Positive Periodic Solutions of Cooperative Systems with Delays and Feedback Controls

Lemma 1 (see [35].)Suppose τ ∈ C¹(R, R) with τ(t + ω) ≡ τ(t) and τ^′(t) < 1, ∀t ∈ [0, ω]. Then the function t − τ(t) has a unique inverse function μ(t) satisfying μ ∈ C(R, R), μ(u + ω) = μ(u) + ω, ∀u ∈ R.

Lemma 2 (see [36].)Let L be a Fredholm operator of index zero and let N be L-compact on $$. If

• (a)

  for each λ ∈ (0,1) and x ∈ ∂Ω  ∩  Dom  L, Lx ≠ λNx;

• (b)

  for each x ∈ ∂Ω  ∩  Ker  L, QNx ≠ 0;

• (c)

  deg   {JQN, Ω  ∩  Ker  L, 0} ≠ 0,

then the operator equation Lx = Nx has at least one solution lying in Dom $$.

Lemma 3. Suppose that $$,$$, $$, $$,  $$ is an ω-periodic solution of (4) and (5); then $$,  $$,  $$ satisfies the system

where The converse is also true.

Proof. By (4), (5), and the variation constants formula in ordinary differential equations, we have

From (9), we obtain Considering that $$,  $$, $$,   $$,   …,$$ is an ω-periodic solution of system (4) and (5), we obtain Then which implies That is, Hence On the other hand, assume that $$, $$, $$, $$ is an ω-periodic solution of system (7), then By a direct calculation, we have This completes the proof.

Theorem 4. Suppose that assumption (H1) holds and there exists a constant θ_i > 0, ζ_i > 0, i = 1,2, …, n, such that

where and the algebraic equation where has a unique positive solution. Then system (4) has at least one positive ω-periodic solution.

Proof. For system (18) we introduce new variables y_i(t)  (i = 1,2, …, n) such that

Then system (18) is rewritten in the following form: where In order to apply Lemma 2 to system (26), we introduce the normed vector spaces X and Z as follows. Let C(R, Rⁿ) denote the space of all continuous functions  y(t) = (y₁(t), y₂(t), …, y_n(t)) : R → Rⁿ. We take with norm It is obvious that X and Z are the Banach spaces. We define a linear operator L: Dom L ⊂ X → Z and a continuous operator N : X → Z as follows: where Further, we define continuous projectors P : X → X and Q : Z → Z as follows: We easily see $$ and Ker  L = Rⁿ. It is obvious that Im L is closed in Z and dimKer L = n. Since for any v ∈ Z there are unique v₁ ∈ Rⁿ and v₂ ∈ Im L with such that v(t) = v₁ +  v₂(t), we have codimIm L = n. Therefore, L is a Fredholm mapping of index zero. Furthermore, the generalized inverse (to L) K_p: Im L → Ker P∩Dom L is given in the following form: For convenience, we denote F(t) = (F₁(t), F₂(t), …, F_n(t)) as follows: Thus, we have From formulas (36), we easily see that QN and K_p(I − Q)N are continuous operators. Furthermore, it can be verified that $$ is compact for any open bounded set Ω ⊂ X by using the Arzela-Ascoli theorem and $$ is bounded. Therefore, N is L-compact on $$ for any open bounded subset Ω ⊂ X.

Now, we reach the position to search for an appropriate open bounded subset Ω for the application of the continuation theorem (Lemma 2) to system (26).

Corresponding to the operator equation Ly(t) = λNy(t) with parameter λ ∈ (0,1), we have

where F_i(t)  (i = 1,2, …, n) are given in (35).

Assume that y(t) = (y₁(t), y₂(t), …, y_n(t)) ∈ X is a solution of system (37) for some parameter λ ∈ (0,1). By integrating system (37) over the interval [0, ω], we obtain

Consequently, From the continuity of y(t) = (y₁(t), y₂(t), …, y_n(t)), there exist constants ξ_i, η_i ∈ [0, ω]  (i = 1,2, …, n) such that By (39) and (40) we obtain where Therefore, we further have Let s_ijl(t) = t − τ_ijl(t)  (i, j = 1,2, …, n,  l = 1,2, …, m); then from Lemma 1 and (H1) we get that function s_ijl(t) has a unique ω periodic inverse function φ_ijl(t); then, for every i, j = 1,2, …, n,   l = 1,2, …, m, we have One can see that are ω periodic functions. Then, for every i, j = 1,2, …, n,   l = 1,2, …, m, we have From (39) and (46) we further obtain From the above equality we have From the assumptions of Theorem 4 and (48) we can obtain Consequently, From (40) and (50), we further obtain On the other hand, directly from system (26) we have where From (43), (51), and (52), we have for any t ∈ [0, ω] Therefore, from (54) and (55), we have It can be seen that the constants B_i  (i = 1,2, …, n) are independent of parameter λ ∈ (0,1). For any y = (y₁, y₂, …, y_n) ∈ Rⁿ, from (31) we can obtain We consider the following algebraic equation:

From the assumption of Theorem 4, the equation has a unique positive solution $$. Hence, the equation QNy = 0 has a unique solution $$.

Choosing constant B > 0 large enough such that $$ and B > B₁ + B₂ + ⋯+B_n, we define a bounded open set Ω ⊂ X as follows:

It is clear that Ω satisfies conditions (a) and (b) of Lemma 2. On the other hand, by directly calculating we can obtain where From the assumption of Theorem 4, we have From this, we finally have This shows that Ω satisfies condition (c) of Lemma 2. Therefore, system (26) has an ω-periodic solution $$. Finally, we have system that (4) has a positive ω-periodic solution. This completes the proof.