The Existence and Uniqueness of Positive Periodic Solutions for a Class of Nicholson-Type Systems with Impulses and Delays

Definition 1. A map y(t) = (y₁(t), y₂(t)) ^T : [−τ, ∞)→(0, ∞)×(0, ∞) is said to be a solution of (3) on [τ, ∞) satisfying the initial value condition (4), if

• (i)

  y_i(t) is absolutely continuous on each interval (0, t₁] and (t_k, t_k+1],   k = 1, 2, …;

• (ii)

  for any t_k, k = 1, 2, …, $$ and $$ exist, and $$;

• (iii)

  y_i(t) satisfies (3), i = 1, 2.

Under the previous hypotheses (A₁)–(A₃), we consider the following delay differential equation without impulses:

with initial condition (4) where

By a solution y(t) of (6) on [−τ, ∞), we mean that an absolutely continuous function on [−τ, ∞) satisfies (6) for t ≥ 0 and initial condition (4). Similar to the method of [18], we have the following.

Lemma 2. Assume that (A₁)–(A₃) hold. Then

• (i)

  if $$ is a solution (or positive ω-periodic solution) of (6) on [−τ, ∞), then $$ is a solution (or positive ω-periodic solution) of (3) on [−τ, ∞);

• (ii)

  if $$ is a solution (or positive ω-periodic solution) of (3) on [−τ, ∞), then $$ is a solution (or positive ω-periodic solution) of (6) on [−τ, ∞).

Proof. First, we prove (i). If $$ is a solution (or positive ω-periodic solution) of (6) on [−τ, ∞), then it is easy to see that y_i(t)  (i = 1, 2) is absolutely continuous on each interval (0, t₁] and (t_k, t_k+1], k = 1, 2, …, and, for any t ≠ t_k,

Similarly, we have On the other hand, for every t_k ∈ {t_k}, i = 1, 2, Thus, for every k = 1, 2, …, It follows from (9) and (12) that x(t) is the solution (or positive ω-periodic solution) of (3) corresponding to initial condition (4).

Next, we prove (ii). If $$ is a solution (or positive ω-periodic solution) of (3) on [−τ, ∞), then y_i(t)  (i = 1, 2) is absolutely continuous on each interval (0, t₁] and (t_k, t_k+1], and in view of (12), it follows that, for any k = 1, 2, …, i = 1, 2,

Equations in (13) imply that x_i(t) is continuous on [0, ∞). It is easy to prove that x(t) is also absolutely continuous on [0, ∞). Now, one can easily check that $$ is the solution (or positive ω-periodic solution) of (6) corresponding to initial condition (4). This completes the proof.

Lemma 3. Assume that (A₁)–(A₃) hold. Then every solution of (3) is defined and positive on [−τ, ∞).

Proof. Clearly, by Lemma 2, we only need to prove that every solution of (6) is defined and positive on [−τ, ∞). In order to show that, we only need to see Lemma 2.3 in [13].

Lemma 4 (see [19].)Let X be a Banach space. Suppose that L : D(L) ⊂ X → X is a Fredholm operator with index zero and $$ is L-compact on $$ with Ω open bounded in X. Moreover, assume that all the following conditions are satisfied:

• (i)

  Lx ≠ λNx,  for all x ∈ ∂Ω∩D(L),  λ ∈ (0,1);

• (ii)

  QNx ≠ 0, for all x ∈ ∂Ω∩Ker L;

• (iii)

  the Brouwer degree

  $$ ()

Then equation Lx = Nx has at least one solution in $$.

Theorem 5. Assume that (A₁)–(A₃) hold. Moreover, the following condition is satisfied:

•  

  (A₄)   $$

Then (3) has at least one positive ω-periodic solution.

Proof. Based on Lemma 4, what we need to do is just to search for an appropriate open bounded subset Ω for applying Mawhin’s continuous theorem. To do this, it suffices to prove that the set of all possible positive ω-periodic solution of (6) is bounded.

Let u(t) = (x₁(t), x₂(t)) ^T be an arbitrary positive ω-periodic solution of (6). Corresponding to the operator equation (21), we have

Multiplying x₁(t) and the first formula of (24), and then integrating from 0 to ω, we obtain hence, Furthermore, from the Hölder inequality and e^x > x (or x · e^−x < 1) for x > 0, we have that is, Similarly, we have Combining (28) and (29), we get Since $$, (30) implies that $$ and $$ are bounded. Therefore, according to the previous proof, there exists a positive constant D₁ (independent of λ) such that From (24), together with the Hölder inequality and xe^−x < 1 for x > 0, we can obtain (31)–(32) imply that there exist two positive constants μ_i, i = 1, 2, such that Equations in (31) imply that there exist two points ξ_i ∈ [0, ω], i = 1, 2, and two positive constants d_i, i = 1, 2, such that Since, for ∀t ∈ [0, ω], from (33), it follows that there exists a positive constant ζ such that Clearly, ζ is independent of λ. Let H^* = ζ + C, where C > 1 is taken sufficiently large so that Now, we take $$. This satisfied condition (i) of Lemma 4.

When $$, $$ is a constant vector $$ in R² with $$, then

In view of (38), we have Equations in (39), together with (37), imply that Consequently, condition (ii) of Lemma 4 is satisfied.

Furthermore, we define a continuous function ψ : Dom L × [0,1] → X by

When $$ and μ ∈ [0,1], $$ is a constant vector $$ in R² with $$, then, from (37), we obtain It follows that Hence, using the homotopy invariance theorem, we obtain Condition (iii) of Lemma 4 is also satisfied. Thus, by Lemma 4, we conclude that $$; that is, (6) has at least one solution in X. Then, by Lemma 2, we immediately obtain that (3) has at least one positive ω-periodic solution. This completes the proof.

Theorem 6. Let (A₁)–(A₃) hold; furthermore, assume that the following conditions are satisfied:

•  

  (A₅) $$,

•  

  (A₆) τ_ij(t) ∈ C¹([0, ∞), (0, ∞)) and $$, i = 1, 2, j = 1, 2, …, m.

Then (3) has a unique positive ω-periodic solution.

Proof. By Lemma 2, it suffices to prove the uniqueness of positive ω-periodic solutions for system (6). According to Theorem 5, we know that (6) has at least a positive ω-periodic solution $$ with initial condition (4). Suppose that $$ is an arbitrary positive ω-periodic solution of (6) with initial condition (4). Then it follows from (6) that

Calculating the upper-right derivative, we have For y = xe^−rx, r > 0 is a real number, 0 < a ≤ x ≤ b, and by mean value theorem, we have where a < ξ < b. Thus, for any fixed j = 1, 2, …, m, we also have Hence, Similarly, we have We define a Lyapunov function V(·) by for t > 0, and by virtue of (49), (50), and assumption (A₆), we get According to (A₅), and it follows that Hence, we obtain So In view of (56) and periodicity of $$, we have Then by Lemma 2, we conclude that (3) has a unique positive ω-periodic solution. This completes the proof.