We give very short and elegant proofs of the main results in the work of Yalcinkaya et al. (2008).

1. Introduction and a Proof of Some Resent Results

Motivated by our paper [1], the authors of [2] studied the following two systems of difference equations:

xn+1(i)=xn(i+1(modk))xn(i+1(modk))-1,i=1,…,k,n∈ℕ0,xn+1(i)=xn(i-1(modk))xn(i-1(modk))-1,i=1,…,k,n∈ℕ0,
where we regard that 0(modk)=k(modk)=k.

Following line by line the proofs of the main results in [1] they proved the following result (see Theorems 2.1 and 2.4 in [2])

Theorem 1 A.

Assume k∈ℕ, then the following statements are true.

If k=0(mod2), then every (well-defined) solution of systems (1.1) and (1.2) is periodic with period k.

If k=1(mod2), then every (well-defined) solution of systems (1.1) and (1.2) is periodic with period 2k.

Here we give a very short and elegant proof of Theorem A.

Proof of Theorem A.

By using the change yn(i)=xn(i)-1, i=1,…,k, system (1.1) becomes
yn(i)=(yn-1(i+1(modk)))-1,i=1,…,k,n∈ℕ,
while system (1.2) becomes
yn(i)=(yn-1(i-1(modk)))-1,i=1,…,k,n∈ℕ.

From (1.3) and (1.4), for each i∈{1,…,k}, and n≥k, we obtain correspondingly that
yn(i)=(yn-k(i+1+k-1(modk))(-1)k=(yn-k(i))(-1)k,yn(i)=(yn-k(i-1-(k-1)(modk))(-1)k=(yn-k(i))(-1)k.
From (1.5), with k=0(mod2), it follows that
yn(i)=yn-k(i),i=1,…,k,
from which the statement in (a) easily follows.

If k=1(mod2), we have that
yn(i)=(yn-k(i))-1,i=1,…,k,
from which it follows that
yn(i)=yn-2k(i),i=1,…,k,n≥2k, implying the statement in (b), as desired.

2. An Extension on Theorem A

Here we extend Theorem A in a natural way. Let gcd(k,l) denote the greatest common divisor of the integers k and l, lcm(k,l) the least common multiple of k and l, and for r∈ℕ let f[r](x)=f(f[r-1](x)), where f[1](x)=f(x).

Theorem 2.1.

Assume that f is a real function such that f[r](x)≡x on its domain of definition, for some r∈ℕ, then all well-defined solutions of the system of difference equations
xn(1)=f(xn-1(2)),xn(2)=f(xn-1(3)),…,xn(k)=f(xn-1(1)),n∈ℕ0,
are periodic with period T=lcm(k,r).

Proof.

We use our method of “prolongation" described in [1]. Note that for each s∈ℕ, system (2.1) is equivalent to a system of ks difference equations of the same form, where
xn(i)=xn(jk+i),
for every n∈ℕ0, i∈{1,…,k} and j=1,…,s.

From (2.1) and since f[r](x)≡x, for n≥r-1 we have
xn(i+1)=f(xn-1(i+2))=f[2](xn-2(i+3))=⋯=f[r](xn-r(i+r+1))=xn-r(i+r+1).
for each i∈{1,2,…,k}, and every n≥r-1.

It is clear that
T=k·r1=k1·r,
where r1,k1∈ℕ are such that gcd(k,r1)=1 and gcd(k1,r)=1.

From (2.3) we have
xn(i+1)=xn-r(i+r+1)=⋯=xn-k1r(i+k1r+1)=xn-kr1(i+kr1+1)=xn-T(i+1),
for each i=0,1,…,k-1, and n≥T-1, from which the result follows.

The following result is proved similarly. Hence we omit its proof.

Theorem 2.2.

Assume that f is a real function such that f[r](x)≡x on its domain of definition, for some r∈ℕ, then all well-defined solutions of the system of difference equations
xn(2)=f(xn-1(1)),…,xn(k)=f(xn-1(k-1)),xn(1)=f(xn-1(k)),n∈ℕ0,
are periodic with period T=lcm(k,r).

Remark 2.3.

The proof of Theorem A follows from Theorems 2.1 and 2.2. Indeed, note that the function f(x)=x/(x-1) satisfies the condition f[2](x)≡x on its domain of definition. By Theorems 2.1 and 2.2 we know that all well-defined solutions of systems (1.1) and (1.2) are periodic with period T=lcm(k,2), from which the result follows.

Remark 2.4.

We also have to say that the main result in [3] is a trivial consequence of a result in [1] (see Remark 5 therein). Just note that the simple change of variables xn(i)=ayn(i), i∈{1,…,k}, transforms their system (1.3) satisfying conditions a1=a2=⋯=ak=a and b1=b2=⋯=bk=b=a2, into system (4) in [1].

Acknowledgment

The results in this note were presented at the talk: S. Stević, on a class of max-type difference equations and some of our old results, Progress on Difference Equations 2009, Bedlewo, Poland, May 25–29, 2009.

IričaninB. D.StevićS.Some systems of nonlinear difference equations of higher order with periodic solutionsYalçinkayaİ.ÇinarC.AtalayM.On the solutions of systems of difference equationsPapaschinopoulosG.SchinasC. J.StefanidouG.On a k-order system of Lyness-type difference equations