Positive Almost Periodic Solutions for a Time-Varying Fishing Model with Delay

Theorem 2.1 (see [19].)Let Ω ⊂ X be an open bounded set and let N : X → Y be a continuous operator which is L-compact on $$. Assume that

• (1)

  Ly ≠ λNy for every y ∈ ∂Ω∩Dom  L and λ ∈ (0,1);

• (2)

  QNy ≠ 0 for every y ∈ ∂Ω∩Ker  L;

• (3)

  the Brouwer degree deg {JQN, Ω∩Ker  L, 0} ≠ 0.

Then Ly = Ny has at least one solution in $$.

Definition 3.1. y(t) ∈ C(ℝ, ℝ) is said to be almost periodic on ℝ if for any ɛ > 0 the set K(y, ɛ) = {δ:|y(t + δ) − y(t)| < ɛ, ∀t ∈ ℝ} is relatively dense; that is, for any ɛ > 0 it is possible to find a real number l(ɛ) > 0 for any interval with length l(ɛ); there exists a number δ = δ(ɛ) in this interval such that |y(t + δ) − y(t)| < ɛ for any t ∈ ℝ.

Remark 3.2. If f is ɛ-almost periodic function, then $$ is ɛ-almost periodic if and only if m(f) = 0. Whereas f ∈ AP(ℝ) does not necessarily have an almost periodic primitive, m(f) = 0. That is why we can not make V₁ = {z ∈ AP(ℝ) : m(z) = 0} and let V₁ = {y ∈ AP(ℝ) : mod (y) ⊂ mod (F), ∀μ ∈ Λ(y)  satisfy   | μ | > α}.

Lemma 3.3. X and Y are Banach spaces endowed with the norm ∥·∥.

Proof. If y_n ∈ V₁ and y_n converge to y₀, then it is easy to show that y₀ ∈ AP(ℝ) with mod (y₀) ⊂ mod (F). Indeed, for all $$ we have

Thus which implies that y₀ ∈ V₁. One can easily see that V₁ is a Banach space endowed with the norm ∥·∥. The same can be concluded for the spaces X and Y. The proof is complete.

Lemma 3.4. Let L : X → Y and

where Ly = y′ = dy/dt. Then L is a Fredholm mapping of index zero.

Proof. It is obvious that L is a linear operator and Ker  L = V₂. It remains to prove that Im  L = V₁. Suppose that ϕ(t) ∈ Im  L ⊂ Y. Then, there exist ϕ₁ ∈ V₁ and ϕ₂ ∈ V₂ such that

From the definitions of ϕ(t) and ϕ₁(t), one can deduce that $$ and $$ are almost periodic functions and thus ϕ₂(t) ≡ 0, which implies that ϕ(t) ∈ V₁. This tells us that On the other hand, if φ(t) ∈ V₁∖{0} then we have $$. Indeed, if $$ then we obtain It follows that Thus Note that $$ is the primitive of φ in X; therefore we have φ ∈ Im  L. Hence, we deduce that which completes the proof of our claim. Therefore, Furthermore, one can easily show that Im  L is closed in Y and Therefore, L is a Fredholm mapping of index zero.

Lemma 3.5. Let N : X → Y, P : X → X, and Q : Y → Y such that

Then, N is L-compact on $$ (Ω is an open and bounded subset of X).

Proof. The projections P and Q are continuous such that

It is clear that Therefore In view of we can conclude that the generalized inverse (of L) K_P : Im  L → Ker  P∩Dom  L exists and is given by Thus where f[y(t)] is defined by

The integral form of the terms of both QN and (I − Q)N implies that they are continuous. We claim that K_P is also continuous. By our hypothesis, for any ɛ < 1 and any compact set S ⊂ C((−∞, 0], ℝ), let l(ɛ, S) be the inclusion interval of K(F, ɛ, S). Suppose that {z_n(t)} ⊂ Im L = V₁ and z_n(t) uniformly converges to z₀(t). Because $$, there exists ρ(0 < ρ < ɛ) such that $$. Let l(ρ, S) be the inclusion interval of K(F, ρ, S) and

It is easy to see that l is the inclusion interval of both K(F, ɛ, S) and K(F, ρ, S). Hence, for all t ∉ [0, l], there exists $$ such that t + μ_t ∈ [0, l]. Therefore, by the definition of almost periodic functions we observe that By applying (3.26), we conclude that $$ is continuous and consequently K_P and K_P(I − Q)Ny are also continuous.

From (3.26), we also have that $$ and K_P(I − Q)Ny are uniformly bounded in $$. In addition, it is not difficult to verify that $$ is bounded and K_P(I − Q)Ny is equicontinuous in $$. Hence by the Arzelà-Ascoli theorem, we can immediately conclude that $$ is compact. Thus N is L-compact on $$.

Theorem 3.6. Let condition (H) hold. Then (2.1) has at least one positive almost periodic solution.

Proof. It is easy to see that if (2.1) has one almost periodic solution $$, then $$ is a positive almost periodic solution of (1.3). Therefore, to complete the proof it suffices to show that (2.1) has one almost periodic solution.

In order to use the continuation theorem of coincidence degree theory, we set the Banach spaces X and Y the same as those in Lemma 3.3 and the mappings L, N, P, Q the same as those defined in Lemmas 3.4 and 3.5, respectively. Thus, we can obtain that L is a Fredholm mapping of index zero and N is a continuous operator which is L-compact on $$. It remains to search for an appropriate open and bounded subset Ω. Corresponding to the operator equation

we may write Assume that y = y(t) ∈ X is a solution of (3.28) for a certain λ ∈ (0,1). Denote In view of (3.28), we obtain and consequently, which implies from (H) that Similarly, we can get By inequalities (3.32) and (3.33), we can find that there exists t₁ ∈ ℝ such that where Then from (3.26), we have or Choose the point ν − t₂ ∈ [l, 2l]  ∩K(F, ρ, S), where ρ(0 < ρ < ɛ) satisfies K(F, ρ) ⊂ K(z, ɛ). Integrating (3.28) from t₂ to ν, we get However, from (3.28) and (3.38), we obtain Substituting back in (3.37) and for ν ≥ t₂ + l, we have where Let $$. Obviously, it is independent of λ. Take It is clear that Ω satisfies assumption (1) of Theorem 2.1. If y ∈ ∂Ω  ∩  Ker  L, then y is a constant with $$. It follows that which implies that assumption (2) of Theorem 2.1 is satisfied. The isomorphism J : Im   Q → Ker   L is defined by J(z) = z for z ∈ ℝ. Thus, JQNy ≠ 0. In order to compute the Brouwer degree, we consider the homotopy For any y ∈ ∂Ω∩Ker  L, s ∈ [0,1], we have H(y, s) ≠ 0. By the homotopic invariance of topological degree, we get Therefore, assumption (3) of Theorem 2.1 holds. Hence, Ly = Ny has at least one solution in $$. In other words, (2.1) has at least one positive almost periodic solution. Therefore, (1.3) has at least one positive almost periodic solution. The proof is complete.