On the Dynamics of a Higher-Order 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, x₀ ∈ (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 $$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 $$consists of a “string” of terms {x_l, x_l+1, …, x_m} all greater than or equal to the equilibrium point $$, with l ≥ −k and m ≤ ∞ such that either l = −k or l > −k and $$; and, either m = ∞ or m < ∞ and $$. A negative semicycle of a sequence $$consists of a "string" of terms {x_l,   x_l+1, …, x_m} all less than the equilibrium point $$, with l ≥ −k and m ≤ ∞ such that: either l = −k or l > −k and $$; and, either m = ∞ or m < ∞ and $$.

Definition 1.4 (oscillation). A sequence $$is called nonoscillatory about the point $$ if there is exists N ≥ −k such that either x_n>$$ for all n ≥ N or x_n<$$ for all n ≥ N. Otherwise $$ is called oscillatory about $$.

Theorem 2.1. The following statements are true:

(i) if r < 1, then the equilibrium point $$ of (2.1) is locally asymptotically stable,

(ii) if r > 1, then the equilibrium point $$ of (2.1) is a saddle point.

Proof. The linearized equation of (2.1) about $$ 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 $$ of (2.1) is globally asymptotically stable.

Proof. Let $$ be a solution of (2.1). It was shown by Theorem 2.1 that the equilibrium point $$ of (2.1) is locally asymptotically stable. So, it is suffices to show that

Theorem 2.3. Assume that r > 1. Then every solution of (2.1) is either oscillatory or tends to the equilibrium point $$.

Proof. Let $$ be a solution of (2.1). Without loss of generality assume that $$ is a nonoscillatory solution of (2.1), then it suffices to show that $$. Assume that $$ for n ≥ n₀ (the case where $$ for n ≥ n₀ is similar and will be omitted). It follows from (2.1) that

Theorem 2.4. Assume that $$ is a solution of (2.1) which is strictly oscillatory about the positive equilibrium point $$ of (2.1). Then the extreme point in any semicycle occurs in one of the first 2(k + 1) terms of the semicycle.

Proof. Assume that $$ is a strictly oscillatory solution of (2.1). Let N ≥ 2k + 2 and let {x_N, x_N+1, …, x_M} be a positive semicycle followed by the negative semicycle {x_M, x_M+1, …, x_M}. Now it follows from (2.1) that

Then x_N ≥ x_N+2(k+1) for all N ≥ 2(k + 1).

Similarly, we see from (2.1) that

Therefore x_M+2(k+1) ≥ x_M for all M ≥ 2(k + 1). The proof is so complete.

Theorem 2.5. Equation (2.1) is permanent.

Proof. Let $$ be a solution of (2.1). There are two cases to consider:

(i)$$ is a nonoscillatory solution of (2.1). Then it follows from Theorem 2.3 that

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

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