#### Abstract

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.

#### 1. Introduction

Studying difference equations and systems which possess some kind of symmetry attracted some attention recently (see, e.g., [1–25] and the related references therein).

Paper [23] studied the following system of difference equations:

In [24], authors claim that they study the system while in [25], the authors studied the system where .

Since , it is clear that the change of variables reduces systems (1) and (3) to the case . The authors of [24] claim that the change of variables (4) reduces (2) to the case too; however by using the change system (2) becomes Therefore, in fact, [24] studied only system (2) for the case .

Based on this observation we may, and will, assume that in systems of difference equations (1)–(3).

In the main results in [23–25] it is proved that when , the positive equilibrium point of systems (1)–(3) is globally asymptotically stable.

The authors of [25] finish their paper by the statement that they believe that the results therein can be conveniently extended to the following higher order system of difference equations: when .

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 [23–25]. For related systems see also [2, 5–10, 12, 17–20].

#### 2. Main Results

Let and be the set of all positive -dimensional vectors. The following theorem was proved in [4].

Theorem A. *Let be a complete metric space, where denotes a metric and is an open subset of , and let be a continuous mapping with the unique equilibrium . Suppose that for the discrete dynamic system
**
there is a such that for the th iterate of , the following inequality holds:
**
for all . Then is globally asymptotically stable with respect to metric . *

The part-metric (see [21]) is a metric defined on by for arbitrary vectors and .

It is known that the part-metric is a continuous metric on , is a complete metric space, and that the distances induced by the part-metric and by the Euclidean norm are equivalent on (see, e.g., [4]).

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

Corollary 1. *Let be a continuous mapping with a unique equilibrium . Suppose that for the discrete dynamic system (7), there is some such that for the part-metric inequality
**
holds for all . Then is globally asymptotically stable. *

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 [1, 3–5, 10, 11, 13–16, 22] (see also the related references therein).

In Lemma 2.3 in [10], Papaschinopoulos and Schinas formulated a variant of the following result, without giving a proof. However, the part concerning the equality in inequality (12) below, is not mentioned, but it is crucial in applying Corollary 1 (see inequality (10)). For this reason, the completeness and the benefit of the reader we will give a complete proof of it.

Proposition 2. *Let , be continuous functions. We suppose that the system of two difference equations,
**
has a unique positive equilibrium . Suppose also that there is an such that for any positive solution of system (11), the following inequalities:
**
hold, with the equalities if and only if , for every , and , for every , respectively. Then the equilibrium is globally asymptotically stable. *

*Proof. *First, we prove that for every
if and only if .

To prove (13), it is enough to prove that
if and only if , and
if and only if .

The proofs of inequalities (14) and (15) are the same (up to the interchanging letters and ) so it is enough to prove (14).

Now note that if the equality holds in the first inequality in (12), then we have that
from which, in both cases, it easily follows that
On the other hand, if (17) holds, then we easily obtain that one of the equalities in (16) holds, and consequently it follows that the equality holds in the first inequality in (12). Hence, by one of the assumptions, we have that (17) holds if and only if for every .

Now suppose that the first inequality in (12), is strict. Then, if , directly follows that , while from the first inequality in (12) it follows that . Hence
from which inequality (14) easily follows.

If , then , while from the first inequality in (12), it follows that . From these two inequalities, we have that
and consequently (14).

If (14) and (15) hold then if and , inequality (13) immediately follows by using the following elementary implication: if and , then .

If and , then from the second inequality in (12), we have that . Hence
which along with (14) implies (13). The case and directly follows from the case and , by the symmetry.

Finally, note that if , then from (12), we have that and , so that the first equality in (20) holds and
from which it follows that both minima in (13) are equal, finishing the proof of the claim.

Now we define the map as follows:

Then we get
and by induction
for every .

By using inequality (13) and the fact that the inequalities ,, , along with equalities , , imply the inequality , we have that for each vector such that ,
from which the proof follows by Corollary 1.

It is not difficult to see that the following extension of Proposition 2 can be proved by slight modifications of the proof of Proposition 2.

Proposition 3. *Let ,, be continuous functions. Suppose that the system of difference equations
**
has a unique positive equilibrium , and that there is an such that for any solution of system (26), the following inequalities:
**
hold, with the equalities if and only if , for every , and . Then the equilibrium is globally asymptotically stable. *

Now we use Proposition 2 in proving the results in papers [23–25].

Corollary 4. *Let ,. Consider the system
**
Then the positive equilibrium point of system (28) is globally asymptotically stable with respect to the set , where .*

* Proof. *We may assume that . From system (28), we have that
from which it follows that
so that condition (12) in Proposition 2 is fulfilled with .

Clearly if
then in (31) equalities follow. On the other hand, if equality holds in the first inequality in (31), we have that
If , then from the first equality in (29) we have that , while if , then from the second equality in (29), we have that .

By symmetry (see (30)), we have that if equality holds in the second inequality in (31), then . Therefore, equalities in (31) hold if and only if . Hence all the conditions of Proposition 2 are fulfilled from which it follows that the positive equilibrium is globally asymptotically stable with respect to the set .

*Remark 5. *Corollary 4 extends and gives a very short proof of the main result in [23], which is obtained for and . Further, it also extends and gives a very short proof of the main result in [25], which is obtained for and . Moreover, it confirms the conjecture in [25], which is obtained for and .

Corollary 6. *Let ,. Consider the system
**
Then the positive equilibrium point of system (34) is globally asymptotically stable with respect to the set , where .*

*Proof. *We may assume that . From system (34), we have that
from which it follows that
Hence condition (12) in Proposition 2 is fulfilled with . On the other hand, similarly as in the proof of Corollary 4 it is proved that equalities in (36) hold if and only if . Hence all the conditions of Proposition 2 are fulfilled from which it follows that the positive equilibrium is globally asymptotically stable with respect to the set .

*Remark 7. *Corollary 6 extends and gives a very short proof of the main result in [24], which is obtained for and .

*Remark 8. *Corollary 6 is also a consequence of Corollary 4. Namely, by using the change of variables , system (34) is transformed into the system
which is system (28). In particular, this shows that systems (1) and (2), for the case , are equivalent and consequently the results in [23, 24].

*Remark 9. *Similar type of issues appear in some literature on scalar difference equations (see, e.g., related results in papers [1, 5, 11, 13]).

It is of some interest to extend results in Corollaries 4 and 6 by using Proposition 2. The next result is of this kind and it extends a result in [5].

Corollary 10. *Let and with , and and satisfy the following two conditions:*(*H1*)*,*(*H2*)*,**where .**Then is the unique positive equilibrium of the system of difference equations
**
and it is globally asymptotically stable. *

*Proof. *Let
We should determine the sign of the product of the following expressions:

From (40) and (41), we see if we show that and have the same sign for , then will be nonpositive.

We consider four cases.*Case 1. *, . Clearly in this case . By (*H1*) and (*H2*), we have that

Hence and consequently
*Case 2*. , . Since , we obtain . On the other hand, by (*H1*) and (*H2*), we have
so that . Hence (43) follows in this case.*Case 3*. Case , . Then we have that and consequently . On the other hand, we have
so that . Hence (43) follows in this case too.*Case 4*. Case , . Then . On the other hand, we have
so that . Hence (43) also holds in this case. Thus , for every .

Assume that , then or . Using (40) or (41) along with (43) in any of these two cases, we have that
Hence ,.

Let
Using the following expressions:
it can be proved similarly that , for every , and that , if and only if ,.

Finally, let be a solution of the system

Then we have that
where denotes the vector consisting of copies of . Then similar to the considerations in Cases (i)–(iv), it follows that , so that , and similarly it is obtained that . Hence is a unique positive equilibrium of system (26).

From all above mentioned and by Proposition 2, we get the result.

#### Acknowledgments

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.