Dynamics of a Family of Nonlinear Delay Difference Equations

Theorem 1. Assume that (H₁)–(H₅) hold. Then the unique positive equilibrium $$ of (2) is globally asymptotically stable.

Proof. Let l = max {m_t, k_s}. Since

we have

Claim  1.$$.

Proof of Claim  1. Assume on the contrary that $$. Then it follows from (H₁), (H₃), and (H₄) that

This is a contradiction. Therefore $$. Obviously Claim  1 is proven.

Claim  2. For any M ≥ A, J = [a, M] is an invariable interval of (2).

Proof of Claim  2. For any x₀, x₋₁, …, x_−l ∈ J, we have from (H₄) that

By induction, we may show that x_n ∈ J for any n ≥ 1. Claim  2 is proven.

Let m₀ = a, M₀ = M ≥ A and for any i ≥ 0,

Claim  3. For any n ≥ 0, we have

Proof of Claim  3. From Claim  2, we obtain

By induction, we have that for n ≥ 0, Set Then This with (H₂) and (H₅) implies $$. Claim  3 is proven.

Claim  4. The equilibrium $$ of (2) is locally stable.

Proof of Claim  4. Let M = A and m_n, M_n be the same as Claim  3. For any ε > 0 with $$, there exists n > 0 such that

Set $$. Then for any $$, we have In similar fashion,we can show that for any k ≥ 1, Claim  4 is proven.

Claim  5. $$ is the global attractor of (2).

Proof of Claim  5. Let $$ be a positive solution of (2), and let M = max {x₁, …, x_l+1, A} and m_n, M_n be the same as Claim  3. From Claim  2, we have x_n ∈ [m₀, M₀] = [a, M] for any n ≥ 1. Moreover, we have

In similar fashion, we may show x_n ∈ [m₁, M₁] for any n ≥ l + 2. By induction, we obtain It follows from Claim  3 that $$. Claim  5 is proven.

From Claims  4 and 5, Theorem 1 follows.

Example 2. Consider equation

where 0 ≤ n₁ < n₂ < ⋯<n_s and 0 ≤ m₁ < m₂ < ⋯<m_t with {n₁, n₂, …, n_s}⋂ {m₁, m₂, …, m_t} = ∅, p > 0, a_i > 0 for any i ∈ {1,2, …, t} and b_k > 0 for any k ∈ {1,2, …, s}, and the initial conditions x_−l, …, x₀ ∈ (0, ∞) with l = max {m_t, n_s}. Write $$ and $$. If pB > A, then the unique positive equilibrium $$ of (20) is globally asymptotically stable.

Proof. Let E = (0, +∞). It is easy to verify that (H₁), (H₂), and (H₄) hold for (20). Note that $$. Then

has only solution in the interval (p, +∞), which implies that (H₃) holds for (20). In addition, let then Therefore x/y = 1, which implies that (23) has unique solution Thus (H₅) holds for (20). It follows from Theorem 1 that the equilibrium $$ of (20) is globally asymptotically stable.

Example 3. Consider equation

where 0 ≤ n₁ < n₂ < ⋯<n_s and 0 ≤ m₁ < m₂ < ⋯<m_t with {n₁, n₂, …, n_s}⋂ {m₁, m₂, …, m_t} = ∅, p > 0,   q > 0, a_i > 0 for any 1 ≤ i ≤ t and b_j > 0 for any 1 ≤ j ≤ s, and the initial conditions x_−l, …, x₀ ∈ (0, ∞) with l = max {m_t, n_s}. Write $$ and $$. If p > A, then the unique positive equilibrium $$ of (26) is globally asymptotically stable.

Proof. Let E = [0, +∞). It is easy to verify that (H₁)–(H₄) hold for (26). In addition, the following equation

has unique solution which implies that (H₅) holds for (26). It follows from Theorem 1 that the equilibrium $$ of (26) is globally asymptotically stable.