# Asymptotic Representation of the Solutions of Linear Volterra Difference Equations

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## 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].

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.

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].

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 .

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

Here, we assume

- (H1)
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;

- (H2)
for any fixed Open image in new window the limit Open image in new window is finite and Open image in new window ;

- (H3)the matrix(3.3)
is finite;

- (H4)the matrix(3.4)
is finite and Open image in new window ;

- (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.

_{1})–(H

_{5}) 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.

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 .

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).

- (A)There exists a Open image in new window for which(3.10)(3.11)From the theory of the infinite series, one can easily see that condition (A) yields(3.12)

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

- (B)or there is an Open image in new window with Open image in new window , and
- (i)
Open image in new window defined in (3.12) is finite,

- (ii)or(3.17)
- (iii)or(3.19)

- (i)

*Remark 3.2.*

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.

## 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.

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.

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 .

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

Example 4.2.

and for any sequence Open image in new window holds.

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

Proposition 4.3.

is satisfied.

Example 4.4.

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.

Example 4.6.

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.

Example 4.8.

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

Proposition 4.9.

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.

thus (5.2) is satisfied.

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

Lemma 5.2.

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

Proof.

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

Lemma 5.3.

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.

Proof.

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.

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.

Lemma 5.5.

Proof.

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.*

- (a)
*The series of functions*(5.58)*is convergent on*Open image in new window*and it is divergent on*Open image in new window . - (b)
- (c)
- (d)
Open image in new window

*is strictly decreasing.* - (e)
*If*Open image in new window ,*then the equation*Open image in new window*has a unique solution*.

Proof.

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.

_{2})–(H

_{5}) in Section 3. Since the series Open image in new window is convergent,

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.

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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