On Some Symmetric Systems of Difference Equations

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.


Introduction
Studying difference equations and systems which possess some kind of symmetry attracted some attention recently (see, e.g.,  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  > 0.
Since  > 0, it is clear that the change of variables reduces systems (1) and (3) to the case  = 1.The authors of [24] claim that the change of variables (4) reduces (2) to the case  = 1 too; however by using the change system (2) becomes 2 Abstract and Applied Analysis Based on this observation we may, and will, assume that  = 1 in systems of difference equations ( 1)- (3).
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  ∈ N \ {1}.
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][24][25].For related systems see also [2, 5-10, 12, 17-20].

Main Results
Let R + = (0,+∞) and R  + 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 R  , and let  :  →  be a continuous mapping with the unique equilibrium  * ∈ .Suppose that for the discrete dynamic system there is a  ∈ N 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 R  + by for arbitrary vectors  = ( 1 ,  2 , . . .,   )  ∈ R  + and  = ( 1 ,  2 , . . .,   )  ∈ R  + .It is known that the part-metric  is a continuous metric on R  + , (R  + , ) is a complete metric space, and that the distances induced by the part-metric and by the Euclidean norm are equivalent on R  + (see, e.g., [4]).Based on these properties and Theorem A, the following corollary follows.
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  : R 2 + → R + ,  : R 2 + → R + be continuous functions.We suppose that the system of two difference equations, has a unique positive equilibrium (, ).Suppose also that there is an  ∈ N such that for any positive solution (  , V  ) ∈N 0 of system (11), the following inequalities: hold, with the equalities if and only if   = , for every  ∈ N 0 , and V  = , for every  ∈ N 0 , respectively.Then the equilibrium (, ) is globally asymptotically stable.
Proof.First, we prove that for every if and only if (  , V  ) ̸ = (, ).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 V) 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  ∈ N 0 .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   ̸ =  and V  = , then from the second inequality in (12), we have that which along with ( 14) implies (13).The case   =  and V  ̸ =  directly follows from the case   ̸ =  and V  = , by the symmetry.
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  =  + 1.
Clearly if then in (31) equalities follow.On the other hand, if equality holds in the first inequality in (31), we have that If  +1 =  − , then from the first equality in (29) we have that  − = 1, while if  +1 = 1/ − , then from the second equality in (29), we have that  − = 1.By symmetry (see ( 30)), we have that if equality holds in the second inequality in (31), then  − = 1.Therefore, equalities in (31) hold if and only if ( − ,  − ) = (1, 1).Hence all the conditions of Proposition 2 are fulfilled from which it follows that the positive equilibrium (1, 1) is globally asymptotically stable with respect to the set R  + × R  + .
Remark 5. Corollary 4 extends and gives a very short proof of the main result in [23], which is obtained for  = 1 and  = 0. Further, it also extends and gives a very short proof of the main result in [25], which is obtained for  = 0 and  = 1.Moreover, it confirms the conjecture in [25], which is obtained for  = 0 and  ∈ N \ {1}.
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  =  + 1.On the other hand, similarly as in the proof of Corollary 4 it is proved that equalities in (36) hold if and only if ( − ,  − ) = (1, 1).Hence all the conditions of Proposition 2 are fulfilled from which it follows that the positive equilibrium (1, 1) is globally asymptotically stable with respect to the set R  + × R  + .
Remark 7. Corollary 6 extends and gives a very short proof of the main result in [24], which is obtained for  = 1 and  = 0.
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].
We should determine the sign of the product of the following expressions: From ( 40) and ( 41), we see if we show that  −  −1   − 1 and ( −  −1 /  ) − 1 have the same sign for  ∈ N, then     will be nonpositive.
We consider four cases.
From all above mentioned and by Proposition 2, we get the result.