On the Periodicity of General Class of Difference Equations

: In this paper, we are interested in studying the periodic behavior of solutions of nonlinear difference equations. We used a new method to ﬁnd the necessary and sufﬁcient conditions for the existence of periodic solutions. Through examples, we compare the results of this method with the usual method.

where p ≥ 1 is a real number. By a new method, Elsayed [12] and Moaaz [24] studied the existence of the solution of prime period two of equation where α, β and γ are real numbers. Recently, Abdelrahman et al. [1] and Moaaz [25] studied the asymptotic behavior of the solutions of general equation w n+1 = aw n−l + bw n−k + f (w n−l , w n−k ) , where a and b are nonnegative real number. This paper aims to shed light on the study of the existence or nonexistence of periodic solutions for difference equations. We describe and modify the new method in Elsayed [12]. Moreover, we use this new method to study the existence of periodic solutions of the general class of difference equation. Furthermore, we discuss some of the nonexistence cases of periodic solutions. Finally, through examples, we compare the results of this method with the usual method.

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 ..., ρ, σ, ρ, σ, ..., with w n−(2s+1) = ρ and w n−2s = σ. Hence, we get that Next, we let τ = ρ/σ, and substitute into (2). Then, we get that By using the fact ρ − τσ = 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 Φ (u, v) is a homothetic function, that is there exist a strictly increasing function G : R → R and a homogenous function H : R 2 → R with degree β, such that Φ = G (H).

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.

It follows from (4) that
Hence, and so, By dividing (8) by (7), we have that (5) holds. On the other hand, let (5) holds. If we choose 1) and for τ ∈ R + , then we get Similarly, we have that w 2 = w 0 . Hence, it is followed by the induction that 1) and for all n > 0.
Therefore, Equation (4) has a prime period two solution, and the proof is complete.

Proof.
Assume that l > k. Since l odd and k even, we have w n−l = ρ and w n−k = σ. From Equation (9), we get Since τ = ρ/σ, we obtain On the other hand, let (10) holds. Now, we choose where τ ∈ R + . Hence, we see that Similarly, we can proof that w 2 = f (1, τ). Hence, it is followed by the induction that Therefore, Equation (9) has a prime period two solution, and the proof is complete.

Proof.
The proof is similar to that of proof of Theorem 2 and hence is omitted.
Consider the difference equation where β is a positive real number, γ, δ, w −1 and w 0 are arbitrary real numbers and the function g (u, v) is continuous real function and homogenous with degree β where τ = ρ/σ.
Therefore, Equation (12) has a prime period two, and the proof is complete.

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 w n+1 = f (w n , w n−1 ) (18) and where f ∈ C (0, ∞) 2 , (0, ∞) 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)  where r = s. It follows from (18) that Thus, we get r f (r, s) − s f (s, r) = 0. Now, we define the function for v 0 ∈ (0, ∞). Since f > 0, 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 ∈ (0, ∞). But, G (v 0 ) = 0, then he only root of G v 0 (w) 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{l,k} , w − max{l,k}+1 , ..., w 0 are positive real numbers. The difference equation does not have positive period two solutions, in the following cases: (a) l is even, k is odd, f u > 0 and f v < 0; (b) l and k are even, f u > 0 and f v > 0.

Application and Discussion
Next, we -by using Theorem 1-study the periodic character of the positive solutions of equation w n+1 = aw n−1 exp −w n w n−1 bw n + cw n−1 , where a, b, c ∈ (0, ∞). Let G (y) = e y and Φ (w n , w n−1 ) = G (H (u, v)). From (5), if b = c, then (22) has a prime period two solution.
Moreover, by using Theorem 1, the discrete model with two age classes has a prime period two solution if λ = 1.
In [10], El-Dessoky studied the periodic character of the positive solutions of equation where a, b, c, d, δ, w −r , w −r+1 , ..., w 0 are positive real numbers, r = max {k, l, s}, l, k odd and s even. He is proved that the Equation (24) has no prime period two solution if c + δ (a + b − 1) = 0. In the following, by the present method, we will find the necessary and sufficient conditions that this equation has periodic solutions of prime period two. Proof. Assume that there exists a prime period two solution of Equation (24) ..., ρ, σ, ρ, σ, ... Thus, from (24), we find Then, we have and hence c + δ (a + b − 1) = 0. On the other hand, in view of [10] (Theorem 5), if c + δ (a + b − 1) = 0, then (24) has no solutions of prime period two. This completes the proof.
Author Contributions: All authors claim to have contributed equally and significantly in this paper. All authors read and approved the final manuscript.