Bifurcation from Interval and Positive Solutions of a Nonlinear Second-Order Dynamic Boundary Value Problem on Time Scales

Abstract

Let 𝕋 be a time scale with 0,𝑇∈𝕋. We give a global description of the branches of positive solutions to the nonlinear boundary value problem of second-order dynamic equation on a time scale 𝕋, 𝑢^ΔΔ(𝑡)+𝑓(𝑡,𝑢^𝜎(𝑡))=0,𝑡∈[0,𝑇]_𝕋,𝑢(0)=𝑢(𝜎²(𝑇))=0, which is not necessarily linearizable. Our approaches are based on topological degree theory and global bifurcation techniques.

1. Introduction

Let 𝕋 be a time scale with 0,𝑇∈𝕋, we consider the existence of positive solutions, in this paper, for a nonlinear boundary value problem of second-order dynamic equation on a time scale 𝕋 as follows: 𝑢^ΔΔ(𝑡)+𝑓𝑡,𝑢^𝜎(𝑡)=0,𝑡∈[0,𝑇]_𝕋,𝑢(0)=𝑢𝜎²(𝑇)=0.(1.1) Research for the existence of solutions to the dynamic boundary value problem is rapidly growing in recent years. A great many existence results of positive solutions have been established for problem (1.1), see [1–5] and the references therein. The main tool used by them is the fixed point theorem in cones, and the key conditions in these papers do not depend on the first eigenvalue, 𝜆₁(𝑎), of the following linear problem: 𝑢^ΔΔ(𝑡)+𝜆𝑎(𝑡)𝑢^𝜎(𝑡)=0,𝑡∈[0,𝑇]_𝕋,𝑢(0)=𝑢𝜎²(𝑇)=0,(1.2) and the corresponding existence conditions are not optimal.

In 2006, for 0,1∈𝕋, 𝜎(0)=0,  𝜌(1)=1,  𝑓∈𝐶(ℝ) with 𝑠𝑓(𝑠)>0 for 𝑠≠0, Luo and Ma [6] obtained the existence of at least one positive solution to problem: 𝑢^ΔΔ(𝑡)+𝑓𝑢^𝜎(𝑡)=0,𝑡∈[0,1]_𝕋,𝑢(0)=𝑢(1)=0,(1.3) under the condition (H) if either 𝑓₀<𝜇₁<𝑓_∞ or 𝑓_∞<𝜇₁<𝑓₀, where 𝑓₀=lim_|𝑠|→0𝑓(𝑠)𝑠,𝑓_∞=lim_|𝑠|→∞𝑓(𝑠)𝑠,(1.4)and 𝜇₁ is the first eigenvalue of the linear problem: 𝑢^ΔΔ(𝑡)+𝜇𝑢^𝜎(𝑡)=0,𝑢(0)=𝑢(1)=0.(1.5) The approaches adopted by Luo and Ma [6] are based on global bifurcation techniques. They obtained the existence of at least one positive solution by considering the branches of solutions, which bifurcate from one point. The key conditions in [6] depend on the first eigenvalue of the corresponding linear problem and the condition (H) is optimal!

In this paper, we will use the following assumptions. (A1) 𝑓∶[0,𝜎(𝑇)]_𝕋×[0,∞)→[0,∞) is continuous and there exist functions 𝑎₀(⋅),𝑎⁰(⋅),𝑏_∞(⋅),𝑏^∞(⋅)∈𝐶([0,𝜎(𝑇)]_𝕋,(0,∞)), such that 𝑎₀(𝑡)𝑢−𝜉₁(𝑡,𝑢)≤𝑓(𝑡,𝑢)≤𝑎⁰(𝑡)𝑢+𝜉₂(𝑡,𝑢)(1.6) for some functions 𝜉_𝑖∈𝐶([0,𝜎(𝑇)]_𝕋×[0,∞),ℝ) with lim_𝑢→0⁺(𝜉_𝑖(𝑡,𝑢)/𝑢)=0  (𝑖=1,2) uniformly for 𝑡∈[0,𝜎(𝑇)]_𝕋, and 𝑏_∞(𝑡)𝑢−𝜁₁(𝑡,𝑢)≤𝑓(𝑡,𝑢)≤𝑏^∞(𝑡)𝑢+𝜁₂(𝑡,𝑢)(1.7) for some functions 𝜁_𝑖∈𝐶([0,𝜎(𝑇)]_𝕋×[0,∞),ℝ) with lim_𝑢→+∞(𝜁_𝑖(𝑡,𝑢)/𝑢)=0  (𝑖=1,2) uniformly for 𝑡∈[0,𝜎(𝑇)]_𝕋. (A2) 𝑓(𝑡,𝑢)>0 for (𝑡,𝑢)∈[0,𝜎(𝑇)]_𝕋×(0,∞). (A3) There exists a function 𝑐∈𝐶([0,𝜎(𝑇)]_𝕋,(0,∞)) such that 𝑓(𝑡,𝑢)≥𝑐(𝑡)𝑢,(𝑡,𝑢)∈[0,𝜎(𝑇)]_𝕋×[0,∞).(1.8)

Obviously, (A1) means that 𝑓 is not necessarily linearizable at 0 and +∞. We consider the existence of positive solutions of problem (1.1) in this paper by using bifurcation techniques. The difference from [6] is that the branches of positive solutions under consideration now bifurcate from not one point, but an interval. Our main idea is from [7], in which they considered positive solutions of fourth-order boundary value problems for differential equations. The main tool we will use is the following global bifurcation theorem for problems which is not necessarily linearizable.

Theorem A (Rabinowitz, [8]). Let 𝑉 be a real reflexive Banach space. Let 𝐹∶ℝ×𝑉→𝑉 be completely continuous such that 𝐹(𝜆,0)=0, for all 𝜆∈ℝ. Let 𝑎,𝑏∈ℝ(𝑎<𝑏) be such that 𝑢=0 is an isolated solution of the equation 𝑢−𝐹(𝜆,𝑢)=0,𝑢∈𝑉,(1.9) for 𝜆=𝑎, and 𝜆=𝑏, where (𝑎,0), (𝑏,0) are not bifurcation points of (1.9). Furthermore, assuming that deg𝐼−𝐹(𝑎,⋅),𝐵_𝑟(0),0≠deg𝐼−𝐹(𝑏,⋅),𝐵_𝑟(0),0,(1.10) where 𝐵_𝑟(0) is an isolating neighborhood of the nontrivial solution, and deg(𝐼−𝐹,𝐵_𝑟(0),0) denote the degree of 𝐼−𝐹 on 𝐵_𝑟(0) with respect to 0. Let 𝒮={(𝜆,𝑢)∶(𝜆,𝑢)isasolutionof(1.9)with𝑢≠0}∪([𝑎,𝑏]×{0}).(1.11) Then there exists a connected component 𝒞 of 𝒮 containing [𝑎,𝑏]×{0}, and either(i)𝒞 is unbounded, or(ii)𝒞∩[(ℝ⧵[𝑎,𝑏])×{0}]≠∅.

The rest of the paper is organized as follows. In Section 2, we firstly introduce the time scales concepts and notations that we will use in this paper. Next, Section 3 states some notations and proves some necessary preliminary results, and Section 4 studies the bifurcation from the trivial solution for a nonlinear problem which is not necessarily linearizable and then establishes our main result.

2. Introduction for Time Scales

A time scale 𝕋 is a nonempty closed subset of ℝ, assuming that 𝕋 has the topology that it inherits from the standard topology on ℝ. Define the forward and backward jump operators 𝜎,𝜌∶𝕋→𝕋 by 𝜎(𝑡)=inf{𝜏>𝑡∣𝜏∈𝕋},𝜌(𝑡)=sup{𝜏<𝑡∣𝜏∈𝕋}.(2.1) Here, we put inf∅=sup𝕋, sup∅=inf𝕋. Let 𝕋^𝑘 which is derived from the time scale 𝕋 be 𝕋^𝑘∶=𝑡∈𝕋∶𝑡isnonmaximalor𝜌(𝑡)=𝑡,(2.2) and 𝕋^𝑘2∶=𝕋^𝑘𝑘. Define interval 𝐼 on 𝕋 by 𝐼_𝕋=𝐼∩𝕋.

Definition 2.1. If 𝑢∶𝕋→ℝ is a function and 𝑡∈𝕋^𝑘, then the Δ-derivative of 𝑢 at the point 𝑡 is defined to be the number 𝑢^Δ(𝑡) (provided that it exists) with the property that for each 𝜀>0, there is a neighborhood 𝑈 of 𝑡 such that 𝑢(𝜎(𝑡))−𝑢(𝑠)−𝑢^Δ(𝑡)(𝜎(𝑡)−𝑠)⩽𝜀|𝜎(𝑡)−𝑠|(2.3) for all 𝑠∈𝑈. The function 𝑢 is called Δ-differentiable on 𝕋 if 𝑢^Δ(𝑡) exists for all 𝑡∈𝕋^𝑘.

The second Δ-derivative of 𝑢 at 𝑡∈𝕋^𝑘2, if it exists, is defined to be 𝑢^Δ2(𝑡)=𝑢^ΔΔ(𝑡)∶=(𝑢^Δ)^Δ(𝑡). We also define the function 𝑢^𝜎∶=𝑢∘𝜎 and 𝑢^𝜌∶=𝑢∘𝜌.

Definition 2.2. If 𝑈^Δ=𝑢 holds on 𝕋^𝑘, we define the Cauchy Δ-integral by ^𝑡_𝑠𝑢(𝜏)Δ𝜏=𝑈(𝑡)−𝑈(𝑠),𝑠,𝑡∈𝕋^𝑘.(2.4)

Lemma 2.3 (see [2, Theorems 2.7 and 2.8]). Assume 𝑎,𝑏∈𝕋, then ^𝑏_𝑎𝑓^Δ(𝑡)𝑔(𝑡)Δ𝑡=𝑓(𝑡)𝑔(𝑡)^𝑏_𝑎−^𝑏_𝑎𝑓^𝜎(𝑡)𝑔^Δ(𝑡)Δ𝑡.(2.5) Furthermore, if 𝑎≤𝑏, 𝑓(𝑡) is a continuous function on [𝑎,𝑏], then ^𝑏_𝑎𝑓(𝑡)Δ𝑡=_𝑎^𝜌(𝑏)𝑓(𝑡)Δ𝑡+𝑏−𝜌(𝑏)𝑓𝜌(𝑏).(2.6)

Define the Banach space C(𝕋,ℝ)  (denoted by C(𝕋)) to be the set of continuous functions 𝑢∶𝕋→ℝ with the norm ‖𝑢‖₀=sup{|𝑢(𝑡)|∣𝑡∈𝕋}.(2.7) For 𝑖=1,2, we define the Banach space C^𝑖(𝕋) to be the set of the 𝑖th Δ-differential functions 𝑢∶𝕋→ℝ for which 𝑢^Δ𝑖∈C(𝕋^𝑘𝑖) with the norm ‖𝑢‖_𝑖=sup‖𝑢‖_∞,𝑢^Δ₀,…,𝑢^Δ𝑖₀,(2.8) where 𝑢^Δ𝑗₀=sup𝑢^Δ𝑗(𝑡)∣𝑡∈𝕋^𝑘𝑗,𝑗=0,1,…,𝑖.(2.9)

3. Preliminaries and Necessary Lemmas

Assuming that 𝑎∈C([0,𝜎²(𝑇)]_𝕋,(0,∞)), then from [9, Theorem 2.9], linear problem 𝑢^ΔΔ(𝑡)+𝜆𝑎(𝑡)𝑢^𝜎(𝑡)=0,𝑡∈[0,𝑇]_𝕋,𝑢(0)=𝑢𝜎²(𝑇)=0(3.1) has a unique principal eigenvalue 𝜆₁(𝑎), with a corresponding positive eigenfunction.

Let 𝐸=C²[0,𝑇]_𝕋,  𝑋=C[0,𝑇]_𝕋, and 𝑌=𝑦∈C¹0,𝜎²(𝑇)_𝕋∣𝑦(0)=𝑦𝜎²(𝑇)=0.(3.2) We will work essentially in the Banach space 𝑌 with the norm 𝑦=max𝑦₀,𝑦^Δ₀,(3.3) where 𝑦₀=sup𝑦(𝑡)∣𝑡∈0,𝜎²(𝑇)_𝕋,𝑦^Δ₀=sup𝑦^Δ(𝑡)∣𝑡∈[0,𝜎(𝑇)]_𝕋.(3.4) By a positive solution of problem (1.1), we mean 𝑢 is a solution of (1.1) with 𝑢≥0 in (0,𝜎²(𝑇))_𝕋 and 𝑢≢0.

Lemma 3.1. For 𝑦∈𝑌, we have 𝑦₀≤𝜎(𝑇)𝑦^Δ₀1,max1,𝜎(𝑇)𝑦₀≤𝑦≤max{1,𝜎(𝑇)}𝑦^Δ₀.(3.5)

Proof. By 𝑦(0)=0, we have that 𝑦(𝑡)=^𝑡₀𝑦^Δ(𝑠)Δ𝑠,𝑡∈[0,𝜎(𝑇)]_𝕋(3.6) and so 𝑦₀≤𝜎(𝑇)𝑦^Δ₀.(3.7) Therefore, 𝑦=max𝑦₀,𝑦^Δ₀≤max{1,𝜎(𝑇)}𝑦^Δ₀,𝑦≥max𝑦₀,1𝜎(𝑇)𝑦₀1=max1,𝜎(𝑇)𝑦₀.(3.8)

Define the linear operator 𝐿∶𝐷(𝐿)⊂𝐸→𝑋, 𝐿𝑢=−𝑢^ΔΔ,𝑢∈𝐷(𝐿)(3.9) with 𝐷(𝐿)=𝑢∈𝐸∣𝑢(0)=𝑢𝜎²(𝑇)=0.(3.10) Then 𝐿 is a closed operator, and 𝐿⁻¹∶𝑋→𝑌 is completely continuous, see [10, Lemma 3.7].

Let Σ⊂ℝ⁺×𝑌 be the closure of the set of positive solutions to the problem 𝐿𝑢(𝑡)=𝜆𝑓𝑡,𝑢^𝜎(𝑡),𝑡∈[0,𝜎(𝑇)]_𝕋.(3.11) We extend the function 𝑓 to a continuous function 𝑓 defined on [0,𝜎(𝑇)]_𝕋×ℝ by 𝑓()=𝑓(),()∈[(𝑇𝑡,𝑢𝑡,𝑢𝑡,𝑢0,𝜎)]_𝕋×[),𝑓(),()∈[(𝑇0,∞𝑡,0𝑡,𝑢0,𝜎)]_𝕋×(−∞,0).(3.12) Then 𝑓(𝑡,𝑢)≥0 on [0,𝜎(𝑇)]_𝕋×ℝ. For 𝜆≥0, the arbitrary solution 𝑢 to the eigenvalue problem 𝑢^ΔΔ(𝑡)+𝜆𝑓𝑡,𝑢^𝜎(𝑡)=0,𝑡∈[0,𝑇]_𝕋,𝑢(0)=𝑢𝜎²(𝑇)=0(3.13) satisfies that 𝑢^ΔΔ(𝑡)≤0 on [0,𝑇]_𝕋, and consequently, the graph of 𝑢 is concave down on [0,𝜎²(𝑇)]_𝕋. This together with the boundary conditions 𝑢(0)=𝑢(𝜎²(𝑇))=0 imply that 𝑢(𝑡)≥0,𝑡∈0,𝜎²(𝑇)_𝕋.(3.14) Thus, (3.14) implies that 𝑢 is a nonnegative solution of problem (3.13), and the closure of the set of nontrivial solutions (𝜆,𝑢) of (3.13) in ℝ⁺×𝑌 is exactly Σ.

Let 𝑔∈C([0,𝜎(𝑇)]_𝕋×ℝ,ℝ), and let 𝑁∶𝑌→𝑋 be the Nemytskii operator associated with the function 𝑔: 𝑁(𝑢)(𝑡)=𝑔𝑡,𝑢^𝜎(𝑡),𝑢∈𝑌.(3.15)

Lemma 3.2. Let 𝑔(𝑡,𝑢)≥0 on [0,𝜎(𝑇)]_𝕋×ℝ. Let 𝑢∈𝐷(𝐿) be such that 𝐿𝑢≥𝜆𝑁(𝑢) in [0,𝑇]_𝕋, 𝜆≥0. Then 𝑢(𝑡)≥0,𝑡∈0,𝜎²(𝑇)_𝕋.(3.16) Moreover, 𝑢(𝑡)>0,𝑡∈0,𝜎²(𝑇)_𝕋,(3.17) whenever 𝑢≢0.

Let 𝑁∶𝑌→𝑋 be the Nemytskii operator associated with the function 𝑓𝑁(𝑢)(𝑡)=𝑓𝑡,𝑢^𝜎(𝑡),𝑢∈𝑌.(3.18) Then (3.13), with 𝜆≥0, is equivalent to the operator equation 𝑢=𝜆𝐿⁻¹𝑁(𝑢),𝑢∈𝑌.(3.19) In the following we will apply the Brouwer degree theory, mainly to the mapping Φ_𝜆∶𝑌→𝑌, Φ_𝜆(𝑢)=𝑢−𝜆𝐿⁻¹𝑁(𝑢).(3.20) For 𝑅>0, let 𝐵_𝑅={𝑢∈𝑌∶‖𝑢‖<𝑅}.

Lemma 3.3. Let Λ⊂ℝ⁺ be a compact interval with [𝜆₁(𝑎⁰),𝜆₁(𝑎₀)]∩Λ=∅. Then there exists a number 𝛿₁>0 with the property Φ_𝜆(𝑢)≠0,∀𝑢∈𝑌∶0<‖𝑢‖≤𝛿₁,∀𝜆∈Λ.(3.21)

Proof. Suppose to the contrary that there exist sequences {𝜇_𝑛} in Λ and {𝑢_𝑛} in 𝑌∶𝜇_𝑛→𝜇^∗∈Λ,‖𝑢_𝑛‖>0 and 𝑢_𝑛→0  (𝑛→∞) in 𝑌, such that Φ_𝜇𝑛(𝑢_𝑛)=0 for all 𝑛∈ℕ. By Lemma 3.2, 𝑢_𝑛(𝑡)≥0 for 𝑡∈[0,𝜎²(𝑇)]_𝕋.
Set 𝑣_𝑛=𝑢_𝑛/‖𝑢_𝑛‖. Then from 𝐿𝑢_𝑛=𝜇_𝑛𝑁(𝑢_𝑛), we have 𝐿𝑣_𝑛=𝜇_𝑛‖𝑢_𝑛‖⁻¹𝑁(𝑢_𝑛). Since ‖𝑢_𝑛‖⁻¹𝑁(𝑢_𝑛) is bounded in 𝑋, we infer that {𝑣_𝑛} is relatively compact in 𝑌, hence (for a subsequence) 𝑣_𝑛→𝑣(𝑛→∞) with 𝑣≥0 in 𝑌, ‖𝑣‖=1. Let 𝐼^𝜎∶C_rd[0,𝜎²(𝑇)]_𝕋→C_rd[0,𝜎(𝑇)]_𝕋 be defined as 𝐼^𝜎(𝑦)=𝑦^𝜎. Then 𝐼^𝜎 is linear. For 𝑢∈𝑌, ‖𝐼^𝜎𝑢‖₀=sup_{𝑡∈[0,𝜎(𝑇)]𝕋}|𝑢^𝜎(𝑡)|=‖𝑢‖₀.
Now, from condition (A1), we have that 𝑎₀(𝑡)𝑢^𝜎_𝑛(𝑡)−𝜉₁𝑡,𝑢^𝜎_𝑛(𝑡)≤𝑓𝑡,𝑢^𝜎_𝑛(𝑡)≤𝑎⁰(𝑡)𝑢^𝜎_𝑛(𝑡)+𝜉₂𝑡,𝑢^𝜎_𝑛(𝑡).(3.22) According to 𝑢^𝜎_𝑛(𝑡)𝑢_𝑛=𝐼^𝜎𝑢_𝑛𝑢_𝑛=𝐼^𝜎𝑢_𝑛𝑢_𝑛=𝐼^𝜎𝑣_𝑛=𝑣^𝜎_𝑛,(3.23) we get 𝜇_𝑛𝑎₀(𝑡)𝑣^𝜎_𝑛(𝑡)−𝜉₁𝑡,𝑢^𝜎_𝑛(𝑡)𝑢_𝑛≤𝜇_𝑛𝑓𝑡,𝑢^𝜎_𝑛(𝑡)𝑢_𝑛≤𝜇_𝑛𝑎⁰(𝑡)𝑣^𝜎_𝑛(𝑡)+𝜉₂𝑡,𝑢^𝜎_𝑛(𝑡)𝑢_𝑛.(3.24)
Let 𝜑⁰ and 𝜑₀ denote the eigenfunctions corresponding to 𝜆₁(𝑎⁰) and 𝜆₁(𝑎₀), respectively. Denote 𝜉₁(𝑦)∶=max_{(𝑡,𝑠)∈[0,𝜎(𝑇)]𝕋×[0,𝑦]}|𝜉₁(𝑡,𝑠)|. Then 𝜉₁(𝑦) is nondecreasing. From lim_𝑢→0⁺(𝜉₁(𝑡,𝑢)/𝑢)=0 uniformly for 𝑡∈[0,𝜎(𝑇)]_𝕋, we have lim_𝑦→0+𝜉₁𝑦𝑦=0.(3.25) According to 𝐿𝑢_𝑛=𝜇_𝑛𝑓(𝑡,𝑢^𝜎_𝑛(𝑡)) and 𝑢_𝑛(𝑡)≥0, we have from the first inequality in (3.24) that ₀^𝜎(𝑇)𝜇_𝑛𝑎₀(𝑡)𝑣^𝜎_𝑛(𝑡)−𝜉₁𝑡,𝑢^𝜎_𝑛(𝑡)𝑢_𝑛𝜑^𝜎₀(𝑡)Δ𝑡≤₀^𝜎(𝑇)𝐿𝑢_𝑛(𝑡)𝑢_𝑛𝜑^𝜎₀(𝑡)Δ𝑡=₀^𝜎(𝑇)𝐿𝑣_𝑛(𝑡)𝜑^𝜎₀(𝑡)Δ𝑡.(3.26) Notice that 𝜉₁𝑡,𝑢^𝜎_𝑛(𝑡)𝑢_𝑛≤𝜉₁𝐼^𝜎𝑢_𝑛0max{1,1/𝜎(𝑇)}𝑢_𝑛0≤𝜉₁𝐼^𝜎𝑢_𝑛0𝐼^𝜎𝑢_𝑛0→0(𝑛→∞)(3.27) by Lemma 3.1. Let 𝑛→∞, by integration by parts (2.5), we have ^𝜎2₀^(𝑇)𝜇^∗𝑎₀(𝑡)𝑣^𝜎(𝑡)𝜑^𝜎₀(𝑡)Δ𝑡≤^𝜎2₀^(𝑇)𝐿𝑣(𝑡)𝜑^𝜎₀(𝑡)Δ𝑡=^𝜎2₀^(𝑇)𝐿𝜑₀(𝑡)𝑣^𝜎(𝑡)=Δ𝑡_𝜎20(𝑇)𝜆₁𝑎₀𝑎₀(𝑡)𝜑^𝜎₀(𝑡)𝑣^𝜎(𝑡)Δ𝑡,(3.28) and consequently 𝜇^∗≤𝜆₁𝑎₀.(3.29) Similarly, we deduce from the second inequality in (3.24) that 𝜆₁𝑎⁰≤𝜇^∗.(3.30) Thus, 𝜆₁(𝑎⁰)≤𝜇^∗≤𝜆₁(𝑎₀). This contradicts 𝜇^∗∈Λ.

Remark 3.4. If 𝜎(𝑇)<𝜎²(𝑇), then the value of 𝜎²(𝑇) do not contribute to the value of (3.28). Thus we can discuss the integral from 0 to 𝜎²(𝑇). For 𝐿𝑣 and 𝐿𝜑₀ are not defined at 𝜎(𝑇), we may define the values of (𝐿𝑣)𝜑^𝜎₀ and (𝐿𝜑₀)𝑣^𝜎 to be zero at 𝜎(𝑇), since the functions 𝜑^𝜎₀ and 𝑣^𝜎 are zero at 𝜎(𝑇). The details of the discussion can be found in [9, Page 497].

Corollary 3.5. For 𝜆∈(0,𝜆₁(𝑎⁰)) and 𝛿∈(0,𝛿₁), deg(Φ_𝜆,𝐵_𝛿,0)=1.

Proof. Lemma 3.3, applied to the interval Λ=[0,𝜆], guarantees the existence of 𝛿₁>0 such that for 𝛿∈(0,𝛿₁)𝑢−𝜏𝜆𝐿⁻¹𝑁(𝑢)≠0,∀𝑢∈𝑌∶0<‖𝑢‖≤𝛿,𝜏∈[0,1].(3.31) Hence, for any 𝛿∈(0,𝛿₁), degΦ_𝜆,𝐵_𝛿,0=deg𝐼,𝐵_𝛿,0=1,(3.32) which implies the assertion.

On the other hand, we have the following.

Lemma 3.6. Suppose 𝜆>𝜆₁(𝑎₀). Then there exists 𝛿₂>0 such that for all𝑢∈𝑌 with 0<‖𝑢‖≤𝛿₂, for all𝜏≥0, Φ_𝜆(𝑢)≠𝜏𝜑₀,(3.33) where 𝜑₀ is the positive eigenfunction corresponding to 𝜆₁(𝑎₀).

Proof. We assume again to the contrary that there exist 𝜏_𝑛≥0 and a sequence {𝑢_𝑛} with ‖𝑢_𝑛‖>0 and 𝑢_𝑛→0 in 𝑌 such that Φ_𝜆(𝑢_𝑛)=𝜏_𝑛𝜑₀ for all 𝑛∈ℕ. As 𝐿𝑢_𝑛=𝜆𝑁𝑢_𝑛+𝜏_𝑛𝜆₁𝑎₀𝑎₀(𝑡)𝜑^𝜎₀(𝑡)(3.34) and 𝜏_𝑛𝜆₁(𝑎₀)𝑎₀(𝑡)𝜑^𝜎₀(𝑡)≥0 in [0,𝑇]_𝕋, we can conclude from Lemma 3.2 that 𝑢_𝑛(𝑡)≥0 for 𝑡∈[0,𝜎²(𝑇)]_𝕋.
Notice that 𝑢_𝑛∈𝐷(𝐿) has a unique orthogonal decomposition 𝑢_𝑛=𝑤_𝑛+𝑠_𝑛𝜑₀,(3.35) with 𝑠_𝑛∈ℝ. Since 𝑢_𝑛≥0 on [0,𝜎²(𝑇)]_𝕋 and ‖𝑢_𝑛‖>0, we have from (3.35) that 𝑠_𝑛>0.
Choose 𝜎>0 such that 𝜎<𝜆−𝜆₁𝑎₀𝜆.(3.36) By (A1), there exists 𝑟₄>0, such that 𝑓(𝑡,𝑢)≥(1−𝜎)𝑎₀(𝑡)𝑢,∀(𝑡,𝑢)∈[0,𝜎(𝑇)]_𝕋×0,𝑟₄.(3.37) Since ‖𝑢_𝑛‖→0, there exists 𝑁^∗>0, such that 0≤𝑢_𝑛≤𝑟₄,∀𝑛≥𝑁^∗,(3.38) and consequently 𝑓𝑡,𝑢^𝜎_𝑛(𝑡)≥(1−𝜎)𝑎₀(𝑡)𝑢^𝜎_𝑛(𝑡),∀𝑛≥𝑁^∗.(3.39) Applying (3.35) and (3.39), it follows that 𝑠_𝑛𝜆₁𝑎₀^𝜎2₀^(𝑇)𝑎₀(𝑡)𝜑^𝜎₀(𝑡)²Δ𝑡=^𝜎2₀^(𝑇)𝑤^𝜎_𝑛(𝑡)+𝑠_𝑛𝜙^𝜎₀(𝑡)𝐿𝜑₀(𝑡)Δ𝑡=^𝜎2₀^(𝑇)𝑢^𝜎_𝑛(𝑡)𝐿𝜑₀(𝑡)=Δ𝑡_𝜎20(𝑇)𝜑^𝜎₀(𝑡)𝐿𝑢_𝑛(𝑡)Δ𝑡=𝜆^𝜎2₀^(𝑇)𝑁𝑢_𝑛𝜑^𝜎₀(𝑡)Δ𝑡+𝜏_𝑛𝜆₁𝑎₀^𝜎2₀^(𝑇)𝑎₀(𝑡)𝜑^𝜎₀(𝑡)²Δ𝑡≥𝜆^𝜎2₀^(𝑇)𝑁𝑢_𝑛𝜑^𝜎₀(𝑡)()Δ𝑡≥𝜆1−𝜎^𝜎2₀^(𝑇)𝑎₀(𝑡)𝑢^𝜎_𝑛(𝑡)𝜑^𝜎₀(𝑡)()𝑠Δ𝑡=𝜆1−𝜎_𝑛^𝜎2₀^(𝑇)𝑎₀(𝑡)𝜑^𝜎₀(𝑡)²Δ𝑡.(3.40) Thus, 𝜆₁𝑎₀≥𝜆(1−𝜎).(3.41) This contradicts (3.36).

Corollary 3.7. For 𝜆>𝜆₁(𝑎₀) and 𝛿∈(0,𝛿₂), deg(Φ_𝜆,𝐵_𝛿,0)=0.

Proof. Let 0<𝜖≤𝛿₂, where 𝛿₂ is the number asserted in Lemma 3.6. As Φ_𝜆 is bounded in 𝐵_𝜖, there exists 𝑐>0 such that Φ_𝜆(𝑢)≠𝑐𝜑₀, for all𝑢∈𝐵_𝜖. By Lemma 3.6, Φ_𝜆(𝑢)≠𝑡𝑐𝜑₀,𝑢∈𝜕𝐵_𝜖,𝑡∈[0,1].(3.42) Hence, degΦ_𝜆,𝐵_𝜖,0=degΦ_𝜆−𝑐𝜑₀,𝐵_𝜖,0=0.(3.43)

Now, using Theorem A, we may prove the following.

Lemma 3.8. [𝜆₁(𝑎⁰),𝜆₁(𝑎₀)] is a bifurcation interval from the trivial solution for (3.19). There exists an unbounded component 𝒞 of positive solutions of (3.19) which meets [𝜆₁(𝑎⁰),𝜆₁(𝑎₀)]×{0}. Moreover, 𝒞∩ℝ⧵𝜆₁𝑎⁰,𝜆₁𝑎₀×{0}=∅.(3.44)

Proof. For fixed 𝑛∈ℕ with 𝜆₁(𝑎⁰)−1/𝑛>0, set 𝑎_𝑛=𝜆₁(𝑎⁰)−1/𝑛<𝜆₁(𝑎⁰),  𝑏_𝑛=𝜆₁(𝑎₀)+1/𝑛>𝜆₁(𝑎₀) and 𝛿=min{𝛿₁,𝛿₂}. It is easy to check that for 0<𝛿<𝛿, all of the conditions of Theorem A are satisfied. So there exists a connected component 𝒞_𝑛 of solutions of (3.19) containing [𝑎_𝑛,𝑏_𝑛]×{0}, and either(i)𝒞_𝑛 is unbounded, or(ii)𝒞_𝑛∩[(ℝ⧵[𝑎_𝑛,𝑏_𝑛])×{0}]≠∅.
By Lemma 3.3, the case (ii) cannot occur. Thus, 𝒞_𝑛 is unbounded bifurcated from [𝑎_𝑛,𝑏_𝑛]×{0} in ℝ×𝑌. Furthermore, we have also from Lemma 3.3 that for any closed interval 𝐼⊂[𝑎_𝑛,𝑏_𝑛]⧵[𝜆₁(𝑎⁰),𝜆₁(𝑎₀)], if 𝑢∈{𝑦∈𝑌∣(𝜆,𝑦)∈𝒞_𝑛,𝜆∈𝐼}, then the fact that ‖𝑢‖→0 in 𝑌 is impossible. So 𝒞_𝑛 must be bifurcated from [𝜆₁(𝑎⁰),𝜆₁(𝑎₀)]×{0} in ℝ×𝑌.

4. The Main Result

We obtain the following main result in this paper.

Theorem 4.1. Let (A1), (A2), and (A3) hold. Assuming that either 𝜆₁𝑏_∞<1<𝜆₁𝑎⁰(4.1) or 𝜆₁𝑎₀<1<𝜆₁𝑏^∞.(4.2) Then problem (1.1) has at least one positive solution.

Proof. It is clear that any solution to (3.19) of the form (1,𝑢) yields a solution 𝑢 of problem (1.1). We will show that 𝒞 crosses the hyperplane {1}×𝑌 in ℝ×𝑌. To do this, it is enough to show that 𝒞 joins [𝜆₁(𝑎⁰),𝜆₁(𝑎₀)]×{0} to [𝜆₁(𝑏^∞),𝜆₁(𝑏_∞)]×{∞}. Let (𝜂_𝑛,𝑦_𝑛)∈𝒞 satisfy 𝜂_𝑛+𝑦_𝑛→∞.(4.3) We note that 𝜂_𝑛>0 for all 𝑛∈ℕ since (0,0) is the only solution to (3.19) for 𝜆=0, and 𝒞∩({0}×𝑌)=∅.
Case  1 (𝜆₁(𝑏_∞)<1<𝜆₁(𝑎⁰)). In this case, we show that the interval𝜆₁𝑏_∞,𝜆₁𝑎⁰⊆{𝜆∈ℝ∣(𝜆,𝑢)∈𝒞}.(4.4) We divide the proof into two steps.
Step  1. We show that {𝜂_𝑛} is bounded.
Since (𝜂_𝑛,𝑦_𝑛)∈𝒞,  𝐿𝑦_𝑛=𝜂_𝑛𝑓(𝑡,𝑦^𝜎_𝑛(𝑡)). From (A3), we have 𝐿𝑦_𝑛≥𝜂_𝑛𝑐(𝑡)𝑦^𝜎_𝑛(𝑡).(4.5)
Let 𝜑 denote the nonnegative eigenfunction corresponding to 𝜆₁(𝑐). From (4.5), by integration by parts formula (2.5), we have 𝜆₁(𝑐)^𝜎2₀^(𝑇)𝑦^𝜎_𝑛(𝑡)𝑐(𝑡)𝜑^𝜎(𝑡)Δ𝑡=^𝜎2₀^(𝑇)𝐿𝜑(𝑡)𝑦^𝜎_𝑛(𝑡)Δ𝑡=^𝜎2₀^(𝑇)𝐿𝑦_𝑛(𝑡)𝜑^𝜎(𝑡)Δ𝑡≥𝜂_𝑛^𝜎2₀^(𝑇)𝑐(𝑡)𝑦^𝜎_𝑛(𝑡)𝜑^𝜎(𝑡)Δ𝑡.(4.6) Thus, 𝜂_𝑛≤𝜆₁(𝑐).(4.7)
Step  2. We show that 𝒞 joins [𝜆₁(𝑎⁰),𝜆₁(𝑎₀)]×{0} to [𝜆₁(𝑏^∞),𝜆₁(𝑏_∞)]×{∞}.
From (4.3) and (4.7), we have that ‖𝑦_𝑛‖→∞. Notice that (3.19) is equivalent to the equation 𝑦_𝑛(𝑡)=𝜂_𝑛^𝜎2₀^(𝑇)𝐻(𝑡,𝑠)𝑓𝑠,𝑦^𝜎_𝑛(𝑠)Δ𝑠,(4.8) where 𝐻(𝑡,𝑠) is the Green’s function for problem −𝑢^ΔΔ=0,  𝑢(0)=𝑢(𝜎²(𝑇))=0. So we have from (A1), 𝜂_𝑛^𝜎2₀^(𝑇)𝐻(𝑡,𝑠)𝑏^∞(𝑠)𝑦^𝜎_𝑛(𝑠)+𝜁₂𝑠,𝑦^𝜎_𝑛(𝑠)Δ𝑠≥𝑦_𝑛(𝑡)≥𝜂_𝑛^𝜎2₀^(𝑇)𝐻(𝑡,𝑠)𝑏_∞(𝑠)𝑦^𝜎_𝑛(𝑠)+𝜁₁𝑠,𝑦^𝜎_𝑛(𝑠)Δ𝑠.(4.9) We divide the both sides of (4.9) by ‖𝑦_𝑛‖ and set 𝑣_𝑛=𝑦_𝑛/‖𝑦_𝑛‖. Since 𝑣_𝑛 is bounded in 𝑌, choosing a subsequence and relabelling if necessary, we see that 𝑣_𝑛→𝑣  (𝑛→∞) for some 𝑣∈𝑌 with 𝑣≥0 in 𝑌, ‖𝑣‖=1 and 𝜂_𝑛→𝜂^∗  (𝑛→∞). Similar to the proof of Lemma 3.3, we have that lim_𝑛→∞𝜁_𝑖𝑠,𝑦^𝜎_𝑛(𝑠)𝑦_𝑛=0,𝑖=1,2,(4.10) and 𝑛→∞, it is easy to verify that 𝜂^∗𝜎2₀^(𝑇)𝐻(𝑡,𝑠)𝑏^∞(𝑠)𝑣^𝜎(𝑠)Δ𝑠≥𝑣(𝑡)≥𝜂^∗𝜎2₀^(𝑇)𝐻(𝑡,𝑠)𝑏_∞(𝑠)𝑣^𝜎(𝑠)Δ𝑠,(4.11) which implies that 𝜂^∗𝑏_∞(𝑡)𝑣^𝜎(𝑡)≤𝐿𝑣≤𝜂^∗𝑏^∞(𝑡)𝑣^𝜎(𝑡).(4.12)
Let 𝜑^∞ and 𝜑_∞ denote the nonnegative eigenfunctions corresponding to 𝜆₁(𝑏^∞) and 𝜆₁(𝑏_∞), respectively. Then we have from the first inequality in (4.12) that ^𝜎2₀^(𝑇)𝜂^∗𝑏_∞(𝑡)𝑣^𝜎(𝑡)𝜑^𝜎_∞(𝑡)Δ𝑡≤^𝜎2₀^(𝑇)𝐿𝑣(𝑡)𝜑^𝜎_∞(𝑡)Δ𝑡.(4.13) By integration by parts formula (2.5), we obtain that 𝜂^∗𝜎2₀^(𝑇)𝑏_∞(𝑡)𝜑^𝜎_∞(𝑡)𝑣^𝜎(𝑡)Δ𝑡≤^𝜎2₀^(𝑇)𝐿𝜑_∞(𝑡)𝑣^𝜎(𝑡)Δ𝑡=𝜆₁𝑏_∞^𝜎2₀^(𝑇)𝑏_∞(𝑡)𝜑^𝜎_∞(𝑡)𝑣^𝜎(𝑡)Δ𝑡,(4.14) and consequently 𝜂^∗≤𝜆₁𝑏_∞.(4.15) Similarly, we deduce from the second inequality in (4.12) that 𝜆₁𝑏^∞≤𝜂^∗.(4.16) Thus, 𝜆₁𝑏^∞≤𝜂^∗≤𝜆₁𝑏_∞.(4.17) So 𝒞 joins [𝜆₁(𝑎⁰),𝜆₁(𝑎₀)]×{0} to [𝜆₁(𝑏^∞),𝜆₁(𝑏_∞)]×{∞}.
Case  2 (𝜆₁(𝑎₀)<1<𝜆₁(𝑏^∞)). In this case, if (𝜂_𝑛,𝑦_𝑛)∈𝒞 is such that lim_𝑛→∞𝜂_𝑛+𝑦_𝑛=∞,lim_𝑛→∞𝜂_𝑛=∞,(4.18) then 𝜆₁𝑎₀,𝜆₁𝑏^∞⊆{𝜆∈(0,∞)∣(𝜆,𝑢)∈𝒞}(4.19) and, moreover, ({1}×𝑌)∩𝒞≠∅.(4.20)
Assuming that {𝜂_𝑛} is bounded, applying a similar argument to that used in Step  2 of Case  1, after taking a subsequence and relabelling if necessary, it follows that 𝜂_𝑛→𝜂^∗∈𝜆₁𝑎₀,𝜆₁𝑏^∞,𝑦_𝑛→∞,as𝑛→∞.(4.21) Again 𝒞 joins [𝜆₁(𝑎⁰),𝜆₁(𝑎₀)]×{0} to [𝜆₁(𝑏^∞),𝜆₁(𝑏_∞)]×{∞} and the result follows.

Acknowledgments

This work supported by China Postdoctoral Science Foundation funded Project (no. 201104602 and no. 20100481239), General Project for Scientific Research of Liaoning Educational Committee (no. L2011200), Teaching and Research Project of DUFE (no. YY12012), and the NSFC (no. 71201019).