On the Nonhomogeneous Fourth-Order 𝑝-Laplacian Generalized Sturm-Liouville Nonlocal Boundary Value Problems

Abstract

We study the nonlinear nonhomogeneous 𝑛-point generalized Sturm-Liouville fourth-order 𝑝-Laplacian boundary value problem by using Leray-Schauder nonlinear alternative and Leggett-Williams fixed-point theorem.

1. Introduction

In this paper, we prove the existence of one and multiple positive solutions of the following differential equations: 𝜙_𝑝𝑢^′′(𝑡)^′′−𝑘²𝜙_𝑝𝑢^′′(𝑡)(𝑡)𝑓((𝑡[],𝑢=𝑔𝑡,𝑢)),𝑡∈0,1_′′(0)=𝜙_𝑞(𝑎),𝑢^′′(1)=𝜙_𝑞(𝑏),𝛼𝑢(0)−𝛽𝑢^′(0)=^𝑛−2_𝑖=1𝑎_𝑖𝑢𝜉_𝑖,𝛾𝑢(1)+𝛿𝑢^′(1)=^𝑛−2_𝑖=1𝑏_𝑖𝑢𝜉_𝑖, (1.1) where 𝜙_𝑝 is 𝑝-Laplacian operator, that is, 𝜙_𝑝(𝑢)=|𝑢|^𝑝−2𝑢, 𝑝>1, 𝜙_𝑝⁻¹=𝜙_𝑞, 1/𝑝+1/𝑞=1𝑚, 𝑘≠0, 𝛼,𝛽,𝛾,𝛿⩾0, 𝜉_𝑖∈(0,1), 𝑎,𝑏,𝑎_𝑖,𝑏_𝑖∈(0,∞)(𝑖=1,2,…,𝑛−2), 𝑓∈𝐶([0,1]×[0,∞),[0,∞)), 𝑓(𝑡,0)≢0, 𝑔(𝑡)∈𝐶([0,1],[0,∞)).

Recently, much attention has been paid to the existence of positive solutions for nonlocal nonlinear boundary value problems (BVPs for short), see [1–4] and references therein. Such problems have potential applications in physics, biology, chemistry, and so forth. For example, a second-order three-point is used as a model for the membrane response of a spherical cap in nonlinear diffusion generated by nonlinear sources and in chemical reactor theory.

At the same time, the boundary value problems with 𝑝-Laplacian operator have been discussed extensively, for example, see [1–3, 5–7].

In [1], Feng et al. researched the boundary value problem 𝜙_𝑝𝑢^′′(𝑡)(𝑡)𝑓((𝑡𝑢(0)==𝑞𝑡,𝑢)),0⩽𝑡⩽1,_𝑚𝑖=1𝑎_𝑖𝑢𝜉_𝑖,𝑢(1)=^𝑚_𝑖=1𝑏_𝑖𝑢𝜉_𝑖; (1.2) they obtained at least one or two positive solutions under some assumptions imposed on the nonlinearity of 𝑓 by applying Krasnoselskii fixed-point theorem.

Zhou and Ma studied the existence and iteration of positive solutions for the following third-order generalized right-focal boundary value problem with 𝑝-Laplacian operator in [3]: 𝜙_𝑝𝑢^′′′(𝑡)(𝑡)𝑓((𝑡𝑢(0)==𝑞𝑡,𝑢)),0⩽𝑡⩽1,_𝑚𝑖=1𝛼_𝑖𝑢𝜉_𝑖,𝑢^′𝜂=0,𝑢^′′(1)=^𝑛_𝑖=1𝛽_𝑖𝑢^′′𝜃_𝑖; (1.3) they established a corresponding iterative scheme for (1.4) by employing the monotone iterative technique.

We would also like to mention the work of Zhang and Liu in [7], in which they considered the existence of positive solutions for 𝜙_𝑝𝑢^′′(𝑡)^′′((𝑡𝑢(0)==𝑓𝑡,𝑢)),0<𝑡<1,_{𝑛−2𝑖=1}𝑎_𝑖𝑢𝜉_𝑖,𝑢(1)=0,𝑢^′′(0)=^𝑛−2_𝑖=1𝑏_𝑖𝑢^′′𝜉_𝑖,𝑢^′′(1)=0, (1.4) by virtue of monotone iterative techniques, and they established a necessary and sufficient condition of positive solutions for their problem.

However, to the best of our knowledge, there are not many results concerning about the existence and multiple solutions of fourth-order 𝑝-Laplacian generalized Sturm-Liouville 𝑛-point boundary value problems. In this paper, motivated by the study of [4, 8], we committed to consider the fourth-order 𝑝-Laplacian generalized Sturm-Liouville nonlocal boundary value problem without assuming any monotonicity condition on the nonlinearity 𝑓.

The rest of the paper is arranged as follows. We state some definitions and several preliminary results in Section 2 that we will use in the sequel. Then in Section 3 we present the existence of one positive solution of BVP (1.1) by Leray-Schauder nonlinear alternative. In Section 4 we get three solutions by Leggett-Williams fixed-point theorem.

2. Preliminaries and Some Lemmas

The basic space used in this paper is 𝐸=𝐶[0,1]. It is well known that 𝐸 is a real Banach space with the norm ‖𝑢‖=max_𝑡∈[0,1]|𝑢(𝑡)|.

Denote 𝜑(𝑡)(𝑡)[]−=𝛽+𝛼𝑡,𝜓=𝛾+𝛿−𝛾𝑡,𝑡∈0,1,𝜌=𝛼𝛾+𝛽𝛾+𝛼𝛿,Δ=^𝑛−2_𝑖∑₌₁𝑎_𝑖𝜑𝜉_𝑖𝜌−∑^𝑛−2_𝑖=1𝑎_𝑖𝜓𝜉_𝑖𝜌−∑^𝑛−2_𝑖=1𝑏_𝑖𝜑𝜉_𝑖−∑^𝑛−2_𝑖=1𝑏_𝑖𝜓𝜉_𝑖. (2.1)

Definition 2.1. A function 𝑢 is said to be a solution of the boundary value problem (1.1) if 𝑢∈𝐶²[0,1] satisfies (1.1) and 𝜙_𝑝(𝑢)∈𝐶²[0,1]. In addition, 𝑢 is said to be a positive solution if 𝑢(𝑡)>0 for 𝑡∈(0,1), and 𝑢 is a solution of BVP (1.1).

Throughout the paper, we assume the following condition is satisfied:(H0)𝜌>0,𝜌−∑^𝑛−2_𝑖=1𝑎_𝑖𝜓(𝜉_𝑖)>0,𝜌−∑^𝑛−2_𝑖=1𝑏_𝑖𝜑(𝜉_𝑖)>0,Δ<0,𝑓(𝑡,𝑢(𝑡))⩾𝑘²(𝑎+𝑏)/min_𝑡∈[0,1]𝑔(𝑡). Let 𝑦(𝑡)=−𝜙_𝑝(𝑢^′′(𝑡)), then BVP (1.1) is divided into the following two parts: −𝑦^′′+𝑘²(𝑡)𝑓((𝑡(),𝑦(0)(1)𝑦=𝑔𝑡,𝑢)),𝑡∈0,1=𝑎,𝑦=𝑏,𝑢^′′+𝜙_𝑞𝑦(),(0)=0,𝑡∈0,1𝛼𝑢−𝛽𝑢^′(0)=^𝑛−2_𝑖=1𝑎_𝑖𝑢𝜉_𝑖,𝛾𝑢(1)+𝛿𝑢^′(1)=^𝑛−2_𝑖=1𝑏_𝑖𝑢𝜉_𝑖. (2.2)(2.3)

It is not difficult that we can transform (2.2) into the following differential equations: −𝑦^′′+𝑘²𝑦=𝑔(𝑡)𝑓(𝑡,𝑢(𝑡))−𝑘²𝑎(1−𝑡)−𝑘²(),𝑦(0)(1)𝑏𝑡,𝑡∈0,1=0,𝑦=0. (2.4)

By routine calculations we can get the following three Lemmas.

Lemma 2.2. The BVP (2.4) has a unique solution 𝑦(𝑡)=¹₀𝐺₁(𝑡,𝑠)𝑔(𝑠)𝑓(𝑠,𝑢(𝑠))−𝑘²𝑎(1−𝑠)−𝑘²𝑏𝑠𝑑𝑠, (2.5) where 𝐺₁()=1𝑡,𝑠𝜌sinh𝑘𝑠⋅sinh𝑘(1−𝑡)()𝑘sinh𝑘,0⩽𝑠⩽𝑡⩽1,sinh𝑘𝑡⋅sinh𝑘1−𝑠𝑘sinh𝑘0⩽𝑡⩽𝑠⩽1. (2.6)

Lemma 2.3. The BVP (2.3) has a unique solution 𝑢(𝑡)=¹₀𝐺₂(𝑡,𝑠)𝜙_𝑞𝑦(𝑠)𝑑𝑠+𝐴𝜑(𝑡)+𝐵𝜓(𝑡), (2.7) where 𝐺₂()=1𝑡,𝑠𝜌𝜓(𝑡)𝜑(𝑠)𝜓(𝑠)𝜑(𝑡)1,0⩽𝑠⩽𝑡⩽1,,0⩽𝑡⩽𝑠⩽1,𝐴=Δ^𝑛−2_𝑖∑₌₁𝑎_𝑖∫¹₀𝐺₂𝜉_𝑖,𝑠𝜙_𝑞𝑦(𝑠)𝑑𝑠𝜌−∑^𝑛−2_𝑖=1𝑎_𝑖𝜓𝜉_{𝑖𝑛−2𝑖}∑₌₁𝑏_𝑖∫¹₀𝐺₂𝜉_𝑖,𝑠𝜙_𝑞𝑦(𝑠)𝑑𝑠−∑^𝑛−2_𝑖=1𝑎_𝑖𝜓𝜉_𝑖,1𝐵=Δ−∑^𝑛−2_𝑖=1𝑎_𝑖𝜑𝜉∑_𝑖^𝑛−2_𝑖=1𝑎_𝑖∫¹₀𝐺₂𝜉_𝑖,𝑠𝜙_𝑞𝑦(𝑠)𝑑𝑠𝜌−∑^𝑛−2_𝑖=1𝑎_𝑖𝜑𝜉∑_𝑖^𝑛−2_𝑖=1𝑏_𝑖∫¹₀𝐺₂𝜉_𝑖,𝑠𝜙_𝑞𝑦(𝑠).𝑑𝑠 (2.8)

The proof of Lemma 2.3 is similar to that of Lemma 5.5.1 in [8], so we omit it here. From Lemmas 2.2 and 2.3 we can get that 𝑢(𝑡) is a solution of BVP (1.1) if and only if 𝑢(𝑡)=¹₀𝐺₂(𝑡,𝑠)𝜙_𝑞¹₀𝐺₁(𝑠,𝜏)𝑔(𝜏)𝑓(𝜏,𝑢(𝜏))−𝑘²𝑎(1−𝜏)−𝑘²𝜙𝑏𝜏𝑑𝑠+𝐴_𝑞𝑦𝜑(𝑡)+𝐵𝜙_𝑞𝑦𝜓(𝑡), (2.9) where 𝐴𝜙_𝑞𝑦=1Δ^𝑛−2_𝑖∑₌₁𝑎_𝑖∫¹₀𝐺₂𝜉_𝑖,𝑠𝜙_𝑞∫¹₀𝐺₁(𝑠,𝜏)𝑔(𝜏)𝑓(𝜏,𝑢(𝜏))−𝑘²𝑎(1−𝜏)−𝑘²𝑏𝜏𝑑𝜏𝑑𝑠𝜌−∑^𝑛−2_𝑖=1𝑎_𝑖𝜓𝜉_{𝑖𝑛−2𝑖}∑₌₁𝑏_𝑖∫¹₀𝐺₂𝜉_𝑖,𝑠𝜙_𝑞∫¹₀𝐺₁(𝑠,𝜏)𝑔(𝜏)𝑓(𝜏,𝑢(𝜏))−𝑘²𝑎(1−𝜏)−𝑘²𝑏𝜏𝑑𝜏𝑑𝑠−∑^𝑛−2_𝑖=1𝑎_𝑖𝜓𝜉_𝑖,𝐵𝜙_𝑞𝑦=1Δ−∑^𝑛−2_𝑖=1𝑎_𝑖𝜑𝜉∑_𝑖^𝑛−2_𝑖=1𝑎_𝑖∫¹₀𝐺₂𝜉_𝑖,𝑠𝜙_𝑞∫¹₀𝐺₁(𝑠,𝜏)𝑔(𝜏)𝑓(𝜏,𝑢(𝜏))−𝑘²𝑎(1−𝜏)−𝑘²𝑏𝜏𝑑𝜏𝑑𝑠𝜌−∑^𝑛−2_𝑖=1𝑎_𝑖𝜑𝜉∑_𝑖^𝑛−2_𝑖=1𝑏_𝑖∫¹₀𝐺₂𝜉_𝑖,𝑠𝜙_𝑞∫¹₀𝐺₁(𝑠,𝜏)𝑔(𝜏)𝑓(𝜏,𝑢(𝜏))−𝑘²𝑎(1−𝜏)−𝑘².𝑏𝜏𝑑𝜏𝑑𝑠 (2.10)

Lemma 2.4. Consider, 𝐺_𝑖(𝑡,𝑠)⩽𝐺_𝑖()[]𝐺𝑠,𝑠,𝑡,𝑠∈0,1,𝑖=1,2,₁(𝑡,𝑠)⩾Λ₀𝐺₁()[][],𝐺𝑠,𝑠,𝑡∈𝜖,1−𝜖,𝑠∈0,1₂(𝑡,𝑠)⩾Λ𝐺₂(𝑠,𝑠),𝑡∈[𝜖,1−𝜖],𝑠∈[0,1], (2.11) where Λ₀=sinh𝑘𝜖1sinh𝑘,𝜖∈0,2,𝜑(𝜖)Λ=min𝜑(1),𝜓(1−𝜖)𝜓(0)1,𝜖∈0,2. (2.12)

Denote Λ₁=max𝜑,𝜓,Λ₂=minmin_{[]𝑡∈𝜖,1−𝜖}𝜑(𝑡),min_{[]𝑡∈𝜖,1−𝜖}𝜓(𝑡Λ),𝜆=minΛ,²Λ₁. (2.13)

Lemma 2.5. Let 𝐾=𝑢∣𝑢∈𝐶[0,1],𝑢(𝑡)⩾0,∀𝑡∈[0,1],min_{[]𝑡∈𝜖,1−𝜖}𝑢(𝑡)‖𝑢,(𝑡)=⩾𝜆‖𝑇𝑢¹₀𝐺₂(𝑡,𝑠)𝜙_𝑞¹₀𝐺₁(𝑠,𝜏)𝑔(𝜏)𝑓(𝜏,𝑢(𝜏))−𝑘²𝑎(1−𝜏)−𝑘²𝜙𝑏𝜏𝑑𝑠+𝐴_𝑞𝑦𝜑(𝑡)+𝐵𝜙_𝑞𝑦𝜓(𝑡), (2.14) then 𝑇(𝐾)⊆𝐾. (2.15)

Proof. Firstly, we prove that 𝑇𝑢(𝑡)⩾0. For 𝑓∈𝐶([0,1]×[0,∞),[0,∞), 𝑔(𝑡)∈𝐶([0,1],[0,∞)), then we can get 𝜙_𝑞(𝑦)=𝜙_𝑞(∫¹₀𝐺₁(𝑠,𝜏)[𝑔(𝜏)𝑓(𝜏,𝑢(𝜏))−𝑘²𝑎(1−𝜏)−𝑘²𝑏𝜏]𝑑𝜏)>0, for all 𝑢∈𝐾. Furthermore, condition (H0) leads to 𝐴(𝜙_𝑞(𝑦))⩾0, and 𝐵(𝜙_𝑞(𝑦))⩾0, thus, we get 𝑇𝑢(𝑡)⩾0.
Secondly, for 𝑡∈[𝜖,1−𝜖], we can get min_{[]𝑡∈𝜖,1−𝜖}𝑇𝑢(𝑡)=min_{[]𝑡∈𝜖,1−𝜖}¹₀𝐺₂(𝑡,𝑠)𝜙_𝑞¹₀𝐺₁(𝑠,𝜏)𝑔(𝜏)𝑓(𝜏,𝑢(𝜏))−𝑘²𝑎(1−𝜏)−𝑘²𝜙𝑏𝜏𝑑𝜏𝑑𝑠+𝐴_𝑞𝑦𝜑(𝑡)+𝐵𝜙_𝑞𝑦𝜓(𝑡)⩾Λ¹₀𝐺₂(𝑠,𝑠)𝜙_𝑞¹₀𝐺₁(𝑠,𝜏)𝑔(𝜏)𝑓(𝜏,𝑢(𝜏))−𝑘²𝑎(1−𝜏)−𝑘²𝜙𝑏𝜏𝑑𝜏𝑑𝑠+𝐴_𝑞𝑦𝜑(𝑡)+𝐵𝜙_𝑞𝑦𝜓(𝑡)⩾Λ¹₀𝐺₂(𝑠,𝑠)𝜙_𝑞¹₀𝐺₁(𝑠,𝜏)𝑔(𝜏)𝑓(𝜏,𝑢(𝜏))−𝑘²𝑎(1−𝜏)−𝑘²+Λ𝑏𝜏𝑑𝜏𝑑𝑠₂Λ₁Λ₁𝐴𝜙_𝑞𝑦+𝐵𝜙_𝑞𝑦⩾𝜆¹₀𝐺₂(𝑠,𝑠)𝜙_𝑞¹₀𝐺₁(𝑠,𝜏)𝑔(𝜏)𝑓(𝜏,𝑢(𝜏))−𝑘²𝑎(1−𝜏)−𝑘²𝑏𝜏𝑑𝜏𝑑𝑠+Λ₁𝐴𝜙_𝑞𝑦+𝐵𝜙_𝑞𝑦=𝜆‖𝑇𝑢‖. (2.16)
Thus we can get that min_{𝑡∈[𝜖,1−𝜖]}𝑇𝑢(𝑡)⩾𝜆‖𝑇𝑢‖, which means 𝑇(𝐾)⊆𝐾.

We present here several definitions.

Given a cone 𝐾 in a real Banach space 𝐸, a map 𝛼 is said to be a nonnegative continuous concave (resp., convex) functional on 𝐾 provided that 𝛼∶𝐾→[0,+∞) is continuous and 𝛼𝑡𝑥+(1−𝑡)𝑦⩾𝑡𝛼(𝑥)+(1−𝑡)𝛼𝑦,resp.,𝛼𝑡𝑥+(1−𝑡)𝑦⩽𝑡𝛼(𝑥)+(1−𝑡)𝛼𝑦, (2.17) for all, 𝑥,𝑦∈𝐾 and 𝑡∈[0,1].

Let 0<𝑎<𝑏 be given, and let 𝛼 be a nonnegative continuous concave functional on 𝐾. Define the convex sets 𝑃_𝑟 and 𝑃(𝛼,𝑎,𝑏) by 𝑃_𝑟={𝑥∈𝐾∣‖𝑥‖<𝑟},𝑃(𝛼,𝑎,𝑏)={𝑥∈𝐾∣𝑎⩽𝛼(𝑥),‖𝑥‖⩽𝑏}. (2.18)

For the convenience of the reader, we present here the Leggett-Williams fixed-point theorem and the Leray-Schauder nonlinear alternative theorem.

Lemma 2.6 (see [9], Leggett-Williams fixed-point theorem). Let 𝐴∶𝑃_𝑐→𝑃_𝑐 be a completely continuous operator, and let 𝛼 be a nonnegative continuous concave functional on 𝐾 such that 𝛼(𝑥)⩽‖𝑥‖  for all 𝑥∈𝑃_𝑐. Suppose there exist 0<𝑎<𝑏<𝑑⩽𝑐 such that(A1){𝑥∈𝑃(𝛼,𝑏,𝑑)∶𝛼(𝑥)>𝑏}≠∅, and 𝛼(𝐴𝑥)>𝑏 for 𝑥∈𝑃(𝛼,𝑏,𝑑);(A2)‖𝐴𝑥‖<𝑎 for ‖𝑥‖⩽𝑎;(A3)𝛼(𝐴𝑥)>𝑏 for 𝑥∈𝑃(𝛼,𝑏,𝑐) with ‖𝐴𝑥‖>𝑑.
Then 𝐴 has at least three fixed points 𝑥₁, 𝑥₂, and 𝑥₃ and such that ‖𝑥₁‖<𝑎, 𝑏<𝛼(𝑥₂) and ‖𝑥₃‖>𝑎, with 𝛼(𝑥₃)<𝑏.

Now we cite the Leray-Schauder nonlinear alternative.

Lemma 2.7 (see [10]). Let 𝐹 be a Banach space and Ω a bounded open subset of 𝐹,0∈Ω. 𝑇∶Ω→𝐹 be a completely continuous operator. Then, either there exists 𝑥∈𝜕Ω,𝜆>1 such that 𝑇(𝑥)=𝜆𝑥, or there exists a fixed point 𝑥^∗∈Ω.

3. Results of One Nontrivial Solution

In this section, we study the existence of one nontrivial solution of BVP (1.1) by Leray-Schauder nonlinear alternative.

Denote 𝐻₁=𝜙_𝑞¹₀𝐺₁()𝑔(𝜏)𝑝(𝜏),𝐻𝜏,𝜏𝑑𝜏₂=𝜙_𝑞¹₀𝐺₁()𝑔(𝜏)𝑟(𝜏),1𝜏,𝜏𝑑𝜏𝐴=Δ^𝑛−2_𝑖∑₌₁𝑎_𝑖𝜌−∑^𝑛−2_𝑖=1𝑎_𝑖𝜓𝜉_{𝑖𝑛−2𝑖}∑₌₁𝑏_𝑖−∑^𝑛−2_𝑖=1𝑎_𝑖𝜓𝜉_𝑖,1𝐵=Δ−∑^𝑛−2_𝑖=1𝑎_𝑖𝜑𝜉∑_𝑖^𝑛−2_𝑖=1𝑎_𝑖𝜌−∑^𝑛−2_𝑖=1𝑎_𝑖𝜑𝜉∑_𝑖^𝑛−2_𝑖=1𝑏_𝑖,𝑁=𝑀⋅2^𝑞−11+𝐴Λ₁+𝐵Λ₁,𝑙=𝑁𝐻₂1−𝑁𝐻₁,Ω={𝑢∈𝐶[0,1],‖𝑢‖<𝑙}. (3.1)

Theorem 3.1. Assume 𝑁𝐻₁<1,𝑓(𝑡,0)≢0, and there exist nonnegative functions 𝑝,𝑟∈𝐿¹[0,1] such that |𝑓(𝑡,𝑢)|⩽𝑝(𝑡)|𝑢|^𝑝−1+𝑟(𝑡),a.e.(𝑡,𝑢)∈[0,1]×[0,+∞), then BVP (1.1) has a nontrivial solution 𝑢^∗∈Ω.

Proof. If there exist two nonnegative functions 𝑝,𝑟∈𝐿¹[0,1] such that |𝑓(𝑡,𝑢)|⩽𝑝(𝑡)|𝑢|^𝑝−1+𝑟(𝑡),a.e.(𝑡,𝑢)∈[0,1]×[0,+∞), we can get that 𝜙_𝑞¹₀𝐺₁(𝑠,𝜏)𝑔(𝜏)𝑓(𝜏,𝑢(𝜏))−𝑘²𝑎(1−𝜏)−𝑘²𝑏𝜏𝑑𝜏⩽𝜙_𝑞¹₀𝐺₁(𝑠,𝜏)𝑔(𝜏)𝑓(𝜏,𝑢(𝜏))𝑑𝜏⩽𝜙_𝑞¹₀𝐺₁(𝜏,𝜏)𝑔(𝜏)𝑝(𝜏)|𝑢|^𝑝−1+𝑟(𝜏)𝑑𝜏=2^𝑞−1‖𝑢‖𝜙_𝑞¹₀𝐺₁(𝜏,𝜏)𝑔(𝜏)𝑝(𝜏)𝑑𝜏+𝜙_𝑞¹₀𝐺₁(𝜏,𝜏)𝑎(𝜏)𝑟(𝜏)𝑑𝜏=2^𝑞−1‖𝑢‖𝐻₁+𝐻₂, (3.2) thus, we get ¹₀𝐺₂𝜉_𝑖,𝑠𝜙_𝑞¹₀𝐺₁(𝑠,𝜏)𝑔(𝜏)𝑓(𝜏,𝑢(𝜏))−𝑘²𝑎(1−𝜏)−𝑘²⩽𝑏𝜏𝑑𝜏𝑑𝑠₁₀𝐺₂𝜉_𝑖,𝑠𝜙_𝑞¹₀𝐺₁(𝑠,𝜏)𝑔(𝜏)𝑓(𝜏,𝑢(𝜏))𝑑𝜏𝑑𝑠⩽𝑀⋅2^𝑞−1‖𝑢‖𝐻₁+𝐻₂. (3.3) In the same way, we obtain 𝐴𝜙_𝑞𝑦⩽𝑀⋅2^𝑞−1‖𝑢‖𝐻₁+𝐻₂𝐵𝜙𝐴,_𝑞𝑦⩽𝑀⋅2^𝑞−1‖𝑢‖𝐻₁+𝐻₂𝐵. (3.4) Thus we have ‖𝑇𝑢‖=max_0⩽𝑡⩽1|𝑇𝑢(𝑡)|=max_0⩽𝑡⩽1¹₀𝐺₂(𝑡,𝑠)𝜙_𝑞¹₀𝐺₁(𝑠,𝜏)𝑔(𝜏)𝑓(𝜏,𝑢(𝜏))−𝑘²𝑎(1−𝜏)−𝑘²𝜙𝑏𝜏𝑑𝑠+𝐴_𝑞𝑦𝜑(𝑡)+𝐵𝜙_𝑞𝑦𝜓(𝑡)⩽max_0⩽𝑡⩽1¹₀𝐺₂(𝑡,𝑠)𝜙_𝑞¹₀𝐺₁()𝑔(𝜏)𝑓((𝜏𝜙𝑠,𝜏𝜏,𝑢))𝑑𝜏𝑑𝑠+𝐴_𝑞𝑦𝜑(𝑡)+𝐵𝜙_𝑞𝑦𝜓(𝑡)⩽𝑀⋅2^𝑞−1‖𝑢‖𝐻₁+𝐻₂+𝑀⋅2^𝑞−1‖𝑢‖𝐻₁+𝐻₂𝐴Λ₁+𝐵Λ₁=𝑀⋅2^𝑞−1‖𝑢‖𝐻₁+𝐻₂1+𝐴Λ₁+𝐵Λ₁=𝑁‖𝑢‖𝐻₁+𝐻₂. (3.5)
Suppose that there exists 𝜇>1 such that 𝑇𝑢=𝜇𝑢,𝑢∈𝜕Ω. (3.6) Therefore, 𝜇𝑙=𝜇‖𝑢‖=‖𝑇𝑢‖⩽𝑙𝑁𝐻₁+𝑁𝐻₂𝑙, (3.7) which leads to 𝜇⩽𝑁𝐻₁+𝑁𝐻₂/𝑙=1, and this contradicts 𝜇>1, then by Lemma 2.7, 𝑇 has a fixed point 𝑢^∗∈Ω; since 𝑓(𝑡,0)≢0, the BVP (1.1) has a nontrivial solution 𝑢^∗∈Ω. This completes the proof of Theorem 3.1.

4. Results of Multiple Positive Solutions

In the following parts, we will study the existence of multiple positive solutions of BVP (1.1) by using Leggett-Williams fixed-point theorem.

Denote 𝑃_𝑐={𝑢∈𝐾∣‖𝑢‖<𝑐}. (4.1) Define the nonnegative continuous concave functional on 𝐾 by 𝛼(𝑢)=min_{𝜖⩽𝑡⩽1−𝜖}𝑢(𝑡). (4.2) It is obvious that for each 𝑢∈𝐾,𝛼(𝑢)⩽‖𝑢‖.

Let 𝑀=max_0⩽𝑡⩽1∫¹₀𝐺₂(𝑡,𝑠)𝑑𝑠, 𝑚=∫_𝜖^1−𝜖𝐺₂(𝑠,𝑠)𝑑𝑠, ℎ=𝜙_𝑞(∫_𝜖^1−𝜖𝑘²(𝑎+𝑏)𝐺₁(𝜏,𝜏)𝑑𝜏), and 𝐴,𝐵,Λ,Λ₀,Λ₁ be defined in Sections 2 and 3.

We list the following three hypotheses:(H1)𝑓(𝑡,𝑢)<𝜙_𝑝(𝑐/𝑀(1+Λ₁𝐴+Λ₁𝐵))/∫¹₀𝐺₁(𝜏,𝜏)𝑔(𝜏)𝑑𝜏, for all 𝑡∈[0,1], 0⩽𝑢⩽𝑐;(H2)𝑓(𝑡,𝑢)<𝜙_𝑝(𝑎/𝑀(1+Λ₁𝐴+Λ₁𝐵))/∫¹₀𝐺₁(𝜏,𝜏)𝑔(𝜏)𝑑𝜏, for all 𝑡∈[0,1], 0⩽𝑢⩽𝑎;(H3)𝑓(𝑡,𝑢)>𝜙_𝑝(2^𝑞−1(𝑏/𝑚Λ(1+Λ₁𝐴+Λ₁𝐵)+ℎ))/Λ₀∫_𝜖^1−𝜖𝐺₁(𝜏,𝜏)𝑔(𝜏)𝑑𝜏, for all 𝑡∈[0,1], 𝑏⩽𝑢⩽𝑏/𝜆.

Theorem 4.1. Assume (𝐻1)−(𝐻3) hold, then BVP (1.1) has at least three positive solutions 𝑢₁, 𝑢₂, and 𝑢₃, such that ‖𝑢₁‖<𝑎,𝑏<min_{[𝜖,1−𝜖]}𝑢₂(𝑡), and ‖𝑢₃‖>𝑎, with min_{[𝜖,1−𝜖]}𝑢₃(𝑡)<𝑏.

Proof. Firstly, we prove that 𝑇∶𝑃_𝑐→𝑃_𝑐. The operator 𝑇 is completely continuous.
From condition (H1), we can get ¹₀𝐺₂𝜉_𝑖,𝑠𝜙_𝑞¹₀𝐺₁(𝑠,𝜏)𝑔(𝜏)𝑓(𝜏,𝑢(𝜏))−𝑘²𝑎(1−𝜏)−𝑘²𝑏𝜏𝑑𝜏𝑑𝑠⩽𝑀𝜙_𝑞¹₀𝐺₁()𝑔(𝜏)𝑓((𝜏⩽𝑐𝜏,𝜏𝜏,𝑢))𝑑𝜏1+Λ₁𝐴+Λ₁𝐵. (4.3)
Hence, ‖𝑇𝑢‖=max_0⩽𝑡⩽1|𝑇𝑢(𝑡)|=max_0⩽𝑡⩽1¹₀𝐺₂(𝑡,𝑠)𝜙_𝑞¹₀𝐺₁(𝑠,𝜏)𝑔(𝜏)𝑓(𝜏,𝑢(𝜏))−𝑘²𝑎(1−𝜏)−𝑘²𝜙𝑏𝜏𝑑𝜏𝑑𝑠+𝐴_𝑞𝑦𝜑(𝑡)+𝐵𝜙_𝑞𝑦𝜓(𝑡)⩽max_0⩽𝑡⩽1¹₀𝐺₂(𝑡,𝑠)𝜙_𝑞¹₀𝐺₁(𝑔(𝜏)𝑓((𝜏𝜙𝑠,𝜏)𝜏,𝑢)]𝑑𝜏𝑑𝑠+𝐴_𝑞𝑦𝜑(𝑡)+𝐵𝜙_𝑞𝑦𝜓(𝑡⩽𝑐)1+Λ₁𝐴+Λ₁𝐵+𝑐1+Λ₁𝐴+Λ₁𝐵Λ₁𝑐𝐴+1+Λ₁𝐴+Λ₁𝐵Λ₁𝐵=𝑐. (4.4) Thus we get ‖𝑇𝑢‖⩽𝑐; therefore, 𝑇∶𝑃_𝑐→𝑃_𝑐. The operator 𝑇 is completely continuous by an application of Ascoli-Arzela theorem.
In the same way, condition (H2) implies that condition (A2) of Lemma 2.6 is satisfied. In the following, we show that condition (A1) of Lemma 2.6 is satisfied.
Let 𝑢₀(𝑡)=𝑏𝜆,𝑡∈[0,1], (4.5) then 𝑢₀𝑏∈𝑃𝛼,𝑏,𝜆,𝛼𝑢₀=𝑏𝜆>𝑏, (4.6) thus, {𝑢∈𝑃(𝛼,𝑏,𝑏/𝜆)∣𝛼(𝑢)>𝑏}≠∅.
If 𝑢∈𝑃(𝛼,𝑏,𝑏/𝜆), then 𝑏⩽𝑢(𝑠)⩽𝑏/𝜆, 𝑠∈[𝜖,1−𝜖].
By condition (H3), we obtain ¹₀𝐺₂𝜉_𝑖,𝑠𝜙_𝑞¹₀𝐺₁(𝑠,𝜏)𝑔(𝜏)𝑓𝜏,𝑢(𝜏)−𝑘²𝑎(1−𝜏)−𝑘²𝑏𝜏𝑑𝜏𝑑𝑠⩾Λ_𝜖^1−𝜖𝐺₂(1𝑠,𝑠)2^𝑞−1𝜙_𝑞𝜖^1−𝜖𝐺₁(𝑠,𝜏)𝑔(𝜏)𝑓(𝜏,𝑢(𝜏))𝑑𝜏𝑑𝑠−𝜙_𝑞𝜖^1−𝜖𝐺₁(𝑠,𝜏)𝑘²𝑎(1−𝜏)+𝑘²𝑏𝜏𝑑𝜏𝑑𝑠⩾Λ_𝜖^1−𝜖𝐺₂(1𝑠,𝑠)2^𝑞−1𝜙_𝑞𝜖^1−𝜖𝐺₁(𝑠,𝜏)𝑔(𝜏)𝑓(𝜏,𝑢(𝜏))𝑑𝜏𝑑𝑠−𝜙_𝑞𝜖^1−𝜖𝐺₁(𝜏,𝜏)𝑘²𝑎(1−𝜏)+𝑘²𝑏𝜏𝑑𝜏𝑑𝑠⩾Λ_𝜖^1−𝜖𝐺₂(1𝑠,𝑠)2^𝑞−1𝜙_𝑞𝜖^1−𝜖𝐺₁(𝑠,𝜏)𝑔(𝜏)𝑓(𝜏,𝑢(𝜏))𝑑𝜏−𝜙_𝑞𝜖^1−𝜖𝐺₁(𝜏,𝜏)𝑘²𝑎+𝑘²𝑏⩾𝑏𝑑𝜏𝑑𝑠1+Λ₁𝐴+Λ₁𝐵. (4.7)
Thus we get 𝛼(𝑇𝑢(𝑡))=min_{[]𝑡∈𝜖,1−𝜖}𝑇𝑢(𝑡)⩾min_{[]𝑡∈𝜖,1−𝜖𝜖}^1−𝜖𝐺₂(𝑠,𝑠)𝜙_𝑞¹₀𝐺₁(𝑠,𝜏)𝑔(𝜏)𝑓𝜏,𝑢(𝜏)−𝑘²𝑎(1−𝜏)−𝑘²𝜙𝑏𝜏𝑑𝜏𝑑𝑠+𝐴_𝑞𝑦𝜑(𝑡)+𝐵𝜙_𝑞𝑦𝜓𝑡⩾𝑏(())1+Λ₂𝐴+Λ₂𝐵+𝑏1+Λ₂𝐴+Λ₂𝐵Λ₂𝑏𝐴+1+Λ₂𝐴+Λ₂𝐵Λ₂𝐵=𝑏. (4.8)
Therefore, condition (A1) of Lemma 2.6 is satisfied.
Finally, we show that condition (A3) of Lemma 2.6 is satisfied.
If 𝑢∈𝑃(𝛼,𝑏,𝑐), and ‖𝑇𝑢‖>𝑏/𝜆, then 𝛼(𝑇𝑢(𝑡))=min_{𝜖⩽𝑡⩽1−𝜖}𝑇𝑢(𝑡)⩾𝜆‖𝑇𝑢‖>𝑏.
Therefore, condition (A3) of Lemma 2.6 is also satisfied. By Lemma 2.6, there exist three positive solutions 𝑢₁, 𝑢₂, and 𝑢₃ such that ‖𝑢₁‖<𝑎,𝑏<min_{𝑡∈[𝜖,1−𝜖]}𝑢₂(𝑡), and ‖𝑢₃‖>𝑎, with min_{𝑡∈[𝜖,1−𝜖]}𝑢₃(𝑡)<𝑏. Thus we completed the proof.

Acknowledgments

The first author is supported by the Natural Science Foundation of Shandong Province of China (ZR2012AQ024) and the University Science and Technology Foundation of Shandong Provincial Education Department (J10LA62). The second author is supported by the National Natural Science Foundation of China (10871116), the Natural Science Foundation of Shandong Province of China (ZR2010AM005), and the Doctoral Program Foundation of Education Ministry of China (200804460001).