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

Abstract

This paper is concerned with a time-varying fishing model with delay. By means of the continuation theorem of coincidence degree theory, we prove that it has at least one positive almost periodic solution.

1. Introduction

Consider the following differential equation which is widely used in fisheries [1–4]: 𝑁̇=𝑁[𝐿(𝑡,𝑁)−𝑀(𝑡,𝑁)]−𝑁𝐹(𝑡), (1.1) where 𝑁=𝑁(𝑡) is the population biomass, 𝐿(𝑡,𝑁) is the per capita fecundity rate, 𝑀(𝑡,𝑁) is the per capita mortality rate, and 𝐹(𝑡) is the harvesting rate per capita.

In (1.1), let 𝐿(𝑡,𝑁) be a Hills' type function ([1, 2])𝐿()=𝑎𝑡,𝑁1+(𝑁/𝐾)^𝛾 (1.2) and take into account the delay and the varying environments; Berezansky and Idels [5] proposed the following time-lag model based on (1.1) [1–6] 𝑁̇(𝑡)=𝑁(𝑡)𝑎(𝑡)1+(𝑁(𝜃(𝑡))/𝐾(𝑡))^𝛾−𝑏(𝑡), (1.3) where 𝑏(𝑡)=𝑀(𝑡,𝑁)+𝐹(𝑡).

The model (1.3) has recently attracted the attention of many mathematicians and biologists; see the differential equations which are widely used in fisheries [1, 2]. However, one can easily see that all equations considered in the above-mentioned papers are subject to periodic assumptions, and the authors, in particular, studied the existence of their periodic solutions. On the other hand, ecosystem effects and environmental variability are very important factors and mathematical models cannot ignore, for example, reproduction rates, resource regeneration, habitat destruction and exploitation, the expanding food surplus, and other factors that affect the population growth. Therefore it is reasonable to consider the various parameters of models to be changing almost periodically rather than periodically with a common period. Thus, the investigation of almost periodic behavior is considered to be more accordant with reality. Although it has widespread applications in real life, the generalization to the notion of almost periodicity is not as developed as that of periodic solutions; we refer the reader to [7–18].

Recently, the authors of [19] proved the persistence and almost periodic solutions for a discrete fishing model with feedback control. In [20, 21], the contraction mapping principle and the continuation theorem of coincidence degree have been employed to prove the existence of positive almost periodic exponential stable solutions for logarithmic population model, respectively. A primary purpose of this paper, nevertheless, is to utilize the continuation theorem of coincidence degree for this purpose. To the best of the authors’ observation, there exists no paper dealing with the proof of the existence of positive almost periodic solutions for (1.3) using the continuation theorem of coincidence degree. Therefore, our result is completely different and presents a new approach.

2. Preliminaries

Our first observation is that under the invariant transformation 𝑁(𝑡)=𝑒^𝑦(𝑡), (1.3) reduces to𝑦̇(𝑡)=𝑎(𝑡)1+𝑒^{𝑦(𝜃(𝑡))}/𝐾(𝑡)^𝛾−𝑏(𝑡) (2.1) for 𝛾>0, with the initial function and the initial value 𝑦(𝑡)=𝜙(𝑡),𝑦(0)=𝑦₀,𝑡∈(−∞,0). (2.2) For (2.1) and (2.2), we assume the following conditions: (A1)𝑎(𝑡),𝑏(𝑡)∈𝐶([0,+∞),[0,+∞)) and 𝐾(𝑡)∈𝐶([0,+∞),(0,+∞));(A2)𝜃(𝑡) is a continuous function on [0,+∞) that satisfies 𝜃(𝑡)≤𝑡; (A3)𝜙(𝑡)∶(−∞,0)→[0,∞) is a continuous bounded function, 𝜙(𝑡)≥0,𝑦₀>0.

By a solution of (2.1) and (2.2) we mean an absolutely continuous function 𝑦(𝑡) defined on (−∞,+∞) satisfying (2.1) almost everywhere for 𝑡≥0 and (2.2). As we are interested in solutions of biological significance, we restrict our attention to positive ones.

According to [22], the initial value problem (2.1) and (2.2) has a unique solution defined on (−∞,∞).

Let 𝑋,𝑌 be normed vector spaces, 𝐿∶Dom𝐿⊂𝑋→𝑌 be a linear mapping, and 𝑁∶𝑋→𝑌 be a continuous mapping. The mapping 𝐿 will be called a Fredholm mapping of index zero if dimKer𝐿=codimIm𝐿<+∞ and Im𝐿 is closed in 𝑌. If 𝐿 is a Fredholm mapping of index zero and there exist continuous projectors 𝑃∶𝑋→𝑋 and 𝑄∶𝑌→𝑌 such that Im𝑃=Ker𝐿, Ker𝑄=Im𝐿=Im(𝐼−𝑄), it follows that the mapping 𝐿|_{Dom𝐿∩Ker𝑃}∶(𝐼−𝑃)𝑋→Im𝐿 is invertible. We denote the inverse of that mapping by 𝐾_𝑃. If Ω is an open bounded subset of 𝑋, then the mapping 𝑁 will be called 𝐿-compact on Ω, if 𝑄𝑁(Ω) is bounded and 𝐾_𝑃(𝐼−𝑄)𝑁∶Ω→𝑋 is compact. Since Im𝑄 is isomorphic to Ker𝐿, there exists an isomorphism 𝐽∶Im𝑄→Ker𝐿.

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 Ω. Assume that (1)𝐿𝑦≠𝜆𝑁𝑦 for every 𝑦∈𝜕Ω∩Dom𝐿 and 𝜆∈(0,1); (2)𝑄𝑁𝑦≠0 for every 𝑦∈𝜕Ω∩Ker𝐿; (3)the Brouwer degree deg{𝐽𝑄𝑁,Ω∩Ker𝐿,0}≠0. Then 𝐿𝑦=𝑁𝑦 has at least one solution in Dom𝐿∩Ω.

3. Existence of Almost Periodic Solutions

Let 𝐴𝑃(ℝ) denote the set of all real valued almost periodic functions on ℝ, for 𝑓∈𝐴𝑃(ℝ) we denote by Λ𝑓=𝜆∈ℝ∶lim_𝑇→∞1𝑇^𝑇₀𝑓(𝑠)𝑒^−𝑖𝜆_𝑠,𝑓=d𝑠≠0mod^𝑚_𝑗=1𝑛_𝑗𝜆_𝑗∶𝑛_𝑗∈ℤ,𝑚∈ℕ,𝜆_𝑗∈Λ𝑓,𝑗=1,2,…,𝑚, (3.1) the set of Fourier exponents and the module of 𝑓, respectively. Let 𝐾(𝑓,𝜀,𝑆) denote the set of 𝜀-almost periods for 𝑓 with respect to 𝑆⊂𝐶((−∞,0],ℝ), 𝑙(𝜀) denote the length of the inclusion interval, and 𝑚(𝑓)=lim_𝑇→∞(1/𝑇)∫^𝑇₀𝑓(𝑠)d𝑠 denote the mean value of 𝑓.

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

Throughout the rest of the paper we assume the following condition for (2.1): (H)𝑎(𝑡),𝑏(𝑡),𝐾(𝑡),𝑡−𝜃(𝑡)∈𝐴𝑃(ℝ),𝑚(b/𝐾^𝛾)≠0and𝑚(𝑎)≠𝑚(𝑏).

In our case, we set 𝑋=𝑌=𝑉₁⊕𝑉₂, (3.2) where 𝑉₁=(ℝ)𝑦(𝐹)𝑦𝜇,𝑉𝑦∈𝐴𝑃∶mod⊆mod,∀𝜇∈Λsatisfies>𝛼₂=𝑦(𝑡)≡𝑘,𝑘∈ℝ, (3.3) where =𝑎(𝑡)𝐹=𝐹𝑡,𝜙1+𝑒^{𝜙(𝜃(0))}/𝐾(𝑡)^𝛾−𝑏(𝑡),𝜙∈𝐶([−∞,0],ℝ) (3.4) and 𝛼 is a given constant; define the norm 𝑦=sup_𝑡∈ℝ𝑦(𝑡),𝑦∈𝑋(orY). (3.5)

Remark 3.2. If 𝑓 is 𝜀-almost periodic function, then ∫^𝑡𝑓(𝑠)d𝑠 is 𝜀-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}.

We start with the following lemmas.

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 |𝜆|≤𝛼 we have lim_𝑇→∞1𝑇^𝑡₀𝑦_𝑛(𝑠)𝑒^−𝑖𝜆_𝑠d𝑠=0. (3.6) Thus lim_𝑇→∞1𝑇^𝑇₀𝑦₀(𝑠)𝑒^−𝑖𝜆_𝑠d𝑠=0, (3.7) 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 𝑎(𝑡)𝐿𝑦=1+𝑒^{𝑦(𝜃(𝑡))}/𝐾(𝑡)^𝛾−𝑏(𝑡), (3.8) 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 Ker𝐿=𝑉₂. It remains to prove that Im𝐿=𝑉₁. Suppose that 𝜙(𝑡)∈Im𝐿⊂𝑌. Then, there exist 𝜙₁∈𝑉₁ and 𝜙₂∈𝑉₂ such that 𝜙=𝜙₁+𝜙₂. (3.9) From the definitions of 𝜙(𝑡) and 𝜙₁(𝑡), one can deduce that ∫^𝑡𝜙(𝑠)d𝑠 and ∫^𝑡𝜙₁(𝑠)d𝑠 are almost periodic functions and thus 𝜙₂(𝑡)≡0, which implies that 𝜙(𝑡)∈𝑉₁. This tells us that Im𝐿⊂𝑉₁. (3.10) On the other hand, if 𝜑(𝑡)∈𝑉₁⧵{0} then we have ∫^𝑡₀𝜑(𝑠)d𝑠∈𝐴𝑃(ℝ). Indeed, if 𝜆≠0 then we obtain lim_𝑇→∞1𝑇^𝑇₀^𝑡₀𝜑(𝑠)d𝑠𝑒^−𝑖𝜆_𝑡1d𝑡=𝑖𝜆lim_𝑇→∞1𝑇^𝑇₀𝜑(𝑡)𝑒^−𝑖𝜆_𝑡d𝑡. (3.11) It follows that Λ^𝑡₀𝜑(𝑠)d𝑠−𝑚^𝑡₀𝜑(𝑠)d𝑠=Λ𝜑(𝑡). (3.12) Thus ^𝑡₀𝜑(𝑠)d𝑠−𝑚^𝑡₀𝜑(𝑠)d𝑠∈𝑉₁⊂𝑋. (3.13) Note that ∫^𝑡₀𝜑(𝑠)d𝑠−𝑚(∫^𝑡₀𝜑(𝑠)d𝑠) is the primitive of 𝜑 in 𝑋; therefore we have 𝜑∈Im𝐿. Hence, we deduce that 𝑉₁⊂Im𝐿, (3.14) which completes the proof of our claim. Therefore, Im𝐿=𝑉₁. (3.15) Furthermore, one can easily show that Im𝐿 is closed in 𝑌 and dimKer𝐿=1=codimIm𝐿. (3.16) 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 𝑎(𝑡)𝑁𝑦=1+𝑒^{𝑦(𝜃(𝑡))}/𝐾(𝑡)^𝛾(𝑡)𝑦(𝑧)−𝑏,y∈𝑋,𝑃𝑦=𝑚,𝑦∈𝑋,𝑄𝑧=𝑚,𝑧∈𝑌. (3.17) Then, 𝑁 is L-compact on Ω (Ω is an open and bounded subset of 𝑋).

Proof. The projections 𝑃 and 𝑄 are continuous such that Im𝑃=Ker𝐿,Im𝐿=Ker𝑄. (3.18) It is clear that (𝐼−𝑄)𝑉₂={0},(𝐼−𝑄)𝑉₁=𝑉₁. (3.19) Therefore Im(𝐼−𝑄)=𝑉₁=Im𝐿. (3.20) In view of (),Im𝑃=Ker𝐿,Im𝐿=Ker𝑄=Im𝐼−𝑄 (3.21) we can conclude that the generalized inverse (of 𝐿) 𝐾_𝑃∶Im𝐿→Ker𝑃∩Dom𝐿 exists and is given by 𝐾_𝑃(𝑧)=^𝑡₀𝑧(𝑠)d𝑠−𝑚^𝑡₀𝑧(𝑠)d𝑠. (3.22) Thus 𝑎(𝑡)𝑄𝑁𝑦=𝑚1+𝑒^{𝑦(𝜃(𝑡))}/𝐾(𝑡)^𝛾(𝑡,𝐾−𝑏)_𝑃(𝐼−𝑄)𝑁𝑦=𝑓𝑦(𝑡)−𝑄𝑓𝑦(𝑡), (3.23) where 𝑓[𝑦(𝑡)] is defined by 𝑓𝑦(𝑡)=^𝑡₀𝑁𝑦(𝑠)−𝑄𝑁𝑦(𝑠)d𝑠. (3.24)
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 𝜀<1 and any compact set 𝑆⊂𝐶((−∞,0],ℝ), let 𝑙(𝜀,𝑆) be the inclusion interval of 𝐾(𝐹,𝜀,𝑆). Suppose that {𝑧_𝑛(𝑡)}⊂Im𝐿=𝑉₁ and 𝑧_𝑛(𝑡) uniformly converges to 𝑧₀(𝑡). Because ∫^𝑡₀𝑧_𝑛(𝑠)d𝑠∈𝑌(𝑛=0,1,2,3,…), there exists 𝜌(0<𝜌<𝜀) such that 𝐾(𝐹,𝜌,𝑆)⊂𝐾(∫^𝑡_𝑜𝑧_𝑛(𝑠)d𝑠,𝜀). Let 𝑙(𝜌,𝑆) be the inclusion interval of 𝐾(𝐹,𝜌,𝑆) and 𝑙=max𝑙𝜌,𝑆,𝑙(𝜀,𝑆). (3.25) It is easy to see that 𝑙 is the inclusion interval of both 𝐾(𝐹,𝜀,𝑆) and 𝐾(𝐹,𝜌,𝑆). Hence, for all 𝑡∉[0,𝑙], there exists 𝜇_𝑡∈𝐾(𝐹,𝜌,S)⊂𝐾(∫^𝑡₀𝑧_𝑛(𝑠)d𝑠,𝜀) such that 𝑡+𝜇_𝑡∈[0,𝑙]. Therefore, by the definition of almost periodic functions we observe that ^𝑡₀𝑧_𝑛(𝑠)d𝑠=sup_𝑡∈ℝ^𝑡₀𝑧_𝑛(𝑠)d𝑠≤sup_{[]𝑡∈0,𝑙}^𝑡₀𝑧_𝑛(𝑠)d𝑠+sup_{[]𝑡∉0,𝑙}^𝑡₀𝑧_𝑛(𝑠)d𝑠−^{𝑡+𝜇𝑡}₀𝑧_𝑛(𝑠)d𝑠+^{𝑡+𝜇𝑡}₀𝑧_𝑛(𝑠)d𝑠≤2sup_{[]𝑡∈0,𝑙}^𝑡₀𝑧_𝑛(𝑠)d𝑠+sup_{[]𝑡∉0,𝑙}^𝑡₀𝑧_𝑛(𝑠)d𝑠−^{𝑡+𝜇𝑡}₀𝑧_𝑛(𝑠)d𝑠≤2^𝑙₀𝑧_𝑛(𝑠)d𝑠+𝜀. (3.26) By applying (3.26), we conclude that ∫^𝑡₀𝑧(𝑠)d𝑠(𝑧∈Im𝐿) is continuous and consequently 𝐾_𝑃 and 𝐾_𝑃(𝐼−𝑄)𝑁𝑦 are also continuous.
From (3.26), we also have that ∫^𝑡₀𝑧(𝑠)d𝑠 and 𝐾_𝑃(𝐼−𝑄)𝑁𝑦 are uniformly bounded in Ω. In addition, it is not difficult to verify that 𝑄𝑁(Ω) is bounded and 𝐾_𝑃(𝐼−𝑄)𝑁𝑦 is equicontinuous in Ω. Hence by the Arzelà-Ascoli theorem, we can immediately conclude that 𝐾_𝑃(𝐼−𝑄)𝑁(Ω) is compact. Thus 𝑁 is 𝐿-compact on Ω.

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 𝑦, then 𝑁=𝑒^𝑦 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 Ω. It remains to search for an appropriate open and bounded subset Ω. Corresponding to the operator equation 𝐿𝑦=𝜆𝑁𝑦,𝜆∈(0,1), (3.27) we may write 𝑦̇(𝑡)𝑎(𝑡)=𝜆1+𝑒^{𝑦(𝜃(𝑡))}/𝐾(𝑡)^𝛾−𝑏(𝑡). (3.28) Assume that 𝑦=𝑦(𝑡)∈𝑋 is a solution of (3.28) for a certain 𝜆∈(0,1). Denote 𝑦^∗=sup_𝑡∈ℝ𝑦(𝑡),𝑦_∗=inf_𝑡∈ℝ𝑦(𝑡). (3.29) In view of (3.28), we obtain 𝑚(𝑎(𝑡)(𝑡𝑏(𝑡)−𝑏))=𝑚𝐾^𝛾(𝑡)𝑒^{𝑦(𝜃(𝑡))𝛾} (3.30) and consequently, 𝑚(𝑎(𝑡)(𝑡𝑏(𝑡)−𝑏))≥𝑚𝐾^𝛾(𝑡)𝑒^𝛾𝑦∗, (3.31) which implies from (H) that 𝑦_∗≤𝛾⁻¹𝑚[𝑎(𝑡)(𝑡ln−𝑏)]𝑚(𝑏(𝑡)/𝐾^𝛾(𝑡.)) (3.32) Similarly, we can get 𝑦^∗≥𝛾⁻¹𝑚(𝑎(𝑡)(𝑡ln−𝑏))𝑚(𝑏(𝑡)/𝐾^𝛾(𝑡.)) (3.33) By inequalities (3.32) and (3.33), we can find that there exists 𝑡₁∈ℝ such that 𝑦𝑡₁≤𝑀, (3.34) where 𝑀=𝛾⁻¹𝑚(𝑎(𝑡)(𝑡ln−𝑏))𝑚(𝑏(𝑡)/𝐾^𝛾(𝑡))+1. (3.35) Then from (3.26), we have 𝑦(𝑡)≤𝑦𝑡₁+sup_𝑡∈ℝ^𝑡_𝑡2𝑦^′(𝑠)d𝑠≤𝑀+2sup_{𝑡𝑡∈2,𝑡2+𝑙}^𝑡_𝑡2𝑦^′(𝑠)d𝑠+𝜀 (3.36) or 𝑦(𝑡)≤𝑀+2^𝑡2_𝑡^+𝑙₂𝑦^′(𝑠)d𝑠+1. (3.37) Choose the point 𝜈−𝑡₂∈[𝑙,2𝑙]∩𝐾(𝐹,𝜌,𝑆), where 𝜌(0<𝜌<𝜀) satisfies 𝐾(𝐹,𝜌)⊂𝐾(𝑧,𝜀). Integrating (3.28) from 𝑡₂ to 𝜈, we get 𝜆^𝜈_𝑡2𝑎(𝑡)1+[𝐾(𝑠)]^−𝛾𝑒^{𝛾𝑦(𝜃(𝑠))}d𝑠=𝜆^𝜈_𝑡2|𝑏(𝑠)|d𝑠+^𝜈_𝑡2𝑦^′(𝑠)d𝑠≤𝜆^𝜈_𝑡2|𝑏(𝑠)|d𝑠+𝜀. (3.38) However, from (3.28) and (3.38), we obtain ^𝜈_𝑡2𝑦^′(𝑠)d𝑠≤𝜆^𝜈_𝑡2|𝑏(𝑠)|d𝑠+𝜆^𝜈_𝑡2𝑎(𝑡)1+[𝐾(𝑠)]^−𝛾𝑒^{𝛾𝑦(𝜃(𝑠))}d𝑠,≤2^𝜈_𝑡2|𝑏(𝑠)|d𝑠+𝜀≤2^𝜈_𝑡2|𝑏(𝑠)|d𝑠+1. (3.39) Substituting back in (3.37) and for 𝜈≥𝑡₂+𝑙, we have 𝑦(𝑡)≤𝑀^′, (3.40) where 𝑀^′=𝑀+4^𝜈₀|𝑏(𝑠)|d𝑠+3. (3.41) Let 𝑀=𝑀+𝑀^′. Obviously, it is independent of 𝜆. Take Ω=𝑦∈𝑋∶𝑦<𝑀. (3.42) It is clear that Ω satisfies assumption (1) of Theorem 2.1. If 𝑦∈𝜕Ω∩Ker𝐿, then 𝑦 is a constant with ‖𝑦‖=𝑀. It follows that 𝑎(𝑡)𝑄𝑁𝑦=𝑚1+𝑒^{𝑦(𝜃(𝑡))}/𝐾(𝑡)^𝛾−𝑏(𝑡)≠0, (3.43) which implies that assumption (2) of Theorem 2.1 is satisfied. The isomorphism 𝐽∶Im𝑄→Ker𝐿 is defined by 𝐽(𝑧)=𝑧 for 𝑧∈ℝ. Thus, 𝐽𝑄𝑁𝑦≠0. In order to compute the Brouwer degree, we consider the homotopy 𝐻𝑦,𝑠=−𝑠𝑦+(1−𝑠)𝐽𝑄𝑁𝑦,0≤𝑠≤1. (3.44) For any 𝑦∈𝜕Ω∩Ker𝐿, 𝑠∈[0,1], we have 𝐻(𝑦,𝑠)≠0. By the homotopic invariance of topological degree, we get deg{𝐽𝑄𝑁,Ω∩Ker𝐿,0}=deg−𝑦,Ω∩Ker𝐿,0≠0. (3.45) Therefore, assumption (3) of Theorem 2.1 holds. Hence, 𝐿𝑦=𝑁𝑦 has at least one solution in Dom𝐿∩Ω. 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.

4. An Example

Let 𝑎(𝑡)=𝑒^𝜋(3+cos√2𝑡),𝑏(𝑡)=(1/2)𝑒^𝜋(3+cos√2𝑡),𝐾(𝑡)=4+sin√𝑡,𝛾>0,𝜃(𝑡)=𝑡−2−sin√3𝑡. Then (1.3) has the form𝑁̇(𝑡)(𝑡)𝑒=𝑁^𝜋3+cos√2𝑡1+𝑁𝑡−2−sin√3𝑡/4+sin√𝑡^𝛾−12𝑒^𝜋3+cos√2𝑡. (4.1) One can easily realize that 𝑚(𝑏(𝑡)/[𝐾(𝑡)]^𝛾)>0 and 𝑚(𝑎(𝑡))>𝑚(𝑏(𝑡)); thus condition (H) holds. Therefore, by the consequence of Theorem 3.6, (4.1) has at least one positive almost periodic solution (Figure 1).

Figure 1 

Transient response of state 𝑁(𝑡) when 𝛾=2.

Acknowledgment

This work is supported by the National Natural Sciences Foundation of People’s Republic of China under Grant 10971183.