difference equations, we choose to refer to the books by Agarwal [1], Elaydi [2], and Kelley and Peterson [3]. Recent contribution to the asymptotic theory of difference equations is given in the papers by Kolmanovskii et al. [4], Medina [5], Medina and Gil [6], and Song and Baker [7]; see [8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19] for related results.

The results obtained in this paper are motivated by the results of two papers by Applelby et al. [20], and Philos and Purnaras [21].

This paper studies the asymptotic constancy of the solution of the system of nonconvolution Volterra difference equation
with the initial condition

where Open image in new window , Open image in new window and Open image in new window are sequences with elements in Open image in new window and Open image in new window , respectively.

Under appropriate assumptions, it is proved that the solution converges to a finite limit which obeys a limit formula. Our paper develops further the recent work [20]. The distinction between the works is explained as follows. For large enough Open image in new window , in fact Open image in new window , the sum in (1.1) can be split into three terms
In [20, Theorem 3.1] the middle sum in (1.3) contributed nothing to the limit Open image in new window , since it was assumed that

In our case, we split the sum in (1.1) only into two terms, and the condition (1.5) is not assumed. In fact, we show an example in Section 4, where (1.5) does not hold and hence in [20, Theorem 3.1] is not applicable. At the same time our main theorem gives a limit formula. It is also interesting to note that our proof is simpler than it was applied in [20].

Once our main result for, the general equation, (1.1) has been proven, we may use it for the scalar convolution Volterra difference equation with infinite delay,
with the initial condition,

where Open image in new window , and Open image in new window and Open image in new window are real sequences.

Here, Open image in new window denotes the forward difference operator to be defined as usual, that is, Open image in new window .

If we look for a solution Open image in new window of the homogeneous equation associated with (1.6), we see that Open image in new window is a root of the characteristic equation
We immediately observe that Open image in new window is a simple root if

In the paper [21] (see also [22]), it is shown that if Open image in new window satisfies (1.8) and (1.9), and the initial sequence Open image in new window is suitable, then for the solution Open image in new window of (1.6) and (1.7) the sequence Open image in new window , Open image in new window is bounded. Furthermore, some extra conditions guarantee that the limit Open image in new window is finite and satisfies a limit formula.

In our paper, we improve considerably the result in [21]. First, we give explicit necessary and sufficient conditions for the existence of a Open image in new window for which (1.8) and (1.9) are satisfied. Second, we prove the existence of the limit Open image in new window and give a limit formula for Open image in new window under the condition only Open image in new window . These two statements are formulated in our second main theorem stated in Section 3. The proof of the existence of Open image in new window is based on our first main result.

The article is organized as follows. In Section 2, we briefly explain some notation and definitions which are used to state and to prove our results. In Section 3, we state our two main results, whose proofs are relegated to Section 5.

Our theory is illustrated by examples in Section 4, including an interesting nonconvolution equation. This example shows the significance of the middle sum in (1.3), since only this term contributes to the limit of the solution of (1.1) in this case.

2. Mathematical Preliminaries

In this section, we briefly explain some notation and well-known mathematical facts which are used in this paper.

Let Open image in new window be the set of integers, Open image in new window and Open image in new window . Open image in new window stands for the set of all Open image in new window -dimensional column vectors with real components and Open image in new window is the space of all Open image in new window by Open image in new window real matrices. The zero matrix in Open image in new window is denoted by Open image in new window , and the identity matrix by Open image in new window . Let Open image in new window be the matrix in Open image in new window whose elements are all Open image in new window . The absolute value of the vector Open image in new window and the matrix Open image in new window is defined by Open image in new window and Open image in new window , respectively. The vector Open image in new window and the matrix Open image in new window is nonnegative if Open image in new window and Open image in new window , Open image in new window , respectively. In this case, we write Open image in new window and Open image in new window . Open image in new window can be endowed with any norms, but they are equivalent. A vector norm is denoted by Open image in new window and the norm of a matrix in Open image in new window induced by this vector norm is also denoted by Open image in new window . The spectral radius of the matrix Open image in new window is given by Open image in new window , which is independent of the norm employed to calculate it.

A partial ordering is defined on Open image in new window Open image in new window by letting Open image in new window Open image in new window if and only if Open image in new window Open image in new window . The partial ordering enables us to define the Open image in new window , and so forth for the sequences of vectors and matrices, which can also be determined componentwise and elementwise, respectively. It is known that Open image in new window for Open image in new window , and Open image in new window if Open image in new window and Open image in new window .

3. The Main Results

First, consider the nonconvolutional linear Volterra difference equation
with initial condition

Here, we assume

  1. (H1)
     
  2. (H2)
     
  3. (H3)
    the matrix

    is finite;

     
  4. (H4)
    the matrix

    is finite and Open image in new window ;

     
  5. (H5)

    the limit Open image in new window is finite.

     

By a solution of (3.1), we mean a sequence Open image in new window in Open image in new window satisfying (3.1) for any Open image in new window . It is clear that (3.1) with initial condition (3.2) has a unique solution.

Now, we are in a position to state our first main result.

Theorem 3.1.

Assume (H1)–(H5) are satisfied. Then for any Open image in new window the unique solution Open image in new window of (3.1) and (3.2) has a finite limit at Open image in new window and it satisfies

and hence Open image in new window yields Open image in new window , thus Open image in new window is invertible. On the other hand under our conditions the unique solution Open image in new window of (3.1) and (3.2) is a bounded sequence, therefore Open image in new window is finite, and (3.5) makes sense.

The second main result is dealing with the scalar Volterra difference equation
with the initial condition

where Open image in new window , and Open image in new window are given.

By a solution of the Volterra difference equation (3.7) we mean a sequence Open image in new window satisfies (3.7) for any Open image in new window .

In what follows, by Open image in new window we will denote the set of all initial sequence Open image in new window such that for each Open image in new window

exists.

It can be easily seen that for any initial sequence Open image in new window , (3.7) has exactly one solution satisfying (3.8). This unique solution is denoted by Open image in new window and it is called the solution of the initial value problem (3.7), (3.8).

The asymptotic representation of the solutions of (3.7) is given under the next condition.
  1. (A)
    There exists a Open image in new window for which
    From the theory of the infinite series, one can easily see that condition (A) yields
     
is finite. Moreover, the mapping Open image in new window defined by
is real valued on Open image in new window . It is also clear (see Section 5) that if there is an Open image in new window such that Open image in new window , and if Open image in new window then the equation

has a unique solution, say Open image in new window .

Now we formulate the following more explicit condition:
  1. (B)
    or there is an Open image in new window with Open image in new window , and
    1. (i)

      Open image in new window defined in (3.12) is finite,

       
    2. (ii)
       
    3. (iii)
       
     

Remark 3.2.

Let Open image in new window be a sequence such that Open image in new window for some Open image in new window . It will be proved in Lemma 5.7 that there is at most one Open image in new window satisfying (3.10) and (3.11). It is an easy consequence of this statement that if Open image in new window satisfies (3.10) and (3.11), and Open image in new window Open image in new window is a solution of (3.10), then Open image in new window , thus Open image in new window is the leading root of (3.10). Really, from the condition Open image in new window we have

that is (3.11) holds for Open image in new window instead of Open image in new window , and this contradicts the uniqueness of Open image in new window .

Now, we are ready to state our second result which will be proved in Section 5. This result shows that the implicit condition (A) and the explicit condition (B) are equivalent and the solutions of (3.7) can be asymptotically characterized by Open image in new window as Open image in new window .

Theorem 3.3.

Let Open image in new window , and Open image in new window be given. Then

Open image in new window Condition (A) holds if and only if condition (B) is satisfied.

Open image in new window If condition (A) or equivalently condition (B) holds, moreover
is finite, then for the solution Open image in new window of (3.7), (3.8) the limit Open image in new window is finite and it obeys

4. Examples and the Discussion of the Results

In this section, we illustrate our results by examples and the interested reader could also find some discussions.

Example 4.1.

Our Theorem 3.1 is given for system of equations, however the next example shows that this result is also new even in scalar case.

Let us consider the scalar nonconvolution Volterra difference equation
with the initial condition

where Open image in new window and real, and Open image in new window is a real sequence such that its limit Open image in new window is finite.

Now, let the values Open image in new window and the sequence Open image in new window be defined by

Then, it can be easily seen that problem (4.1), (4.2) is equivalent to problem (3.1), (3.2).

We find that Open image in new window for any fixed Open image in new window .

It is known that
where Open image in new window is the well-known Beta function at Open image in new window defined by
Using the nonnegativity of Open image in new window , and Lemma 5.2 we have that
Now, one can easily see that for the sequences Open image in new window and Open image in new window all of the conditions of Theorem 3.1 are satisfied. Thus, by Theorem 3.1 we get that the solution Open image in new window of the initial value problem (4.1), (4.2) satisfies
On the other hand, we know (see [23]) that

and hence in [20, Theorem 3.1] is not applicable.

Example 4.2.

Thus, (3.7) reduces to the delay difference equation

and for any sequence Open image in new window holds.

Since Open image in new window for any large enough Open image in new window , Open image in new window , moreover the function Open image in new window defined in (3.13) satisfies
Let Open image in new window be the unique value satisfying

Now, statement( Open image in new window ) of Theorem 3.3 is applicable and so the next statement is valid.

Proposition 4.3.

hold if and only if either

is satisfied.

Now, let Open image in new window and Open image in new window . Then Open image in new window , moreover (4.14) and (4.15) reduce to
If especially Open image in new window and Open image in new window , then Open image in new window , moreover (4.14) and (4.15) are equivalent to the condition

Example 4.4.

Let Open image in new window and Open image in new window . Then, (3.7) has the following form:
It is clear that Open image in new window , and the function Open image in new window defined in (3.13) is given by

moreover Open image in new window is the unique positive root of Open image in new window .

Thus statement ( Open image in new window ) in Theorem 3.3 is applicable and as a corollary of it we obtain the following.

Proposition 4.5.

There is a Open image in new window such that (3.10) and (3.11) hold with the sequence Open image in new window , Open image in new window , if and only if either

Example 4.6.

In this case, Open image in new window and by using the well-known properties of the binomial series, we find

Thus, Open image in new window , therefore by statement ( Open image in new window ) of Theorem 3.3 we get the following.

Proposition 4.7.

There is a Open image in new window such that (3.10) and (3.11) hold with the sequence Open image in new window , if and only if either
where Open image in new window is the unique positive solution of the equation

Example 4.8.

Let Open image in new window and Open image in new window , and Open image in new window . Then, (3.7) reduces to the special form
It is not difficult to see that Open image in new window ,

and Open image in new window . From statement Open image in new window of Theorem 3.3 we have the following.

Proposition 4.9.

There is a Open image in new window such that (3.10) and (3.11) hold with the sequence Open image in new window , if and only if either

where Open image in new window is the well-known Riemann function.

5. Proofs of the Main Theorems

5.1. Proof of Theorem 3.1

To prove Theorem 3.1 we need the next result from [20].

Theorem A.

Let us consider the initial value problem ( 3.1 ), ( 3.2 ). Suppose that there are Open image in new window such that
Assume also that Open image in new window . Then, there is a nonnegative matrix Open image in new window , independent of Open image in new window and Open image in new window , such that the solution Open image in new window of ( 3.1 ), ( 3.2 ) satisfies

Now, we prove some lemmas.

Lemma 5.1.

The hypotheses of Theorem 3.1 imply that the hypotheses of Theorem A are satisfied, and hence the solution Open image in new window of (3.1), (3.2) is bounded.

Proof.

Let Open image in new window be such that Open image in new window . This can be satisfied because Open image in new window . Then, there is an Open image in new window for which
and hence for an Open image in new window , we have
therefore,
But the matrices are nonnegative in the above inequality, thus
and this shows (5.1). Since condition Open image in new window holds, we get
therefore,

thus (5.2) is satisfied.

In the next lemma we give an equivalent form of Open image in new window .

Lemma 5.2.

if and only if
is finite. In both cases

If Open image in new window satisfies Open image in new window too, and (5.11) holds, then Open image in new window .

Proof.

First we show that
is finite if and only if
is finite, and in both cases
These come from Open image in new window , since
and hence
Now, suppose that (5.11) holds. Then by (5.19), either
Both of the previous cases implies that
which shows that
is finite and
As we have seen, this is equivalent with (5.12). If (5.12) is true or equivalently
is finite, then by (5.19)

satisfies (5.11). Open image in new window follows from Open image in new window . The proof is now complete.

Lemma 5.3.

The hypotheses of Theorem 3.1 imply that
is the only vector satisfying the equation

Proof.

Since Open image in new window the matrix Open image in new window is invertible, which shows the uniqueness part of the lemma. On the other hand, by Lemma 5.1 we have that Open image in new window is a bounded sequence, and hence Open image in new window is finite. Thus, Open image in new window is well defined and satisfies (5.28). The proof is complete.

Lemma 5.4.

The vector defined by (5.27) satisfies the relation

Proof.

Let Open image in new window be arbitrarily fixed and
But under the hypotheses of Theorem 3.1 we find
Now, by Lemma 5.2

and hence (5.30) holds. On the other hand, it can be easily seen that by the above definition of Open image in new window the relation (5.29) also holds. The proof is complete.

Now, we prove Theorem 3.1.

Proof.

Let Open image in new window be arbitrarily fixed. Then, (3.1) can be written in the form
Subtracting (5.29) from the above equation, we get
On the other hand, by Lemma 5.1, Open image in new window is bounded and hence
is finite. Let Open image in new window be arbitrarily fixed and Open image in new window . Then there is an Open image in new window such that
Thus, (5.35) yields
From this it follows:
and hence Lemma 5.4 implies that
Since Open image in new window is a nonnegative matrix with Open image in new window , we have that Open image in new window . Thus,

and hence the proof of Theorem 3.1 is complete.

5.2. Proof of Theorem 3.3

Theorem 3.3 will be proved after some preparatory lemmas.

In the next lemma, we show that (3.7) can be transformed into an equation of the form (3.1) by using the transformation

Lemma 5.5.

Under the conditions of Theorem 3.3, the sequence Open image in new window defined by (5.43) satisfies (3.1), where the sequences Open image in new window and Open image in new window are defined by

Proof.

Let Open image in new window be defined by (5.43). Then,
On the other hand, from (3.7) it follows:
and hence
Therefore,
where Open image in new window is defined in (5.45). By interchanging the order of the summation we get
This and (5.52) yield
By using the definition Open image in new window , we have
and hence

But by using the definition of Open image in new window in (5.44) the proof of the lemma is complete.

In the next lemma, we collect some properties of the function Open image in new window defined in (3.13).

Lemma 5.6.

Then, the function Open image in new window defined in (3.13) has the following properties.

  1. (a)
    The series of functions

    is convergent on Open image in new window and it is divergent on Open image in new window .

     
  2. (b)
     
  3. (c)
     
  4. (d)

    Open image in new window is strictly decreasing.

     
  5. (e)

    If Open image in new window , then the equation Open image in new window has a unique solution.

     

Proof.

(a) The root test can be applied. (b) The series of functions (5.58) is uniformly convergent on Open image in new window for every Open image in new window , and this, together with Open image in new window , implies the result. (c) If Open image in new window is finite, then the series of functions (5.58) is uniformly convergent on Open image in new window , hence Open image in new window is continuous on Open image in new window . Suppose now that Open image in new window and Open image in new window . Let Open image in new window be fixed and let Open image in new window such that

whenever Open image in new window , and this shows Open image in new window . Finally, we consider the case Open image in new window . Then, Open image in new window follows from the condition Open image in new window . (d) The series of functions (5.58) can be differentiated term-by-term within Open image in new window , and therefore Open image in new window , Open image in new window . Together with (c) this gives the claim. (e) We have only to apply (d), (c), and (b). The proof is complete.

We are now in a position to prove Theorem 3.3.

Proof.

(a) Let Open image in new window for all Open image in new window . Then, it is easy to see that there is a Open image in new window such that (3.10) holds if and only if (3.15) is true, and in this case (3.11) is also satisfied. Suppose Open image in new window for some Open image in new window . Let Open image in new window be finite. By the root test, the series
are convergent for all Open image in new window . Moreover, it can be easily verified that the series
are absolutely convergent, whenever Open image in new window is finite. Define the functions Open image in new window by
where Open image in new window if Open image in new window is finite and Open image in new window , otherwise. The series of functions in (5.64) are uniformly convergent on Open image in new window for every Open image in new window , hence Open image in new window and Open image in new window are continuous. Further,
Let Open image in new window if Open image in new window , and let Open image in new window if Open image in new window . It now follows from the previous inequalities and Lemma 5.6(d) that
and hence Open image in new window is strictly increasing on Open image in new window . It is immediate that Open image in new window . If (3.10) is hold for some Open image in new window , then the convergence of the series Open image in new window implies Open image in new window . Suppose Open image in new window . It is simple to see that there is a Open image in new window satisfying (3.10) if and only if either Open image in new window (in case Open image in new window ) or Open image in new window (in case Open image in new window ). Moreover, the existence of a Open image in new window satisfying (3.11) is equivalent to either Open image in new window (in case Open image in new window ) or Open image in new window (in case Open image in new window ). Now, the result follows from the properties of the functions Open image in new window . The proof of (a) is complete. (b) In virtue of Lemma 5.5 the proof of the theorem will be complete if we show that the sequences Open image in new window and Open image in new window satisfy the conditions (H2)–(H5) in Section 3. Since the series Open image in new window is convergent,
is finite and Open image in new window . In a similar way, one can easily prove that
is also finite. It is also clear that
is finite. Thus, by Theorem 3.1 we get that the limit

is finite and satisfies the required relation (3.22). The proof is now complete.

Lemma 5.7.

Let Open image in new window be a sequence such that Open image in new window for some Open image in new window . Then, there is at most one Open image in new window satisfying (3.10) and (3.11).

Proof.

Suppose on the contrary that there exist two different numbers from Open image in new window , denoted by Open image in new window and Open image in new window , such that (3.10) and (3.11) hold. Then,
It follows from (5.74), the mean value inequality and (5.75) that

and this is a contradiction.

Notes

Acknowledgment

This work was supported by Hungarian National Foundation for Scientific Research Grant no. K73274.

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Copyright information

© Győri and L. Horváth. 2008

This article is published under license to BioMed Central Ltd. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Authors and Affiliations

  1. 1.Department of Mathematics and ComputingUniversity of PannoniaEgyetemHungary

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