Positive Solutions of a Nonlinear Fourth-Order Dynamic Eigenvalue Problem on Time Scales

Abstract

Let 𝕋 be a time scale and 𝑎,𝑏∈𝕋, 𝑎<𝜌²(𝑏). We study the nonlinear fourth-order eigenvalue problem on 𝕋, 𝑢^Δ4(𝑡)=𝜆ℎ(𝑡)𝑓(𝑢(𝑡),𝑢^Δ2(𝑡)), 𝑡∈[𝑎,𝜌²(𝑏)]_𝕋, 𝑢(𝑎)=𝑢^Δ(𝜎(𝑏))=𝑢^Δ2(𝑎)=𝑢^Δ3(𝜌(𝑏))=0 and obtain the existence and nonexistence of positive solutions when 0<𝜆≤𝜆^∗ and 𝜆>𝜆^∗, respectively, for some 𝜆^∗. The main tools to prove the existence results are the Schauder fixed point theorem and the upper and lower solution method.

1. Introduction

The deformation of an elastic beam with one end fixed and the other end free can be described by the nonlinear fourth-order dynamic eigenvalue problem on𝕋𝑢^Δ4(𝑡)=𝜆ℎ(𝑡)𝑓𝑢(𝑡),𝑢^Δ2(𝑡),𝑡∈𝑎,𝜌²(𝑏)_𝕋,𝑢(𝑎)=𝑢^Δ(𝜎(𝑏))=𝑢^Δ2(𝑎)=𝑢^Δ3𝜌(𝑏)=0, (1.1) where 𝕋 is a time scale, 𝜆>0 is a parameter, 𝑎,𝑏∈𝕋, and 𝑎<𝜌²(𝑏).

Nonlinear dynamic eigenvalue problems of the above type have been studied by some authors, but most of them study only second-order dynamic equations. In 2000, Chyan and Henderson [1] obtained the existence of at least one positive solution for some 𝜆 to second-order case of the dynamic equation in problem (1.1) under conjugate boundary value condition and right focal boundary value condition, respectively. Anderson [2] discussed the same second-order dynamic equation under the Sturm-Liouville boundary value condition and directly generalized the result of [1]. Erbe et al. [3] then studied the general second-order Sturm-Liouville dynamic boundary value problem and obtained the existence, nonexistence, and multiplicity results of positive solutions. In 2005, Li and Liu [4] further studied the dependence of positive solutions on the parameter 𝜆>0 for the second-order dynamic equation under the right focal boundary value condition. Luo and Ma [5] in 2006 were concerned with the existence and multiplicity of nodal solutions and obtained eigenvalue intervals of the nonlinear second-order dynamic eigenvalue problem under conjugate boundary value condition by using bifurcation methods. In 2007, Sun et al. [6] obtained some sufficient conditions for the nonexistence and existence of at least one or two positive solutions for the 𝑝-Laplacian three-point dynamic eigenvalue problem with mixed derivatives by using the Krasnosel’skii’s fixed point theorem in a cone. In 2009, Luo [7] derived the eigenvalue intervals in which there exist positive solutions of a singular second-order multipoint dynamic eigenvalue problem with mixed derivatives by making use of the fixed point index theory.

As for the nonlinear higher-order dynamic eigenvalue problems, few papers can be found in the literature to the best of our knowledge. L. Kong and Q. Kong [8], and Boey and Wong [9] discussed the even-order dynamic eigenvalue problem and the right focal eigenvalue problem, respectively, but their problems do not contain (1.1). Particularly for fourth-order problems and special case 𝜆≡1, Wang and Sun [10] studied the existence of positive solutions for dynamic equations under nonhomogeneous boundary-value conditions describing an elastic beam that is simply supported at its two ends. And both Karaca [11] and Pang and Bai [12] obtained the existence of a solution for two classes of fourth-order four-point problems on time scales by the Leray-Schauder fixed point theorem and the upper and lower solution method, respectively, but the problems they studied are different to (1.1).

This paper studies the relationship between the existence and nonexistence of positive solutions and the value of parameter 𝜆>0. We find the existence of a 𝜆^∗ such that problem (1.1) has positive solutions for 0<𝜆≤𝜆^∗ and no positive solutions for 𝜆>𝜆^∗.

The rest of this paper is organized as follows: in Section 2, we firstly introduce the time scales concepts and notations and present some basic properties on time scales which are needed later. Next, Section 3 gives some preliminary results relevant to our discussion, and Section 4 is devoted to establish our main theorems.

2. Introduction to Time Scales

The calculus theory on time scales, which unifies continuous and discrete analysis, is now still an active area of research. We refer the reader to [13–16] and the references therein for introduction on this theory. For the convenience of readers, we present some necessary definitions and results here.

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 𝜇(𝑡)=𝜎(𝑡)−𝑡,𝑡∈𝕋 be the graininess function. And 𝕋^𝑘 which are derived from the time scale 𝕋 is 𝕋^𝑘∶=𝑡∈𝕋∶𝑡isnonmaximalor𝜌(𝑡)=𝑡, (2.2) and 𝕋^𝑘𝑛∶=(𝕋^𝑘𝑛−1)^𝑘,𝑛>1,𝑛∈ℕ. 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 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). Similarly, 𝑢^Δ𝑖(𝑡)∶=𝑢^Δ𝑖−1Δ(𝑡),𝑖>2,𝑖∈ℕ (2.4) is called the 𝑖th Δ-derivative of 𝑢 at 𝑡∈𝕋^𝑘𝑖. We also define the function 𝑢^𝜎∶=𝑢∘𝜎.

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. (2.5)

Lemma 2.3. If 𝑎,𝑏∈𝕋, 𝑎<𝜌(𝑏), ℎ∶[𝑎,𝑏]_𝕋→[0,+∞) is continuous and ℎ(𝑡)≢0 on [𝑎,𝑏)_𝕋, then ^𝑏_𝑎ℎ(𝑡)Δ𝑡>0. (2.6)

Proof. From [15, Theorems  1.28 (Viii) and  1.29], it is clear.

Lemma 2.4 (See [14, Theorem  1.16]). If the Δ-derivative of 𝑢 exists at 𝑡∈𝕋^𝑘, then 𝑢(𝜎(𝑡))=𝑢(𝑡)+𝜇(𝑡)𝑢^Δ(𝑡). (2.7)

Define the Banach space 𝐶(𝕋) to be the set of continuous functions 𝑢∶𝕋→ℝ with the norm ‖𝑢‖_∞=max{|𝑢(𝑡)|∣𝑡∈𝕋}. (2.8) For 𝑖∈ℕ, we define the Banach space 𝐶^𝑖_Δ(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=max‖𝑢‖_∞,𝑢^Δ_∞,…,𝑢^Δ𝑖_∞, (2.9) where 𝑢^Δ𝑗_∞=max𝑢^Δ𝑗(𝑡)∣𝑡∈𝕋^𝑘𝑗,𝑗=0,1,…,𝑖. (2.10)

3. Preliminaries

Throughout this paper, we assume that both 𝜎𝜉=min𝑡∈𝕋∣𝑡≥²(𝑏)+3𝑎4,𝜔=max𝑡∈𝕋∣𝑡≤3𝜎²(𝑏)+𝑎4 (3.1) exist and 𝑎<𝜉<𝜌(𝜔)≤𝜔≤𝜌(𝑏). So there exists a number 𝑚>0 such that 𝜔−𝜉≥𝑚. (3.2)

We also make the following assumptions:(𝐻1)ℎ∶[𝑎,𝜎(𝑏)]_𝕋→[0,+∞) is continuous and ℎ(𝑡)≢0 on [𝜉,𝜔)_𝕋;(𝐻2)𝑓∶[0,+∞)×(−∞,0]→[0,+∞) is continuous. 𝑓(𝑢,𝑤) is nondecreasing in 𝑢, nonincreasing in 𝑤 and 𝑓(0,0)>0.

Set 𝑣(𝑡)=𝑢^Δ2(𝑡),𝑡∈[𝑎,𝑏]_𝕋. Then problem (1.1) is equivalent to the system𝑢^Δ2(𝑡)=𝑣(𝑡),𝑡∈[𝑎,𝑏]_𝕋,𝑣^Δ2(𝑡)=𝜆ℎ(𝑡)𝑓(𝑢(𝑡),𝑣(𝑡)),𝑡∈𝑎,𝜌²(𝑏)_𝕋,𝑢(𝑎)=𝑢^Δ(𝜎(𝑏𝑣(𝑎)))=0,=𝑣^Δ𝜌(𝑏)=0. (3.3) According to [14, Corollary  4.84 and Theorem  4.70], the Green’s function of problems 𝑢^Δ2(𝑡)=0,𝑡∈[𝑎,𝑏]_𝕋,𝑢(𝑎)=𝑢^Δ(𝜎(𝑏𝑣))=0,_Δ2(𝑡)=0,𝑡∈𝑎,𝜌²(𝑏)_𝕋,𝑣(𝑎)=𝑣^Δ𝜌(𝑏)=0, (3.4) is of the same form 𝐺()=𝜎(𝑠)(𝑠)𝑡,𝑠𝑡−𝑎,𝑡≤𝑠,−𝑎,𝜎≤𝑡, (3.5) and the solution of system (3.3) is 𝑢(𝑡)=−_𝑎^𝜎(𝑏)𝐺(𝑡,𝑠)𝑣(𝑠)Δ𝑠,𝑡∈𝑎,𝜎²(𝑏)_𝕋,𝑣(𝑡)=−𝜆_𝑎^𝜌(𝑏)𝐺(𝑡,𝑠)ℎ(𝑠)𝑓(𝑢(𝑠),𝑣(𝑠))Δ𝑠,𝑡∈[𝑎,𝑏]_𝕋. (3.6) Therefore, the solution of problem (1.1) is 𝑢(𝑡)=𝜆_𝑎^𝜎(𝑏)_𝑎^𝜌(𝑏)𝐺(𝑡,𝑠)𝐺𝑠,𝑗ℎ𝑗𝑓𝑢𝑗,𝑢^Δ2𝑗Δ𝑗Δ𝑠,𝑡∈𝑎,𝜎²(𝑏)_𝕋. (3.7)

Lemma 3.1. Green’s function (3.5) is of the following properties: 0≤𝐺(𝑡,𝑠)≤min{𝑡−𝑎,𝜎(𝑠)−𝑎},(𝑡,𝑠)∈𝑎,𝜎²(𝑏)_𝕋×[𝑎,𝜎(𝑏)]_𝕋,𝐺()≥1𝑡,𝑠4(𝜎(𝑠)−𝑎),(𝑡,𝑠)∈𝜉,𝜔_𝕋×[𝑎,𝜎(𝑏)]_𝕋𝐺()≥,𝑡,𝑠𝑡−𝑎𝜎²(𝑏)𝐺(),()∈−𝑎𝑥,𝑠𝑡,𝑠𝑎,𝜎²(𝑏)_𝕋×[𝑎,𝜎(𝑏)]_𝕋,𝑥∈𝑎,𝜎²(𝑏)_𝕋𝐺()≥1,𝑡,𝑠4𝐺(𝑥,𝑠),(𝑡,𝑠)∈𝜉,𝜔_𝕋×[𝑎,𝜎(𝑏)]_𝕋,𝑥∈𝑎,𝜎²(𝑏)_𝕋𝐺()≥𝜎,𝑡,𝑠²(𝑏)−𝑎4,(𝑡,𝑠)∈𝜉,𝜔_𝕋×𝜉,𝜔_𝕋. (3.8)(3.9)(3.10)(3.11)(3.12)

Proof. We here only give the proof of (3.10), and the others can be obtained easily. We divide the proof into the following four cases.
Case 1 (𝑡≤𝑠,𝑥≤𝑠). We have 𝐺(𝑡,𝑠)=𝑡−𝑎≥𝑥−𝑎𝜎²(𝑏)()=−𝑎𝑡−𝑎𝑡−𝑎𝜎²(𝑏)()=−𝑎𝑥−𝑎𝑡−𝑎𝜎²(𝑏)𝐺().−𝑎𝑥,𝑠 (3.13)
Case 2 (𝑡≤𝑠,𝜎(𝑠)≤𝑥). We have 𝐺()𝜎(𝑠)𝑡,𝑠=𝑡−𝑎≥−𝑎𝜎²(𝑏)()=−𝑎𝑡−𝑎𝑡−𝑎𝜎²(𝑏)(𝜎(𝑠))=−𝑎−𝑎𝑡−𝑎𝜎²(𝑏)𝐺().−𝑎𝑥,𝑠 (3.14)
Case 3 (𝑥≤𝑠,𝜎(𝑠)≤𝑡). We have 𝐺(𝑡,𝑠)=𝜎(𝑠)−𝑎≥𝑥−𝑎≥𝑡−𝑎𝜎²(𝑏)()=−𝑎𝑥−𝑎𝑡−𝑎𝜎²(𝑏)𝐺().−𝑎𝑥,𝑠 (3.15)
Case 4 (𝜎(𝑠)≤𝑡,𝜎(𝑠)≤𝑥). We have 𝐺(𝑡,𝑠)=𝜎(𝑠)−𝑎≥𝑡−𝑎𝜎²(𝑏)(𝜎(𝑠))=−𝑎−𝑎𝑡−𝑎𝜎²(𝑏)𝐺().−𝑎𝑥,𝑠 (3.16)

Define 𝐸=𝑢∈𝐶²_Δ𝑎,𝜎²(𝑏)∣𝑢(𝑎)=𝑢^Δ(𝜎(𝑏))=𝑢^Δ2(𝑎)=𝑢^Δ3𝜌(𝑏)=0. (3.17)

Lemma 3.2. For 𝑢∈𝐸, one has ‖𝑢‖_∞≤2𝜎²(𝑏)−𝑎𝑢^Δ_∞,𝑢^Δ_∞≤2𝜎²(𝑏)−𝑎𝑢^Δ2_∞, (3.18)(3.19) where ‖𝑢‖_∞=max{|𝑢(𝑡)||𝑡∈[𝑎,𝜎²(𝑏)]_𝕋}, ‖𝑢^Δ‖_∞=max{|𝑢^Δ(𝑡)||𝑡∈[𝑎,𝜎(𝑏)]_𝕋}, ‖𝑢^Δ2‖_∞=max{|𝑢^Δ2(𝑡)||𝑡∈[𝑎,𝑏]_𝕋}.

Proof. Firstly, we show that (3.18) holds.
For all𝑡∈[𝑎,𝜎(𝑏)], we have from 𝑢(𝑎)=0 that |𝑢(𝑡)|=𝑢(𝑎)+^𝑡_𝑎𝑢^Δ(𝑠)≤Δ𝑠_𝑡𝑎𝑢^Δ(𝑠≤𝑢)Δ𝑠_Δ∞()≤𝜎𝑡−𝑎₂(𝑏)−𝑎𝑢^Δ_∞. (3.20) Combining this with 𝑢𝜎²(𝑏)=𝑢(𝜎(𝑏))+𝜇(𝜎(𝑏))𝑢^Δ(𝜎(𝑏≤𝜎))₂(𝑏)−𝑎𝑢^Δ_∞+𝜎²(𝑏)−𝑎𝑢^Δ_∞=2𝜎²(𝑏)−𝑎𝑢^Δ_∞, (3.21) we have ‖𝑢‖_∞≤2[𝜎²(𝑏)−𝑎]‖𝑢^Δ‖_∞.
Secondly, we show that (3.19) holds.
For all𝑡∈[𝑎,𝑏], we have from 𝑢^Δ(𝜎(𝑏))=0 that 𝑢^Δ(𝑡)=−^𝑏_𝑡𝑢^Δ2(𝑠)Δ𝑠+𝑢^Δ(𝑏=−_𝑏𝑡)𝑢^Δ2(𝑠)Δ𝑠+𝑢^Δ(𝜎(𝑏))−𝜇(𝑏)𝑢^Δ2(𝑏≤𝑢_Δ2∞)(𝑏−𝑡)+𝑢^Δ2_∞𝜇(𝑏)≤2𝜎²(𝑏)−𝑎𝑢^Δ2_∞. (3.22) Combining this with 𝑢^Δ(𝜎(𝑏))=0≤2𝜎²(𝑏)−𝑎𝑢^Δ2_∞, (3.23) we have ‖𝑢^Δ‖_∞≤2[𝜎²(𝑏)−𝑎]‖𝑢^Δ2‖_∞.

Define ‖𝑢‖_𝐸=max‖𝑢‖_∞,2𝜎²(𝑏)−𝑎𝑢^Δ_∞,4𝜎²(𝑏)−𝑎²𝑢^Δ2_∞=4𝜎²(𝑏)−𝑎²𝑢^Δ2_∞. (3.24) Then 𝐸 is a Banach space under the norm ‖⋅‖_𝐸. Set 𝐴_𝜆𝑢(𝑡)=𝜆_𝑎^𝜎(𝑏)_𝑎^𝜌(𝑏)𝐺(𝑡,𝑠)𝐺𝑠,𝑗ℎ𝑗𝑓𝑢𝑗,𝑢^Δ2𝑗Δ𝑗Δ𝑠,𝑡∈𝑎,𝜎²(𝑏)_𝕋. (3.25) Then 𝐴_𝜆∶𝐸→𝐸. Since 𝐴_𝜆∶𝐶²_Δ([𝑎,𝜎²(𝑏)])→𝐶⁴_Δ([𝑎,𝜎²(𝑏)])↪𝐶²_Δ([𝑎,𝜎²(𝑏)]), we have that 𝐴_𝜆 is completely continuous.

Lemma 3.3. Suppose (𝐻1) and (𝐻2) hold, and 𝑢(𝑡) is a solution of problem (1.1), then 𝑢(𝑡)≥0,𝑡∈𝑎,𝜎²(𝑏)_𝕋,𝑢^Δ2(𝑡)≤0,𝑡∈[𝑎,𝑏]_𝕋,𝑢(𝑡)>0,𝑢^Δ2(𝑡)<0,𝑡∈𝜉,𝜔_𝕋,min_{𝑡∈[𝜉,𝜔]𝕋}−𝑢^Δ2(𝑡)≥14𝑢^Δ2_∞=116𝜎²(𝑏)−𝑎²‖𝑢‖_𝐸,min_{𝑡∈[𝜉,𝜔]𝕋}𝑢(𝑡)≥14‖𝑢‖_∞,min_{𝑡∈[𝜉,𝜔]𝕋}𝑢(𝑡)≥𝑚64𝜎²(𝑏)‖𝑢‖−𝑎^𝐸. (3.26)(3.27)(3.28)(3.29)(3.30)

Proof. By 𝜆>0, (3.7), (3.8) and the fact that ℎ, 𝑓 are nonnegative functions, (3.26) holds. For 𝑡∈[𝜉,𝜔]_𝕋, from (3.7), (3.26), (3.2) and (3.12), we have 𝑢(𝑡)=𝜆_𝑎^𝜎(𝑏)_𝑎^𝜌(𝑏)𝐺(𝑡,𝑠)𝐺𝑠,𝑗ℎ𝑗𝑓𝑢𝑗,𝑢^Δ2𝑗()Δ𝑗Δ𝑠≥𝜆𝑓0,0^𝜔_𝜉^𝜔_𝜉𝐺()𝐺ℎ𝑗𝑚𝜎𝑡,𝑠𝑠,𝑗Δ𝑗Δ𝑠≥𝜆²(𝑏)−𝑎²𝑓()160,0^𝜔_𝜉ℎ𝑗Δ𝑗>0,−𝑢^Δ2(𝑡)=𝜆_𝑎^𝜌(𝑏)𝐺(𝑡,𝑠)ℎ(𝑠)𝑓𝑢(𝑠),𝑢^Δ2(𝑠)()Δ𝑠≥𝜆𝑓0,0^𝜔_𝜉𝐺()ℎ(𝑠)𝜎𝑡,𝑠Δ𝑠≥𝜆²(𝑏)−𝑎4𝑓(0,0)^𝜔_𝜉ℎ(𝑠)Δ𝑠>0 (3.31) Therefor (3.27) holds.
For all𝑡∈[𝜉,𝜔]_𝕋,forall𝑥∈[𝑎,𝜎²(𝑏)]_𝕋, from (3.7), (3.11), and (3.12), we have −𝑢^Δ2(𝑡)=𝜆_𝑎^𝜌(𝑏)𝐺(𝑡,𝑠)ℎ(𝑠)𝑓𝑢(𝑠),𝑢^Δ2(𝑠)≥1Δ𝑠4𝜆_𝑎^𝜌(𝑏)𝐺(𝑥,𝑠)ℎ(𝑠)𝑓𝑢(𝑠),𝑢^Δ2(𝑠)=1Δ𝑠4−𝑢^Δ2(𝑥,𝑢(𝑡))=𝜆_𝑎^𝜎(𝑏)_𝑎^𝜌(𝑏)𝐺(𝑡,𝑠)𝐺𝑠,𝑗ℎ𝑗𝑓𝑢𝑗,𝑢^Δ2𝑗≥1Δ𝑗Δ𝑠4𝜆_𝑎^𝜎(𝑏)_𝑎^𝜌(𝑏)𝐺(𝑥,𝑠)𝐺𝑠,𝑗ℎ𝑗𝑓𝑢𝑗,𝑢^Δ2𝑗=1Δ𝑗Δ𝑠4𝑢(𝑥),𝑢(𝑡)=𝜆_𝑎^𝜎(𝑏)_𝑎^𝜌(𝑏)𝐺(𝑡,𝑠)𝐺𝑠,𝑗ℎ𝑗𝑓𝑢𝑗,𝑢^Δ2𝑗≥𝜎Δ𝑗Δ𝑠₂(𝑏)−𝑎4𝜆^𝜔_𝜉𝑎^𝜌(𝑏)𝐺𝑠,𝑗ℎ𝑗𝑓𝑢𝑗,𝑢^Δ2𝑗≥𝜎Δ𝑗Δ𝑠₂(𝑏)−𝑎𝜆16^𝜔_𝜉𝑎^𝜌(𝑏)𝐺𝑥,𝑗ℎ𝑗𝑓𝑢𝑗,𝑢^Δ2𝑗≥𝑚𝜎Δ𝑗Δ𝑠₂(𝑏)−𝑎16−𝑢^Δ2(𝑥=𝑚)64𝜎²(𝑏)4𝜎−𝑎²(𝑏)−𝑎²−𝑢^Δ2(𝑥). (3.32) Thus (3.28), (3.29), and (3.30) hold.

At the end of this section, we state a lemma of the upper and lower solution method, which is needed for some proofs in next section.

Lemma 3.4 (See [17, Theorem  3.3.8]). Let 𝑃 be a cone with nonempty interior in Banach space 𝐸, and 𝐴∶𝑃→𝑃 a completely continuous and increasing operator. Suppose the following conditions hold:
(i)there exist 𝑥₀,𝑦₀∈𝑃, such that 𝑥₀≤𝐴𝑥₀,𝐴𝑦₀≤𝑦₀,𝑥₀≰𝑦₀;(ii)there exist 𝜙∈𝑃 and a constant 𝛼>0, such that 𝐴𝑥≥𝛼‖𝐴𝑥‖𝜙,forall𝑥∈𝑃;(iii)𝑦₀ is in the interior of 𝑃, and there exists 𝛽>0, such that 𝜙≥𝛽𝑥₀. Then 𝐴 has at least one fixed point 𝑥≠0 in 𝑃.

4. The Main Result

Our main result is the following existence theorem.

Theorem 4.1. Suppose (𝐻1) and (𝐻2) hold, and either (𝐻3) or (𝐻4) holds. Here(𝐻3)lim_{(𝑢,𝑤)→(+∞,−∞)}(𝑓(𝑢,𝑤)/𝑢)=+∞;(𝐻4)lim_{(𝑢,𝑤)→(+∞,−∞)}(𝑓(𝑢,𝑤)/(−𝑤))=+∞.Then there exists 𝜆^∗>0, such that problem (1.1) has at least one positive solution for 𝜆∈(0,𝜆^∗], and has no positive solution for 𝜆∈(𝜆^∗,+∞).

To prove Theorem 4.1, first, we show that problem (1.1) has positive solutions for some 𝜆 small enough.

Theorem 4.2. Suppose (𝐻1) and (𝐻2) hold. Then there exists 𝜆_∗>0, such that problem (1.1) has at least one positive solution for 𝜆∈(0,𝜆_∗].

Proof. The fixed point of 𝐴_𝜆 defined in (3.25) is the solution of problem (1.1), so it will be enough to find the fixed point of 𝐴_𝜆.
Set 𝑀=max_{𝑡∈[𝑎,𝑏]𝕋𝑎}^𝜌(𝑏)𝐺(𝑡,𝑠)ℎ(𝑠)Δ𝑠>0. (4.1) Let 𝜆_∗=1𝑀𝑓4𝜎²(𝑏)−𝑎².,−1 (4.2) For 𝜆∈(0,𝜆_∗], we have 𝑓4𝜎²(𝑏)−𝑎²≤1,−1.𝑀𝜆 (4.3) Set 𝐷₁=𝑢∈𝐸∣‖𝑢‖_𝐸≤4𝜎²(𝑏)−𝑎². (4.4) If 𝑢∈𝐷₁, then ‖𝑢‖_∞≤‖𝑢‖_𝐸≤4𝜎²(𝑏)−𝑎²,𝑢^Δ2_∞=‖𝑢‖_𝐸4𝜎²(𝑏)−𝑎²≤1. (4.5) Consequently, for 𝑡∈[𝑎,𝑏]_𝕋, 𝐴_𝜆𝑢^Δ2(𝑡)=−𝜆_𝑎^𝜌(𝑏)𝐺(𝑡,𝑠)ℎ(𝑠)𝑓𝑢(𝑠),𝑢^Δ2(𝑠)≤‖𝑢‖Δ𝑠𝑀𝜆𝑓_∞,−𝑢^Δ2_∞≤𝑀𝜆𝑓4𝜎²(𝑏)−𝑎²,−1≤1. (4.6) Then 𝐴_𝜆𝑢_𝐸=4𝜎²(𝑏)−𝑎²𝐴_𝜆𝑢^Δ2_∞≤4𝜎²(𝑏)−𝑎², (4.7) that is, 𝐴_𝜆∶𝐷₁→𝐷₁. By the Schauder fixed point theorem, 𝐴_𝜆 has at least one fixed point 𝑢_𝜆 in 𝐸 satisfying ‖𝑢_𝜆‖_𝐸≤4[𝜎²(𝑏)−𝑎]². From Lemma 3.3, 𝑢_𝜆 is a positive solution of problem (1.1).

Next, we show that there exist no positive solution for some 𝜆 large enough.

Theorem 4.3. Suppose that (𝐻1) and (𝐻2), hold, and either (𝐻3) or (𝐻4) holds. Then problem (1.1) has no positive solution for 𝜆≫1.

Proof. Suppose 𝑢_𝜆∈𝐸 is a solution to problem (1.1) for some 𝜆>0. We divide our discussions into two cases.
Case 1 ((𝐻1), (𝐻2) and (𝐻3) hold). By (𝐻3), for a fixed 𝑀₁>0, there is 𝐻>0 such that 𝑓(𝑢,𝑤)≥𝑀₁𝑢,𝑢≥𝑚𝐻64𝜎²(𝑏)−𝑎,𝑤≤−𝐻16𝜎²(𝑏)−𝑎². (4.8) If 𝑢∈𝐸 with ‖𝑢_𝜆‖_𝐸≥𝐻, then from (3.30) and (3.28), we have min_{𝑡∈[𝜉,𝜔]𝕋}𝑢_𝜆(𝑡)≥𝑚64𝜎²(𝑏)𝑢−𝑎^𝑛𝐸≥𝑚𝐻64𝜎²(𝑏),−𝑎min_{𝑡∈[𝜉,𝜔]𝕋}−𝑢^Δ2_𝜆(𝑡)≥14𝑢^Δ2_𝜆∞=𝑢_𝜆𝐸16𝜎²(𝑏)−𝑎²≥𝐻16𝜎²(𝑏)−𝑎². (4.9) Further by (4.8), 𝑓𝑢_𝜆(𝑡),𝑢^Δ2_𝜆(𝑡)≥𝑀₁𝑢_𝜆(𝑡),𝑡∈𝜉,𝜔_𝕋. (4.10) If 𝑢∈𝐸 with ‖𝑢_𝜆‖_𝐸<𝐻, then from (3.27), there is 𝑓𝑢_𝜆(𝑡),𝑢^Δ2_𝜆(𝑡)≥𝑀₂𝑢_𝜆(𝑡),𝑡∈𝜉,𝜔_𝕋 (4.11) with 𝑀₂=min_{𝑡∈[𝜉,𝜔]𝕋}𝑓𝑢_𝜆(𝑡),𝑢^Δ2_𝜆(𝑡)𝑢_𝜆(𝑡)>0. (4.12) Set 𝑀=min{𝑀₁,𝑀₂}, we have from (4.10) and (4.11) that 𝑓𝑢_𝜆(𝑡),𝑢^Δ2_𝜆(𝑡)≥𝑀𝑢_𝜆(𝑡),𝑡∈𝜉,𝜔_𝕋. (4.13) Combining this with (3.7), (4.13), (3.29), (3.12), and (3.2), we have for 𝑡∈[𝜉,𝜔]_𝕋, 𝑢_𝜆(𝑡)=𝜆_𝑎^𝜎(𝑏)_𝑎^𝜌(𝑏)𝐺(𝑡,𝑠)𝐺𝑠,𝑗ℎ𝑗𝑓𝑢_𝜆𝑗,𝑢^Δ2_𝜆𝑗𝑀Δ𝑗Δ𝑠≥𝜆_𝑎^{𝜎(𝑏)𝜔}_𝜉𝐺(𝑡,𝑠)𝐺𝑠,𝑗ℎ𝑗𝑢_𝜆𝑗≥𝜆𝑀𝑢Δ𝑗Δ𝑠_𝜆∞4^𝜔_𝜉^𝜔_𝜉𝐺()𝐺ℎ𝑗𝜎𝑡,𝑠𝑠,𝑗Δ𝑗Δ𝑠≥𝜆𝑀𝑚²(𝑏)−𝑎²𝑢64^𝜆∞𝜔_𝜉ℎ𝑗Δ𝑗. (4.14) If we choose 𝜆 such that 𝜆𝑀𝑚𝜎²(𝑏)−𝑎²64^𝜔_𝜉ℎ𝑗Δ𝑗>1, (4.15) then 𝑢_𝜆(𝑡)>𝑢_𝜆∞,𝑡∈𝜉,𝜔_𝕋, (4.16) which is a contradiction.
Case 2 ((𝐻1), (𝐻2), and (𝐻4) hold). Similar to Case 1, by (𝐻4), there is also an 𝑀, such that 𝑓𝑢_𝜆(𝑡),𝑢^Δ2_𝜆(𝑡)≥𝑀−𝑢^Δ2_𝜆(𝑡),𝑡∈𝜉,𝜔_𝕋. (4.17) Thus for 𝑡∈[𝜉,𝜔]_𝕋, we have from (3.7), (4.17), (3.28), and (3.12) that −𝑢^Δ2_𝜆(𝑡)=𝜆_𝑎^𝜌(𝑏)𝐺(𝑡,𝑠)ℎ(𝑠)𝑓𝑢_𝜆(𝑠),𝑢^Δ2_𝜆(𝑠)𝑀Δ𝑠≥𝜆^𝜔_𝜉𝐺(𝑡,𝑠)ℎ(𝑠)−𝑢^Δ2_𝜆(𝑠𝑀𝜎)Δ𝑠≥𝜆²(𝑏)−𝑎𝑢16^{Δ2𝜆∞𝜔}_𝜉ℎ(𝑠)Δ𝑠. (4.18) If we take 𝜆 such that 𝜆𝑀𝜎²(𝑏)−𝑎16^𝜔_𝜉ℎ(𝑠)Δ𝑠>1, (4.19) then −𝑢^Δ2_𝜆(𝑡)>𝑢^Δ2_𝜆∞,𝑡∈𝜉,𝜔_𝕋, (4.20) which is a contradiction.
Therefore, problem (1.1) has no positive solution for 𝜆≫1.

Define the set𝐵={𝜆>0∶problem(1.1)hasatleastonepositivesolutionin𝐸}.

Theorem 4.4. One has 𝜆∈𝐵⟹𝜆∈𝐵,∀𝜆∈0,𝜆. (4.21)

Proof. Let 𝜆_∗ be defined as Theorem 4.2. For 𝜆≤𝜆_∗, the result holds from Theorem 4.2. So, we discuss the case that 𝜆>𝜆_∗.
For 𝜆∈(0,𝜆], three cases will be discussed.(1)𝜆∈(0,𝜆_∗];(2)𝜆∈(𝜆_∗,𝜆);(3)𝜆=𝜆.Cases  (1) and  (3) are clear from Theorem 4.2 and the assumption 𝜆∈𝐵, respectively. Now, we deal with Case  (2).
Define 𝐷₂=𝑢∈𝐸∣𝑢(𝑡)≥0,𝑡∈𝑎,𝜎²(𝑏),𝑢^Δ2(𝑡)≤0,𝑡∈[𝑎,𝑏]. (4.22) Then 𝐷₂ is a cone with nonempty interior in 𝐸. For 𝜆∈(𝜆_∗,𝜆), let 𝐴_𝜆 be defined as (3.25). Then 𝐴_𝜆∶𝐷₂→𝐷₂ and 𝐴_𝜆 is an increasing operator from (𝐻2). Set 𝑢_∗ and 𝑢 as positive solutions of problem (1.1) at 𝜆_∗ and 𝜆, respectively. Then 𝐴_𝜆𝑢_∗(𝑡)>𝐴_𝜆∗𝑢_∗(𝑡)=𝑢_∗(𝑡),𝐴_𝜆𝑢(𝑡)<𝐴_𝜆𝑢(𝑡)=𝑢(𝑡). (4.23) So 𝑢 is an upper solution of the operator 𝐴_𝜆 and 𝑢_∗ is a lower solution. If 𝑢_∗≤𝑢, then there exists a positive solution 𝑢 satisfying 𝑢_∗(𝑡)≤𝑢(𝑡)≤𝑢(𝑡) for 𝑡∈[𝑎,𝜎²(𝑏)]_𝕋 by the upper and lower solution method. If 𝑢_∗≰𝑢, we verify the conditions of Lemma 3.4.
Clearly, the condition (i) in Lemma 3.4 holds for 𝑢_∗=𝑥₀,𝑢=𝑦₀.
For all𝑢∈𝐷₂,forall𝑡,𝑧∈[𝑎,𝜎²(𝑏)]_𝕋,𝑥∈[𝜉,𝜔]_𝕋, we have from (3.10), (3.12), (3.11), and (3.2) that 𝐴_𝜆𝑢(𝑡)=𝜆_𝑎^𝜎(𝑏)_𝑎^𝜌(𝑏)𝐺(𝑡,𝑠)𝐺𝑠,𝑗ℎ𝑗𝑓𝑢𝑗,𝑢^Δ2𝑗Δ𝑗Δ𝑠≥𝜆𝑡−𝑎𝜎²(𝑏)−𝑎_𝑎^𝜎(𝑏)_𝑎^𝜌(𝑏)𝐺(𝑥,𝑠)𝐺𝑠,𝑗ℎ𝑗𝑓𝑢𝑗,𝑢^Δ2𝑗Δ𝑗Δ𝑠≥𝜆𝑡−𝑎4^𝜔_𝜉𝑎^𝜌(𝑏)𝐺𝑠,𝑗ℎ𝑗𝑓𝑢𝑗,𝑢^Δ2𝑗Δ𝑗Δ𝑠≥𝜆𝑡−𝑎4^𝜔_𝜉𝑎^𝜌(𝑏)14𝐺𝑧,𝑗ℎ𝑗𝑓𝑢𝑗,𝑢^Δ2𝑗≥𝑚Δ𝑗Δ𝑠(−𝐴16𝑡−𝑎)_𝜆𝑢^Δ2(𝑧). (4.24) Then 𝐴_𝜆𝑚𝑢≥𝐴16^𝜆𝑢^Δ2_∞()=𝑚𝑡−𝑎64𝜎²(𝑏)−𝑎²𝐴_𝜆𝑢_𝐸(𝑡−𝑎), (4.25) and the condition (ii) of Lemma 3.4 is satisfied for 𝜙=𝑡−𝑎,𝛼=𝑚/64[𝜎²(𝑏)−𝑎]²>0.
From (3.26) and (3.27), we have that 𝑢 is in the interior of 𝐷₂. From (3.8), there is 𝑢_∗(𝑡)=𝜆_∗𝑎^𝜎(𝑏)_𝑎^𝜌(𝑏)𝐺(𝑡,𝑠)𝐺𝑠,𝑗ℎ𝑗𝑓𝑢_∗𝑗,𝑢^Δ2_∗𝑗Δ𝑗Δ𝑠≤𝜆_∗(𝑡−𝑎)_𝑎^{𝜎(𝑏)𝜌(𝑏)}_a𝐺𝑠,𝑗ℎ𝑗𝑓𝑢_∗𝑗,𝑢^Δ2_∗𝑗Δ𝑗Δ𝑠≤𝜆_∗(𝑡−𝑎)_𝑎^𝜎(𝑏)(𝑠−𝑎)Δ𝑠_𝑎^𝜌(𝑏)ℎ𝑗𝑓𝑢_∗𝑗,𝑢^Δ2_∗𝑗Δ𝑗. (4.26) This implies 𝜙(𝑡)=𝑡−𝑎≥𝛽𝑢_∗(𝑡),𝑡∈[𝑎,𝜎²(𝑏)]_𝕋 for 𝛽=𝜆_∗𝑎^𝜎(𝑏)(𝑠−𝑎)Δ𝑠_𝑎^𝜌(𝑏)ℎ𝑗𝑓𝑢_∗𝑗,𝑢^Δ2_∗𝑗Δ𝑗⁻¹>0, (4.27) and the condition (iii) of Lemma 3.4 is satisfied.
By Lemmas 3.4 and 3.3, we get a positive solution 𝑢_𝜆 in 𝐷₂∈𝐸. That is, 𝜆∈𝐵.

Now, we give the proof of Theorem 4.1.

Proof of Theorem 4.1. From Theorems 4.2 and 4.3, 𝐵≠∅ is bounded. Thus, we can define 𝜆^∗=sup𝐵. Firstly, we show that 𝜆^∗∈𝐵.
Choose a sequence {𝜆_𝑛}^∞_𝑛=1, 𝜆_𝑛∈𝐵(𝑛=1,2,…) which belongs to a compact subinterval in (0,∞), and 𝜆_𝑛→𝜆^∗(𝑛→∞). Then there exists 𝑁₁>0, for 𝑛≥𝑁₁, 𝜆_𝑛>𝜆^∗2. (4.28) Let 𝑢_𝑛∈𝐸 satisfy 𝐴_𝜆𝑛𝑢_𝑛=𝑢_𝑛. (4.29) If {𝑢_𝑛}^∞_𝑛=1 is uniformly bounded, then there exists 𝑢^∗ such that 𝑢_𝑛→𝑢^∗(𝑛→∞) and 𝑢_𝑛=𝐴_𝜆𝑛𝑢_𝑛⟶𝐴_𝜆^∗𝑢^∗(𝑛⟶∞). (4.30) Consequently, 𝑢^∗=𝐴_𝜆^∗𝑢^∗. (4.31) By Lemma 3.3, 𝑢^∗ is a positive solution of problem (1.1), and 𝜆^∗∈𝐵.
Next, we will prove that {𝑢_𝑛}^∞_𝑛=1 is uniformly bounded and the discussions will be divided into two cases.
Case 1 ((𝐻1), (𝐻2), and (𝐻3) hold). From (𝐻3), there exists 𝐻₁>0, 𝑓(𝑢,𝑤)≥𝑀₁𝑢,for𝑢≥𝑚𝐻₁64𝜎²(𝑏)𝐻−𝑎,−𝑤≥¹16𝜎²(𝑏)−𝑎², (4.32) where 𝑀₁>0 satisfies 𝑀₁𝜎²(𝑏)−𝑎²𝑚𝜆^∗128^𝜔_𝜉ℎ𝑗Δ𝑗>1. (4.33) Suppose on the contrary that {𝑢_𝑛}^∞_𝑛=1 is unbounded. Then 𝑢_𝑛𝐸⟶∞,𝑛⟶∞, (4.34) which implies that there exists 𝑁₂>0, ‖𝑢_𝑛‖_𝐸≥𝐻₁ for 𝑛≥𝑁₂. From (3.30) and (3.28), we have min_{𝑡∈[𝜉,𝜔]𝕋}𝑢_𝑛(𝑡)≥𝑚64𝜎²(𝑏)𝑢−𝑎^𝑛𝐸≥𝑚𝐻₁64𝜎²(𝑏),−𝑎min_{𝑡∈[𝜉,𝜔]𝕋}−𝑢^Δ2_𝑛(𝑡)≥116𝜎²(𝑏)−𝑎²𝑢_𝑛𝐸≥𝐻₁16𝜎²(𝑏)−𝑎². (4.35) Subsequently, for 𝑡∈[𝜉,𝜔]_𝕋, 𝑛≥max{𝑁₁,𝑁₂}, 𝑢_𝑛(𝑡)=𝜆_𝑛𝑎^𝜎(𝑏)_𝑎^𝜌(𝑏)𝐺(𝑡,𝑠)𝐺𝑠,𝑗ℎ𝑗𝑓𝑢_𝑛𝑗,𝑢^Δ2_𝑛𝑗Δ𝑗Δ𝑠≥𝜆_𝑛𝑀_1𝑎^{𝜎(𝑏)𝜔}_𝜉𝐺(𝑡,𝑠)𝐺𝑠,𝑗ℎ𝑗𝑢_𝑛𝑗≥𝜆Δ𝑗Δ𝑠_𝑛𝑀₁4𝑢_𝑛∞^𝜔_𝜉^𝜔_𝜉𝐺()𝐺ℎ𝑗≥𝑀𝑡,𝑠𝑠,𝑗Δ𝑗Δ𝑠₁𝜎²(𝑏)−𝑎²𝑚𝜆^∗𝑢128^𝑛∞𝜔_𝜉ℎ𝑗>𝑢Δ𝑗_𝑛∞ (4.36) by (4.32), (3.29), (3.12), (4.28), (3.2), and (4.33). This is a contradiction.
Case 2 ((𝐻1), (𝐻2), and (𝐻4) hold). From (𝐻4), there exists 𝐻₂>0, 𝑓(𝑢,𝑤)≥𝑀₂(−𝑤),for𝑢≥𝑚𝐻₂64𝜎²(𝑏)𝐻−𝑎,−𝑤≥²16𝜎²(𝑏)−𝑎², (4.37) where 𝑀₂>0 satisfies 𝑀₂𝜎²(𝑏)−𝑎𝜆^∗32^𝜔_𝜉ℎ(𝑠)Δ𝑠>1. (4.38) Suppose on the contrary that {𝑢_𝑛}^∞_𝑛=1 is unbounded. Then similar to Case 1, there exists 𝑁₃>0 such that ‖𝑢_𝑛‖_𝐸≥𝐻₂ for 𝑛≥𝑁₃. Thus for 𝑡∈[𝜉,𝜔]_𝕋, 𝑛≥max{𝑁₁,𝑁₃}, we have −𝑢^Δ2_𝑛(𝑡)=𝜆_𝑛𝑎^𝜌(𝑏)𝐺(𝑡,𝑠)ℎ(𝑠)𝑓𝑢_𝑛(𝑠),𝑢^Δ2_𝑛(𝑠)Δ𝑠≥𝜆_𝑛𝑀₂^𝜔_𝜉𝐺(𝑡,𝑠)ℎ(𝑠)−𝑢^Δ2_𝑛(𝑠>𝑀)Δ𝑠₂4𝜆^∗2𝑢^Δ2_𝑛∞𝜎²(𝑏)−𝑎4^𝜔_𝜉ℎ(𝑠)>𝑢Δ𝑠_Δ2𝑛∞ (4.39) by (4.37), (3.28), (3.12), (4.28) and (4.38). This is also a contradiction.
According to the definition of 𝜆^∗ and 𝐵, and Theorem 4.4, we complete the proof.

Acknowledgments

The authors would like to thank editor and the referees for carefully reading this paper and suggesting many valuable comments. This paper is supported by China Postdoctoral Science Foundation Funded Projects (no. 201104602 and no. 20100481239) and General Project for Scientific Research of Liaoning Educational Committee (no. L2011200).