On the Dynamics of a Higher-Order Difference Equation

This paper deals with the investigation of the following more general rational difference equation: yn 1 αyn/ β γ ∑k i 0 y p n− 2i 1 ∏k i 0yn− 2i 1 , n 0, 1, 2, . . . ,where α, β, γ, p ∈ 0,∞ with the initial conditions x0, x−1, . . . , x−2k, x−2k−1 ∈ 0,∞ . We investigate the existence of the equilibrium points of the considered equation and then study their local and global stability. Also, some results related to the oscillation and the permanence of the considered equation have been presented.

In general, there are a lot of interest in studying the global attractivity, boundedness character, and periodicity of the solutions of nonlinear difference equations.In particular there are many papers that deal with the rational difference equations and that is because many researchers believe that the results about this type of difference equations are of paramount importance in their own right, and furthermore they believe that these results offer prototype towards the development of the basic theory of the global behavior of solutions of nonlinear difference equations of order greater than one.
Kulenović and Ladas 2 presented some known results and derived several new ones on the global behavior of the difference equation x n 1 α βx n γx n−1 / A Bx n Cx n−1 and of its special cases.Elabbasy et al. 3-5 established the solutions form and then investigated the global stability and periodicity character of the obtained solutions of the following difference equations: El-Metwally 6 gave some results about the global behavior of the solutions of the following more general rational difference equations 1.3 C ¸inar 7-9 obtained the solutions form of the difference equations Other related results on rational difference equations can be found in 14-19 .
Let I be some interval of real numbers and let be a continuously differentiable function.Then for every set of initial conditions x −k , x −k 1 , . . ., x 0 ∈ I, the difference equation Definition 1.1 permanence .The difference equation 1.5 is said to be permanent if there exist numbers m and M with 0 < m ≤ M < ∞ such that for any initial conditions x −k , x −k 1 , . . ., x −1 , x 0 ∈ 0, ∞ there exists a positive integer N which depends on the initial conditions such that m ≤ x n ≤ M for all n ≥ N.
Definition 1.2 periodicity .A sequence {x n } ∞ n −k is said to be periodic with period p if x n p x n for all n ≥ −k.

Definition 1.3 semicycles . A positive semicycle of a sequence {x n } ∞
n −k consists of a "string" of terms {x l , x l 1 , . . ., x m } all greater than or equal to the equilibrium point x, with l ≥ −k and m ≤ ∞ such that either l −k or l > −k and x l−1 < x; and, either m ∞ or m < ∞ and x m 1 < x.A negative semicycle of a sequence {x n } ∞ n −k consists of a "string" of terms {x l , x l 1 , . . ., x m } all less than the equilibrium point x, with l ≥ −k and m ≤ ∞ such that: either l −k or l > −k and x l−1 ≥ x; and, either m ∞ or m < ∞ and x m 1 ≥ x.
Recall that the linearized equation of 1.5 about the equilibrium x is the linear difference equation
In this section we study the local stability character and the global stability of the equilibrium points of the solutions of 2.1 .Also we give some results about the oscillation and the permanence of 2.1 .
Recall that the equilibrium point of 2.1 are given by x rx Then 2.1 has the equilibrium points x 0 and whenever r > 1, 2.1 possesses the unique equilibrium point x r − 1 / k 1 1/ p k 1 .The following theorem deals with the local stability of the equilibrium point x 0 of 2.1 .

Theorem 2.1. The following statements are true:
i if r < 1, then the equilibrium point x 0 of 2.1 is locally asymptotically stable, ii if r > 1, then the equilibrium point x 0 of 2.1 is a saddle point.
Proof.The linearized equation of 2.1 about x 0 is u n 1 − ru n 0. Then the associated eigenvalues are λ 0 and λ r.Then the proof is complete.Theorem 2.2.Assume that r < 1, then the equilibrium point x 0 of 2.1 is globally asymptotically stable.
Proof.Let {x n } ∞ n −2k 1 be a solution of 2.1 .It was shown by Theorem 2.1 that the equilibrium point x 0 of 2.1 is locally asymptotically stable.So, it is suffices to show that lim x n n → ∞ 0.

2.3
Now it follows from 2.1 that

2.4
Then the sequence {x n } ∞ n 0 is decreasing and this completes the proof.
Theorem 2.3.Assume that r > 1.Then every solution of 2.1 is either oscillatory or tends to the equilibrium point x r − 1 / k 1 1/ p k 1 .
Proof.Let {x n } ∞ n −2k 1 be a solution of 2.1 .Without loss of generality assume that {x n } ∞ n −2k 1 is a nonoscillatory solution of 2.1 , then it suffices to show that lim n → ∞ x n x.Assume that x n ≥ x for n ≥ n 0 the case where x n ≤ x for n ≥ n 0 is similar and will be omitted .It follows from 2.1 that

2.5
Hence {x n } is monotonic for n ≥ n 0 2k 1, therefore it has a limit.Let lim n → ∞ x n μ, and for the sake of contradiction, assume that μ > x.Then by taking the limit of both side of 2.1 , we obtain μ rμ/ 1 k 1 μ p k 1 , which contradicts the hypothesis that x r − 1 / k 1 1/ p k 1 is the only positive solution of 2.2 .that is there is a sufficiently large positive integer N such that |x n − x| < ε for all n ≥ N and for some ε > 0. So, x − ε < x n < x ε, this means that there are two positive real numbers, say C and D, such that 2.9 ii {x n } ∞ n −2k 1 is strictly oscillatory about x r − 1 / k 1 1/ p k 1 .Now let {x s 1 , x s 2 , . . ., x t } be a positive semicycle followed by the negative semicycle {x t 1 , x t 2 , . . ., x u }.If x V and x W are the extreme values in these positive and negative semicycle, respectively, with the smallest possible indices V and W, then by Theorem 2.4 we see that V − s ≤ 2 k 1 and W − u ≤ 2 k 1 .Now for any positive indices μ and L with μ < L, it follows from 2.1 for n μ, μ 1, . . ., L − 1 that

2.10
Therefor for V L and s μ we obtain

2.11
Again whenever W L and μ t, we see that

2.12
That is, G ≤ x n ≤ H.It follows from i and ii that min{C, G} ≤ x n ≤ max{D, H}.

2.13
Then the proof is complete.
Cinar et al. 10 studied the existence and the convergence for the solutions of the difference equation x n 1 x n−3 / −1 x n x n−1 x n−2 x n−3 .Simsek et al. 11 obtained the solution of the difference equation x n 1 x n−3 / 1 x n−1 .In 12 Yalcinkaya got the solution form of the difference equation x n 1 x n− 2k 1 / 1 x n−k x n− 2k 1 .In 13 Stević studied the difference equation x n 1 x n−1 / 1 x n .

Theorem 2 . 4 .
Assume that {x n } ∞ n −2k 1 is a solution of 2.1 which is strictly oscillatory about the positive equilibrium point x r − 1 / k 1 1/ p k 1 of 2.1 .Then the extreme point in any semicycle occurs in one of the first 2 k 1 terms of the semicycle.