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We consider homogeneous linear Volterra Discrete Equations and we study the asymptotic behaviour of their solutions under hypothesis on the sign of the coefficients and of the first- and second-order differences. The results are then used to analyse the numerical stability of some classes of Volterra integrodifferential equations.

Linear Volterra Discrete Equations (VDEs) are usually represented according to two types of formulae (see, e.g., [

To be more specific, a simple numerical method for the VIDE,

Using (

For the sake of completeness, there is also another type of VDE widely used in literature (see, e.g., [

Asymptotic analysis of difference equations of the form (

Since (

In the whole paper it is assumed the empty sum convention

Let

Consider (

Then, for any

or

then, for any

Set

In order to prove the second part of our result, let us proceed by contradiction. Assume that

Now consider hypothesis (

It is well known that one of the most used tools in the stability analysis of VDEs is the Lyapunov approach [

It is easy to see that if hypothesis (iii) in Theorem

Consider (

Furthermore, we point out that checking assumption (

Consider (

From (iv) and the definition of

We want to underline that Theorem

Observe that, when

Consider (

or

Then, for any

First of all observe that (a) assures

As a consequence of this result, the following can be easily proved.

Under assumptions (b)–(f) of Theorem

If, in Theorem

Consider the following theoretical examples of application of Corollaries

It can be easily seen that (

To be more specific (

As a counterexample, consider (

Problem (

Theorem

Problem (

We want explicitly to mention that Theorem

Finally, we observe that, if in (

A more practical application of our results is the study of the longtime behaviour of the numerical solution to VIDEs. Let us consider the homogeneous problem

Consider (

Then the solution

or

then

Note that (

Theorem

As we mentioned in the previous section, the analogue of Theorem

Assume that hypotheses (i)–(iv) of Theorem

From Theorems

Moreover, it is worth noting that if in (

In the literature we sometimes encounter VIDEs with the following structure (see, e.g., [

Consider (

or

Then, for any

A comparison to Theorem

Consider (

Then, for any

We want to prove that all the hypotheses of Theorem

As already mentioned above, in [

Problem (

The authors declare that there are no conflicts of interest regarding the publication of this paper.

The research was supported by GNCS-INdAM.