One-Signed Periodic Solutions of First-Order Functional Differential Equations with a Parameter

Abstract

We study one-signed periodic solutions of the first-order functional differential equation 𝑢′(𝑡)=−𝑎(𝑡)𝑢(𝑡)+𝜆𝑏(𝑡)𝑓(𝑢(𝑡−𝜏(𝑡))),𝑡∈ℝ by using global bifurcation techniques. Where 𝑎,𝑏∈𝐶(ℝ,[0,∞)) are 𝜔−periodic functions with ∫^𝜔₀𝑎(𝑡)𝑑𝑡>0, ∫^𝜔₀𝑏(𝑡)𝑑𝑡>0, 𝜏 is a continuous 𝜔-periodic function, and 𝜆>0 is a parameter. 𝑓∈𝐶(ℝ,ℝ) and there exist two constants 𝑠₂<0<𝑠₁ such that 𝑓(𝑠₂)=𝑓(0)=𝑓(𝑠₁)=0, 𝑓(𝑠)>0 for 𝑠∈(0,𝑠₁)∪(𝑠₁,∞) and 𝑓(𝑠)<0 for 𝑠∈(−∞,𝑠₂)∪(𝑠₂,0).

1. Introduction

In recent years, there has been considerable interest in the existence of periodic solutions of the following equation: 𝑢^′(𝑡)=−𝑎(𝑡)𝑢(𝑡)+𝜆𝑏(𝑡)𝑓(𝑢(𝑡−𝜏(𝑡))), (1.1) where 𝑎,𝑏∈𝐶(ℝ,[0,∞)) are 𝜔-periodic functions, and ∫^𝜔₀𝑎(𝑡)𝑑𝑡>0,∫^𝜔₀𝑏(𝑡)𝑑𝑡>0, 𝜏 is a continuous 𝜔-periodic function, 𝜆>0 is a parameter. (1.1) has been proposed as a model for a variety of physiological processes and conditions including production of blood cells, respiration, and cardiac arrhythmias; see, for example, [1–12] and the references therein. Roughly speaking, 𝑢(𝑡) represents the number of adult (sexually mature) members in a population at time 𝑡,𝑎(𝑡) is the per capita death rate, and 𝑓(𝑢(𝑡−𝜏(𝑡))) is the rate at which new members are recruited into the population at time 𝑡 ( 𝜏 is the age at which members mature, and it is assumed that the birth rate at a given time depends only on the adult population size). The most famous models of this type are(i)the Nicholson's blowflies equation proposed in [1] to explain the oscillatory population fluctuations observed by A. J. Nicholson in 1957 in his studies of the sheep blowfly Lucilia cuprina: 𝑢^′(𝑡)=−𝑎𝑢(𝑡)+𝑝⋅𝑢(𝑡−ℎ)𝑒^{−𝛾𝑢(𝑡−ℎ)},𝑎,𝑝,𝛾,ℎ>0; (1.2)(ii)the model for blood cell populations proposed by Mackey and Glass in [2] 𝑢^′(𝑡)(𝑡)𝑢()=−𝑎𝑢+𝑝𝑡−ℎ1+[𝑢(𝑡−ℎ)]^𝑛,𝑎,𝑝,𝛾,ℎ>0,𝑛>1; (1.3)(iii)the model for the survival of red blood cells in an animal proposed by Wazewska-Czyzewska and Lasota in [3] 𝑢^′(𝑡)=−𝑎𝑢(𝑡)+𝑝⋅𝑒^{−𝛾𝑢(𝑡−ℎ)},𝑎,𝑝,𝛾,ℎ>0. (1.4)

Recently, Cheng and Zhang [7] studied the existence of positive 𝜔-periodic solutions of the functional equation (1.1) under the assumptions:(H1)𝑓∈𝐶([0,∞),[0,∞)), and 𝑓(𝑠)>0 for 𝑠>0;(H2)𝑎,𝑏∈𝐶(ℝ,[0,∞)) are 𝜔−periodic functions, ∫^𝜔₀𝑎(𝑡)𝑑𝑡>0, ∫^𝜔₀𝑏(𝑡)𝑑𝑡>0, 𝜏∈𝐶(ℝ,ℝ) is a 𝜔-periodic function;(H3)there exist 𝑓₀,𝑓_∞∈(0,∞) such that 𝑓₀=lim_|𝑠|→0𝑓(𝑠)𝑠,𝑓_∞=lim_|𝑠|→∞𝑓(𝑠)𝑠. (1.5) They proved the following.

Theorem A. Assume (H1)–(H3)hold. Then for each 𝜆 satisfying 1𝜎𝐵𝑓_∞1<𝜆<𝐴𝑓₀1,or𝜎𝐴𝑓₀1<𝜆<𝐵𝑓_∞, (1.6) equation (1.1) has a positive periodic solution, where 𝐴=max_{[]𝑡∈0,𝜔}^𝜔₀𝐺(𝑡,𝑠)𝑏(𝑠)𝑑𝑠,𝐵=min_{[]𝑡∈0,𝜔}^𝜔₀𝐺(𝑡,𝑠)𝑏(𝑠)𝑑𝑠,𝜎=𝑒^{∫𝜔0𝑎(𝑡)𝑑𝑡}. (1.7)

However, the condition used in [7] is not sharp, and the main results in [7] give no any information about the global structure of the set of positive periodic solutions. Moreover, 𝑓 satisfied (H1) in [7], so a natural question is what would happen if 𝑓 is allowed to have some zeros in ℝ? The purpose of this work is to study the global behavior of the components of one-signed solutions of (1.1) under the condition(H4)𝑓∈𝐶(ℝ,ℝ); there exist two constants 𝑠₂<0<𝑠₁ such that 𝑓(𝑠₂)=𝑓(0)=𝑓(𝑠₁)=0, 𝑓(𝑠)>0 for 𝑠∈(0,𝑠₁)∪(𝑠₁,∞), and 𝑓(𝑠)<0 for 𝑠∈(−∞,𝑠₂)∪(𝑠₂,0).

The rest of this paper is organized as follows. In Section 2, we give some notations and the main results. Section 3 is devoted to proving the main results.

2. Statement of the Main Results

Let 𝑌={𝑢∈𝐶(ℝ,ℝ)∶𝑢(𝑡)=𝑢(𝑡+𝜔)} with the norm ‖𝑢‖_∞=max_{[]𝑡∈0,𝜔}|𝑢(𝑡)|. (2.1) Then (𝑌,‖⋅‖_∞) is a Banach space. Let 𝐸=𝑢∈𝐶¹(ℝ,ℝ)∶𝑢(𝑡)=𝑢(𝑡+𝜔) (2.2) be the Banach space with the norm ‖𝑢‖=max{‖𝑢‖_∞,‖𝑢^′‖_∞}.

It is well known that (1.1) is equivalent to 𝑢(𝑡)=𝜆_𝑡^𝑡+𝜔𝐺(𝑡,𝑠)𝑏(𝑠)𝑓(𝑢(𝑠−𝜏(𝑠)))𝑑𝑠∶=(𝐴𝑢)(𝑡), (2.3) where 𝐺()=𝑒𝑡,𝑠^{∫𝑠𝑡𝑎(𝜃)𝑑𝜃}𝑒^{∫𝜔0𝑎(𝜃)𝑑𝜃}[]−1,𝑠∈𝑡,𝑡+𝜔. (2.4) Notice that ∫^𝑤₀𝑎(𝑡)𝑑𝑡>0, we have 1()≤𝜎𝜎−1≤𝐺𝑡,𝑠𝜎−1, (2.5) where 𝜎=𝑒^{∫𝜔0𝑎(𝑡)𝑑𝑡}, and 0<1/𝜎<1.

Define that 𝐾 is a cone in 𝑌 by (𝑡)(𝑡)≥1𝐾=𝑢∈𝑌∶𝑢≥0,𝑢𝜎‖𝑢‖. (2.6) It is not difficult to prove that 𝐴(𝐾)⊂𝐾 and 𝐴∶𝐾→𝐾 is completely continuous.

Let us consider the spectrum of the linear eigenvalue problem 𝑢^′(𝑡)=−𝑎(𝑡)𝑢(𝑡)+𝜆𝑏(𝑡)𝑢(𝑡−𝜏(𝑡)),𝑡∈ℝ. (2.7)

Lemma 2.1. Assume that (H2) holds. Then the linear problem (2.7) has a unique eigenvalue 𝜆₁, which is positive and simple, and the corresponding eigenfunction 𝜑 is of one sign.

Proof. It is a direct consequence of the Krein-Rutman Theorem [13, Theorem 19.3].

In the rest of the paper, we always assume that 𝜑=1,𝜑(𝑡)>0,𝑡∈ℝ. (2.8)

Define 𝐿∶𝐸→𝑌 by setting 𝐿𝑢∶=𝑢^′(𝑡)+𝑎(𝑡)𝑢(𝑡),𝑢∈𝐸. (2.9) Then 𝐿⁻¹∶𝑌→𝐸 is completely continuous.

Let 𝜁,𝜉∈𝐶(ℝ,ℝ) be such that 𝑓(𝑠)=𝑓₀𝑠+𝜁(𝑠),𝑓(𝑠)=𝑓_∞𝑠+𝜉(𝑠). (2.10) Clearly, lim_|𝑠|→0𝜁(𝑠)𝑠=0,lim_|𝑠|→∞𝜉(𝑠)𝑠=0. (2.11) Let us consider 𝐿𝑢(𝑡)−𝜆𝑏(𝑡)𝑓₀𝑢(𝑡−𝜏(𝑡))=𝜆𝑏(𝑡)𝜁(𝑢(𝑡−𝜏(𝑡))) (2.12) as a bifurcation problem from the trivial solution 𝑢≡0 and 𝐿𝑢(𝑡)−𝜆𝑏(𝑡)𝑓_∞𝑢(𝑡−𝜏(𝑡))=𝜆𝑏(𝑡)𝜉(𝑢(𝑡−𝜏(𝑡))) (2.13) as a bifurcation problem from infinity. We note that (2.12) and (2.13) are the same and each of them is equivalent to (1.1).

Let 𝔼=ℝ×𝐸 under the product topology. We add the points {(𝜆,∞)∣𝜆∈ℝ} to our space 𝔼. Let 𝑆⁺ denote the set of positive functions in 𝐸 and 𝑆⁻=−𝑆⁺, and 𝑆=𝑆⁻∪𝑆⁺. They are disjoint and open in 𝐸. Finally, let Φ^±=ℝ×𝑆^± and Φ=ℝ×𝑆.

Remark 2.2. It is worth remaking that if 𝑢 is a nontrivial solution of (1.1) and 𝑎,𝑏, and 𝑓 satisfy (H2)–(H4), then 𝑢∈𝑆^𝜈 for some 𝜈={+,−}. To see this, define 𝑞(𝑡)=𝑓(𝑢(𝑡))𝑢(𝑡)(𝑡)𝑓,𝑢≠0,₀,𝑢(𝑡)=0. (2.14) Thus (1.1) is equivalent to 𝑢^′(𝑡)=−𝑎(𝑡)𝑢(𝑡)+𝜆𝑏(𝑡)𝑞(𝑡−𝜏(𝑡))𝑢(𝑡−𝜏(𝑡)),𝑡∈ℝ. (2.15) Obviously, 𝑏(⋅)𝑞(⋅−𝜏(⋅)) satisfies (H2). From Lemma 2.1, the nontrivial solution 𝑢∈𝑆^𝜈 for some 𝜈∈{+,−}.

The result of Rabinowitz [14] for (2.12) can be stated as follows: for each 𝜈∈{+,−}, there exists a continuum 𝒞^𝜈 of solutions of (2.12) joining (𝜆₁/𝑓₀,0) to infinity, and 𝒞^𝜈⧵{(𝜆₁/𝑓₀,0)}⊂Φ^𝜈.

The result of Rabinowitz [15] for (2.13) can be stated as follows: for each 𝜈∈{+,−}, there exists a continuum 𝒟^𝜈 of solutions of (2.13) meeting (𝜆₁/𝑓_∞,∞), and 𝒟^𝜈⧵{(𝜆₁/𝑓_∞,∞)}⊂Φ^𝜈.

Our main result is the following.

Theorem 2.3. Assume (H2)–(H4) hold. Moreover, suppose that(H5)𝑓 satisfies the Lipschitz condition in [𝑠₂,𝑠₁]. Then(i)for (𝜆,𝑢)∈𝒞⁺∪𝒞⁻, 𝑠₂<𝑢(𝑡)<𝑠₁,𝑡∈[0,𝜔]; (2.16)(ii)for (𝜆,𝑢)∈𝒟⁺∪𝒟⁻, we have that either max_{[]𝑡∈0,𝜔}𝑢(𝑡)>𝑠₁ (2.17) or min_{[]𝑡∈0,𝜔}𝑢(𝑡)<𝑠₂. (2.18)

Corollary 2.4. Let (H2)–(H5) hold. Then(i)if 𝜆∈(𝜆₁/𝑓_∞,𝜆₁/𝑓₀], then (1.1) has at least two solutions 𝑢⁺_∞ and 𝑢⁻_∞, such that 𝑢⁺_∞ is positive on [0,𝜔] and 𝑢⁻_∞ is negative on [0,𝜔];(ii)if 𝜆∈(𝜆₁/𝑓₀,∞), then (1.1) has at least four solutions 𝑢⁺_∞, 𝑢⁻_∞, 𝑢⁺₀, and 𝑢⁻₀, such that 𝑢⁺_∞, 𝑢⁺₀ are positive on [0,𝜔] and 𝑢⁻_∞, 𝑢⁻₀ are negative on [0,𝜔].

Corollary 2.5. Let (H2)–(H5) hold. Then(i)if 𝜆∈(𝜆₁/𝑓₀,𝜆₁/𝑓_∞], then (1.1) has at least two solutions 𝑢⁺₀ and 𝑢⁻₀, such that 𝑢⁺₀ is positive on [0,𝜔] and 𝑢⁻₀ is negative on [0,𝜔];(ii)if 𝜆∈(𝜆₁/𝑓_∞,∞), then (1.1) has at least four solutions 𝑢⁺_∞, 𝑢⁻_∞, 𝑢⁺₀, and 𝑢⁻₀, such that 𝑢⁺_∞, 𝑢⁺₀ are positive on [0,𝜔] and 𝑢⁻_∞, 𝑢⁻₀ are negative on [0,𝜔].

3. Proof of the Main Results

To prove Theorem 2.3, we give a Proposition.

Proposition 3.1. (i) The first-order boundary value problem 𝑢^′(𝑡)+𝑎(𝑡)𝑢(𝑡)=ℎ(𝑡),𝑢(0)=𝑢(𝜔) (3.1) has a unique solution for all ℎ∈𝐿¹[0,𝜔] if and only if ∫^𝜔₀𝑎(𝑠)𝑑𝑠≠0.
(ii) Assume that 𝑢 is a solution of (3.1). If ℎ≥0 and ℎ(⋅)≢0 on any subinterval of [0,𝜔], then 𝑢(𝑡)∫^𝜔₀𝑎(𝑠)𝑑𝑠>0 on [0,𝜔].

Proof. (i) The equation 𝑢′(𝑡)+𝑎(𝑡)𝑢(𝑡)=0 has a solution 𝑢(𝑡)=𝐶𝑒^{−∫𝑡0𝑎(𝑠)𝑑𝑠}, where 𝐶 is a constant. If 𝑢(𝑡) is a nontrivial solution, then by 𝑢(0)=𝐶,𝑢(𝜔)=𝐶𝑒^{−∫𝜔0𝑎(𝑠)𝑑𝑠}, we can get that ∫^𝜔₀𝑎(𝑠)𝑑𝑠=0.
On the other hand, from ∫^𝜔₀𝑎(𝑠)𝑑𝑠=0, we can get that 𝑢′(𝑡)+𝑎(𝑡)𝑢(𝑡)=0 has a nontrivial solution 𝑢(𝑡)=𝐶𝑒^{−∫𝑡0𝑎(𝑠)𝑑𝑠}, where 𝐶∈ℝ⧵{0}.
(ii) We claim that 𝑢(𝑡)≠0,𝑡∈[0,𝜔]. Suppose on the contrary that there exists 𝑡₀∈[0,𝜔], such that 𝑢(𝑡₀)=0; it is not difficult to compute that 𝑢^′(𝑡)+𝑎(𝑡)𝑢(𝑡)=ℎ(𝑡),𝑢𝑡₀=𝑢₀ (3.2) has a solution 𝑢(𝑡)=^𝑡_𝑡0ℎ(𝑠)𝑒^{∫𝑠𝑡𝑎(𝜏)𝑑𝜏}𝑑𝑠. (3.3) Since ℎ≥0, we have 𝑢(0)≤𝑢𝑡₀≤𝑢(𝜔). (3.4)
If ℎ(𝑡)̂>0,𝑡̂∈[0,𝑡₀), then there exists a neighborhood 𝑈(𝑡)̂⊂[0,𝑡₀) of 𝑡̂, such that ℎ(𝑡)>0 on 𝑈(𝑡)̂. Thus, 𝑢(0)=∫⁰_𝑡0ℎ(𝑠)𝑒^{∫𝑠0𝑎(𝜏)𝑑𝜏}𝑑𝑠<0; this contradicts with 𝑢(0)=𝑢(𝜔).
If ℎ(𝑡)>0,𝑡∈(𝑡₀,𝜔], then there exists a neighborhood 𝑈(𝑡)⊂(𝑡₀,𝜔] of 𝑡, such that ℎ(𝑡)>0 on 𝑈(𝑡)̂. By using a similar way, we can prove that 𝑢(𝜔)>0, which also contradicts with 𝑢(0)=𝑢(𝜔).
Hence 𝑢(𝑡)≠0 on [0,𝜔]. Moreover, it follows that ^𝜔₀𝑢^′(𝑡)𝑢(𝑡)𝑑𝑡+^𝜔₀𝑎(𝑡)𝑑𝑡=^𝜔₀ℎ(𝑡)𝑢(𝑡)𝑑𝑡, (3.5) that is, (ln𝑢(𝑡))∣^𝜔₀+^𝜔₀𝑎(𝑡)𝑑𝑡=^𝜔₀ℎ(𝑡)𝑢(𝑡)𝑑𝑡. (3.6) Thus ∫^𝜔₀𝑎(𝑡)𝑑𝑡=∫^𝜔₀ℎ(𝑡)/𝑢(𝑡)𝑑𝑡, that is 𝑢∫^𝜔₀𝑎(𝑠)𝑑𝑠>0.

Next, we prove Theorem 2.3 and Corollaries 2.4 and 2.5.

Proof of Theorem 2.3. Suppose on the contrary that there exists (𝜆,𝑢)∈𝒞⁺∪𝒞⁻∪𝒟⁺∪𝒟⁻ such that either max{𝑢(𝑡)∣𝑡∈[0,𝜔]}=𝑠₁ (3.7) or min{𝑢(𝑡)∣𝑡∈[0,𝜔]}=𝑠₂. (3.8) We divide the proof into two cases.
Case 1 (max{𝑢(𝑡)|𝑡∈[0,𝜔]}=𝑠₁). In this case, we know that 0≤𝑢(𝑡)≤𝑠₁,0≤𝑢(𝑡−𝜏(𝑡))≤𝑠₁,𝑡∈[0,𝜔]. (3.9) Let us consider the functional differential equation 𝑢^′(𝑡)+𝑎(𝑡)𝑢(𝑡)=𝜆𝑏(𝑡)𝑓(𝑢(𝑡−𝜏(𝑡))),𝑡∈ℝ. (3.10) By (H2), (H4) and (H5), there exists 𝑚≥0 such that 𝑏(𝑡)𝑓(𝑠)+𝑚𝑠 is strictly increasing on 𝑠 for 𝑠∈[𝑠₂,𝑠₁]. Then (3.10) can be rewritten to the form 𝐿𝑢+𝜆𝑚𝑢(𝑡−𝜏(𝑡))=𝜆𝑏(𝑡)𝑓(𝑢(𝑡−𝜏(𝑡)))+𝑚𝑢(𝑡−𝜏(𝑡)), (3.11) and since 𝐿𝑠₁−𝑎(𝑡)𝑠₁=0=𝑓(𝑠₁), 𝐿𝑠₁−𝑎(𝑡)𝑠₁+𝜆𝑚𝑠₁=𝜆𝑏(𝑡)𝑓𝑠₁+𝑚𝑠₁. (3.12) Subtracting, we get 𝐿𝑠₁−𝑢+𝜆𝑚𝑠₁−𝑢(𝑡−𝜏(𝑡))−𝑎(𝑡)𝑠₁≥0. (3.13) That is, 𝐿𝑠₁−𝑢+𝜆𝑚𝑠₁[],𝑠≥0,𝑡∈0,𝜔₁−𝑢(0)=𝑠₁−𝑢(𝜔)>0. (3.14) From Proposition 3.1, we deduce that 𝑠₁>𝑢(𝑡),𝑡∈[0,𝜔], which contradicts with that max{𝑢(𝑡)|𝑡∈[0,𝜔]}=𝑠₁. Hence, 𝑢(𝑡)<𝑠₁,𝑡∈[0,𝜔]. (3.15)
Case 2 (min{𝑢(𝑡)|𝑡∈[0,𝜔]}=𝑠₂). In this case, we know that 𝑠₂≤𝑢(𝑡)≤0,𝑠₂≤𝑢(𝑡−𝜏(𝑡))≤0,𝑡∈[0,𝜔]. (3.16) Let us consider (3.10); by (H2), (H4), and (H5), there exists 𝑚≥0 such that 𝑏(𝑡)𝑓(𝑠)+𝑚𝑠 is strictly increasing in 𝑠 for 𝑠∈[𝑠₂,𝑠₁]. Then 𝐿𝑢+𝜆𝑚𝑢(𝑡−𝜏(𝑡))=𝜆𝑏(𝑡)𝑓(𝑢(𝑡−𝜏(𝑡)))+𝑚𝑢(𝑡−𝜏(𝑡)) (3.17) and since 𝐿𝑠₂−𝑎(𝑡)𝑠₂=0=𝑓(𝑠₂), 𝐿𝑠₂−𝑎(𝑡)𝑠₂+𝜆𝑚𝑠₂=𝜆𝑏(𝑡)𝑓𝑠₂+𝑚𝑠₂. (3.18) Subtracting, we get 𝐿𝑠₂−𝑢+𝜆𝑚𝑠₂−𝑢(𝑡−𝜏(𝑡))−𝑎(𝑡)𝑠₂≤0. (3.19) That is 𝐿𝑠₂−𝑢+𝜆𝑚𝑠₂[],𝑠≤0,𝑡∈0,𝜔₂−𝑢(0)=𝑠₂−𝑢(𝜔)<0. (3.20) From Proposition 3.1, we deduce that 𝑠₂−𝑢(𝑡)<0,𝑡∈[0,𝜔], this contradicts with that min{𝑢(𝑡)∣𝑡∈[0,𝜔]}=𝑠₂. Therefore, 𝑠₂<𝑢(𝑡),𝑡∈[0,𝜔]. (3.21)

Proof of Corollaries 2.4 and 2.5. Since boundary value problem 𝑢^′(𝑡)+𝑎(𝑡)𝑢(𝑡)=0,𝑢(0)=𝑢(𝜔) (3.22) has a unique solution 𝑢≡0, we get 𝒞⁺∪𝒞⁻∪𝒟⁺∪𝒟⁻⊂{(𝜆,𝑢)∈ℝ×𝐸∣𝜆≥0}. (3.23) Take Λ∈ℝ as an interval such that Λ∩{𝜆₁/𝑓_∞}={𝜆₁/𝑓_∞} and ℳ as a neighborhood of (𝜆₁/𝑓_∞,∞) whose projection on ℝ lies in Λ and whose projection on 𝐸 is bounded away from 0. Then by [15, Theorem 1.6, and Corollary 1.8], we have that for each 𝜈∈{+,−}, either(1)𝒟^𝜈⧵ℳ is bounded in ℝ×𝐸 in which case 𝒟^𝜈⧵ℳ meets {(𝜆,0)|𝜆∈ℝ}, or(2)𝒟^𝜈⧵ℳ is unbounded.
Moreover, if (1) occurs and 𝒟^𝜈⧵ℳ has a bounded projection on ℝ, then 𝒟^𝜈⧵ℳ meets (𝜆_𝑘/𝑓_∞,∞), where 𝜆_𝑘≠𝜆₁ is another eigenvalue of (2.7).
Obviously, Theorem 2.3 (ii) implies that (1) does not occur. So 𝒟⁺⧵ℳ is unbounded.
Remark 2.2 guarantees that 𝒟⁺ is a component of solutions of (2.12) in 𝑆⁺ which meets (𝜆₁/𝑓_∞,∞), and consequently Proj_ℝ(𝒟⁺⧵ℳ) is unbounded. Thus Proj_ℝ𝒟⁺⊃𝜆₁𝑓_∞,+∞. (3.24) Similarly, we get Proj_ℝ(𝒟⁻)⊃𝜆₁𝑓_∞,+∞. (3.25) By Theorem 2.3, for any (𝜆,𝑢)∈(𝒞⁺∪𝒞⁻), ‖𝑢‖_∞<max𝑠₁,𝑠₂∶=𝑠^∗. (3.26)  (3.26) and (2.12) imply that ‖𝑢‖<max𝑠^∗,‖𝑎‖_∞𝑠^∗+𝜆‖𝑏‖_∞max_{|𝑠|≤𝑠∗}𝑓(𝑠), (3.27) which means that the sets {(𝜆,𝑢)∈𝒞⁺∣𝜆∈[0,𝑑]} and {(𝜆,𝑢)∈𝒞⁻∣𝜆∈[0,𝑑]} are bounded for any fixed 𝑑∈(0,∞). This together with the fact that 𝒞⁺ and 𝒞⁻ join (𝜆₁/𝑓₀,0) to infinity yields, respectively, that Proj_ℝ𝒞⁺⊃𝜆₁𝑓₀,+∞,Proj_ℝ(𝒞⁻)⊃𝜆₁𝑓₀,+∞. (3.28) Combining (3.24), (3.25), and (3.28), we conclude the desired results.

Remark 3.2. The methods used in the proof of Theorem 2.3, Corollaries 2.4, and 2.5 have been used in the study of other kinds of boundary value problems; see [16–18] and the references therein.

Remark 3.3. The conditions in Corollaries 2.4 and 2.5 are sharp. Let us take 𝑎(𝑡)≡𝑎>0,𝜆=𝑎,𝑏(𝑡)=1,𝑓(𝑠)=𝑠+ℎ(𝑠),𝜏(𝑡)≡0. (3.29) Let ℎ(𝑠)=−2𝑠𝑠²()∪(),−+1,𝑠∈−∞,−11,+∞2𝑠³𝑠²[],+1,𝑠∈−1,1 (3.30) and consider problem 𝑢^′(𝑡)=−𝑎(𝑡)𝑢(𝑡)+𝑎[𝑢(𝑡)+ℎ(𝑢(𝑡))],𝑡∈[0,𝜔],𝑢(0)=𝑢(𝜔). (3.31) It is easy to see that 𝜆₁=𝑎,𝑓₀=𝑓_∞=1. Since 𝜆₁𝑓_∞𝜆=𝑎=¹𝑓₀, (3.32) the conditions of Corollaries 2.4 and 2.5 are not valid. In this case, (3.31) has no nontrivial solution. In fact, if 𝑢 is a nontrivial solution of (3.31), then 0=^𝜔₀𝑢^′(𝑡)𝑑𝑡=𝑎^𝜔₀ℎ(𝑢(𝑡))𝑑𝑡≠0, (3.33) which is a contradiction.

Acknowledgment

The paper is supported by the NSFC (no. 11061030), the Fundamental Research Funds for the Gansu Universities.