^{1,2}

^{3}

^{4,5}

^{2}

^{1}

^{2}

^{3}

^{4}

^{5}

Here we show that the main results in the papers by Yalcinkaya (2008), Yalcinkaya and Cinar (2010), and Yalcinkaya, Cinar, and Simsek (2008), as well as a conjecture from the last mentioned paper, follow from a slight modification of a result by G. Papaschinopoulos and C. J. Schinas. We also give some generalizations of these results.

Studying difference equations and systems which possess some kind of symmetry attracted some attention recently (see, e.g., [

Paper [

In [

Since

Based on this observation we may, and will, assume that

In the main results in [

The authors of [

Here, among others, we show that all the results and conjectures mentioned above follow from a slight modification of a result in the literature published before papers [

Let

Let

The part-metric (see [

It is known that the part-metric

Based on these properties and Theorem A, the following corollary follows.

Let

Some applications of various part-metric-related inequalities and some asymptotic methods in studying difference equations related to symmetric ones can be found, for example, in [

In Lemma 2.3 in [

Let

First, we prove that for every

To prove (

The proofs of inequalities (

Now note that if the equality holds in the first inequality in (

Now suppose that the first inequality in (

If

If (

If

Finally, note that if

Now we define the map

Then we get

By using inequality (

It is not difficult to see that the following extension of Proposition

Let

Now we use Proposition

Let

We may assume that

Clearly if

By symmetry (see (

Corollary

Let

We may assume that

Corollary

Corollary

Similar type of issues appear in some literature on scalar difference equations (see, e.g., related results in papers [

It is of some interest to extend results in Corollaries

Let

Then

Let

From (

We consider four cases.

Hence

Assume that

Let

Finally, let

Then we have that

From all above mentioned and by Proposition

S. Stević would like to express his sincere thanks to Professors G. Papaschinopoulos and C. J. Schinas for useful conversations and their help during writing this paper. The first author is supported by the Grant P201/10/1032 of the Czech Grant Agency (Prague). The fourth author is supported by the grant FEKT-S-11-2-921 of Faculty of Electrical Engineering and Communication, Brno University of Technology. This paper is also supported by the Serbian Ministry of Science Projects III 41025, III 44006, and OI 171007.