Positive Solutions for Third-Order Nonlinear 𝑝-Laplacian 𝑚-Point Boundary Value Problems on Time Scales

Abstract

We study the following third-order 𝑝-Laplacian 𝑚-point boundary value problems on time scales: (𝜙_𝑝(𝑢^Δ∇))^∇+𝑎(𝑡)𝑓(𝑡,𝑢(𝑡))=0, 𝑡∈[0,𝑇]_𝐓, 𝛽𝑢(0)−𝛾𝑢^Δ(0)=0, 𝑢(𝑇)=∑^𝑚−2_𝑖=1𝑎_𝑖𝑢(𝜉_𝑖), 𝜙_𝑝(𝑢^Δ∇(0))=∑^𝑚−2_𝑖=1𝑏_𝑖𝜙_𝑝(𝑢^Δ∇(𝜉_𝑖)), where 𝜙_𝑝(𝑠) is 𝑝-Laplacian operator, that is, 𝜙_𝑝(𝑠)=|𝑠|^𝑝−2𝑠, 𝑝>1,𝜙_𝑝⁻¹=𝜙_𝑞, 1/𝑝+1/𝑞=1,0<𝜉₁<⋯<𝜉_𝑚−2<𝜌(𝑇). We obtain the existence of positive solutions by using fixed-point theorem in cones. The conclusions in this paper essentially extend and improve the known results.

1. Introduction

The theory of time scales was initiated by Hilger [1] as a means of unifying and extending theories from differential and difference equations. The study of time scales has lead to several important applications in the study of insect population models, neural networks, heat transfer, and epidemic models, see, for example [2–6]. Recently, the boundary value problems with 𝑝-Laplacian operator have also been discussed extensively in the literature, for example, see [7–13].

A time scale 𝐓 is a nonempty closed subset of ℝ. We make the blanket assumption that 0, T are points in 𝐓. By an interval (0, T), we always mean the intersection of the real interval (0, T) with the given time scale; that is (0, T)∩𝐓.

In [14], Anderson considered the the following third-order nonlinear boundary value problem (BVP): 𝑥^′′′(𝑡)=𝑓(𝑡,𝑥(𝑡)),𝑡₁≤𝑡≤𝑡₃,𝑥(𝑡₁)=𝑥^′(𝑡₂)=0,𝛾𝑥(𝑡₃)+𝛿𝑥^′′(𝑡₃)=0. (1.1)Author studied the existence of solutions for the nonlinear boundary value problem by using the Krasnoselskii's fixed point theorem and Leggett and Williams fixed point theorem, respectively.

In [8, 9], He considered the existence of positive solutions of the 𝑝-Laplacian dynamic equations on time scales(𝜙_𝑝(𝑢^Δ))^∇+𝑎(𝑡)𝑓(𝑢(𝑡))=0,𝑡∈[0,𝑇]_𝐓, (1.2)satisfying the boundary conditions𝑢(0)−𝐵₀(𝑢^Δ(𝜂))=0,𝑢^Δ(𝑇)=0, (1.3)or𝑢^Δ(0)=0,𝑢(𝑇)−𝐵₁(𝑢^Δ(𝜂))=0, (1.4)where 𝜂∈(0,𝜌(𝑇)). He obtained the existence of at least double and triple positive solutions of the boundary value problems by using a new double fixed point theorem and triple fixed point theorem, respectively.

In [13], Zhou and Ma firstly studied the existence and iteration of positive solutions for the following third-order generalized right-focal boundary value problem with 𝑝-Laplacian operator:(𝜙_𝑝(𝑢^′′))^′(𝑡)=𝑞(𝑡)𝑓(𝑡,𝑢(𝑡)),0≤𝑡≤1,𝑢(0)=∑^𝑚_𝑖=1𝛼_𝑖𝑢(𝜉_𝑖),𝑢^′(𝜂)=0,𝑢^′′(1)=∑^𝑛_𝑖=1𝛽_𝑖𝑢^′′(𝜃_𝑖). (1.5)They established a corresponding iterative scheme for the problem by using the monotone iterative technique.

However, to the best of our knowledge, little work has been done on the existence of positive solutions for third-order 𝑝-Laplacian 𝑚-point boundary value problems on time scales. This paper attempts to fill this gap in the literature.

In this paper, by using different method, we are concerned with the existence of positive solutions for the following third-order 𝑝-Laplacian 𝑚-point boundary value problems on time scales:(𝜙_𝑝(𝑢^Δ∇))^∇+𝑎(𝑡)𝑓(𝑡,𝑢(𝑡))=0,𝑡∈[0,𝑇]_𝐓,𝛽𝑢(0)−𝛾𝑢^Δ(0)=0,𝑢(𝑇)=∑^𝑚−2_𝑖=1𝑎_𝑖𝑢(𝜉_𝑖),𝜙_𝑝(𝑢^Δ∇(0))=∑^𝑚−2_𝑖=1𝑏_𝑖𝜙_𝑝(𝑢^Δ∇(𝜉_𝑖)), (1.6)where 𝜙_𝑝(𝑠) is 𝑝-Laplacian operator, that is, 𝜙_𝑝(𝑠)=|𝑠|^𝑝−2𝑠,𝑝>1,𝜙_𝑝⁻¹=𝜙_𝑞,(1/𝑝)+(1/𝑞)=1, and 𝑎_𝑖,𝑏_𝑖,𝑎,𝑓 satisfy

(H₁)𝛽,𝛾≥0,𝛽+𝛾>0, 𝑎_𝑖∈[0,+∞),𝑖=1,2,…,𝑚−3,𝑎_𝑚−2>0,0<𝜉₁<⋯<𝜉_𝑚−2<𝜌(𝑇), 0<∑^𝑚−2_𝑖=1𝑏_𝑖<1,0<∑^𝑚−2_𝑖=1𝑎_𝑖𝜉_𝑖<𝑇, 𝑑=𝛽(𝑇−∑^𝑚−2_𝑖=1𝑎_𝑖𝜉_𝑖)+𝛾(1−∑^𝑚−2_𝑖=1𝑎_𝑖)>0;(H₂)𝑓∶[0,𝑇]_𝐓×[0,+∞)→𝑅⁺ is continuous, 𝑎∈𝐶_ld((0,𝑇)_𝐓,ℝ⁺) and there exists 𝑡₀∈[𝜉_𝑚−2,𝑇)_𝐓 such that 𝑎(𝑡₀)>0, where ℝ⁺=[0,+∞).

2. Preliminaries and Lemmas

For convenience, we list the following definitions which can be found in [1–5].

Definition 2.1. A time scale 𝐓 is a nonempty closed subset of real numbers ℝ. For 𝑡<sup𝐓 and 𝑟>inf𝐓, define the forward jump operator 𝜎 and backward jump operator 𝜌, respectively, by𝜎(𝑡)=inf{𝜏∈𝐓∣𝜏>𝑡}∈𝐓,𝜌(𝑟)=sup{𝜏∈𝐓∣𝜏<𝑟}∈𝐓, (2.1) for all 𝑡,𝑟∈𝐓. If 𝜎(𝑡)>𝑡, 𝑡 is said to be right scattered; if 𝜌(𝑟)<𝑟, 𝑟 is said to be left scattered; if 𝜎(𝑡)=𝑡, 𝑡 is said to be right dense; if 𝜌(𝑟)=𝑟, 𝑟 is said to be left dense. If 𝐓 has a right scattered minimum 𝑚, define 𝐓_𝑘=𝐓−{𝑚}, otherwise set 𝐓_𝑘=𝐓. If 𝐓 has a left scattered maximum 𝑀, define 𝐓^𝑘=𝐓−{𝑀}, otherwise set 𝐓^𝑘=𝐓.

Definition 2.2. For 𝑓∶𝐓→ℝ and 𝑡∈𝐓^𝑘, the delta derivative of 𝑓 at the point 𝑡 is defined to be the number 𝑓^Δ(𝑡), (provided it exists), with the property that for each 𝜖>0, there is a neighborhood 𝑈 of 𝑡 such that|𝑓(𝜎(𝑡))−𝑓(𝑠)−𝑓^Δ(𝑡)(𝜎(𝑡)−𝑠)|≤𝜖|𝜎(𝑡)−𝑠|, (2.2)for all 𝑠∈𝑈.
For 𝑓∶𝐓→ℝ and 𝑡∈𝐓_𝑘, the nabla derivative of 𝑓 at 𝑡, denoted by 𝑓^∇(𝑡) (provided it exists), with the property that for each 𝜖>0, there is a neighborhood 𝑈 of 𝑡 such that|𝑓(𝜌(𝑡))−𝑓(𝑠)−𝑓^∇(𝑡)(𝜌(𝑡)−𝑠)|≤𝜖|𝜌(𝑡)−𝑠|, (2.3)for all 𝑠∈𝑈.

Definition 2.3. A function 𝑓 is left-dense continuous (i.e., ld-continuous), if 𝑓 is continuous at each left-dense point in 𝐓 and its right-sided limit exists at each right-dense point in 𝐓.

Definition 2.4. If 𝜙^Δ(𝑡)=𝑓(𝑡), then one defines the delta integral by^𝑏_𝑎𝑓(𝑡)Δ𝑡=𝜙(𝑏)−𝜙(𝑎). (2.4)If 𝐹^∇(𝑡)=𝑓(𝑡), then one defines the nabla integral by^𝑏_𝑎𝑓(𝑡)∇𝑡=𝐹(𝑏)−𝐹(𝑎). (2.5)

Lemma 2.5. If 𝑑=𝛽(𝑇−∑^𝑚−2_𝑖=1𝑎_𝑖𝜉_𝑖)+𝛾(1−∑^𝑚−2_𝑖=1𝑎_𝑖)>0, then for ℎ∈𝐶_𝑙𝑑[0,𝑇]_𝐓, the boundary value problem (BVP) 𝑢^Δ∇+ℎ(𝑡)=0,𝑡∈[0,𝑇]_𝐓,𝛽𝑢(0)−𝛾𝑢^Δ(0)=0,𝑢(𝑇)=∑^𝑚−2_𝑖=1𝑎_𝑖𝑢(𝜉_𝑖) (2.6)has the unique solution𝑢(𝑡)=−^𝑡₀(𝑡−𝑠)ℎ(𝑠)∇𝑠+𝛽𝑡+𝛾𝑑^𝑇₀−(𝑇−𝑠)ℎ(𝑠)∇𝑠𝛽𝑡+𝛾𝑑^𝑚−2_𝑖=1𝑎_𝑖^𝜉𝑖₀(𝜉_𝑖−𝑠)ℎ(𝑠)∇𝑠. (2.7)

Proof. By direct computation, we can easily get (2.7). So, we omit it.

Lemma 2.6. If 0<∑^𝑚−2_𝑖=1𝑏_𝑖<1,0<∑^𝑚−2_𝑖=1𝑎_𝑖𝜉_𝑖<𝑇,𝑑=𝛽(𝑇−∑^𝑚−2_𝑖=1𝑎_𝑖𝜉_𝑖)+𝛾(1−∑^𝑚−2_𝑖=1𝑎_𝑖)>0, then for ℎ∈𝐶_ld[0,𝑇]_𝐓, the boundary value problem (BVP) (𝜙_𝑝(𝑢^Δ∇))^∇+ℎ(𝑡)=0,𝑡∈[0,𝑇]_𝐓,𝛽𝑢(0)−𝛾𝑢^Δ(0)=0,𝑢(𝑇)=∑^𝑚−2_𝑖=1𝑎_𝑖𝑢(𝜉_𝑖),𝜙_𝑝(𝑢^Δ∇(0))=∑^𝑚−2_𝑖=1𝑏_𝑖𝜙_𝑝(𝑢^Δ∇(𝜉_𝑖)) (2.8)has the unique solution𝑢(𝑡)=−^𝑡₀(𝑡−𝑠)𝜙_𝑞(^𝑠₀ℎ(𝑟)∇𝑟+𝐵)∇𝑠+𝛽𝑡+𝛾𝑑^𝑇₀(𝑇−𝑠)𝜙_𝑞(^𝑠₀−ℎ(𝑟)∇𝑟+𝐵)∇𝑠𝛽𝑡+𝛾𝑑^𝑚−2_𝑖=1𝑎_𝑖^𝜉𝑖₀(𝜉_𝑖−𝑠)𝜙_𝑞(^𝑠₀ℎ(𝑟)∇𝑟+𝐵)∇𝑠, (2.9)where 𝐵=∑^𝑚−2_𝑖=1𝑏_𝑖∫^𝜉𝑖₀ℎ(𝑟)∇𝑟/(1−∑^𝑚−2_𝑖=1𝑏_𝑖).

Proof. Integrating both sides of (1.6) on [0,𝑡], we have𝜙_𝑝(𝑢^Δ∇(𝑡))=𝜙_𝑝(𝑢^Δ∇(0))−^𝑡₀ℎ(𝑟)∇𝑟. (2.10)So𝜙_𝑝(𝑢^Δ∇(𝜉_𝑖))=𝜙_𝑝(𝑢^Δ∇(0))−^𝜉𝑖₀ℎ(𝑟)∇𝑟. (2.11)By boundary value condition 𝜙_𝑝(𝑢^Δ∇(0))=∑^𝑚−2_𝑖=1𝑏_𝑖𝜙_𝑝(𝑢^Δ∇(𝜉_𝑖)), we have𝜙_𝑝(𝑢^Δ∇∑(0))=−^{𝑚−2𝑖=1}𝑏_𝑖∫^𝜉𝑖₀ℎ(𝑟)∇𝑟1−∑^𝑚−2_𝑖=1𝑏_𝑖. (2.12)By (2.10) and (2.12), we know𝑢^Δ∇(𝑡)=−𝜙_𝑞(∑^𝑚−2_𝑖=1𝑏_𝑖∫^𝜉𝑖₀ℎ(𝑟)∇𝑟1−∑^𝑚−2_𝑖=1𝑏_𝑖+^𝑡₀ℎ(𝑟)∇𝑟). (2.13)This together with Lemma 2.5 implies that𝑢(𝑡)=−^𝑡₀(𝑡−𝑠)𝜙_𝑞(^𝑠₀ℎ(𝑟)∇𝑟+𝐵)∇𝑠+𝛽𝑡+𝛾𝑑^𝑇₀(𝑇−𝑠)𝜙_𝑞(^𝑠₀−ℎ(𝑟)∇𝑟+𝐵)∇𝑠𝛽𝑡+𝛾𝑑^𝑚−2_𝑖=1𝑎_𝑖^𝜉𝑖₀(𝜉_𝑖−𝑠)𝜙_𝑞(^𝑠₀ℎ(𝑟)∇𝑟+𝐵)∇𝑠, (2.14)where 𝐵=∑^𝑚−2_𝑖=1𝑏_𝑖∫^𝜉𝑖₀ℎ(𝑟)∇𝑟/(1−∑^𝑚−2_𝑖=1𝑏_𝑖). The proof is complete.

Lemma 2.7. Let 0<∑^𝑚−2_𝑖=1𝑎_𝑖𝜉_𝑖<1,𝑑>0. If ℎ∈𝐶_𝑙𝑑[0,𝑇]_𝐓 and ℎ(𝑡)≥0, then the unique solution 𝑢 of (2.8) satisfies 𝑢(𝑡)≥0. (2.15)

Proof. By 𝑢^Δ∇(𝑡)=−𝜙_𝑞(∑^𝑚−2_𝑖=1𝑏_𝑖∫^𝜉𝑖₀ℎ(𝑟)∇𝑟/(1−∑^𝑚−2_𝑖=1𝑏_𝑖))+∫^𝑡₀ℎ(𝑟)∇𝑟)≤0, we can know that the graph of 𝑢(𝑡) is concave down on (0,𝑇)_𝐓. So we only prove 𝑢(0)≥0,𝑢(𝑇)≥0.
Firstly, we shall prove 𝑢(0)≥0 by the following two perspectives.
(i) If 0<∑^𝑚−2_𝑖=1𝑎_𝑖≤1, we have𝛾𝑢(0)=𝑑[^𝑇₀(𝑇−𝑠)𝜙_𝑞(^𝑠₀ℎ(𝑟)∇𝑟+𝐵)∇𝑠−^𝑚−2_𝑖=1𝑎_𝑖^𝜉𝑖₀(𝜉_𝑖−𝑠)𝜙_𝑞(^𝑠₀≥𝛾ℎ(𝑟)∇𝑟+𝐵)∇𝑠]𝑑[^𝑇₀(𝑇−𝑠)𝜙_𝑞(^𝑠₀ℎ(𝑟)∇𝑟+𝐵)∇𝑠−^𝑚−2_𝑖=1𝑎_𝑖^𝑇₀(𝑇−𝑠)𝜙_𝑞(^𝑠₀=𝛾ℎ(𝑟)∇𝑟+𝐵)∇𝑠]𝑑(1−^𝑚−2_𝑖=1𝑎_𝑖)^𝑇₀(𝑇−𝑠)𝜙_𝑞(^𝑠₀ℎ(𝑟)∇𝑟+𝐵)∇𝑠≥0. (2.16)(ii)If ∑^𝑚−2_𝑖=1𝑎_𝑖>1, by (2.8), we have𝛾𝑢(0)=𝑑[^𝑇₀(𝑇−𝑠)𝜙_𝑞(^𝑠₀ℎ(𝑟)∇𝑟+𝐵)∇𝑠−^𝑚−2_𝑖=1𝑎_𝑖^𝜉𝑖₀(𝜉_𝑖−𝑠)𝜙_𝑞(^𝑠₀≥𝛾ℎ(𝑟)∇𝑟+𝐵)∇𝑠]𝑑[^𝑇₀(𝑇−𝑠)𝜙_𝑞(^𝑠₀ℎ(𝑟)∇𝑟+𝐵)∇𝑠−^𝑚−2_𝑖=1𝑎_𝑖^𝑇₀(𝜉_𝑖−𝑠)𝜙_𝑞(^𝑠₀=𝛾ℎ(𝑟)∇𝑟+𝐵)∇𝑠]𝑑^𝑇₀[(𝑇−^𝑚−2_𝑖=1𝑎_𝑖𝜉_𝑖)+(^𝑚−2_𝑖=1𝑎_𝑖−1)𝑠]𝜙_𝑞(^𝑠₀ℎ(𝑟)∇𝑟+𝐵)∇𝑠≥0. (2.17)On the other hand, we have𝑢(𝑇)=−^𝑇₀(𝑇−𝑠)𝜙_𝑞(^𝑠₀ℎ(𝑟)∇𝑟+𝐵)∇𝑠+𝛽+𝛾𝑑^𝑇₀(𝑇−𝑠)𝜙_𝑞(^𝑠₀−ℎ(𝑟)∇𝑟+𝐵)∇𝑠𝛽+𝛾𝑑^𝑚−2_𝑖=1𝑎_𝑖^𝜉𝑖₀(𝜉_𝑖−𝑠)𝜙_𝑞(^𝑠₀≥𝛽ℎ(𝑟)∇𝑟+𝐵)∇𝑠𝑑[^𝑚−2_𝑖=1𝑎_𝑖^𝜉𝑖₀(𝜉_𝑖(𝑇−𝑠)−𝑇(𝜉_𝑖−𝑠))𝜙_𝑞(^𝑠₀+ℎ(𝑟)∇𝑟+𝐵)∇𝑠_{𝑚−2𝑖=1}𝑎_𝑖𝜉_𝑖^𝑇_𝜉𝑖(𝑇−𝑠)𝜙_𝑞(^𝑠₀+𝛾ℎ(𝑟)∇𝑟+𝐵)∇𝑠]𝑑^𝑚−2_𝑖=1𝑎_𝑖[^𝑇₀(𝑇−𝑠)𝜙_𝑞(^𝑠₀ℎ(𝑟)∇𝑟+𝐵)∇𝑠−^𝑇₀(𝜉_𝑖−𝑠)𝜙_𝑞(^𝑠₀=𝛽ℎ(𝑟)∇𝑟+𝐵)∇𝑠]𝑑^𝑚−2_𝑖=1𝑎_𝑖[^𝜉𝑖₀(𝑇−𝜉_𝑖)𝑠𝜙_𝑞(^𝑠₀ℎ(𝑟)∇𝑟+𝐵)∇𝑠+𝜉_𝑖^𝑇_𝜉𝑖(𝑇−𝑠)𝜙_𝑞(^𝑠₀+𝛾ℎ(𝑟)∇𝑟+𝐵)∇𝑠]𝑑^𝑚−2_𝑖=1𝑎_𝑖[^𝑇₀(𝑇−𝜉_𝑖)𝜙_𝑞(^𝑠₀ℎ(𝑟)∇𝑟+𝐵)∇𝑠]≥0. (2.18)The proof is completed.

Lemma 2.8. Let 𝑎_𝑖≥0,𝑖=1,…,𝑚−2,0<∑^𝑚−2_𝑖=1𝑎_𝑖𝜉_𝑖<𝑇,𝑑>0. If ℎ∈𝐶_ld[0,𝑇]_𝐓 and ℎ(𝑡)≥0, then the unique positive solution 𝑢(𝑡) of (BVP) (2.8) satisfies inf_{𝑡∈[𝜉𝑚−2,𝑇]𝐓}𝑢(𝑡)≥𝜎‖𝑢‖, (2.19)where 𝜎=min{𝑎_𝑚−2(𝑇−𝜉_𝑚−2)/(𝑇−𝑎_𝑚−2𝜉_𝑚−2),𝑎_𝑚−2𝜉_𝑚−2/𝑇,𝜉_𝑚−2/𝑇},‖𝑢‖=sup_{𝑡∈[0,𝑇]𝐓}|𝑢(𝑡)|.

Proof. Let 𝑢(𝑡)=max_{𝑡∈[0,𝑇]𝐓}𝑢(𝑡)=‖𝑢‖, we shall discuss it from the following two perspectives.
Case 1. If 0<∑^𝑚−2_𝑖=1𝑎_𝑖<1.
Firstly, assume 𝑡<𝜉_𝑚−2<𝑇, then min_{𝑡∈[𝜉𝑚−2,𝑇]𝐓}𝑢(𝑡)=𝑢(𝑇). By 𝑢(𝑇)=∑^𝑚−2_𝑖=1𝑎_𝑖𝑢(𝜉_𝑖)≥𝑎_𝑚−2𝑢(𝜉_𝑚−2), we have𝑢(𝑡)≤𝑢(𝑇)+𝑢(𝑇)−𝑢(𝜉_𝑚−2)𝑇−𝜉_𝑚−2𝑇(0−𝑇)=𝑢(𝑇)−𝑇−𝜉_𝑚−2𝑇𝑢(𝑇)+𝑇−𝜉_𝑚−2𝑢(𝜉_𝑚−2)𝑇≤𝑢(𝑇)(1−𝑇−𝜉_𝑚−2+𝑇𝑎_𝑚−2(𝑇−𝜉_𝑚−2))=𝑢(𝑇)𝑇−𝑎_𝑚−2𝜉_𝑚−2𝑎_𝑚−2(𝑇−𝜉_𝑚−2). (2.20)Somin_{𝑡∈[𝜉𝑚−2,𝑇]𝐓}𝑎𝑢(𝑡)≥^𝑚−2(𝑇−𝜉_𝑚−2)𝑇−𝑎_𝑚−2𝜉_𝑚−2‖𝑢‖. (2.21)Secondly, assume 𝜉_𝑚−2<𝑡<𝑇, then min_{𝑡∈[𝜉𝑚−2,𝑇]𝐓}𝑢(𝑡)=𝑢(𝑇). Otherwise, we have min_{𝑡∈[𝜉𝑚−2,𝑇]𝐓}𝑢(𝑡)=𝑢(𝜉_𝑚−2), then 𝑡∈[𝜉_𝑚−2,𝑇]_𝐓,𝑢(𝜉_𝑚−2)≥𝑢(𝜉_𝑚−1)≥⋯≥𝑢(𝜉₂)≥𝑢(𝜉₁). By 0<∑^𝑚−2_𝑖=1𝑎_𝑖<1, we have𝑢(𝑇)=^𝑚−2_𝑖=1𝑎_𝑖𝑢(𝜉_𝑖)≤^𝑚−2_𝑖=1𝑎_𝑖𝑢(𝜉_𝑚−2)<𝑢(𝜉_𝑚−2)≤𝑢(𝑇), (2.22)a contradiction.
By concave of 𝑢(𝑡), we get 𝑢(𝜉_𝑚−2)/𝜉_𝑚−2≥𝑢(𝑡)/𝑡≥𝑢(𝑡)/𝑇. In fact, since 𝑢(𝑇)≥𝑎_𝑚−2𝑢(𝜉_𝑚−2), then 𝑢(𝑇)/𝑎_𝑚−2𝜉_𝑚−2≥𝑢(𝑡)/𝑇, which impliesmin_{𝑡∈[𝜉𝑚−2,𝑇]𝐓}𝑎𝑢(𝑡)≥^𝑚−2𝜉_𝑚−2𝑇‖𝑢‖. (2.23)
Case 2. If ∑^𝑚−2_𝑖=1𝑎_𝑖>1.
Firstly, assume 𝑢(𝜉_𝑚−2)≤𝑢(𝑇), then min_{𝑡∈[𝜉𝑚−2,𝑇]𝐓}𝑢(𝑡)=𝑢(𝜉_𝑚−2). By concave of 𝑢(𝑡), we have 𝑡∈[𝜉_𝑚−2,𝑡]_𝐓, which implies 𝑢(𝜉_𝑚−2)/𝜉_𝑚−2≥𝑢(𝑡)/𝑡≥𝑢(𝑡)/𝑇, thenmin_{𝑡∈[𝜉𝑚−2,𝑇]𝐓}𝜉𝑢(𝑡)≥^𝑚−2𝑇‖𝑢‖. (2.24)
Secondly, assume 𝑢(𝜉_𝑚−2)>𝑢(𝑡), then min_{𝑡∈[𝜉𝑚−2,𝑇]𝐓}𝑢(𝑡)=𝑢(𝑇), and 𝑡∈[𝜉₁,𝑇]_𝐓. If not, 𝑡∈[0,𝜉₁), then 𝑢(𝜉₁)≥⋯≥𝑢(𝜉_𝑚−2)>𝑢(𝑇). So, we have𝑢(𝑇)=^𝑚−2_𝑖=1𝑎_𝑖𝑢(𝜉_𝑖)>𝑢(𝑇)^𝑚−2_𝑖=1𝑎_𝑖≥𝑢(𝑇), (2.25)a contradiction. By ∑^𝑚−2_𝑖=1𝑎_𝑖>1, there exists 𝜉∈{𝜉₁,𝜉₂,…,𝜉_𝑚−2} such that 𝑢(𝜉)≤𝑢(𝑇), then 𝑢(𝜉₁)≤𝑢(𝜉₂)≤⋯≤𝑢(𝜉_𝑚−2)≤𝑢(1). By concave of 𝑢(𝑡), we have 𝑢(1)/𝜉₁≥𝑢(𝜉₁)/𝜉₁≥𝑢(𝑡)/𝑡≥𝑢(𝑡)/𝑇, thenmin_{𝑡∈[𝜉𝑚−2,𝑇]𝐓}𝑢(𝑡)≥𝜉₁‖𝑢‖. (2.26)Therefore, by (2.21)–(2.26), we haveinf_{𝑡∈[𝜉𝑚−2,𝑇]𝐓}𝑢(𝑡)≥𝜎‖𝑢‖, (2.27)where 𝜎=min{𝑎_𝑚−2(𝑇−𝜉_𝑚−2)/(𝑇−𝑎_𝑚−2𝜉_𝑚−2),𝑎_𝑚−2𝜉_𝑚−2/𝑇,𝜉_𝑚−2/𝑇}. The proof is complete.

Let 𝐸=𝐶_ld[0,𝑇]_𝐓 be endowed with the ordering 𝑥≤𝑦 if 𝑥(𝑡)≤𝑦(𝑡), for all 𝑡∈[0,𝑇]_𝐓, and ‖𝑢‖=max_{𝑡∈[0,𝑇]𝐓}|𝑢(𝑡)| is defined as usual by maximum norm. Clearly, it follows that (𝐸,‖𝑢‖) is a Banach space.

We define a cone by𝐾={𝑢∶𝑢∈𝐸,𝑢(𝑡)isconcave,nonnegativeon[0,𝑇]_𝐓,inf_{𝑡∈[𝜉𝑚−2,𝑇]𝐓}𝑢(𝑡)≥𝜎‖𝑢‖}. (2.28)Define an operator 𝑆∶𝐾→𝐸 by setting𝑆𝑢(𝑡)=−^𝑡₀(𝑡−𝑠)𝜙_𝑞(^𝑠₀+𝑎(𝑟)𝑓(𝑟,𝑢(𝑟))∇𝑟+𝐴)∇𝑠𝛽𝑡+𝛾𝑑^𝑇₀(𝑇−𝑠)𝜙_𝑞(^𝑠₀−𝑎(𝑟)𝑓(𝑟,𝑢(𝑟))∇𝑟+𝐴)∇𝑠𝛽𝑡+𝛾𝑑^𝑚−2_𝑖=1𝑎_𝑖^𝜉𝑖₀(𝜉_𝑖−𝑠)𝜙_𝑞(^𝑠₀𝑎(𝑟)𝑓(𝑟,𝑢(𝑟))∇𝑟+𝐴)∇𝑠, (2.29)where 𝐴=∑^𝑚−2_𝑖=1𝑏_𝑖∫^𝜉𝑖₀𝑎(𝑟)𝑓(𝑟,𝑢(𝑟))∇𝑟/(1−∑^𝑚−2_𝑖=1𝑏_𝑖). Obviously, 𝑢 is a solution of boundary value problem (1.6) if and only if 𝑢 is a fixed point of operator 𝑆.

Lemma 2.9. 𝑆∶𝐾→𝐾 is completely continuous.

Proof. By (𝐻₂) and Lemmas 2.7-2.8, we easily get 𝑆𝐾⊂𝐾. By Arzela-Ascoli theorem and Lebesgue dominated convergence theorem, we can easily prove 𝑆 is completely continuous.

Lemma 2.10 (see [15]). Let 𝐾 be a cone in a Banach space 𝑋. Let 𝐷 be an open bounded subset of 𝑋 with 𝐷_𝐾=𝐷∩𝐾≠∅ and 𝐷_𝐾≠𝐾. Assume that 𝐴∶𝐷_𝐾→𝐾 is a compact map such that 𝑥≠𝐴𝑥 for 𝑥∈𝜕𝐷_𝐾. Then the following results hold.
(1)If ‖𝐴𝑥‖≤‖𝑥‖, 𝑥∈𝜕𝐷_𝐾, then 𝑖_𝐾(𝐴,𝐷_𝐾)=1.(2)If there exists 𝑥₀∈𝐾⧵{0} such that 𝑥≠𝐴𝑥+𝜆𝑥_0, for all 𝑥∈𝜕𝐷_𝐾 and all 𝜆>0, then 𝑖_𝐾(𝐴,𝐷_𝐾)=0.(3)Let 𝑈 be open in 𝑋 such that 𝑈⊂𝐷_𝐾. If 𝑖_𝐾(𝐴,𝐷_𝐾)=1 and 𝑖_𝐾(𝐴,𝑈_𝐾)=0, then 𝐴 has a fixed point in 𝐷_𝐾⧵𝑈_𝐾. The same result holds if 𝑖_𝐾(𝐴,𝐷_𝐾)=0 and 𝑖_𝐾(𝐴,𝑈_𝐾)=1, where 𝑖_𝐾(𝐴,𝐷_𝐾) denotes fixed point index.
We define𝐾_𝜌={𝑢(𝑡)∈𝐾∶‖𝑢‖<𝜌},Ω_𝜌={𝑢(𝑡)∈𝐾∶min_{𝜉𝑚−2≤𝑡≤𝑇}𝑥(𝑡)<𝜎𝜌}. (2.30)

Lemma 2.11 (see [15]). Ω_𝜌 defined above has the following properties:
(a)𝐾_𝜎𝜌⊂Ω_𝜌⊂𝐾_𝜌;(b)Ω_𝜌 is open relative to K;(c)𝑥∈𝜕Ω_𝜌 if and only if min_{𝜉𝑚−2≤𝑡≤𝑇}𝑥(𝑡)=𝜎𝜌;(d)if 𝑥∈𝜕Ω_𝜌, then 𝜎𝜌≤𝑥(𝑡)≤𝜌, for 𝑡∈[𝜉_𝑚−2,𝑇]_𝐓.
For the convenience, we introduce the following notations:𝑓^𝜌_𝜎𝜌=min{min_{𝜉𝑚−2≤𝑡≤𝑇}𝑓(𝑡,𝑢)𝜙_𝑝(𝜌)∶𝑢∈[𝜎𝜌,𝜌]},𝑓^𝜌₀=max{max_{0≤𝑡≤𝑇}𝑓(𝑡,𝑢)𝜙_𝑝𝑓(𝜌)∶𝑢∈[0,𝜌]},_𝛼=lim_𝑢→𝛼supmax_{0≤𝑡≤𝑇}𝑓(𝑡,𝑢)𝜙_𝑝(𝑢),𝑓_𝛼=lim_𝑢→𝛼infmax_{𝜉𝑚−2≤𝑡≤𝑇}𝑓(𝑡,𝑢)𝜙_𝑝(𝑢)(𝛼∶=∞or0⁺1),𝑚=(𝛽𝑇+𝛾)𝑑∫^𝑇₀(𝑇−𝑠)∇𝑠𝜙_𝑞(∫^𝑇₀∑𝑎(𝑟)∇𝑟+^{𝑚−2𝑖=1}𝑏_𝑖∫^𝜉𝑖₀𝑎(𝑟)∇𝑟1−∑^𝑚−2_𝑖=1𝑏_𝑖1),𝑀=1𝑑∫^𝑇_𝜉𝑚−2(𝑇−𝑠)𝜙_𝑞(∫^𝑠_𝜉𝑚−2𝑎(𝑟)∇𝑟)∇𝑠min{𝛽𝜉_𝑚−2+𝛾,𝛽max{∑^𝑚−2_𝑖=1𝑎_𝑖𝜉₁,𝑎_𝑚−2𝜉_𝑚−2}+𝛾∑^𝑚−2_𝑖=1𝑎_𝑖}. (2.31)

Lemma 2.12. If 𝑓 satisfies the following condition: 𝑓^𝜌₀≤𝜙_𝑝(𝑚),𝑢≠𝑆𝑢,𝑢∈𝜕𝐾_𝜌, (2.32)then𝑖_𝐾(𝑆,𝐾_𝜌)=1. (2.33)

Proof. For 𝑢∈𝜕𝐾_𝜌, then from (2.32), we have^𝑠₀𝑎(𝑟)𝑓(𝑟,𝑢(𝑟))∇𝑟+𝐴=^𝑠₀∑𝑎(𝑟)𝑓(𝑟,𝑢(𝑟))∇𝑟+^{𝑚−2𝑖=1}𝑏_𝑖∫^𝜉𝑖₀𝑎(𝑟)𝑓(𝑟,𝑢(𝑟))∇𝑟1−∑^𝑚−2_𝑖=1𝑏_𝑖≤^𝑇₀∑𝑎(𝑟)𝑓(𝑟,𝑢(𝑟))∇𝑟+^{𝑚−2𝑖=1}𝑏_𝑖∫^𝜉𝑖₀𝑎(𝑟)𝑓(𝑟,𝑢(𝑟))∇𝑟1−∑^𝑚−2_𝑖=1𝑏_𝑖≤𝜙_𝑝(𝑚𝜌)(^𝑇₀∑𝑎(𝑟)∇𝑟+^{𝑚−2𝑖=1}𝑏_𝑖∫^𝜉𝑖₀𝑎(𝑟)∇𝑟1−∑^𝑚−2_𝑖=1𝑏_𝑖). (2.34)So that𝜙_𝑞(^𝑠₀𝑎(𝑟)𝑓(𝑟,𝑢(𝑟))∇𝑟+𝐴)≤𝑚𝜌𝜙_𝑞(^𝑇₀∑𝑎(𝑟)∇𝑟+^{𝑚−2𝑖=1}𝑏_𝑖∫^𝜉𝑖₀𝑎(𝑟)∇𝑟1−∑^𝑚−2_𝑖=1𝑏_𝑖). (2.35)Therefore,𝑆𝑢(𝑡)≤𝛽𝑡+𝛾𝑑^𝑇₀(𝑇−𝑠)𝜙_𝑞(^𝑠₀≤𝑎(𝑟)𝑓(𝑟,𝑢(𝑟))∇𝑟+𝐴)∇𝑠(𝛽𝑇+𝛾)𝑚𝜌𝑑^𝑇₀(𝑇−𝑠)∇𝑠𝜙_𝑞(^𝑇₀∑𝑎(𝑟)∇𝑟+^{𝑚−2𝑖=1}𝑏_𝑖∫^𝜉𝑖₀𝑎(𝑟)∇𝑟1−∑^𝑚−2_𝑖=1𝑏_𝑖)=𝜌. (2.36)This implies that ‖𝑆𝑢‖≤‖𝑢‖ for 𝑢∈𝜕𝐾_𝜌. Hence, by Lemma 2.10(1) it follows that 𝑖_𝐾(𝑆,𝐾_𝜌)=1.

Lemma 2.13. If 𝑓 satisfies the following condition: 𝑓^𝜌_𝜎𝜌≥𝜙_𝑝(𝑀𝜎),𝑢≠𝑆𝑢,𝑢∈𝜕Ω_𝜌, (2.37)then𝑖_𝐾(𝑆,Ω_𝜌)=0. (2.38)

Proof. Let 𝑒(𝑡)≡1 for 𝑡∈[0,𝑇]_𝐓. Then 𝑒∈𝜕𝐾₁. We claim that𝑢≠𝑆𝑢+𝜆𝑒,𝑢∈𝜕Ω_𝜌,𝜆>0. (2.39)In fact, if not, there exist 𝑢₀∈𝜕Ω_𝜌 and 𝜆₀>0 such that 𝑢₀=𝑆𝑢₀+𝜆₀𝑒. By 𝑓^𝜌_𝜎𝜌≥𝜙_𝑝(𝑀𝜎), we have^𝑠₀𝑎(𝑟)𝑓(𝑟,𝑢₀(𝑟))∇𝑟+𝐴=^𝑠₀𝑎(𝑟)𝑓(𝑟,𝑢₀∑(𝑟))∇𝑟+^{𝑚−2𝑖=1}𝑏_𝑖∫^𝜉𝑖₀𝑎(𝑟)𝑓(𝑟,𝑢₀(𝑟))∇𝑟1−∑^𝑚−2_𝑖=1𝑏_𝑖≥^𝑠_𝜉𝑚−2𝑎(𝑟)𝑓⁺(𝑟,𝑢(𝑟))∇𝑟≥𝜙_𝑝(𝑀𝜎𝜌)(^𝑠_𝜉𝑚−2𝑎(𝑟)∇𝑟). (2.40)So that𝜙_𝑞(^𝑠₀𝑎(𝑟)𝑓(𝑟,𝑢(𝑟))∇𝑟+𝐴)≥𝑀𝜎𝜌𝜙_𝑞(^𝑠_𝜉𝑚−2𝑎(𝑟)∇𝑟). (2.41)By [16, Theorem 2.2(iv)], for 𝑡>0, we have(∫^𝑡₀(𝑡−𝑠)𝜙_𝑞(∫^𝑠₀𝑎(𝑟)𝑓(𝑟,𝑢₀(𝑟))∇𝑟+𝐴)∇𝑠𝑡)^Δ=∫^𝑡₀𝑠𝜙_𝑞(∫^𝑠₀𝑎(𝑟)𝑓(𝑟,𝑢₀(𝑟))∇𝑟+𝐴)∇𝑠𝑡𝜎(𝑡)≥0. (2.42)So, for 𝑖=1,2,…,𝑚−2, we have∫^𝜉𝑚−2₀(𝜉_𝑚−2−𝑠)𝜙_𝑞(∫^𝑠₀𝑎(𝑟)𝑓(𝑟,𝑢₀(𝑟))∇𝑟+𝐴)∇𝑠𝜉_𝑚−2≥∫^𝜉𝑖₀(𝜉_𝑖−𝑠)𝜙_𝑞(∫^𝑠₀𝑎(𝑟)𝑓(𝑟,𝑢₀(𝑟))∇𝑟+𝐴)∇𝑠𝜉_𝑖. (2.43)Therefore,𝑆𝑢₀(𝜉_𝑚−2𝛽)≥𝑑[𝜉_𝑚−2^𝑇₀(𝑡−𝑠)𝜙_𝑞(^𝑠₀𝑎(𝑟)𝑓(𝑟,𝑢₀(𝑟))∇𝑟+𝐴)∇𝑠−𝑇^𝜉𝑚−2₀(𝜉_𝑚−2−𝑠)𝜙_𝑞(^𝑠₀𝑎(𝑟)𝑓(𝑟,𝑢₀+𝛾(𝑟))∇𝑟+𝐴)∇𝑠]𝑑[^𝑇₀(𝑡−𝑠)𝜙_𝑞(^𝑠₀𝑎(𝑟)𝑓(𝑟,𝑢₀−(𝑟))∇𝑟+𝐴)∇𝑠_{𝜉𝑚−20}(𝜉_𝑚−2−𝑠)𝜙_𝑞(^𝑠₀𝑎(𝑟)𝑓(𝑟,𝑢₀≥(𝑟))∇𝑟+𝐴)∇𝑠]𝛽𝜉_𝑚−2+𝛾𝑑^𝑇_𝜉𝑚−2(𝑇−𝑠)𝜙_𝑞(^𝑠₀𝑎(𝑟)𝑓(𝑟,𝑢₀≥(𝑟))∇𝑟+𝐴)∇𝑠(𝛽𝜉_𝑚−2+𝛾)𝑀𝜎𝜌𝑑^𝑇_𝜉𝑚−2(𝑇−𝑠)𝜙_𝑞(^𝑠_𝜉𝑚−2𝑎(𝑟)∇𝑟)∇𝑠, (2.44)𝑆𝑢₀𝛽(𝑇)≥𝑑^𝑚−2_𝑖=1𝑎_𝑖[^𝜉𝑖₀(𝑡−𝜉_𝑖)𝑠𝜙_𝑞(^𝑠₀𝑎(𝑟)𝑓(𝑟,𝑢₀(𝑟))∇𝑟+𝐴)∇𝑠+𝜉_𝑖^𝑇_𝜉𝑖(𝑇−𝑠)𝜙_𝑞(^𝑠₀𝑎(𝑟)𝑓(𝑟,𝑢₀+𝛾(𝑟))∇𝑟+𝐴)∇𝑠]𝑑^𝑚−2_𝑖=1[^𝑇₀(𝑡−𝑠)𝜙_𝑞(^𝑠₀𝑎(𝑟)𝑓(𝑟,𝑢₀−(𝑟))∇𝑟+𝐴)∇𝑠_𝜉𝑖0(𝑇−𝑠)𝜙_𝑞(^𝑠₀𝑎(𝑟)𝑓(𝑟,𝑢₀≥𝛽(𝑟))∇𝑟+𝐴)∇𝑠]𝑑^𝑚−2_𝑖=1𝑎_𝑖𝜉_𝑖^𝑇_𝜉𝑖(𝑡−𝑠)𝜙_𝑞(^𝑠₀𝑎(𝑟)𝑓(𝑟,𝑢₀+𝛾(𝑟))∇𝑟+𝐴)∇𝑠𝑑^𝑚−2_𝑖=1^𝑇_𝜉𝑖(𝑡−𝑠)𝜙_𝑞(^𝑠₀𝑎(𝑟)𝑓(𝑟,𝑢₀≥(𝑟))∇𝑟+𝐴)∇𝑠𝑀𝜎𝜌𝑑(𝛽max{^𝑚−2_𝑖=1𝑎_𝑖𝜉₁,𝑎_𝑚−2𝜉_𝑚−2}+𝛾^𝑚−2_𝑖=1)^𝑇_𝜉𝑚−2(𝑇−𝑠)𝜙_𝑞(^𝑠_𝜉𝑚−2𝑎(𝑟)∇𝑟)∇𝑠. (2.45)Obviously, we can knowmin_{𝑡∈[𝜉𝑚−2,𝑇]𝐓}𝑆𝑢₀(𝑡)=min{𝑆𝑢₀(𝜉_𝑚−2),𝑆𝑢₀≥(𝑇)}𝑀𝜎𝜌𝑑^𝑇_𝜉𝑚−2(𝑇−𝑠)𝜙_𝑞(^𝑠_𝜉𝑚−2𝑎(𝑟)∇𝑟)∇𝑠×min{𝛽𝜉_𝑚−2+𝛾,𝛽max{^𝑚−2_𝑖=1𝑎_𝑖𝜉₁,𝑎_𝑚−2𝜉_𝑚−2}+𝛾^𝑚−2_𝑖=1}≥𝜎𝜌. (2.46)For 𝑡∈[𝜉_𝑚−2,𝑇]_𝐓, then𝑢₀(𝑡)=𝑆𝑢₀(𝑡)+𝜆₀𝑒(𝑡)≥min_{𝑡∈[𝜉𝑚−2,𝑇]}𝑆𝑢₀(𝑡)+𝜆₀=min{𝑆𝑢₀(𝜉_𝑚−2),𝑆𝑢₀(𝑇)}+𝜆₀≥𝜎𝜌+𝜆₀. (2.47)This together with Lemma 2.11(c) implies that𝜎𝜌≥𝜎𝜌+𝜆₀, (2.48)a contradiction. Hence, by Lemma 2.10(2), it follows that 𝑖_𝐾(𝑆,Ω_𝜌)=0.

3. Main Results

We now give our results on the existence of positive solutions of BVP (1.6).

Theorem 3.1. Suppose conditions (𝐻₁) and (𝐻₂) hold, and assume that one of the following conditions hold.
(H₃) There exist 𝜌₁,𝜌₂∈(0,+∞) with 𝜌₁<𝜎𝜌₂ such that 𝑓^𝜌1₀≤𝜙_𝑝(𝑚),𝑓^𝜌2_𝜎𝜌2≥𝜙_𝑝(𝑀𝜎).(H₄) There exist 𝜌₁,𝜌₂∈(0,+∞) with 𝜌₁<𝜌₂ such that 𝑓^𝜌2₀≤𝜙_𝑝(𝑚),𝑓^𝜌1_𝜎𝜌1≥𝜙_𝑝(𝑀𝜎). Then, the boundary value problem (1.6) has at least one positive solution.

Proof. Assume that (𝐻₃) holds, we show that 𝑆 has a fixed point 𝑢₁ in Ω_𝜌2⧵𝐾_𝜌1. By 𝑓^𝜌1₀≤𝜙_𝑝(𝑚) and Lemma 2.12, we have that𝑖_𝐾(𝑆,𝐾_𝜌1)=1. (3.1)By 𝑓^𝜌2_𝜎𝜌2≥𝜙_𝑝(𝑀𝜎) and Lemma 2.13, we have that𝑖_𝐾(𝑆,𝐾_𝜌2)=0. (3.2)By Lemma 2.11(a) and 𝜌₁<𝜎𝜌₂, we have 𝐾_𝜌1⊂𝐾_𝜎𝜌2⊂Ω_𝜌2. It follows from Lemma 2.10(3) that 𝑆 has a fixed point 𝑢₁ in Ω_𝜌2⧵𝐾_𝜌1. When condition (𝐻₄) holds, the proof is similar to the above, so we omit it here.
As a special case of Theorem 3.1, we obtain the following result.

Corollary 3.2. Suppose conditions (𝐻₁), (𝐻₂) hold, and assume that one of the following conditions holds.
(H₅)0≤𝑓⁰<𝜙_𝑝(𝑚) and 𝜙_𝑝(𝑀)<𝑓_∞≤∞.(H₆)0≤𝑓^∞<𝜙_𝑝(𝑚) and 𝜙_𝑝(𝑀)<𝑓₀≤∞. Then, the boundary value problem (1.6) has at least one positive solution.

Theorem 3.3. Assume conditions (𝐻₁) and (𝐻₂) hold, and suppose that one of the following conditions holds.
(H₇) There exist 𝜌₁,𝜌₂, and 𝜌₃∈(0,+∞) with 𝜌₁<𝜎𝜌₂ and 𝜌₂<𝜌₃ such that 𝑓^𝜌1₀≤𝜙_𝑝(𝑚),𝑓^𝜌2_𝜎𝜌2≥𝜙_𝑝(𝑀𝜎),𝑢≠𝑆𝑢,∀𝑢∈𝜕Ω_𝜌2,𝑓^𝜌3₀≤𝜙_𝑝(𝑚). (3.3)(H₈) There exist 𝜌₁,𝜌₂, and 𝜌₃∈(0,+∞) with 𝜌₁<𝜌₂<𝜎𝜌₃ such that 𝑓^𝜌2₀≤𝜙_𝑝(𝑚),𝑓^𝜌1_𝜎𝜌1≥𝜙_𝑝(𝑀𝜎),𝑢≠𝑆𝑢,∀𝑢∈𝜕𝐾_𝜌2,𝑓^𝜌3_𝜎𝜌3≥𝜙_𝑝(𝑀𝜎). (3.4) Then, the boundary value problem (1.6) has at least two positive solutions. Moreover, if in (𝐻₇)𝑓^𝜌1₀≤𝜙_𝑝(𝑚) is replaced by 𝑓^𝜌1₀<𝜙_𝑝(𝑚), then the BVP (1.6) has a third positive solution 𝑢₃∈𝐾_𝜌1.

Proof. Assume that condition (𝐻₇) holds, we show that either 𝑆 has a fixed point 𝑢₁ in 𝜕𝐾_𝜌1 or Ω_𝜌2⧵𝐾_𝜌1. If 𝑢≠𝑆𝑢 for 𝑢∈𝜕𝐾_𝜌1∪𝜕𝐾_𝜌3. By Lemma 2.12 and Lemma 2.13, we have that𝑖_𝐾(𝑆,𝐾_𝜌1𝑖)=1,_𝐾(𝑆,𝐾_𝜌3𝑖)=1,_𝐾(𝑆,𝐾_𝜌2)=0. (3.5) By Lemma 2.11(a) and 𝜌₁<𝜎𝜌₂, we have 𝐾_𝜌1⊂𝐾_𝜎𝜌2⊂Ω_𝜌2. It follows from Lemma 2.10(3) that 𝑆 has a fixed point 𝑢₁ in Ω_𝜌2⧵𝐾_𝜌1. Similarly, 𝑆 has a fixed point 𝑢₂ in 𝐾_𝜌3⧵Ω_𝜌2. When condition (𝐻₈) holds, the proof is similar to the above, so we omit it here.
As a special case of Theorem 3.3, we obtain the following result.

Corollary 3.4. Assume conditions (𝐻₁) and (𝐻₂) hold, if there exists 𝜌>0 such that one of the following conditions holds.
(H₉)0≤𝑓⁰<𝜙_𝑝(𝑚),𝑓^𝜌_𝜎𝜌≥𝜙_𝑝(𝑀𝜎),𝑢≠𝑆𝑢,∀𝑢∈𝜕Ω_𝜌 and 0≤𝑓^∞<𝜙_𝑝(𝑚).(H₁₀)𝜙_𝑝(𝑀)<𝑓₀≤∞,𝑓^𝜌₀≤𝜙_𝑝(𝑚),𝑢≠𝑆𝑢,∀𝑢∈𝜕𝐾_𝜌 and 𝜙_𝑝(𝑀)<𝑓_∞≤∞. Then, the boundary value problem (1.6) has at least two positive solutions.

4. Some Examples

In this section, we present some simple examples to explain our results. We only study the case 𝐓=ℝ,(0,𝑇)_𝐓=(0,1).

Example 4.1. Consider the following three-point boundary value problem with 𝑝-Laplacian:(𝜙_𝑝(𝑢^′′))^′𝑢+𝑎(𝑡)𝑓(𝑡,𝑢)=0,0<𝑡<1,_′1(0)=0,𝑢(1)=21𝑢(3),(𝜙_𝑝(𝑢^′′1)(0))=4(𝜙_𝑝(𝑢^′′1)(3)), (4.1)where 𝛽=0,𝛾=1,𝑎₁=1/2,𝑏₁=1/4,𝜉₁=1/3,𝑎(𝑡)=1,𝑝=𝑞=2. By computing, we can know 𝜎=1/6,𝑀=819/16,𝑚=9/10. Let 𝜌₁=1,𝜌₂=208, then 𝜎𝜌₁<𝜌₁<𝜎𝜌₂<𝜌₂. We define a nonlinearity 𝑓 as follows:𝑓(𝑡,𝑢)=9𝑡³(1106−𝑢)³1,0<𝑡<1,𝑢∈[0,6],9𝑡³610sin(5𝜋21𝑢−5𝜋21),0<𝑡<1,𝑢∈[6,1],9𝑡³(10208−6202202𝑢)[phone]𝑢−202),0<𝑡<1,𝑢∈[1,2086],81996+𝑡³(𝑢−2086)²,0<𝑡<1,𝑢∈[2086,208],81996+𝑡³(208−2086)²[1+(𝑢−208)],0<𝑡<1,𝑢∈[208,+∞]. (4.2)Then, by the definition of 𝑓, we have
(i)𝑓^𝜌1₀≤𝜙_𝑝(𝑚)=9/10;(ii)𝑓^𝜌2_𝜎𝜌2≥𝜙_𝑝(𝑀𝜎)=819/19968.
So condition (𝐻₃) holds, by Theorem 3.1, boundary value problem (4.1) has at least one positive solution.

Acknowledgment

Project supported by the National Natural Science Foundation of China (10471075).