On Some Symmetric Systems of Difference Equations

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: 𝑥_𝑛+1=𝑦_𝑛𝑥_𝑛−1+𝑎𝑦_𝑛+𝑥_𝑛−1,𝑦_𝑛+1=𝑥_𝑛𝑦_𝑛−1+𝑎𝑥_𝑛+𝑦_𝑛−1,𝑛∈ℕ₀. (1)

In [24], authors claim that they study the system 𝑥_𝑛+1=𝑦_𝑛+𝑥_𝑛−1𝑦_𝑛𝑥_𝑛−1,𝑦+𝑎_𝑛+1=𝑥_𝑛+𝑦_𝑛−1𝑥_𝑛𝑦_𝑛−1+𝑎,𝑛∈ℕ₀, (2) while in [25], the authors studied the system 𝑥_𝑛+1=𝑥_𝑛𝑦_𝑛−1+𝑎𝑥_𝑛+𝑦_𝑛−1,𝑦_𝑛+1=𝑦_𝑛𝑥_𝑛−1+𝑎𝑦_𝑛+𝑥_𝑛−1,𝑛∈ℕ₀, (3) where 𝑎>0.

Since 𝑎>0, it is clear that the change of variables 𝑥_𝑛,𝑦_𝑛⟶√𝑎𝑥_𝑛√𝑎𝑦_𝑛, (4) reduces systems (1) and (3) to the case 𝑎=1. The authors of [24] claim that the change of variables (4) reduces (2) to the case 𝑎=1 too; however by using the change system (2) becomes 𝑥_𝑛+1=𝑦_𝑛+𝑥_𝑛−1𝑎𝑦_𝑛𝑥_𝑛−1,𝑦+1_𝑛+1=𝑥_𝑛+𝑦_𝑛−1𝑎𝑥_𝑛𝑦_𝑛−1+1,𝑛∈ℕ₀. (5) Therefore, in fact, [24] studied only system (2) for the case 𝑎=1.

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

In the main results in [23–25] it is proved that when 𝑎=1, the positive equilibrium point (𝑥,𝑦)=(1,1) 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: 𝑥_𝑛+1=𝑥_𝑛𝑦_𝑛−𝑙+𝑎𝑥_𝑛+𝑦_𝑛−𝑙,𝑦_𝑛+1=𝑦_𝑛𝑥_𝑛−𝑙+𝑎𝑦_𝑛+𝑥_𝑛−𝑙,𝑛∈ℕ₀, (6) when 𝑙∈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 [23–25]. For related systems see also [2, 5–10, 12, 17–20].

2. Main Results

Let ℝ₊=(0,+∞) and ℝ^𝑛₊ be the set of all positive 𝑛-dimensional vectors. The following theorem was proved in [4].

Theorem A. Let (𝑀,𝑑) be a complete metric space, where 𝑑 denotes a metric and 𝑀 is an open subset of ℝ^𝑛, and let 𝑇∶𝑀→𝑀 be a continuous mapping with the unique equilibrium 𝑥^∗∈𝑀. Suppose that for the discrete dynamic system 𝑥_𝑛+1=𝑇𝑥_𝑛,𝑛∈ℕ₀, (7) there is a 𝑘∈ℕ such that for the 𝑘th iterate of 𝑇, the following inequality holds: 𝑑𝑇^𝑘𝑥,𝑥^∗<𝑑𝑥,𝑥^∗, (8) for all 𝑥≠𝑥^∗. Then 𝑥^∗ is globally asymptotically stable with respect to metric 𝑑.

The part-metric (see [21]) is a metric defined on ℝ^𝑛₊ by 𝑝(𝑋,𝑌)=−logmin_{1≤𝑖≤𝑛}𝑥_𝑖𝑦_𝑖,𝑦_𝑖𝑥_𝑖, (9) for arbitrary vectors 𝑋=(𝑥₁,𝑥₂,…,𝑥_𝑛)^𝑇∈ℝ^𝑛₊ and 𝑌=(𝑦₁,𝑦₂,…,𝑦_𝑛)^𝑇∈ℝ^𝑛₊.

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

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

Corollary 1. Let 𝑇∶ℝ^𝑛₊→ℝ^𝑛₊ be a continuous mapping with a unique equilibrium 𝑥^∗∈ℝ^𝑛₊. Suppose that for the discrete dynamic system (7), there is some 𝑘∈ℕ such that for the part-metric 𝑝 inequality 𝑝𝑇^𝑘𝑥,𝑥^∗<𝑝𝑥,𝑥^∗ (10) holds for all 𝑥≠𝑥^∗. Then 𝑥^∗ 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 𝑓∶ℝ₊^2𝑚→ℝ₊, 𝑔∶ℝ₊^2𝑚→ℝ₊ be continuous functions. We suppose that the system of two difference equations, 𝑢_𝑛+𝑚=𝑓𝑢_𝑛,𝑢_𝑛+1,…,𝑢_{𝑛+𝑚−1},𝑣_𝑛,𝑣_𝑛+1,…,𝑣_{𝑛+𝑚−1},v_𝑛+𝑚=𝑔𝑢_𝑛,𝑢_𝑛+1,…,𝑢_{𝑛+𝑚−1},𝑣_𝑛,𝑣_𝑛+1,…,𝑣_{𝑛+𝑚−1}, (11) has a unique positive equilibrium (𝑤,𝑤). Suppose also that there is an 𝑟∈ℕ such that for any positive solution (𝑢_𝑛,𝑣_𝑛)_𝑛∈ℕ0 of system (11), the following inequalities: 𝑢_𝑛−𝑢_𝑛+𝑟𝑤²𝑢_𝑛−𝑢_𝑛+𝑟𝑣≤0,_𝑛−𝑣_𝑛+𝑟𝑤²𝑣_𝑛−𝑣_𝑛+𝑟≤0,𝑛∈ℕ₀, (12) hold, with the equalities if and only if 𝑢_𝑛=𝑤, for every 𝑛∈ℕ₀, and 𝑣_𝑛=𝑤, for every 𝑛∈ℕ₀, respectively. Then the equilibrium (𝑤,𝑤) is globally asymptotically stable.

Proof. First, we prove that for every 𝑛∈ℕ₀𝑢min^𝑛+𝑟𝑤,𝑣_𝑛+𝑟𝑤,𝑤𝑢_𝑛+𝑟,𝑤𝑣_𝑛+𝑟𝑢>min^𝑛𝑤,𝑣_𝑛𝑤,𝑤𝑢_𝑛,𝑤𝑣_𝑛, (13) if and only if (𝑢_𝑛,𝑣_𝑛)≠(𝑤,𝑤).
To prove (13), it is enough to prove that 𝑢min^𝑛+𝑟𝑤,𝑤𝑢_𝑛+𝑟𝑢>min^𝑛𝑤,𝑤𝑢_𝑛, (14) if and only if 𝑢_𝑛≠𝑤, and 𝑣min^𝑛+𝑟𝑤,𝑤𝑣_𝑛+𝑟𝑣>min^𝑛𝑤,𝑤𝑣_𝑛, (15) if and only if 𝑣_𝑛≠𝑤.
The proofs of inequalities (14) and (15) are the same (up to the interchanging letters 𝑢 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 𝑢_𝑛=𝑢_𝑛+𝑟𝑤or𝑢_𝑛=𝑢_𝑛+𝑟𝑤, (16) from which, in both cases, it easily follows that 𝑢min^𝑛+𝑟𝑤,𝑤𝑢_𝑛+𝑟𝑢=min^𝑛𝑤,𝑤𝑢_𝑛. (17) 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 𝑢_𝑛=𝑤 for every 𝑛∈ℕ₀.
Now suppose that the first inequality in (12), is strict. Then, if 𝑢_𝑛>𝑢_𝑛+𝑟, directly follows that 𝑤/𝑢_𝑛+𝑟>𝑤/𝑢_𝑛, while from the first inequality in (12) it follows that 𝑢_𝑛+𝑟/𝑤>𝑤/𝑢_𝑛. Hence 𝑢min^𝑛+𝑟𝑤,𝑤𝑢_𝑛+𝑟>𝑤𝑢_𝑛, (18) from which inequality (14) easily follows.
If 𝑢_𝑛<𝑢_𝑛+𝑟, then 𝑢_𝑛+𝑟/𝑤>𝑢_𝑛/𝑤, while from the first inequality in (12), it follows that 𝑤/𝑢_𝑛+𝑟>𝑢_𝑛/𝑤. From these two inequalities, we have that 𝑢min^𝑛+𝑟𝑤,𝑤𝑢_𝑛+𝑟>𝑢_𝑛𝑤, (19) and consequently (14).
If (14) and (15) hold then if 𝑢_𝑛≠𝑤 and 𝑣_𝑛≠𝑤, inequality (13) immediately follows by using the following elementary implication: if 𝑎>𝑏 and 𝑐>𝑑, then min{𝑎,𝑐}>min{𝑏,𝑑}.
If 𝑢_𝑛≠𝑤 and 𝑣_𝑛=𝑤, then from the second inequality in (12), we have that 𝑣_𝑛+𝑟=𝑣_𝑛=𝑤. Hence 𝑣min^𝑛+𝑟𝑤,𝑤𝑣_𝑛+𝑟𝑣=min^𝑛𝑤,𝑤𝑣_𝑛𝑢=1>min^𝑛𝑤,𝑤𝑢_𝑛, (20) which along with (14) implies (13). The case 𝑢_𝑛=𝑤 and 𝑣_𝑛≠𝑤 directly follows from the case 𝑢_𝑛≠𝑤 and 𝑣_𝑛=𝑤, by the symmetry.
Finally, note that if 𝑢_𝑛=𝑣_𝑛=𝑤, then from (12), we have that 𝑢_𝑛+𝑟=𝑢_𝑛=𝑤 and 𝑣_𝑛+𝑟=𝑣_𝑛=𝑤, so that the first equality in (20) holds and 𝑢min^𝑛+𝑟𝑤,𝑤𝑢_𝑛+𝑟𝑢=min^𝑛𝑤,𝑤𝑢_𝑛=1, (21) from which it follows that both minima in (13) are equal, finishing the proof of the claim.
Now we define the map 𝑇∶ℝ₊^2𝑚→ℝ₊^2𝑚 as follows: 𝑇𝑥₁,𝑥₂,…,𝑥_𝑚−1,𝑥_𝑚,𝑦₁,𝑦₂,…,𝑦_𝑚−1,𝑦_𝑚=𝑥₂,…,𝑥_𝑚,𝑓𝑥₁,…,𝑥_𝑚,𝑦₁,…,𝑦_𝑚,𝑦₂,…,𝑦_𝑚,𝑔𝑥₁,…,𝑥_𝑚,𝑦₁,…,𝑦_𝑚. (22)
Then we get 𝑇𝑢_𝑛,𝑢_𝑛+1,…,𝑢_{𝑛+𝑚−2},𝑢_{𝑛+𝑚−1},𝑣_𝑛,𝑣_𝑛+1,…,𝑣_{𝑛+𝑚−2},𝑣_{𝑛+𝑚−1}=𝑢_𝑛+1,…,𝑢_{𝑛+𝑚−1},𝑢_𝑛+𝑚,𝑣_𝑛+1,…,𝑣_{𝑛+𝑚−1},𝑣_𝑛+𝑚, (23) and by induction 𝑇^𝑠𝑢_𝑛,𝑢_𝑛+1,…,𝑢_{𝑛+𝑚−2},𝑢_{𝑛+𝑚−1},𝑣_𝑛,𝑣_𝑛+1,…,𝑣_{𝑛+𝑚−2},𝑣_{𝑛+𝑚−1}=𝑢_𝑛+𝑠,…,𝑢_{𝑛+𝑚−2+𝑠},𝑢_{𝑛+𝑚−1+𝑠},𝑣_𝑛+𝑠,…,𝑣_{𝑛+𝑚−2+𝑠},𝑣_{𝑛+𝑚−1+𝑠}, (24) for every s∈ℕ.
By using inequality (13) and the fact that the inequalities 1≥𝑎_𝑖>𝑏_𝑖,𝑖∈𝐼⊆{1,…,𝑚}, 𝐼≠∅, along with equalities 𝑎_𝑖=𝑏_𝑖=1, 𝑖∈{1,…,𝑚}⧵𝐼, imply the inequality min_{1≤𝑖≤𝑚}𝑎_𝑖>min_{1≤𝑖≤𝑚}𝑏_𝑖, we have that for each vector 𝑥∈ℝ₊^2𝑚 such that 𝑥≠(𝑤,𝑤,…,𝑤)=∶𝑤∈ℝ₊^2𝑚, 𝑝𝑇^𝑟(𝑥),𝑤𝑢=−logmin^𝑛+𝑟𝑤,𝑤𝑢_𝑛+𝑟𝑢,…,^{𝑛+𝑟+𝑚−1}𝑤,𝑤𝑢_{𝑛+𝑟+𝑚−1},𝑣_𝑛+𝑟𝑤,𝑤𝑣_𝑛+𝑟𝑣,…,_{𝑛+𝑟+𝑚−1}𝑤,𝑤𝑣_{𝑛+𝑟+𝑚−1}𝑢<−logmin^𝑛𝑤,𝑤𝑢_𝑛𝑢,…,^{𝑛+𝑚−1}𝑤,𝑤𝑢_{𝑛+𝑚−1},𝑣_𝑛𝑤,𝑤𝑣_𝑛𝑣,…,^{𝑛+𝑚−1}𝑤,𝑤𝑣_{𝑛+𝑚−1}=𝑝(𝑥,𝑤), (25) 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 𝑓_𝑖∶ℝ₊^𝑙𝑚→ℝ₊,𝑖=1,…,𝑙, be continuous functions. Suppose that the system of difference equations 𝑢⁽¹⁾_𝑛+𝑚=𝑓₁𝑢_𝑛⁽¹⁾,𝑢⁽¹⁾_𝑛+1,…,𝑢⁽¹⁾_{𝑛+𝑚−1},…,𝑢_𝑛^(𝑙),𝑢^(𝑙)_𝑛+1,…,𝑢^(𝑙)_{𝑛+𝑚−1},⋮𝑢^(𝑖)_𝑛+𝑚=𝑓_𝑖𝑢_𝑛⁽¹⁾,𝑢⁽¹⁾_𝑛+1,…,𝑢⁽¹⁾_{𝑛+𝑚−1},…,𝑢_𝑛^(𝑙),𝑢^(𝑙)_𝑛+1,…,𝑢^(𝑙)_{𝑛+𝑚−1},⋮𝑢^(𝑙)_𝑛+𝑚=𝑓_𝑙𝑢_𝑛⁽¹⁾,𝑢⁽¹⁾_𝑛+1,…,𝑢⁽¹⁾_{𝑛+𝑚−1},…,𝑢_𝑛^(𝑙),𝑢^(𝑙)_𝑛+1,…,𝑢^(𝑙)_{𝑛+𝑚−1} (26) has a unique positive equilibrium (𝑤,…,𝑤)∈ℝ^𝑙₊, and that there is an 𝑟∈ℕ such that for any solution (𝑢_𝑛⁽¹⁾,…,𝑢^(l)_𝑛)_𝑛∈ℕ0⊂ℝ^𝑙₊ of system (26), the following inequalities: 𝑢_𝑛^(𝑖)−𝑢^(𝑖)_𝑛+𝑟𝑤²𝑢_𝑛^(𝑖)−𝑢^(𝑖)_𝑛+𝑟≤0,𝑛∈ℕ₀,𝑖=1,…,𝑙, (27) hold, with the equalities if and only if 𝑢_𝑛^(𝑖)=𝑤, for every 𝑛∈ℕ₀, and 𝑖=1,…,𝑙. Then the equilibrium (𝑤,…,𝑤) is globally asymptotically stable.

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

Corollary 4. Let 𝑘,𝑙∈ℕ₀,𝑘≠𝑙. Consider the system 𝑥_𝑛+1=𝑥_𝑛−𝑘𝑦_𝑛−𝑙+1𝑥_𝑛−𝑘+𝑦_𝑛−𝑙,𝑦_𝑛+1=𝑦_𝑛−𝑘𝑥_𝑛−𝑙+1𝑦_𝑛−𝑘+𝑥_𝑛−𝑙,𝑛∈ℕ₀. (28) Then the positive equilibrium point (𝑥,𝑦)=(1,1) of system (28) is globally asymptotically stable with respect to the set ℝ^𝑚₊×ℝ^𝑚₊, where 𝑚=max{𝑘,𝑙}.

Proof. We may assume that 𝑚=𝑘. From system (28), we have that 𝑥_𝑛+1−𝑥_𝑛−𝑘=1−𝑥²_𝑛−𝑘𝑦_𝑛−𝑙+𝑥_𝑛−𝑘,𝑥_𝑛+1−1𝑥_𝑛−𝑘=𝑦_𝑛−𝑙𝑥²_𝑛−𝑘−1𝑥_𝑛−𝑘𝑦_𝑛−𝑙+𝑥_𝑛−𝑘,𝑦_𝑛+1−𝑦_𝑛−𝑘=1−𝑦²_𝑛−𝑘𝑥_𝑛−𝑙+𝑦_𝑛−𝑘,𝑦_𝑛+1−1𝑦_𝑛−𝑘=𝑥_𝑛−𝑙𝑦²_𝑛−𝑘−1𝑦_𝑛−𝑘𝑥_𝑛−𝑙+𝑦_𝑛−𝑘 (29)(30) from which it follows that 𝑥_𝑛+1−𝑥_𝑛−𝑘𝑥_𝑛+1−1𝑥_𝑛−𝑘𝑦≤0,_𝑛+1−𝑦_𝑛−𝑘𝑦_𝑛+1−1𝑦_𝑛−𝑘≤0 (31) so that condition (12) in Proposition 2 is fulfilled with 𝑟=𝑘+1.
Clearly if 𝑥_𝑛=1=𝑦_𝑛forevery𝑛≥−max{𝑙,𝑘}, (32) then in (31) equalities follow. On the other hand, if equality holds in the first inequality in (31), we have that 𝑥_𝑛+1=𝑥_𝑛−𝑘or𝑥_𝑛+1=1𝑥_𝑛−𝑘. (33) If 𝑥_𝑛+1=𝑥_𝑛−𝑘, then from the first equality in (29) we have that 𝑥_𝑛−𝑘=1, while if 𝑥_𝑛+1=1/𝑥_𝑛−𝑘, then from the second equality in (29), we have that 𝑥_𝑛−𝑘=1.
By symmetry (see (30)), we have that if equality holds in the second inequality in (31), then 𝑦_𝑛−𝑘=1. Therefore, equalities in (31) hold if and only if (𝑥_𝑛−𝑘,𝑦_𝑛−𝑘)=(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^𝑚₊×ℝ^𝑚₊.

Remark 5. Corollary 4 extends and gives a very short proof of the main result in [23], which is obtained for 𝑘=1 and 𝑙=0. Further, it also extends and gives a very short proof of the main result in [25], which is obtained for 𝑘=0 and 𝑙=1. Moreover, it confirms the conjecture in [25], which is obtained for 𝑘=0 and 𝑙∈ℕ⧵{1}.

Corollary 6. Let 𝑘,𝑙∈ℕ₀,𝑘≠𝑙. Consider the system 𝑥_𝑛+1=𝑥_𝑛−𝑘+𝑦_𝑛−𝑙𝑥_𝑛−𝑘𝑦_𝑛−𝑙,𝑦+1_𝑛+1=𝑦_𝑛−𝑘+𝑥_𝑛−𝑙𝑦_𝑛−𝑘𝑥_𝑛−𝑙+1,𝑛∈ℕ₀. (34) Then the positive equilibrium point (𝑥,𝑦)=(1,1) of system (34) is globally asymptotically stable with respect to the set ℝ^m₊×ℝ^𝑚₊, where 𝑚=max{𝑘,𝑙}.

Proof. We may assume that 𝑚=𝑘. From system (34), we have that 𝑥_𝑛+1−𝑥_𝑛−𝑘=𝑦_𝑛−𝑙1−𝑥²_𝑛−𝑘𝑥_𝑛−𝑘𝑦_𝑛−𝑙,𝑥+1_𝑛+1−1𝑥_𝑛−𝑘=𝑥²_𝑛−𝑘−1𝑥_𝑛−𝑘𝑥_𝑛−𝑘𝑦_𝑛−𝑙,𝑦+1_𝑛+1−𝑦_𝑛−𝑘=𝑥_𝑛−𝑙1−𝑦²_𝑛−𝑘𝑥_𝑛−𝑙𝑦_𝑛−𝑘,𝑦+1_𝑛+1−1𝑦_𝑛−𝑘=𝑦²_𝑛−𝑘−1𝑦_𝑛−𝑘𝑥_𝑛−𝑙𝑦_𝑛−𝑘+1 (35) from which it follows that 𝑥_𝑛+1−𝑥_𝑛−𝑘𝑥_𝑛+1−1𝑥_𝑛−𝑘𝑦≤0,_𝑛+1−𝑦_𝑛−𝑘𝑦_𝑛+1−1𝑦_𝑛−𝑘≤0. (36) Hence condition (12) in Proposition 2 is fulfilled with 𝑟=𝑘+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 (𝑥_𝑛−𝑘,𝑦_𝑛−𝑘)=(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^𝑚₊×ℝ^𝑚₊.

Remark 7. Corollary 6 extends and gives a very short proof of the main result in [24], which is obtained for 𝑘=1 and 𝑙=0.

Remark 8. Corollary 6 is also a consequence of Corollary 4. Namely, by using the change of variables (𝑥_𝑛,𝑦_𝑛)=(1/𝑢_𝑛,1/𝑣_𝑛), system (34) is transformed into the system 𝑣_𝑛+1=𝑣_𝑛−𝑘𝑢_𝑛−𝑙+1𝑣_𝑛−𝑘+𝑢_𝑛−𝑙,𝑢_𝑛+1=𝑢_𝑛−𝑘𝑣_𝑛−𝑙+1𝑢_𝑛−𝑘+𝑣_𝑛−𝑙,𝑛∈ℕ₀, (37) which is system (28). In particular, this shows that systems (1) and (2), for the case 𝑎=1, 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 𝑓∈𝐶(ℝ^𝑘₊,ℝ₊) and 𝑔∈𝐶(ℝ^𝑙₊,ℝ₊) with 𝑘,𝑙∈ℕ,0≤𝑟₁<⋯<𝑟_𝑘 and 0≤𝑚₁<⋯<𝑚_𝑙≤𝑟_𝑘 and satisfy the following two conditions:(H1)[𝑓(𝑢₁,𝑢₂,…,𝑢_𝑘)]^∗=𝑓(𝑢^∗₁,𝑢^∗₂,…,𝑢^∗_𝑘),(H2)𝑓(𝑢^∗₁,𝑢^∗₂,…,𝑢^∗_𝑘)≤𝑢^∗_𝑘,where 𝑎^∗∶=max{𝑎,1/a}.
Then (𝑥,𝑦)=(1,1) is the unique positive equilibrium of the system of difference equations 𝑥_𝑛=𝑓𝑥_{𝑛−𝑟1−1},…,𝑥_{𝑛−𝑟𝑘−1}𝑔𝑦_{𝑛−𝑚1−1},…,𝑦_{𝑛−𝑚𝑙−1}+1𝑓𝑥_{𝑛−𝑟1−1},…,𝑥_{𝑛−𝑟𝑘−1}+𝑔𝑦_{𝑛−𝑚1−1},…,𝑦_{𝑛−𝑚𝑙−1},𝑦𝑛∈ℕ,_𝑛=𝑓𝑦_{𝑛−𝑟1−1},…,𝑦_{𝑛−𝑟𝑘−1}𝑔𝑥_{𝑛−𝑚1−1},…,𝑥_{𝑛−𝑚𝑙−1}+1𝑓𝑦_{𝑛−𝑟1−1},…,𝑦_{𝑛−𝑟𝑘−1}+𝑔𝑥_{𝑛−𝑚1−1},…,𝑥_{𝑛−𝑚𝑙−1},𝑛∈ℕ, (38) and it is globally asymptotically stable.

Proof. Let 𝑓_𝑛=𝑓𝑥_{𝑛−𝑟1−1},…,𝑥_{𝑛−𝑟𝑘−1},𝑔_𝑛=𝑔𝑦_{𝑛−𝑚1−1},…,𝑦_{𝑛−𝑚𝑙−1}. (39) We should determine the sign of the product of the following expressions: 𝑃_𝑛𝑓∶=^𝑛𝑔_𝑛+1𝑓_𝑛+𝑔_𝑛−𝑥_{𝑛−𝑟𝑘−1}=1𝑓_𝑛+𝑔_𝑛𝑓_𝑛𝑔_𝑛𝑥1−^{𝑛−𝑟𝑘−1}𝑓_𝑛+1−𝑥_{𝑛−𝑟𝑘−1}𝑓_𝑛,𝑄_𝑛𝑓∶=^𝑛𝑔_𝑛+1𝑓_𝑛+𝑔_𝑛−1𝑥_{𝑛−𝑟𝑘−1}=1𝑥_{𝑛−𝑟𝑘−1}𝑓_𝑛+𝑔_𝑛𝑔_𝑛𝑥_{𝑛−𝑟𝑘−1}𝑓_𝑛−1+𝑓_𝑛𝑥_{𝑛−𝑟𝑘−1}𝑓_𝑛−1. (40)(41)
From (40) and (41), we see if we show that 𝑥_{𝑛−𝑟𝑘−1}𝑓_𝑛−1 and (𝑥_{𝑛−𝑟𝑘−1}/𝑓_𝑛)−1 have the same sign for 𝑛∈ℕ, then 𝑃_𝑛𝑄_𝑛 will be nonpositive.
We consider four cases.
Case  1. 𝑥_{𝑛−𝑟𝑘−1}≥1, 𝑓_𝑛≥1. Clearly in this case 𝑥_{𝑛−𝑟𝑘−1}𝑓_𝑛−1≥0. By (H1) and (H2), we have that 1≤𝑓_𝑛=𝑓_𝑛^∗=𝑓𝑥^∗_{𝑛−𝑟1−1},…,𝑥^∗_{𝑛−𝑟𝑘−1}≤𝑥^∗_{𝑛−𝑟𝑘−1}=𝑥_{𝑛−𝑟𝑘−1}. (42)
Hence (𝑥_{𝑛−𝑟𝑘−1}/𝑓_𝑛)−1≥0 and consequently 𝑥_{𝑛−𝑟𝑘−1}𝑓_𝑛𝑥−1^{𝑛−𝑟𝑘−1}𝑓_𝑛−1≥0. (43)
Case  2. 𝑥_{𝑛−𝑟𝑘−1}≥1, 𝑓_𝑛≤1. Since 1/𝑓_𝑛≥1, we obtain (𝑥_{𝑛−𝑟𝑘−1}/𝑓_𝑛)−1≥0. On the other hand, by (H1) and (H2), we have 1𝑓_𝑛=𝑓_𝑛^∗=𝑓𝑥^∗_{𝑛−𝑟1−1},…,𝑥^∗_{𝑛−𝑟𝑘−1}≤𝑥^∗_{𝑛−𝑟𝑘−1}=𝑥_{𝑛−𝑟𝑘−1}, (44) so that 𝑥_{𝑛−𝑟𝑘−1}𝑓_𝑛−1≥0. Hence (43) follows in this case.
Case  3. Case 𝑥_{𝑛−𝑟𝑘−1}≤1, 𝑓_𝑛≥1. Then we have that 1/𝑓_𝑛≤1 and consequently (𝑥_{𝑛−𝑟𝑘−1}/𝑓_𝑛)−1≤0. On the other hand, we have 𝑓_𝑛=𝑓_𝑛^∗=𝑓𝑥^∗_{𝑛−𝑟1−1},…,𝑥^∗_{𝑛−𝑟𝑘−1}≤𝑥^∗_{𝑛−𝑟𝑘−1}=1𝑥_{𝑛−𝑟𝑘−1}, (45) so that 𝑥_{𝑛−𝑟𝑘−1}𝑓_𝑛−1≤0. Hence (43) follows in this case too.
Case  4. Case 𝑥_{𝑛−𝑟𝑘−1}≤1, 𝑓_𝑛≤1. Then 𝑥_{𝑛−𝑟𝑘−1}𝑓_𝑛−1≤0. On the other hand, we have 1𝑓_𝑛=𝑓_𝑛^∗=𝑓𝑥^∗_{𝑛−𝑟1−1},…,𝑥^∗_{𝑛−𝑟𝑘−1}≤𝑥^∗_{𝑛−𝑟𝑘−1}=1𝑥_{𝑛−𝑟𝑘−1}, (46) so that (𝑥_{𝑛−𝑟𝑘−1}/𝑓_𝑛)−1≤0. Hence (43) also holds in this case. Thus 𝑃_𝑛𝑄_𝑛≤0, for every 𝑛∈ℕ.
Assume that 𝑃_𝑛𝑄_𝑛=0, then 𝑃_𝑛=0 or 𝑄_𝑛=0. Using (40) or (41) along with (43) in any of these two cases, we have that 𝑓_𝑛=1𝑥_{𝑛−𝑟𝑘−1}=𝑥_{𝑛−𝑟𝑘−1},𝑛∈ℕ. (47) Hence 𝑥_{𝑛−𝑟𝑘−1}=1,𝑛∈ℕ.
Let 𝑓_𝑛=𝑓𝑦_{𝑛−𝑟1−1},…,𝑦_{𝑛−𝑟𝑘−1},𝑔_𝑛=𝑔𝑥_{𝑛−𝑚1−1},…,𝑥_{𝑛−𝑚𝑙−1}. (48) Using the following expressions: 𝑃_𝑛𝑓∶=^𝑛𝑔_𝑛+1𝑓_𝑛+𝑔_𝑛−𝑦_{𝑛−𝑟𝑘−1}=1𝑓_𝑛+𝑔_𝑛𝑓_𝑛𝑔_𝑛𝑦1−^{𝑛−𝑟𝑘−1}𝑓_𝑛+1−𝑦_{𝑛−𝑟𝑘−1}𝑓_𝑛,𝑄_𝑛𝑓∶=^𝑛𝑔_𝑛+1𝑓_𝑛+𝑔_𝑛−1𝑦_{𝑛−𝑟𝑘−1}=1𝑦_{𝑛−𝑟𝑘−1}𝑓_𝑛+𝑔_𝑛𝑔_𝑛𝑦_{𝑛−𝑟𝑘−1}𝑓_𝑛+𝑓−1_𝑛𝑦_{𝑛−𝑟𝑘−1}𝑓_𝑛−1, (49) it can be proved similarly that 𝑃_𝑛𝑄_𝑛≤0, for every 𝑛∈ℕ, and that 𝑃_𝑛𝑄_𝑛=0, if and only if 𝑦_{𝑛−𝑟𝑘−1}=1,𝑛∈ℕ.
Finally, let (𝑥^∗,𝑦^∗) be a solution of the system 𝑥^∗=𝑓𝑥^∗_𝑘𝑔𝑦^∗_𝑙+1𝑓𝑥^∗_𝑘+𝑔𝑦^∗_𝑙,𝑦^∗=𝑓𝑦^∗_𝑘𝑔𝑥^∗_𝑙+1𝑓𝑦^∗_𝑘+𝑔𝑥^∗_𝑙. (50)
Then we have that 𝑓𝑥0=^∗𝑘𝑔𝑦^∗_𝑙+1𝑓𝑥^∗_𝑘+𝑔𝑦^∗_𝑙−𝑥^∗=1𝑓𝑥^∗_𝑘+𝑔𝑦^∗_𝑙𝑓𝑥^∗_𝑘𝑔𝑦^∗_𝑙𝑥1−^∗𝑓𝑥^∗_𝑘+1−𝑥^∗𝑓𝑥^∗_𝑘, (51) where 𝑧^∗_𝑗=(𝑧^∗,…,𝑧^∗) denotes the vector consisting of 𝑗 copies of 𝑧^∗. Then similar to the considerations in Cases (i)–(iv), it follows that 𝑓(𝑥^∗_𝑘)=𝑥^∗=1/𝑥^∗, so that 𝑥^∗=1, and similarly it is obtained that 𝑦^∗=1. Hence (𝑥^∗,𝑦^∗)=(1,1) 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.