On Some Symmetric Systems of Difference Equations

Theorem A. Let (M, d) be a complete metric space, where d denotes a metric and M is an open subset of ℝⁿ, and let T : M → M be a continuous mapping with the unique equilibrium x^* ∈ M. Suppose that for the discrete dynamic system

there is a k ∈ ℕ such that for the kth iterate of T, the following inequality holds: for all x ≠ x^*. Then x^* is globally asymptotically stable with respect to metric d.

Corollary 1. Let $$ be a continuous mapping with a unique equilibrium $$. Suppose that for the discrete dynamic system (7), there is some k ∈ ℕ such that for the part-metric p inequality

holds for all x ≠ x^*. Then x^* is globally asymptotically stable.

Proposition 2. Let $$, $$ be continuous functions. We suppose that the system of two difference equations,

has a unique positive equilibrium (w, w). Suppose also that there is an r ∈ ℕ such that for any positive solution $$ of system (11), the following inequalities: 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 ∈ ℕ₀

if and only if (u_n, v_n)≠(w, w).

To prove (13), it is enough to prove that

if and only if u_n ≠ w, and 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

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 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

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

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

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

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 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 $$ such that $$,

from which the proof follows by Corollary 1.

Proposition 3. Let $$, i = 1, …, l, be continuous functions. Suppose that the system of difference equations

has a unique positive equilibrium $$, and that there is an r ∈ ℕ such that for any solution $$ of system (26), the following inequalities: hold, with the equalities if and only if $$, for every n ∈ ℕ₀, and i = 1, …, l. Then the equilibrium (w, …, w) is globally asymptotically stable.

Corollary 4. Let k, l ∈ ℕ₀,  k ≠ l. Consider the system

Then the positive equilibrium point $$ of system (28) is globally asymptotically stable with respect to the set $$, where m = max {k, l}.

Proof. We may assume that m = k. From system (28), we have that

from which it follows that so that condition (12) in Proposition 2 is fulfilled with r = k + 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 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 $$.

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

Then the positive equilibrium point $$ of system (34) is globally asymptotically stable with respect to the set $$, where m = max {k, l}.

Proof. We may assume that m = k. From system (34), we have that

from which it follows that 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 $$.

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

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 $$ and $$ with k, l ∈ ℕ,  0 ≤ r₁ < ⋯<r_k and 0 ≤ m₁ < ⋯<m_l ≤ r_k and satisfy the following two conditions:

• (H1)

  $$,

• (H2)

  $$,

where a^* : = max {a, 1/a}.

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 n ∈ ℕ, then P_nQ_n will be nonpositive.

We consider four cases.

Case  1. $$, f_n ≥ 1. Clearly in this case $$. By (H1) and (H2), we have that

Hence $$ and consequently

Case  2. $$, f_n ≤ 1. Since 1/f_n ≥ 1, we obtain $$. On the other hand, by (H1) and (H2), we have

so that $$. Hence (43) follows in this case.

Case  3. Case $$, f_n ≥ 1. Then we have that 1/f_n ≤ 1 and consequently $$. On the other hand, we have

so that $$. Hence (43) follows in this case too.

Case  4. Case $$, f_n ≤ 1. Then $$. On the other hand, we have

so that $$. 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

Hence $$,  n ∈ ℕ.

Let

Using the following expressions: it can be proved similarly that $$, for every n ∈ ℕ, and that $$, if and only if $$,  n ∈ ℕ.

Finally, let (x^*, y^*) be a solution of the system

Then we have that

where $$ denotes the vector consisting of j copies of z^*. Then similar to the considerations in Cases (i)–(iv), it follows that $$, 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.