Positive Solutions for Second-Order Singular Semipositone Differential Equations Involving Stieltjes Integral Conditions

Abstract

By means of the fixed point theory in cones, we investigate the existence of positive solutions for the following second-order singular differential equations with a negatively perturbed term: −𝑢^′′(𝑡)=𝜆[𝑓(𝑡,𝑢(𝑡))−𝑞(𝑡)], 0<𝑡<1, 𝛼𝑢(0)−𝛽𝑢^′(0)=∫¹₀𝑢(𝑠)𝑑𝜉(𝑠), 𝛾𝑢(1)+𝛿𝑢^′(1)=∫¹₀𝑢(𝑠)𝑑𝜂(𝑠), where 𝜆>0 is a parameter; 𝑓∶(0,1)×(0,∞)→[0,∞) is continuous; 𝑓(𝑡,𝑥) may be singular at 𝑡=0, 𝑡=1, and 𝑥=0, and the perturbed term 𝑞∶(0,1)→[0,+∞) is Lebesgue integrable and may have finitely many singularities in (0,1), which implies that the nonlinear term may change sign.

1. Introduction

In this paper, we are concerned with positive solutions of the following second-order singular semipositone boundary value problem (BVP):−𝑢^′′(𝑡)𝑓((𝑡(𝑡(0)=𝜆𝑡,𝑢))−𝑞),0<𝑡<1,𝛼𝑢−𝛽𝑢^′(0)=¹₀𝑢(𝑠)(𝑠),(1)𝑑𝜉𝛾𝑢+𝛿𝑢^′(1)=¹₀𝑢(𝑠)𝑑𝜂(𝑠), (1.1) where 𝜆>0 is a parameter, 𝛼,𝛾≥0, 𝛽,𝛿>0 are constants such that 𝜌=𝛼𝛾+𝛼𝛿+𝛽𝛾>0, and the integrals in (1.1) are given by Stieltjes integral with a signed measure, that is, 𝜉, 𝜂 are suitable functions of bounded variation, 𝑞∶(0,1)→[0,+∞) is a Lebesgue integral and may have finitely many singularities in [0,1], 𝑓∶(0,1)×(0,∞)→[0,∞) is continuous, 𝑓(𝑡,𝑥) may be singular at 𝑡=0, 𝑡=1, and 𝑥=0.

Semipositone BVPs occur in models for steady-state diffusion with reactions [1], and interest in obtaining conditions for the existence of positive solutions of such problems has been ongoing for many years. For a small sample of such work, we refer the reader to the papers of Agarwal et al. [2, 3], Kosmatov [4], Lan [5–7], Liu [8], Ma et al. [9, 10], and Yao [11]. In [12], the second-order 𝑚-point BVP, −𝑢^′′(𝑡)((𝑡(),(0)==𝜆𝑓𝑡,𝑢)),𝑡∈0,1𝑢′^𝑚−2_𝑖=1𝑎_𝑖𝑢′𝜉_𝑖,𝑢(1)=^𝑚−2_𝑖=1𝑏_𝑖𝑢𝜉_𝑖, (1.2) is studied, where 𝑎_𝑖,𝑏_𝑖>0 (𝑖=1,2,…,𝑚−2), 0<𝜉₁<𝜉₂<⋯<𝜉_𝑚−2<1, 𝜆 is a positive parameter. By using the Krasnosel’skii fixed point theorem in cones, the authors established the conditions for the existence of at least one positive solution to (1.2), assuming that 0<∑^𝑚−2_𝑖=1𝑎_𝑖<1, 0<∑^𝑚−2_𝑖=1𝑏_𝑖<1, 𝑓∶[0,1]×[0,+∞)→(−∞,+∞) is continuous, and there exists 𝐴>0 such that 𝑓(𝑡,𝑢)≥−𝐴 for (𝑡,𝑢)∈[0,1]×[0,+∞). If the constant 𝐴 is replaced by any continuous function 𝐴(𝑡) on [0,1], 𝑓 also has a lower bound and the existence results are still true.

Recently, Webb and Infante [13] studied arbitrary-order semipositone boundary value problems. The existence of multiple positive solutions is established via a Hammerstein integral equation of the form: 𝑢(𝑡)=¹₀𝑘(𝑡,𝑠)𝑔(𝑠)𝑓(𝑠,𝑢(𝑠))𝑑𝑠, (1.3) where 𝑘 is the corresponding Green function, 𝑔∈𝐿¹[0,1] is nonnegative and may have pointwise singularities, 𝑓∶[0,1]×[0,+∞)→(−∞,+∞) satisfies the Carathéodory conditions and 𝑓(𝑡,𝑢)≥−𝐴 for some 𝐴>0. Although 𝐴 is a constant, because of the term 𝑔, [13] includes nonlinearities that are bounded below by an integral function. It is worth mentioning that the boundary conditions cover both local and nonlocal types. Nonlocal boundary conditions are quite general, involving positive linear functionals on the space 𝐶[0,1], given by Stieltjes integrals.

For the cases where the nonlinear term takes only nonnegative values, the existence of positive solutions of nonlinear boundary value problems with nonlocal boundary conditions, including multipoint and integral boundary conditions, has been extensively studied by many researchers in recent years [14–25]. Kong [17] studied the second-order singular BVP:𝑢^′′(𝑡)((𝑡(),𝑢(0)=+𝜆𝑓𝑡,𝑢))=0,𝑡∈0,1₁₀𝑢(𝑠)(𝑠),𝑢(1)=𝑑𝜉₁₀𝑢(𝑠)𝑑𝜂(𝑠), (1.4) where 𝜆 is a positive parameter, 𝑓∶(0,1)×(0,+∞)→[0,+∞) is continuous, 𝜉(𝑠) and 𝜂(𝑠) are nondecreasing, and the integrals in (1.4) are Riemann-Stieltjes integrals. Sufficient conditions are obtained for the existence and uniqueness of a positive solution by using the mixed monotone operator theory.

Inspired by the above work, the purpose of this paper is to establish the existence of positive solutions to BVP (1.1). By using the fixed point theorem on a cone, some new existence results are obtained for the case where the nonlinearity is allowed to be sign changing. We will address here that the problem tackled has several new features. Firstly, as 𝑞∈𝐿¹[0,1], the perturbed effect of 𝑞 on 𝑓 may be so large that the nonlinearity may tend to negative infinity at some singular points. Secondly, the BVP (1.1) possesses singularity, that is, the perturbed term 𝑞 may has finitely many singularities in [0,1], and 𝑓(𝑡,𝑥) is allowed to be singular at 𝑡=0, 𝑡=1, and 𝑥=0. Obviously, the problem in question is different from those in [2–13]. Thirdly, ∫¹₀𝑢(𝑠)𝑑𝜉(𝑠) and ∫¹₀𝑢(𝑠)𝑑𝜂(𝑠) denote the Stieltjes integrals where 𝜉,𝜂 are of bounded variation, that is, 𝑑𝜉 and 𝑑𝜂 can change sign. This includes the multipoint problems and integral problems as special cases.

The rest of this paper is organized as follows. In Section 2, we present some lemmas and preliminaries, and we transform the singularly perturbed problem (1.1) to an equivalent approximate problem by constructing a modified function. Section 3 gives the main results and their proofs. In Section 4, two examples are given to demonstrate the validity of our main results.

Let 𝐾 be a cone in a Banach space 𝐸. For 0<𝑟<𝑅<+∞, let 𝐾_𝑟={𝑥∈𝐾∶‖𝑥‖<𝑟}, 𝜕𝐾_𝑟={𝑥∈𝐾∶‖𝑥‖=𝑟}, and 𝐾_𝑟,𝑅={𝑥∈𝐾∶𝑟≤‖𝑥‖≤𝑅}. The proof of the main theorem of this paper is based on the fixed point theory in cone. We list here one lemma [26, 27] which is needed in our following argument.

Lemma 1.1. Let 𝐾 be a positive cone in real Banach space 𝐸, 𝑇∶𝐾_𝑟,𝑅→𝐾 is a completely continuous operator. If the following conditions hold: (i)‖𝑇𝑥‖≤‖𝑥‖ for 𝑥∈𝜕𝐾_𝑅,(ii)there exists 𝑒∈𝜕𝐾₁ such that 𝑥≠𝑇𝑥+𝑚𝑒 for any 𝑥∈𝜕𝐾_𝑟 and 𝑚>0,then, 𝑇 has a fixed point in 𝐾_𝑟,𝑅.

Remark 1.2. If (i) and (ii) are satisfied for 𝑥∈𝜕𝐾_𝑟 and 𝑥∈𝜕𝐾_𝑅, respectively, then Lemma 1.1 is still true.

2. Preliminaries and Lemmas

Denote 𝜙₁(𝑡)=1𝜌𝛿+𝛾(1−𝑡),𝜙₂(𝑡)=1𝜌(𝑡)()[],𝑘𝛽+𝛼𝑡,𝑒=𝐺𝑡,𝑡,𝑡∈0,1₁=1−¹₀𝜙₁(𝑡)𝑑𝜉(𝑡),𝑘₂=¹₀𝜙₂(𝑡)𝑑𝜉(𝑡),𝑘₃=¹₀𝜙₁(𝑡)(𝑡),𝑘𝑑𝜂₄=1−¹₀𝜙₂(𝑡)𝑑𝜂(𝑡),𝑘=𝑘₁𝑘₄−𝑘₂𝑘₃𝜌,𝜎=,𝛼+𝛽𝛾+𝛿 (2.1) where 𝐺()=1𝑡,𝑠𝜌()(𝛽+𝛼𝑠𝛿+𝛾1−𝑡,0≤𝑠≤𝑡≤1,𝛽+𝛼𝑡𝛿+𝛾1−𝑠),0≤𝑡≤𝑠≤1. (2.2) Obviously, 𝑒(𝑡)=𝜌𝜙₁(𝑡)𝜙₂(𝑡)=1𝜌([],(𝑡)𝑒(𝑠)()(𝑠(𝑡𝛽+𝛼𝑡𝛿+𝛾1−𝑡),𝑡∈0,1𝜎𝑒≤𝐺𝑡,𝑠≤𝑒)(or𝑒))≤𝜎⁻¹,𝑡,𝑠∈[0,1]. (2.3)(2.4)

Throughout this paper, we adopt the following assumptions.(𝐻₁)𝑘₁,𝑘₄∈(0,1], 𝑘₂≥0, 𝑘₃≥0, 𝑘>0, and 𝒢_𝜉(𝑠)=¹₀𝐺(𝑡,𝑠)𝑑𝜉(𝑡)≥0,𝒢_𝜂(𝑠)=¹₀𝐺(𝑡,𝑠)𝑑𝜂(𝑡)≥0,𝑠∈[0,1]. (2.5)(𝐻₂)𝑞∶(0,1)→[0,+∞) is a Lebesgue integral and ∫¹₀𝑞(𝑡)𝑑𝑡>0.(𝐻₃) For any (𝑡,𝑥)∈(0,1)×(0,+∞), 0≤𝑓(𝑡,𝑥)≤𝑝(𝑡)𝑔(𝑥)+ℎ(𝑥), (2.6) where 𝑝∈𝐶(0,1) with 𝑝>0 on (0,1) and ∫¹₀𝑝(𝑡)𝑑𝑡<+∞, 𝑔>0 is continuous and nonincreasing on (0,+∞), ℎ≥0 is continuous on [0,+∞), and for any constant 𝑟>0, 0<¹₀𝑝(𝑠)𝑔(𝑟𝑒(𝑠))𝑑𝑠<+∞. (2.7)

Remark 2.1. If 𝑑𝜉 and 𝑑𝜂 are two positive measures, then the assumption (𝐻₁) can be replaced by a weaker assumption:(𝐻^′₁)𝑘₁>0, 𝑘₄>0, 𝑘>0.

Remark 2.2. It follows from (2.4) and (𝐻₃) that ¹₀𝑒(𝑠)𝑝(𝑠)𝑔(𝑟𝑒(𝑠))𝑑𝑠≤𝜎⁻¹¹₀𝑝(𝑠)𝑔(𝑟𝑒(𝑠))𝑑𝑠<+∞. (2.8) For convenience, in the rest of this paper, we define several constants as follows: 𝐿₁𝑘=1+⁴𝛾+𝛿+𝑘₃𝛼+𝛽𝜌𝑘¹₀(𝜏)+𝑘𝑑𝜉²𝛾+𝛿+𝑘₁𝛼+𝛽𝜌𝑘¹₀(𝜏),𝐿𝑑𝜂₂𝑘=𝜎1+⁴𝛾+𝛿+𝑘₃𝛼+𝛽𝜌𝑘¹₀𝑒(𝜏)(𝜏)+𝑘𝑑𝜉²𝛾+𝛿+𝑘₁𝛼+𝛽𝜌𝑘¹₀𝑒(𝜏)(𝜏,𝐿𝑑𝜂)₃𝑘=1+⁴𝛿+𝑘₃𝛽𝑘𝛽𝛿¹₀𝑒(𝜏)(𝜏)+𝑘𝑑𝜉²𝛿+𝑘₁𝛽𝑘𝛽𝛿¹₀𝑒(𝜏)𝑑𝜂(𝜏). (2.9)

Remark 2.3. If 𝑥∈𝐶[0,1]∩𝐶²(0,1) satisfies (1.1), and 𝑥(𝑡)>0 for any 𝑡∈(0,1), then we say that 𝑥 is a 𝐶[0,1]∩𝐶²(0,1) positive solution of BVP (1.1).

Lemma 2.4. Assume that (𝐻₁)  holds. Then, for any 𝑦∈𝐿¹[0,1], the problem, −𝑢^′′(𝑡)(𝑡)(),(0)=𝑦,𝑡∈0,1𝛼𝑢−𝛽𝑢^′(0)=¹₀𝑢(𝑠)(𝑠),(1)𝑑𝜉𝛾𝑢+𝛿𝑢^′(1)=¹₀𝑢(𝑠)𝑑𝜂(𝑠), (2.10) has a unique solution 𝑢(𝑡)=¹₀𝐻(𝑡,𝑠)𝑦(𝑠)𝑑𝑠, (2.11) where 𝐻()()+𝑘𝑡,𝑠=𝐺𝑡,𝑠⁴𝜙₁(𝑡)+𝑘₃𝜙₂(𝑡)𝑘¹₀𝐺()(𝜏)+𝑘𝜏,𝑠𝑑𝜉²𝜙₁(𝑡)+𝑘₁𝜙₂(𝑡)𝑘¹₀𝐺(𝜏,𝑠)𝑑𝜂(𝜏). (2.12)

Proof. The proof is similar to Lemma  2.2 of [28], so we omit it.

Lemma 2.5. Suppose that (𝐻₁) holds, then Green’s function 𝐻(𝑡,𝑠) defined by (2.12) possesses the following properties: (i)𝐻(𝑡,𝑠)≤𝐿₁𝑒(𝑠), 𝑡,𝑠∈[0,1];(ii)𝐿₂𝑒(𝑡)𝑒(𝑠)≤𝐻(𝑡,𝑠)≤𝐿₃𝑒(𝑡), 𝑡,𝑠∈[0,1],where 𝐿₁, 𝐿₂, and 𝐿₃ are defined by (2.9).

Proof. (i) It follows from (2.4) that 𝐻()(𝑠)+𝑘𝑡,𝑠≤𝑒⁴𝛾+𝛿+𝑘₃𝛼+𝛽𝜌𝑘¹₀𝑒(𝑠)(𝜏)+𝑘𝑑𝜉₂𝛾+𝛿+𝑘₁𝛼+𝛽𝜌𝑘¹₀𝑒(𝑠)𝑑𝜂(𝜏)=𝐿₁𝑒(𝑠),𝑡,𝑠∈[0,1]. (2.13)
(ii) By the monotonicity of 𝜙₁, 𝜙₂ and the definition of 𝐺(𝑡,𝑠), we have 𝑒(𝑡)𝛼+𝛽≤𝜙₁(𝑡)=𝑒(𝑡)𝜌𝜙₂(𝑡)=𝑒(𝑡)≤𝑒(𝑡)𝛼𝑡+𝛽𝛽[],𝑒(𝑡),𝑡∈0,1𝛾+𝛿≤𝜙₂(𝑡)=𝑒(𝑡)𝜌𝜙₁(𝑡)=𝑒(𝑡)𝛾()≤𝑒(𝑡)1−𝑡+𝛿𝛿,𝑡∈[0,1]. (2.14) By (2.4) and the left-hand side of inequalities (2.14), we have 𝐻()(𝑡)𝑒(𝑠)(𝑡)𝑒(𝑠)𝑘𝑡,𝑠≥𝜎𝑒+𝜎𝑒⁴/𝛼+𝛽+𝑘₃/𝛾+𝛿𝑘¹₀𝑒(𝜏)(𝜏)(𝑡)𝑒(𝑠)𝑘𝑑𝜉+𝜎𝑒²/𝛼+𝛽+𝑘₁/𝛾+𝛿𝑘¹₀𝑒(𝜏)(𝜏)(𝑡)𝑒(𝑠𝑘𝑑𝜂=𝜎𝑒)1+⁴𝛾+𝛿+𝑘₃𝛼+𝛽𝜌𝑘¹₀𝑒(𝜏)(𝜏)+𝑘𝑑𝜉₂𝛾+𝛿+𝑘₁𝛼+𝛽𝜌𝑘¹₀𝑒(𝜏)𝑑𝜂(𝜏)=𝐿₂𝑒(𝑡)𝑒(𝑠),𝑡,𝑠∈[0,1]. (2.15) Similarly, by (2.4) and the right-hand side of inequalities (2.14), we have 𝐻()()+𝑘𝑡,𝑠=𝐺𝑡,𝑠⁴𝜙₁(𝑡)+𝑘₃𝜙₂(𝑡)𝑘¹₀𝐺()(𝜏)+𝑘𝜏,𝑠𝑑𝜉²𝜙₁(𝑡)+𝑘₁𝜙₂(𝑡)𝑘¹₀𝐺()(𝜏)(𝑡)(𝑡𝑘𝜏,𝑠𝑑𝜂≤𝑒+𝑒)⁴/𝛽+𝑘₃/𝛿𝑘¹₀𝐺()(𝜏)+𝑘𝜏,𝑠𝑑𝜉²/𝛽+𝑘₁/𝛿𝑘¹₀𝐺()(𝜏(𝑡)(𝑡𝑘𝜏,𝑠𝑑𝜂)≤𝑒+𝑒)⁴𝛿+𝑘₃𝛽𝑘𝛽𝛿¹₀𝑒(𝜏)(𝜏)+𝑘𝑑𝜉²𝛿+𝑘₁𝛽𝑘𝛽𝛿¹₀𝑒(𝜏)(𝜏(𝑡𝑘𝑑𝜂)=𝑒)1+⁴𝛿+𝑘₃𝛽𝑘𝛽𝛿¹₀𝑒(𝜏)(𝜏)+𝑘𝑑𝜉²𝛿+𝑘₁𝛽𝑘𝛽𝛿¹₀𝑒(𝜏)𝑑𝜂(𝜏)=𝐿₃𝑒(𝑡),𝑡,𝑠∈[0,1]. (2.16) The proof of Lemma 2.5 is completed.

Lemma 2.6. Suppose that (𝐻₁) and (𝐻₂) hold. Then, the boundary value problem, −𝑤^′′(𝑡)(𝑡)(),(0)=2𝜆𝑝,𝑡∈0,1𝛼𝑤−𝛽𝑤^′(0)=¹₀𝑤(𝑠)(𝑠),(1)𝑑𝜉𝛾𝑤+𝛿𝑤^′(1)=¹₀𝑤(𝑠)𝑑𝜂(𝑠), (2.17) has unique solution 𝑤(𝑡)=2𝜆¹₀𝐻(𝑡,𝑠)𝑝(𝑠)𝑑𝑠, (2.18) which satisfies 𝑤(𝑡)≤2𝜆𝐿₃𝑒(𝑡)¹₀𝑝(𝑠)𝑑𝑠,𝑡∈[0,1]. (2.19)

Proof. It follows from (2.11), Lemma 2.5, (𝐻₁) and (𝐻₂) that (2.18) and (2.19) hold.

Let 𝑋=𝐶[0,1] be a real Banach space with the norm ‖𝑥‖=max_𝑡∈[0,1]|𝑥(𝑡)| for 𝑥∈𝑋. We let 𝐾={𝑥∈𝑋∶𝑥isconcaveon[0,1],𝑥(𝑡)≥Λ𝑒(𝑡)‖𝑥‖for𝑡∈[0,1]}, (2.20) where Λ=𝐿₂/𝐿₁. Clearly, 𝐾 is a cone of 𝑋.

For any 𝑢∈𝑋, let us define a function [⋅]⁺: [𝑢(𝑡)]⁺=𝑢(𝑡)(𝑡)(𝑡),𝑢≥0,0,𝑢<0. (2.21)

Next, we consider the following approximate problem of (1.1): −𝑥^′′(𝑡)=𝜆𝑓𝑡,[𝑥(𝑡)−𝑤(𝑡)]⁺(𝑡(),(0)+𝑞),𝑡∈0,1𝛼𝑥−𝛽𝑥^′(0)=¹₀𝑥(𝑠)(𝑠),(1)𝑑𝜉𝛾𝑥+𝛿𝑥^′(1)=¹₀𝑥(𝑠)𝑑𝜂(𝑠). (2.22)

Lemma 2.7. If 𝑥∈𝐶[0,1]∩𝐶²(0,1) is a positive solution of problem (2.22) with 𝑥(𝑡)≥𝑤(𝑡) for any 𝑡∈[0,1], then 𝑥−𝑤 is a positive solution of the singular semipositone differential equation (1.1).

Proof. If 𝑥 is a positive solution of (2.22) such that 𝑥(𝑡)≥𝑤(𝑡) for any 𝑡∈[0,1], then from (2.22) and the definition of [𝑢(𝑡)]⁺, we have −𝑥^′′(𝑡)𝑓((𝑡)(𝑡(𝑡(),(0)=𝜆𝑡,𝑥−𝑤))+𝑞),𝑡∈0,1𝛼𝑥−𝛽𝑥^′(0)=¹₀𝑥(𝑠)(𝑠),(1)𝑑𝜉𝛾𝑥+𝛿𝑥^′(1)=¹₀𝑥(𝑠)𝑑𝜂(𝑠). (2.23) Let 𝑢=𝑥−𝑤, then 𝑢^′′=𝑥^′′−𝑤^′′, which implies that −𝑥^′′=−𝑢^′′−𝑤^′′=−𝑢^′′+2𝜆𝑞(𝑡). (2.24) Thus, (2.23) becomes −𝑢^′′(𝑡)𝑓((𝑡(𝑡(),(0)=𝜆𝑡,𝑢))−𝑞),𝑡∈0,1𝛼𝑢−𝛽𝑢^′(0)=¹₀𝑢(𝑠)(𝑠),(1)𝑑𝜉𝛾𝑢+𝛿𝑢^′(1)=¹₀𝑢(𝑠)𝑑𝜂(𝑠), (2.25) that is, 𝑥−𝑤 is a positive solution of (1.1). The proof is complete.

To overcome singularity, we consider the following approximate problem of (2.22): −𝑥^′′(𝑡)=𝜆𝑓𝑡,[𝑥(𝑡)−𝑤(𝑡)]⁺+𝑛⁻¹(𝑡(),(0)+𝑞),𝑡∈0,1𝛼𝑥−𝛽𝑥^′(0)=¹₀𝑥(𝑠)(𝑠),(1)𝑑𝜉𝛾𝑥+𝛿𝑥^′(1)=¹₀𝑥(𝑠)𝑑𝜂(𝑠), (2.26) where 𝑛 is a positive integer. For any 𝑛∈ℕ, let us define a nonlinear integral operator 𝑇^𝜆_𝑛∶𝐾→𝑋 as follows: 𝑇^𝜆_𝑛𝑥(𝑡)=𝜆¹₀𝐻(𝑡,𝑠)𝑓𝑠,[𝑥(𝑠)−𝑤(𝑠)]⁺+𝑛⁻¹+𝑞(𝑠)𝑑𝑠. (2.27) It is obvious that solving (2.26) in 𝐶[0,1]∩𝐶²(0,1) is equivalent to solving the fixed point equation 𝑇^𝜆_𝑛𝑥=𝑥 in the Banach space 𝐶[0,1].

Lemma 2.8. Assume that (𝐻₁)–(𝐻₃) hold, then for each 𝑛∈ℕ, 𝜆>0, 𝑅>𝑟≥4𝜆𝐿₃Λ⁻¹∫¹₀𝑞(𝑠)𝑑𝑠, 𝑇^𝜆_𝑛∶𝐾_𝑟,𝑅→𝐾 is a completely continuous operator.

Proof. Let 𝑛∈ℕ be fixed. For any 𝑥∈𝐾, by (2.27) we have 𝑇^𝜆_𝑛𝑥^′′(𝑡)=−𝜆𝑓𝑠,[𝑥(𝑠)−𝑤(𝑠)]⁺+𝑛⁻¹(𝑠𝑇+𝑞)≤0,_𝜆𝑛𝑥(0)=𝜆¹₀𝐻(0,𝑠)𝑓𝑠,[𝑥(𝑠)−𝑤(𝑠)]⁺+𝑛⁻¹(𝑠𝑇+𝑞)𝑑𝑠≥0,_𝜆𝑛𝑥(1)=𝜆¹₀𝐻(1,𝑠)𝑓𝑠,[𝑥(𝑠)−𝑤(𝑠)]⁺+𝑛⁻¹+𝑞(𝑠)𝑑𝑠≥0, (2.28) which implies that 𝑇^𝜆_𝑛 is nonnegative and concave on [0,1]. For any 𝑥∈𝐾 and 𝑡∈[0,1], it follows from Lemma 2.5 that 𝑇^𝜆_𝑛𝑥(𝑡)=𝜆¹₀𝐻(𝑡,𝑠)𝑓𝑠,[𝑥(𝑠)−𝑤(𝑠)]⁺+𝑛⁻¹+𝑞(𝑠)𝑑𝑠≤𝜆𝐿₁¹₀𝑒(𝑠)𝑓𝑠,[𝑥(𝑠)−𝑤(𝑠)]⁺+𝑛⁻¹+𝑞(𝑠)𝑑𝑠. (2.29) Thus, 𝑇^𝜆_𝑛𝑥≤𝜆𝐿₁¹₀𝑒(𝑠)𝑓𝑠,[𝑥(𝑠)−𝑤(𝑠)]⁺+𝑛⁻¹+𝑞(𝑠)𝑑𝑠. (2.30) On the other hand, from Lemma 2.5, we also obtain 𝑇^𝜆_𝑛𝑥(𝑡)=𝜆¹₀𝐻(𝑡,𝑠)𝑓𝑠,[𝑥(𝑠)−𝑤(𝑠)]⁺+𝑛⁻¹+𝑞(𝑠)𝑑𝑠≥𝜆𝐿₂𝑒(𝑡)¹₀𝑒(𝑠)𝑓𝑠,[𝑥(𝑠)−𝑤(𝑠)]⁺+𝑛⁻¹+𝑞(𝑠)𝑑𝑠. (2.31) So, 𝑇^𝜆_𝑛𝑥(𝑡)≥Λ𝑒(𝑡)𝑇^𝜆_𝑛𝑥,𝑡∈[0,1]. (2.32) This yields that 𝑇^𝜆_𝑛(𝐾)⊂𝐾.
Next, we prove that 𝑇^𝜆_𝑛∶𝐾_𝑟,𝑅→𝐾 is completely continuous. Suppose 𝑥_𝑚∈𝐾_𝑟,𝑅 and 𝑥₀∈𝐾_𝑟,𝑅 with ‖𝑥_𝑚−𝑥₀‖→0(𝑚→∞). Notice that 𝑥_𝑚(𝑠)−𝑤(𝑠)⁺−𝑥₀(𝑠)−𝑤(𝑠)⁺=𝑥_𝑚(𝑠)−𝑤(𝑠)+𝑥_𝑚(𝑠)−𝑤(𝑠)2−𝑥₀(𝑠)−𝑤(𝑠)+𝑥₀(𝑠)−𝑤(𝑠)2=𝑥_𝑚(𝑠)−𝑤(𝑠)−𝑥₀(𝑠)−𝑤(𝑠)2+𝑥_𝑚(𝑠)−𝑥₀(𝑠)2≤𝑥_𝑚(𝑠)−𝑥₀(𝑠). (2.33) This, together with the continuity of 𝑓, implies 𝑓𝑠,𝑥_𝑚(𝑠)−𝑤(𝑠)⁺+𝑛⁻¹+𝑞(𝑠)−𝑓𝑠,𝑥₀(𝑠)−𝑤(𝑠)⁺+𝑛⁻¹(𝑠=𝑓𝑥+𝑞)𝑠,_𝑚(𝑠)−𝑤(𝑠)⁺+𝑛⁻¹−𝑓𝑠,𝑥₀(𝑠)−𝑤(𝑠)⁺+𝑛⁻¹⟶0,𝑚⟶∞. (2.34) Using the Lebesgue dominated convergence theorem, we have 𝑇^𝜆_𝑛𝑥_𝑚−𝑇^𝜆_𝑛𝑥₀≤𝜆𝐿₁¹₀𝑒(𝑠)𝑓𝑠,𝑥_𝑚(𝑠)−𝑤(𝑠)⁺+𝑛⁻¹−𝑓𝑠,𝑥₀(𝑠)−𝑤(𝑠)⁺+𝑛⁻¹𝑑𝑠⟶0,𝑚⟶∞. (2.35) So, 𝑇^𝜆_𝑛∶𝐾_𝑟,𝑅→𝐾 is continuous.
Let 𝐵⊂𝐾_𝑟,𝑅 be any bounded set, then for any 𝑥∈𝐵, we have 𝑥∈𝐾, 𝑟≤‖𝑥‖≤𝑅. Therefore, we have [𝑥(𝑡)−𝑤(𝑡)]⁺(𝑡)≤‖𝑥‖[],𝑥(𝑡)(𝑡)(𝑡)≤𝑥≤𝑅,𝑡∈0,1−𝑤≥𝑥−2𝜆𝐿₃𝑒(𝑡)¹₀𝑞(𝑠)𝑑𝑠≥𝑥(𝑡)−2𝜆𝐿₃𝑥(𝑡)Λ𝑟¹₀𝑞(𝑠)≥1𝑑𝑠2𝑥(𝑡)≥12𝑟Λ𝑒(𝑡)>0,𝑡∈[0,1]. (2.36) By (𝐻₃), we have 𝐿_𝑟∶=¹₀𝑝(𝑠)𝑔12Λ𝑟𝑒(𝑠)𝑑𝑠<+∞. (2.37) It is easy to show that 𝑇^𝜆_𝑛(𝐵) is uniformly bounded. In order to show that 𝑇^𝜆_𝑛 is a compact operator, we only need to show that 𝑇^𝜆_𝑛(𝐵) is equicontinuous. By the continuity of 𝐻(𝑡,𝑠) on [0,1]×[0,1], for any 𝜀>0, there exists 𝛿₁>0 such that for any 𝑡₁,𝑡₂,𝑠∈[0,1] and |𝑡₁−𝑡₂|<𝛿₁, we have 𝐻𝑡₁,𝑠−𝐻𝑡₂,𝑠<𝜀. (2.38) By (2.36)–(2.37), and (2.27), we have 𝑇^𝜆_𝑛𝑥𝑡₁−𝑇^𝜆_𝑛𝑥𝑡₂≤𝜆¹₀𝐻𝑡₁,𝑠−𝐻𝑡₂,𝑠𝑓𝑠,[𝑥(𝑠)−𝑤(𝑠)]⁺+𝑛⁻¹+𝑞(𝑠)𝑑𝑠<𝜀𝜆¹₀𝑓𝑠,[𝑥(𝑠)−𝑤(𝑠)]⁺+𝑛⁻¹+𝑞(𝑠)𝑑𝑠≤𝜀𝜆¹₀𝑝(𝑠)𝑔[𝑥(𝑠)−𝑤(𝑠)]⁺+𝑛⁻¹+ℎ[𝑥(𝑠)−𝑤(𝑠)]⁺+𝑛⁻¹(𝑠𝐿+𝑞)𝑑𝑠≤𝜀𝜆_𝑟+1+ℎ^∗(𝑅)¹₀𝑝(𝑠)+𝑞(𝑠)𝑑𝑠, (2.39) where ℎ^∗(𝑟)=max_{𝑦∈[0,1+𝑟]}ℎ𝑦. (2.40) This means that 𝑇^𝜆_𝑛(𝐵) is equicontinuous. By the Arzela-Ascoli theorem, 𝑇^𝜆_𝑛(𝐵) is a relatively compact set. Now since 𝜆 and 𝑛 are given arbitrarily, the conclusion of this lemma is valid.

3. Main Results

Theorem 3.1. Assume that conditions (𝐻₁)–(𝐻₃) are satisfied. Further assume that the following condition holds.(𝐻₄) There exists an interval [𝑎,𝑏]⊂(0,1) such thatliminf_𝑢→+∞min_{𝑡∈[𝑎,𝑏]}𝑓(𝑡,𝑢)𝑢=+∞. (3.1) Then, there exists 𝜆^∗>0 such that the BVP (1.1) has at least one positive solution 𝑢(𝑡)∈𝐶[0,1]∩𝐶²(0,1) provided 𝜆∈(0,𝜆^∗). Furthermore, the solution also satisfies 𝑢(𝑡)≥𝑙𝑒̃(𝑡) for some positive constant 𝑙̃.

Proof. Take 𝑟>4𝐿₃Λ⁻¹∫¹₀𝑞(𝑠)𝑑𝑠. Let 𝜆^∗𝑟=min1,2𝐿₁∫¹₀𝑒(𝑠)𝑝(𝑠)𝑔)(𝑠,𝑟((1/2𝑟Λ𝑒))𝑑𝑠2𝐿₁(ℎ^∗(𝑟)+1)∫¹₀𝑒(𝑠)𝑝(𝑠)+𝑞(𝑠)𝑑𝑠, (3.2) where ℎ^∗ is defined by (2.40). For any 𝜆∈(0,𝜆^∗), 𝑥∈𝜕𝐾_𝑟, noticing that 𝜆^∗≤1, we have [𝑥(𝑡)−𝑤(𝑡)]⁺(𝑡)≤‖𝑥‖[],𝑥(𝑡)(𝑡)(𝑡)≤𝑥≤𝑟,𝑡∈0,1−𝑤≥𝑥−2𝜆𝐿₃𝑒(𝑡)¹₀𝑞(𝑠)𝑑𝑠≥𝑥(𝑡)−2𝐿₃𝑒(𝑡)¹₀𝑞(𝑠)(𝑡)𝑑𝑠≥𝑥−2𝐿₃𝑥(𝑡)Λ𝑟¹₀𝑞(𝑠)1𝑑𝑠≥2𝑥(𝑡)≥12𝑟Λ𝑒(𝑡)>0,𝑡∈[0,1]. (3.3) For any 𝜆∈(0,𝜆^∗), by (3.3), we have 𝑇^𝜆_𝑛𝑥(𝑡)=𝜆¹₀𝐻(𝑡,𝑠)𝑓𝑠,[𝑥(𝑠)−𝑤(𝑠)]⁺+𝑛⁻¹+𝑞(𝑠)𝑑𝑠≤𝜆𝐿₁¹₀𝑒(𝑠)𝑝(𝑠)𝑔[𝑥(𝑠)−𝑤(𝑠)]⁺+𝑛⁻¹+ℎ[𝑥(𝑠)−𝑤(𝑠)]⁺+𝑛⁻¹+𝑞(𝑠)𝑑𝑠≤𝜆𝐿₁¹₀𝑒(𝑠𝑝(𝑠)𝑔1)2𝑟Λ𝑒(𝑠)+ℎ^∗(𝑟)+𝑞(𝑠)𝑑𝑠≤𝜆𝐿₁¹₀𝑒(𝑠)𝑝(𝑠)𝑔12𝑟Λ𝑒(𝑠)𝑑𝑠+𝜆𝐿₁ℎ^∗(𝑟)+1¹₀𝑒(𝑠𝑝(𝑠)(𝑠≤𝑟)+𝑞)𝑑𝑠2+𝑟2=𝑟, (3.4) which means that 𝑇^𝜆_𝑛𝑥≤‖𝑥‖,𝑥∈𝜕𝐾_𝑟. (3.5)
On the other hand, choose a real number 𝑀>0 such that 𝜆𝑀𝐿₂𝑙²₁Λ∫^𝑏_𝑎𝑒(𝑠)𝑑𝑠>2, where 𝑙₁=min_0≤𝑡≤1𝑒(𝑡)>0, 𝐿₂ is defined by (2.9). By (𝐻₄), there exists 𝑁>0 such that for any 𝑡∈[𝑎,𝑏], we have 𝑓(𝑡,𝑢)≥𝑀𝑢,𝑢≥𝑁. (3.6) Take 𝑅>max{𝑟,2𝑁/𝑙₁Λ}. Next, we take 𝜑₁≡1∈𝜕𝐾₁={𝑥∈𝐾∶‖𝑥‖=1}, and for any 𝑥∈𝜕𝐾_𝑅, 𝑚>0, 𝑛∈ℕ, we will show 𝑥≠𝑇^𝜆_𝑛𝑥+𝑚𝜑₁. (3.7) Otherwise, there exist 𝑥₀∈𝜕𝐾_𝑅 and 𝑚₀>0 such that 𝑥₀=𝑇^𝜆_𝑛𝑥₀+𝑚₀𝜑₁. (3.8) From 𝑥₀∈𝜕𝐾_𝑅, we know that ‖𝑥₀‖=𝑅. Then, for 𝑡∈[𝑎,𝑏], we have 𝑥₀(𝑡)−𝑤(𝑡)≥𝑥₀(𝑡)−2𝜆𝐿₃𝑒(𝑡)¹₀𝑞(𝑠)𝑑𝑠≥𝑥₀(𝑡)−2𝐿₃𝑥₀(𝑡)Λ𝑅¹₀𝑞(𝑠)≥1𝑑𝑠2𝑥₀(𝑡)≥12(𝑡)≥𝑙𝑅Λ𝑒¹𝑅Λ2≥𝑁>0. (3.9) So, by (3.6), (3.9), we have 𝑥₀(𝑡)=𝜆¹₀𝐻(𝑡,𝑠)𝑓𝑠,𝑥₀(𝑠)−𝑤(𝑠)⁺+𝑛⁻¹+𝑞(𝑠)𝑑𝑠+𝑚₀≥𝜆𝐿₂𝑒(𝑡)¹₀𝑒(𝑠)𝑓𝑠,𝑥₀(𝑠)−𝑤(𝑠)⁺+𝑛⁻¹𝑑𝑠+𝑚₀≥𝜆𝐿₂𝑒(𝑡)^𝑏_𝑎𝑒(𝑠)𝑓𝑠,𝑥₀(𝑠)−𝑤(𝑠)⁺+𝑛⁻¹𝑑𝑠+𝑚₀≥𝜆𝑀𝐿₂𝑒(𝑡)^𝑏_𝑎𝑒(𝑠)𝑥₀(𝑠)−𝑤(𝑠)⁺+𝑛⁻¹𝑑𝑠+𝑚₀≥12𝜆𝑀𝐿₂𝑙²₁Λ𝑅^𝑏_𝑎𝑒(𝑠)𝑑𝑠+𝑚₀≥𝑅+𝑚₀>𝑅. (3.10) This implies that 𝑅>𝑅, which is a contradiction. This yields that (3.7) holds. By (3.5), (3.7), and Lemma 1.1, for any 𝑛∈ℕ and 𝜆∈(0,𝜆^∗), we obtain that 𝑇^𝜆_𝑛 has a fixed point 𝑥_𝑛 in 𝐾_𝑟,𝑅.
Let {𝑥_𝑛}^∞_𝑛=1 be the sequence of solutions of the boundary value problems (2.26). It is easy to see that they are uniformly bounded. Next, we show that {𝑥_𝑛}^∞_𝑛=1 are equicontinuous on [0,1]. From 𝑥_𝑛∈𝐾_𝑟,𝑅, we know that 𝑥_𝑛(𝑡)−𝑤(𝑡)⁺≤𝑥_𝑛(𝑡)≤𝑥_𝑛[],𝑥≤𝑅,𝑡∈0,1_𝑛(𝑡)−𝑤(𝑡)≥𝑥_𝑛(𝑡)−2𝜆𝐿₃𝑒(𝑡)¹₀𝑞(𝑠)𝑑𝑠≥𝑥_𝑛(𝑡)−2𝜆𝐿₃𝑥_𝑛(𝑡)Λ𝑥_𝑛¹₀𝑞(𝑠)≥1𝑑𝑠2𝑥_𝑛(𝑡)≥12Λ𝑒(𝑡)𝑥_𝑛≥12Λ𝑟𝑒(𝑡)>0,𝑡∈[0,1]. (3.11) For any 𝜀>0, by the continuity of 𝐻(𝑡,𝑠) in [0,1]×[0,1], there exists 𝛿₂>0 such that for any 𝑡₁,𝑡₂,𝑠∈[0,1] and |𝑡₁−𝑡₂|<𝛿₂, we have 𝐻𝑡₁,𝑠−𝐻𝑡₂,𝑠<𝜀. (3.12) This, combined with (2.11) and (2.37), implies that for any 𝑡₁,𝑡₂∈[0,1] and |𝑡₁−𝑡₂|<𝛿₂, we have 𝑥_𝑛𝑡₁−𝑥_𝑛𝑡₂≤¹₀𝐻𝑡₁,𝑠−𝐻𝑡₂,𝑠𝑓𝑠,[𝑥(𝑠)−𝑤(𝑠)]⁺+𝑛⁻¹+𝑞(𝑠)𝑑𝑠<𝜀¹₀𝑓𝑠,[𝑥(𝑠)−𝑤(𝑠)]⁺+𝑛⁻¹+𝑞(𝑠)𝑑𝑠≤𝜀¹₀𝑝(𝑠)𝑔[𝑥(𝑠)−𝑤(𝑠)]⁺+𝑛⁻¹+ℎ[𝑥(𝑠)−𝑤(𝑠)]⁺+𝑛⁻¹(𝑠𝐿+𝑞)𝑑𝑠≤𝜀_𝑟+1+ℎ^∗(𝑅)¹₀𝑝(𝑠)+𝑞(𝑠)𝑑𝑠. (3.13) By the Ascoli-Arzela theorem, the sequence {𝑥_𝑛}^∞_𝑛=1 has a subsequence being uniformly convergent on [0,1]. Without loss of generality, we still assume that {𝑥_𝑛}^∞_𝑛=1 itself uniformly converges to 𝑥 on [0,1]. Since {𝑥_𝑛}^∞_𝑛=1∈𝐾_𝑟,𝑅⊂𝐾, we have 𝑥_𝑛≥0. By (2.26), we have 𝑥_𝑛(𝑡)=𝑥_𝑛12+1𝑡−2𝑥^′_𝑛12−𝜆^𝑡_1/2𝑑𝑠^𝑠_1/2𝑓𝜏,𝑥_𝑛(𝜏)−𝑤(𝜏)+𝑛⁻¹+𝑞(𝜏)𝑑𝜏,𝑡∈(0,1). (3.14) From (3.14), we know that {𝑥^′_𝑛(1/2)}^∞_𝑛=1 is bounded sets. Without loss of generality, we may assume 𝑥^′_𝑛(1/2)→𝑐₁ as 𝑛→∞. Then, by (3.14) and the Lebesgue dominated convergence theorem, we have 𝑥(𝑡)1=𝑥2+𝑐₁1𝑡−2−𝜆^𝑡_1/2𝑑𝑠^𝑠_1/2𝑓(𝜏,𝑥(𝜏)−𝑤(𝜏))+𝑞(𝜏)𝑑𝜏,𝑡∈(0,1). (3.15) By (3.15), direct computation shows that −𝑥^′′(𝑡)=𝜆𝑓(𝑡,𝑥(𝑡)−𝑤(𝑡))+𝑞(𝑡),0<𝑡<1. (3.16) On the other hand, letting 𝑛→∞ in the following boundary conditions: 𝛼𝑥_𝑛(0)−𝛽𝑥^′_𝑛(0)=¹₀𝑥_𝑛(𝑠)𝑑𝜉(𝑠),𝛾𝑥_𝑛(1)+𝛿𝑥^′_𝑛(1)=¹₀𝑥_𝑛(𝑠)𝑑𝜂(𝑠), (3.17) we deduce that 𝑥 is a positive solution of BVP (2.22).
Let 𝑢(𝑡)=𝑥(𝑡)−𝑤(𝑡) and 𝑙̃=(1/2)Λ𝑟. By (3.11) and the convergence of sequence {𝑥_𝑛}^∞_𝑛=1, we have 𝑢(𝑡)≥𝑙𝑒̃(𝑡)>0, 𝑡∈[0,1]. It then follows from Lemma 2.7 that BVP (1.1) has at least one positive solution 𝑢 satisfying 𝑢≥𝑙𝑒̃(𝑡) for any 𝑡∈[0,1]. The proof is completed.

Theorem 3.2. Assume that conditions (𝐻₁)–(𝐻₃) are satisfied. In addition, assume that the following condition holds.(𝐻₅) There exists an interval [𝑐,𝑑]⊂(0,1) such that liminf_𝑢→+∞min_{𝑡∈[𝑐,𝑑]}𝑓(𝑡,𝑢)>4𝐿₃∫¹₀𝑞(𝑠)𝑑𝑠Λ𝐿₂𝑙₂∫^𝑑_𝑐𝑒(𝑠)𝑑𝑠, (3.18) where 𝑙₂=min_{𝑐≤𝑡≤𝑑}𝑒(𝑡) and lim_𝑢→+∞ℎ(𝑢)𝑢=0. (3.19) Then there exists 𝜆^∗>0 such that the BVP (1.1) has at least one positive solution 𝑢(𝑡)∈𝐶[0,1]∩𝐶²(0,1) provided 𝜆∈(𝜆^∗,+∞). Furthermore, the solution also satisfies 𝑢(𝑡)≥𝑙𝑒̃(𝑡) for some positive constant 𝑙̃.

Proof. By (3.18), there exists 𝑁₀>0 such that, for any 𝑡∈[𝑐,𝑑], 𝑢≥𝑁₀, we have 𝑓(𝑡,𝑢)≥4𝐿₃∫¹₀𝑞(𝑠)𝑑𝑠Λ𝐿₂𝑙₂∫^𝑑_𝑐𝑒(𝑠)𝑑𝑠. (3.20) Choose 𝜆^∗=𝑁₀/2𝑙₂𝐿₃∫¹₀𝑞(𝑠)𝑑𝑠. Let 𝑟=4𝜆𝐿₃Λ⁻¹∫¹₀𝑞(𝑠)𝑑𝑠 as 𝜆>𝜆^∗. Next, we take 𝜑₁≡1∈𝜕𝐾₁={𝑥∈𝐾∶‖𝑥‖=1}, and for any 𝑥∈𝜕𝐾_𝑟, 𝑚>0, 𝑛∈ℕ, we will show that 𝑥≠𝑇^𝜆_𝑛𝑥+𝑚𝜑₁. (3.21) Otherwise, there exist 𝑥₀∈𝜕𝐾_𝑟 and 𝑚₀>0 such that 𝑥₀=𝑇^𝜆_𝑛𝑥₀+𝑚₀𝜑₁. (3.22) From 𝑥₀∈𝜕𝐾_𝑟, we know that ‖𝑥₀‖=𝑟, and 𝑥₀(𝑡)−𝑤(𝑡)≥Λ𝑟𝑒(𝑡)−2𝜆𝐿₃𝑒(𝑡)¹₀𝑞(𝑠)𝑑𝑠=2𝜆𝐿₃𝑒(𝑡)¹₀𝑞(𝑠)𝑁𝑑𝑠≥⁰𝑙₂𝑒(𝑡). (3.23) So, we have 𝑥₀(𝑡)−𝑤(𝑡)≥𝑁₀ on 𝑡∈[𝑐,𝑑], 𝑥₀∈𝜕𝐾_𝑟. Then, by (3.20) we have 𝑥₀(𝑡)=𝜆¹₀𝐻(𝑡,𝑠)𝑓𝑠,𝑥₀(𝑠)−𝑤(𝑠)⁺+𝑛⁻¹+𝑞(𝑠)𝑑𝑠+𝑚₀≥𝜆𝐿₂𝑒(𝑡)¹₀𝑒(𝑠)𝑓𝑠,𝑥₀(𝑠)−𝑤(𝑠)⁺+𝑛⁻¹𝑑𝑠+𝑚₀≥𝜆𝐿₂𝑒(𝑡)^𝑑_𝑐𝑒(𝑠)𝑓𝑠,𝑥₀(𝑠)−𝑤(𝑠)⁺+𝑛⁻¹𝑑𝑠+𝑚₀≥𝜆𝐿₂𝑒(𝑡)4𝐿₃∫¹₀𝑞(𝑠)𝑑𝑠Λ𝐿₂𝑙₂∫^𝑑_𝑐𝑒(𝑠)𝑑𝑠^𝑑𝑐𝑒(𝑠)𝑑𝑠+𝑚₀≥4𝜆Λ⁻¹𝐿₃¹₀𝑞(𝑠)𝑑𝑠+𝑚₀=𝑟+𝑚₀>𝑟. (3.24) This implies that 𝑟>𝑟, which is a contradiction. This yields that (3.21) holds.
On the other hand, by (3.19) and the continuity of ℎ(𝑢) on [0,+∞), we have lim_𝑢→+∞ℎ^∗(𝑢)𝑢=0, (3.25) where ℎ^∗(𝑢) is defined by (2.40). For 𝜖=4𝜆𝐿₁¹₀𝑒(𝑠)[𝑝(𝑠)+𝑞(𝑠)]𝑑𝑠⁻¹, (3.26) there exists 𝑀₀>0 such that when 𝑥>𝑀₀, for any 0≤𝑦≤𝑥, we have ℎ(𝑦)≤𝜖𝑥. Take 𝑅>max2,𝑟,𝑀₀,2𝜆𝐿₁¹₀𝑒(𝑠)𝑝(𝑠)𝑔(Λ𝑒(𝑠))𝑑𝑠. (3.27) Then, for any 𝑥∈𝜕𝐾_𝑅,𝑡∈[0,1], we have [𝑥(𝑡)−𝑤(𝑡)]⁺(𝑡)≤‖𝑥‖𝑥(𝑡)(𝑡)(𝑡)≤𝑥≤𝑅,−𝑤≥𝑥−2𝜆𝐿₃𝑒(𝑡)¹₀𝑞(𝑠)𝑑𝑠≥𝑥(𝑡)−2𝜆𝐿₃𝑥(𝑡)Λ𝑅¹₀𝑞(𝑠)≥1𝑑𝑠2𝑥(𝑡)≥12𝑅Λ𝑒(𝑡)>Λ𝑒(𝑡)>0,𝑡∈[0,1]. (3.28) It follows from (3.19) and (3.28) that 𝑇^𝜆_𝑛𝑥(𝑡)=𝜆¹₀𝐻(𝑡,𝑠)𝑓𝑠,[𝑥(𝑠)−𝑤(𝑠)]⁺+𝑛⁻¹+𝑞(𝑠)𝑑𝑠≤𝜆𝐿₁¹₀𝑒(𝑠)𝑝(𝑠)𝑔[𝑥(𝑠)−𝑤(𝑠)]⁺+𝑛⁻¹𝑑𝑠+𝜆𝐿₁¹₀𝑒(𝑠)𝑝(𝑠)ℎ[𝑥(𝑠)−𝑤(𝑠)]⁺+𝑛⁻¹+𝑞(𝑠)𝑑𝑠≤𝜆𝐿₁¹₀𝑒(𝑠)𝑝(𝑠)𝑔(Λ𝑒(𝑠))𝑑𝑠+𝜆𝐿₁¹₀𝑒(𝑠)𝑝(𝑠)𝜖(𝑅+1)+𝑞(𝑠)𝑑𝑠≤𝜆𝐿₁¹₀𝑒(𝑠)𝑝(𝑠)𝑔(Λ𝑒(𝑠))𝑑𝑠+𝜖𝜆𝐿₁(𝑅+2)¹₀𝑒(𝑠𝑝(𝑠)(𝑠≤𝑅)+𝑞)𝑑𝑠2+𝑅2=𝑅, (3.29) which means that 𝑇^𝜆_𝑛𝑥≤‖𝑥‖,𝑥∈𝜕𝐾_𝑅. (3.30) By (3.21), (3.30), and Lemma 1.1, for any 𝑛∈ℕ and 𝜆>𝜆^∗, we obtain that 𝑇^𝜆_𝑛 has a fixed point 𝑥_𝑛 in 𝐾_𝑟,𝑅 satisfying 𝑟≤‖𝑥_𝑛‖≤𝑅. The rest of proof is similar to Theorem 3.1. The proof is complete.

Remark 3.3. From the proof of Theorem 3.2, we can see that if (𝐻₅) is replaced by the following condition.(𝐻^′₅) There exists an interval [𝑐,𝑑]⊂(0,1) such that liminf_𝑢→+∞min_{[]𝑡∈𝑐,𝑑}𝑓(𝑡,𝑢)=+∞,lim_𝑢→+∞ℎ(𝑢)𝑢=0, (3.31) then, the conclusion of Theorem 3.2 is still true.

4. Applications

In this section, we construct two examples to demonstrate the application of our main results.

Example 4.1. Consider the following 4-point boundary value problem: −𝑢^′′(𝑡)1=𝜆𝑡()11−𝑡𝑢²+𝑢²(𝑡𝑢(0)+1−𝑞),0<𝑡<1,−𝑢^′(0)=14𝑢13+19𝑢23,𝑢(1)+𝑢^′(1)=38𝑢132+𝑢3, (4.1) where 𝜆>0 is a parameter and 𝑞(𝑡)=14+3√³41√𝑡+1√+11−𝑡³(𝑡−1/2)². (4.2) The BVP (4.1) can be regarded as a boundary value problem of the form of (1.1). In this situation, 𝛼=𝛽=𝛾=𝛿=1 and 𝜉(𝑠)=10,𝑠∈0,3,141,𝑠∈3,23,13236,𝑠∈3,𝜂(𝑠)=1,10,𝑠∈0,3,381,𝑠∈3,23,1182,𝑠∈3,1. (4.3) Let 𝑓()=1𝑡,𝑥𝑡()11−𝑡𝑥²+𝑥²+1,for(𝑡,𝑥)∈(0,1)×(0,+∞) (4.4) and let 𝑝(𝑡)=1/𝑡(1−𝑡), 𝑔(𝑥)=1/𝑥², ℎ(𝑥)=𝑥²+1. By direct calculation, we have ∫¹₀𝑝(𝑡)𝑑𝑡=𝜋, ∫¹₀𝑞(𝑡)𝑑𝑡=1, and 3𝜌=3,𝜎=4,𝜙₁(𝑡)=2−𝑡3,𝜙₂(𝑡)=1+𝑡3(𝑡)=1,𝑒3(),𝑘2−𝑡)(1+𝑡₁=263324,𝑘₂=1481,𝑘₃=4772,𝑘₄=518,𝑘=73,𝐿648₁=11,𝐿₂=22736,𝐿₃=1099. (4.5) Clearly, the conditions (𝐻₁)–(𝐻₃) hold. Taking 𝑡∈[1/4,3/4]⊂[0,1], we have liminf_𝑥→+∞min_{𝑡∈[1/4,3/4]}𝑓(𝑡,𝑥)𝑥=liminf_𝑥→+∞min_{𝑡∈[1/4,3/4]}1/𝑡(1−𝑡)1/𝑥²+𝑥²+1𝑥=+∞. (4.6) Thus (𝐻₄) also holds. Consequently, by Theorem 3.1, we infer that the singular BVP (4.1) has at least one positive solution provided 𝜆 is small enough.

Example 4.2. Consider the following problem: −𝑢^′′(𝑡)1=𝜆𝑡()11−𝑡𝑢²+√−√𝑢+1₄2𝑡³()𝑢(0)1−𝑡,0<𝑡<1,−𝑢^′(0)=−¹₀𝑢(𝑡)𝑢(1)cos2𝜋𝑡𝑑𝑡,+𝑢^′(1)=15𝑢143+𝑢4, (4.7) where 𝜆>0 is a parameter. Let 𝑓()=1𝑡,𝑥𝑡()11−𝑡𝑥²+√,()∈()×(),𝑞(𝑡)=√𝑥+1𝑡,𝑥0,10,+∞₄2𝑡³()(𝑡)=11−𝑡,𝑝𝑡()(𝑥)=11−𝑡,𝑔𝑥²,ℎ(𝑥)=√𝑥+1. (4.8) Then, ∫¹₀𝑝(𝑡)𝑑𝑡=𝜋, ∫¹₀𝑞(𝑡)𝑑𝑡=2𝜋. Here, 𝑑𝜉(𝑡)=−cos2𝜋𝑡𝑑𝑡, so the measure 𝑑𝜉 changes sign on [0,1]. By direct calculation, we have 𝑘₁=1,𝑘₂=0,𝑘₃=815,𝑘₄=131,𝑘=3,𝒢_𝜉(𝑠)=14𝜋²(1−cos2𝜋𝑠)≥0,𝒢_𝜂(𝑠)≥0, (4.9) where 𝒢_𝜂(𝑠)=15𝐺₂14,𝑠+𝐺₂341,𝑠,0≤𝑠<4,15𝐺₁14,𝑠+𝐺₂34,1,𝑠43≤𝑠≤4,15𝐺₁14,𝑠+𝐺₁34,3,𝑠4𝐺()=𝐺<𝑠≤1,𝑡,𝑠¹()=()𝑡,𝑠2−𝑠)(1+𝑡3𝐺,0≤𝑡≤𝑠≤1,₂()=()𝑡,𝑠2−𝑡)(1+𝑠3,0≤𝑠≤𝑡≤1. (4.10) Taking 𝑡∈[1/4,3/4]⊂[0,1], we have liminf_𝑥→+∞min_{𝑡∈[1/4,3/4]}𝑓(𝑡,𝑥)=liminf_𝑥→+∞min_{𝑡∈[1/4,3/4]}1𝑡()11−𝑡𝑥²+√𝑥+1=+∞,lim_𝑥→+∞ℎ(𝑥)𝑥=lim_𝑥→+∞√𝑥+1𝑥=0. (4.11) So all assumptions of Theorem 3.2 are satisfied. By Theorem 3.2, we know that BVP (4.7) has at least one positive solution provided 𝜆 is large enough.

Acknowledgments

J. Jiang and L. Liu were supported financially by the National Natural Science Foundation of China (11071141, 11126231), the Natural Science Foundation of Shandong Province of China (ZR2011AQ008), and a Project of Shandong Province Higher Educational Science and Technology Program (J11LA06). Y. Wu was supported financially by the Australia Research Council through an ARC Discovery Project Grant.