# On Two Systems of Difference Equations

Authors: Bratislav Iričanin; Stevo Stević
Journal: Discrete Dynamics in Nature and Society
 (2010)

---

## Abstract

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


---

## Body

## 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:(1.1)xn+1(i)=xn(i+1(modk))xn(i+1(modk))-1,i=1,…,k,n∈ℕ0,(1.2)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.
Assumek∈ℕ, then the following statements are true.(a)
Ifk=0(mod2), then every (well-defined) solution of systems (1.1) and (1.2) is periodic with period k.(b)
Ifk=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 changeyn(i)=xn(i)-1, i=1,…,k, system (1.1) becomes
(1.3)yn(i)=(yn-1(i+1(modk)))-1,i=1,…,k,n∈ℕ,
while system (1.2) becomes
(1.4)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
(1.5)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
(1.6)yn(i)=yn-k(i),i=1,…,k,
from which the statement in (a) easily follows.
Ifk=1(mod2), we have that
(1.7)yn(i)=(yn-k(i))-1,i=1,…,k,
from which it follows that
(1.8)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. Letgcd(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 thatf 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
(2.1)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
(2.2)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
(2.3)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(2.4)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
(2.5)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 thatf 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
(2.6)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 Theorems2.1 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].



---

*Source: 405121-2010-03-24.xml*