#### 2.1. Existence of Periodic Solutions of Period Two

Elsayed in [

12] and Moaaz in [

24] are established a new technique to study the existence of periodic solutions of some rational difference equation. In the following, we describe and modify this method:

Consider the difference equation

where

k is positive integer. Now, we assume that Equation (1) has periodic solutions of period two

with

${w}_{n-\left(2s+1\right)}=\rho $ and

${w}_{n-2s}=\sigma $. Hence, we get that

Next, we let

$\tau =\rho /\sigma $, and substitute into (2). Then, we get that

By using the fact

$\rho -\tau \sigma =0$, we obtain

Finally, by using the relation (3), we can obtain—in most cases—the necessary and sufficient conditions that Equation (1) has periodic solutions of the prime period two.

The effectiveness of this method appears in a study the existence of periodic solutions of some difference equations with real coefficients and initial conditions (not positive only). Besides, we can study the existence of periodic solutions of some difference equations, which have never been done before due to failure while applying the usual method.

Next, we apply the new method to study the existence of periodic solutions of general equations

where

a is positive real number,

${w}_{-1},$ ${w}_{0}$ are positive real numbers and

$\Phi \left(u,v\right)$ is a homothetic function, that is there exist a strictly increasing function

$G:\mathbb{R}\to \mathbb{R}$ and a homogenous function

$H:{\mathbb{R}}^{2}\to \mathbb{R}$ with degree

$\beta ,$ such that

$\Phi =G\left(H\right)$.

**Remark** **1.** In the following proofs, we use induction to prove the relationships. We’ll only take care of the basic step of induction and the rest of the steps directly, so it was ignored.

**Theorem** **1.** Assume that β is a ratios of odd positive integers and ${G}^{-1}\left(1/a\right)$ exists. Equation (4) has a prime period two solution $\dots ,\rho ,\sigma ,\rho ,\sigma ,\dots $ if and only ifwhere $\tau =\rho /\sigma $ and $A={G}^{-1}\left(1/a\right)$. **Proof.** We suppose that Equation (4) has a prime period two solution

By dividing (8) by (7), we have that (5) holds.

On the other hand, let (5) holds. If we choose

for

$\tau \in {\mathbb{R}}^{+}$, then we get

Similarly, we have that

${w}_{2}={w}_{0}$. Hence, it is followed by the induction that

Therefore, Equation (4) has a prime period two solution, and the proof is complete. □

Consider the recursive sequence

where the function

$f\left(u,v\right):{\left(0,\infty \right)}^{2}\to \left(0,\infty \right)$ is continuous real function and homogenous with degree

$zero$.

**Theorem** **2.** Assume that l odd, k even. Equation (9) has a prime period two solution $\dots ,\rho ,\sigma ,\rho ,\sigma ,\dots $ if and only ifwhere $\tau =\rho /\sigma $. **Proof.** Assume that

$l>k$. Since

l odd and

k even, we have

${w}_{n-l}=\rho $ and

${w}_{n-k}=\sigma $. From Equation (9), we get

Since

$\tau =\rho /\sigma $, we obtain

On the other hand, let (10) holds. Now, we choose

where

$\tau \in {\mathbb{R}}^{+}$. Hence, we see that

Similarly, we can proof that

${w}_{2}=f\left(1,\tau \right)$. Hence, it is followed by the induction that

Therefore, Equation (9) has a prime period two solution, and the proof is complete. □

**Theorem** **3.** Assume that l even, k odd. Equation (9) has a prime period two solution $\dots ,\rho ,\sigma ,\rho ,\sigma ,\dots $ if and only ifwhere $\tau =\rho /\sigma $. **Proof.** The proof is similar to that of proof of Theorem 2 and hence is omitted. □

Consider the difference equation

where

$\beta $ is a positive real number,

$\gamma ,$ $\delta ,$ ${w}_{-1}$ and

${w}_{0}$ are arbitrary real numbers and the function

$g\left(u,v\right)$ is continuous real function and homogenous with degree

$\beta $**Theorem** **4.** Equation (12) has a prime period two solution $\dots ,\rho ,\sigma ,\rho ,\sigma ,\dots $ if and only ifwhere $\tau =\rho /\sigma $. **Proof.** Assume that there exists a prime period two solution of Equation (12)

$\dots ,\rho ,\sigma ,\rho ,\sigma ,\dots $ Thus, from (12), we find

${w}_{n-\left(2r+1\right)}=\rho $ and

${w}_{n-2r}=\sigma $ for

$r=0,1,2,\dots ,$ and so

and

Since

$g\left(u,v\right)$ be homogenous of degree

$\beta $, we get

$g\left(u,v\right)={v}^{\beta}g\left(\frac{u}{v},1\right)={u}^{\beta}g\left(1,\frac{v}{u}\right)$ and hence,

Now, let

$\rho =\tau \sigma $. Then, we get

By using the fact

$\rho -\tau \sigma =0$, we obtain

and so

Next, from (14) and (15), we see that

On the other hand, suppose that (13) holds. Let

${w}_{-1}=\rho $ and

${w}_{0}=\sigma $ where

$\rho ,$ $\sigma $ defined as (11) and (17), respectively. Then, from (12) and (13), we find

Similarly, we can proof that

${w}_{2}=\sigma $. Hence, it is followed by the induction that

Therefore, Equation (12) has a prime period two, and the proof is complete. □

#### 2.2. Nonexistence of Periodic Solutions of Period Two

In the following theorems, we study some general cases which there are no periodic solutions with period two of the equations

and

where

$f\in C\left({\left(0,\infty \right)}^{2},\left(0,\infty \right)\right)$ and

${w}_{-1},$${w}_{0}$ are positive real numbers.

**Theorem** **5.** Assume that ${f}_{u}>0$ and ${f}_{v}<0$. Then Equation (18) does not have positive period two solutions.

**Proof.** On the contrary, we assume that Equation (18) has a period two distinct solution

where

$r\ne s$. It follows from (18) that

Now, we define the function

for

${v}_{0}\in \left(0,\infty \right)$. Since

$f>0,\phantom{\rule{3.33333pt}{0ex}}{f}_{u}>0$ and

${f}_{v}<0$, we obtain

Thus, ${G}_{{v}_{0}}$ is an increasing and hence G has at most one root for $u\in \left(0,\infty \right)$. But, $G\left({v}_{0}\right)=0$, then he only root of ${G}_{{v}_{0}}\left(w\right)$ is $u={v}_{0}$. Thus, only solution of (20) is $s=r$, which is a contradiction. This completes the proof. □

**Theorem** **6.** Assume that ${f}_{u}>0$ and ${f}_{v}>0$. Then Equation (19) does not have positive period two solutions.

**Proof.** The proof is similar to the proof of Theorem 5 and hence is omitted. □

Now, assume that

${f}_{u}<0$ and

${f}_{v}>0$. In view of [

21] (Theorem 1.4.6), if Equation (18) has no solutions of prime period two, then every solution of Equation (18) converges to

${w}^{*}$. Therefore, we conclude the following:

**Corollary** **1.** Assume that ${f}_{u}<0$ and ${f}_{v}>0$. Then Equation (18) either every its solutions converges to ${w}^{*}$ or has a prime period two solution.

**Corollary** **2.** Assume that l and k are nonnegative integers and ${w}_{-max\left\{l,k\right\}},$${w}_{-max\left\{l,k\right\}+1},$$\dots ,$${w}_{0}$ are positive real numbers. The difference equation does not have positive period two solutions, in the following cases: