In this section, we formulate and prove the main results in this paper.
Theorem 2.1.
Consider (1.9) where
Then the following statements are true:
(a)if
, then every solution of (1.9) is periodic with period 2k,
(b)if
, then every solution of (1.9) is periodic with period k.
Proof.
First note that the system is cyclic. Hence it is enough to prove that the sequence
satisfies conditions (a) and (b) in the corresponding cases.
Further, note that for every
system (1.9) is equivalent to a system of
difference equations of the same form, where
On the other hand, we have
(a)Let
for
be the rests mentioned in Remark 1.2. Then from (2.2) and Lemma 1.1 we obtain that
Using (2.1) for sufficently large
we obtain that (2.3) is equivalent to (here we use the condition
)
From this and since by Lemma 1.1 the numbers
are pairwise different, the result follows in this case.
(b)Let
for some
By (2.1) we have
which yields the result.
Remark 2.2.
In order to make the proof of Theorem 2.1 clear to the reader, we explain what happens in the cases
and
.
For
, system (1.9) is equivalent to the system
where we consider that
From this and (2.2), we have
Using again (2.2), we get
which means that the sequence
is periodic with period equal to 2. If
, system (1.9) is equivalent to system (2.6) where we consider that
, and
Using this and (2.2) subsequently, it follows that
that is, the sequence
is periodic with period 6.
Remark 2.3.
The fact that every solution of (1.8) is periodic with period two can be considered as the case
in Theorem 2.1, that is, we can take that
Similarly to Theorem 2.1, using Lemma 1.1 with
for
the following theorem can be proved.
Theorem 2.4.
Consider (1.10) where
Then the following statements are true:
(a)if
, then every solution of (1.10) is periodic with period 2k,
(b)if
, then every solution of (1.10) is periodic with period k.
Proof.
First note that the system is cyclic. Hence, it is enough to prove that the sequence
satisfies conditions (a) and (b) in the corresponding cases.
Indeed, similarly to (2.2), we have
(a)Let
for
be the rests mentioned in Remark 1.2. Then from (2.11) and Lemma 1.1 we obtain that
Using (2.1) for sufficiently large
we obtain that (2.12) is equivalent to (here we use the condition
)
From this and since by Lemma 1.1 the numbers
are pairwise different, the result follows in this case.
(b)Let
for some
By (2.1) we have
which yields the result.
Corollary 2.5.
Let
be solutions of (1.9) with the initial values
Assume that
then all solutions of (1.9) are positive.
Proof.
We consider solutions of (1.9) with the initial values
satisfying (2.15). If
, then from (1.9) and (2.15), we have
for
and
.
If
, then from (1.9) and (2.15), we have
for
and
.
From (2.16) and (2.17), all solutions of (1.9) are positive.
Corollary 2.6.
Let
be solutions of (1.9) with the initial values
. Assume that
then
are positive,
are negative for all 
Proof.
From (2.16), (2.17), and (2.18), the proof is clear.
Corollary 2.7.
Let
be solutions of (1.9) with the initial values
. Assume that
then
are negative,
are positive for all 
Proof.
From (2.16), (2.17), and (2.19), the proof is clear.
Corollary 2.8.
Let
be solutions of (1.9) with the initial values
, then the following statements are true (for all
and 
(i)if
then
and
,
(ii)if
then
and 
(iii)if
then
and 
(iv)if
then
and
,
(v)if
then
and
,
(vi)if
then
and
.
Proof.
From (2.16) and (2.17), the proof is clear.
Corollary 2.9.
Let
be solutions of (1.10) with the initial values
. Assume that
then all solutions of (1.10) are positive.
Proof.
We consider solutions of (1.10) with the initial values
satisfying (2.20). If
, then from (1.10) and (2.20), we have
for
and
.
If
, then from (1.10) and (2.20), we have
for
and
.
From (2.21) and (2.22), all solutions of (1.10) are positive.
Corollary 2.10.
Let
be solutions of (1.10) with the initial values
. Assume that
then
are positive,
are negative for all 
Proof.
From (2.21), (2.22) and (2.23), the proof is clear.
Corollary 2.11.
Let
be solutions of (1.10) with the initial values
. Assume that
then
are negative,
are positive for all 
Proof.
From (2.21), (2.22) and (2.24), the proof is clear.
Corollary 2.12.
Let
be solutions of (1.10) with the initial values
, then following statements are true (for all
and 
(i)if
then
and
,
(ii)if
then
and 
(iii)if
then
and 
(iv)if
then
and 
(v)if
then
and 
(vi)if
then
and 
Proof.
From (2.21), (2.22), and (2.24), the proof is clear.
Example 2.13.
Let
. Then the solutions of (1.9), with the initial values
and
in its invertal of periodicity can be represented by Table 1.