The Dynamics of the Solutions of Some Difference Equations

This paper is devoted to investigate the global behavior of the following rational difference equation: yn 1 αyn−t/ β γ ∑k i 0 y p n− 2i 1 ∏k i 0y q n− 2i 1 , n 0, 1, 2, . . ., where α, β, γ, p, q ∈ 0,∞ and k, t ∈ {0, 1, 2, . . .} with the initial conditions x0, x−1, . . . , x−2k, x−2max{k,t}−1 ∈ 0,∞ . We will find and classify the equilibrium points of the equations under studying and then investigate their local and global stability. Also, we will study the oscillation and the permanence of the considered equations.


Introduction
The aim of this paper is to study the dynamics of the solutions of the following recursive sequence: , n 0, 1, 2, . . ., 1.1 where α, β, γ, p, q ∈ 0, ∞ and K ∈ {0, 1, 2, . ..},where K max{k, t}, with the initial conditions x 0 , x −1 , . .., x −2k , x −2K−1 ∈ 0,∞ .We deal with the classification of the equilibrium points of 1.1 in terms of being stable or unstable, where we investigate the global attractor of the solutions of 1.1 as well as the permanence of the equation.Also, we establish some appropriate conditions, which grantee the oscillation of the solutions of 1.1 .For more results in the direction of this study, see, for example, 1-23 and the papers therein.
In the sequel, we present some well-known results and definition that will be useful in our investigation of 1.1 .Let I be some interval of real numbers and let f : I k 1 → I be a continuously differentiable function.Then, for every set of initial conditions x −k , x −k 1 , . . ., x 0 ∈ I, the difference equation 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; 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

Dynamics of 1.1
The change of variables y n β/γ 1/ p k 1 q x n reduces 1.1 to the following difference equation where r α/β.
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 points of 2.1 are given by x rx Then, whenever r ≤ 1, 2.1 has the only equilibrium point x 0, and, while at r > 1, 2.1 possesses the unique positive equilibrium point x r − 1 / k 1 1/ p k 1 q .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.
iii If r 1, then the equilibrium point x 0 of 2.1 is nonhyperbolic with λ 0 < 1 and λ 1.
Proof.The linearized equation of 2.1 about x 0 is u n 1 − ru n−t 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 in Theorem 2.1 that the equilibrium point x 0 of 2.1 is locally asymptotically stable.So it suffices to show that lim n → ∞ x n 0.

2.3
Now, it follows from 2.1 that
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 q .
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 is 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 x n−t .

2.5
Hence, each subsequence {x t 1 n i }, j 0, 1, . . ., t, of {x n } ∞ n 0 is decreasing sequence and therefore it has a limit.Let for some j 0, 1, . . ., t, lim n → ∞ x t 1 n j μ, and, for the sake of contradiction, assume that μ > x.Then, by taking the limit of both sides of , n 0, 1, 2, . . ., 2.6 we obtain μ rμ/ 1 k 1 μ p k 1 q , which contradicts the hypothesis that x r − 1 / k 1 1/ p k 1 q is the only positive solution of 2.2 .Therefore, lim n → ∞ x t 1 n j μ, for all j 0, 1, . . ., t.This means that all the subsequences {x t 1 n i }, j 0, 1, . . ., t, of {x n } ∞ n 0 have the same limit, x, and therefore lim n → ∞ x n x, which completes the proof.Theorem 2.4.Assume that t 0, r > 1, and let {x n } ∞ n −2k−1 be a solution of 2.1 which is strictly oscillatory about the positive equilibrium point x r − 1 / k 1 1/ p k 1 q of 2.1 .Then, the extreme point in any semicycle occurs in one of the first 2 k 1 terms of the semicycle.
and let {x N , x N 1 , . . ., x M } be a positive semicycle followed by the negative semicycle {x M , x M 1 , . . ., x L }. Now, it follows from 2.1 that Similarly, we see from 2.1 that 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

2.14
Therefore, for V L and s μ, we obtain

2.15
Again whenever W L and μ t, we see that

2.16
That is G ≤ x n ≤ H.It follows from i and ii that min{C, G} ≤ x n ≤ max{D, H}.Then, the proof is complete.
Definition 1.1 permanence .The difference equation 1.2 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 1.3 semicycles .A positive semicycle of a sequence {x n } ∞