Asymptotic Representation of the Solutions of Linear Volterra Difference Equations
© Győri and L. Horváth. 2008
Received: 26 February 2008
Accepted: 4 April 2008
Published: 15 April 2008
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.
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 , Elaydi , and Kelley and Peterson . Recent contribution to the asymptotic theory of difference equations is given in the papers by Kolmanovskii et al. , Medina , Medina and Gil , and Song and Baker ; see [8–19] for related results.
where , and are sequences with elements in and , 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 .
where , and and are real sequences.
Here, denotes the forward difference operator to be defined as usual, that is, .
In the paper  (see also ), it is shown that if satisfies (1.8) and (1.9), and the initial sequence is suitable, then for the solution of (1.6) and (1.7) the sequence , is bounded. Furthermore, some extra conditions guarantee that the limit is finite and satisfies a limit formula.
In our paper, we improve considerably the result in . First, we give explicit necessary and sufficient conditions for the existence of a for which (1.8) and (1.9) are satisfied. Second, we prove the existence of the limit and give a limit formula for under the condition only . These two statements are formulated in our second main theorem stated in Section 3. The proof of the existence of 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 be the set of integers, and . stands for the set of all -dimensional column vectors with real components and is the space of all by real matrices. The zero matrix in is denoted by , and the identity matrix by . Let be the matrix in whose elements are all . The absolute value of the vector and the matrix is defined by and , respectively. The vector and the matrix is nonnegative if and , , respectively. In this case, we write and . can be endowed with any norms, but they are equivalent. A vector norm is denoted by and the norm of a matrix in induced by this vector norm is also denoted by . The spectral radius of the matrix is given by , which is independent of the norm employed to calculate it.
A partial ordering is defined on by letting if and only if . The partial ordering enables us to define the , and so forth for the sequences of vectors and matrices, which can also be determined componentwise and elementwise, respectively. It is known that for , and if and .
3. The Main Results
Here, we assume
and are sequences with elements in and , respectively;
for any fixed the limit is finite and ;
- (H3)the matrix(3.3)
- (H4)the matrix(3.4)
is finite and ;
the limit is finite.
By a solution of (3.1), we mean a sequence in satisfying (3.1) for any . 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.
and hence yields , thus is invertible. On the other hand under our conditions the unique solution of (3.1) and (3.2) is a bounded sequence, therefore is finite, and (3.5) makes sense.
where , and are given.
By a solution of the Volterra difference equation (3.7) we mean a sequence satisfies (3.7) for any .
It can be easily seen that for any initial sequence , (3.7) has exactly one solution satisfying (3.8). This unique solution is denoted by and it is called the solution of the initial value problem (3.7), (3.8).
- (A)There exists a 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 .
- (B)either , , and(3.15)or there is an with , and
defined in (3.12) is finite,
- (ii)if , then the constant satisfies either(3.16)or(3.17)
- (iii)if , then the constant satisfies either(3.18)or(3.19)
that is (3.11) holds for instead of , and this contradicts the uniqueness of .
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 as .
Let , and be given. Then
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.
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 and real, and is a real sequence such that its limit 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 for any fixed .
and hence in [20, Theorem 3.1] is not applicable.
and for any sequence holds.
Now, statement( ) of Theorem 3.3 is applicable and so the next statement is valid.
moreover is the unique positive root of .
Thus statement ( ) in Theorem 3.3 is applicable and as a corollary of it we obtain the following.
Thus, , therefore by statement ( ) of Theorem 3.3 we get the following.
and . From statement of Theorem 3.3 we have the following.
where 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 .
Now, we prove some lemmas.
The hypotheses of Theorem 3.1 imply that the hypotheses of Theorem A are satisfied, and hence the solution of (3.1), (3.2) is bounded.
thus (5.2) is satisfied.
In the next lemma we give an equivalent form of .
If satisfies too, and (5.11) holds, then .
satisfies (5.11). follows from . The proof is now complete.
Since the matrix is invertible, which shows the uniqueness part of the lemma. On the other hand, by Lemma 5.1 we have that is a bounded sequence, and hence is finite. Thus, is well defined and satisfies (5.28). The proof is complete.
and hence (5.30) holds. On the other hand, it can be easily seen that by the above definition of the relation (5.29) also holds. The proof is complete.
Now, we prove Theorem 3.1.
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.
But by using the definition of in (5.44) the proof of the lemma is complete.
In the next lemma, we collect some properties of the function defined in (3.13).
Then, the function defined in (3.13) has the following properties.
- (a)The series of functions(5.58)
is convergent on and it is divergent on .
is strictly decreasing.
If , then the equation has a unique solution.
whenever , and this shows . Finally, we consider the case . Then, follows from the condition . (d) The series of functions (5.58) can be differentiated term-by-term within , and therefore , . 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.
is finite and satisfies the required relation (3.22). The proof is now complete.
Let be a sequence such that for some . Then, there is at most one satisfying (3.10) and (3.11).
and this is a contradiction.
This work was supported by Hungarian National Foundation for Scientific Research Grant no. K73274.
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