Triple Positive Solutions for Third-Order 𝑚-Point Boundary Value Problems on Time Scales

Abstract

We study the following third-order 𝑚-point boundary value problems on time scales (𝜑(𝑢^Δ∇))^∇+𝑎(𝑡)𝑓(𝑢(𝑡))=0, 𝑡∈[0,𝑇]_T, 𝑢(0)=∑^𝑚−2_𝑖=1𝑏_𝑖𝑢(𝜉_𝑖), 𝑢^Δ(𝑇)=0, 𝜑(𝑢^Δ∇(0))=∑^𝑚−2_𝑖=1𝑐_𝑖𝜑(𝑢^Δ∇(𝜉_𝑖)), where 𝜑∶𝑅→𝑅 is an increasing homeomorphism and homomorphism and 𝜑(0)=0, 0<𝜉₁<⋯<𝜉_𝑚−2<𝜌(𝑇). We obtain the existence of three 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 mean of unifying and extending theories from differential and difference equations. The study of time scales has led 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 [7–15].

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 [16], Anderson considered 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) He used the Krasnoselskii and Leggett-Williams fixed-point theorems to prove the existence of solutions to the nonlinear boundary value problem.

In [9, 10], 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 conditions 𝑢(0)−𝐵₀(𝑢^Δ(𝜂))=0,𝑢^Δ(𝑇)=0, (1.3) or 𝑢^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 problems by using a new double fixed point theorem and triple fixed point theorem, respectively.

In [15], 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 the increasing homeomorphism and positive homomorphism operator on time scales. So the goal of the present paper is to improve and generate 𝑝-Laplacian operator and establish some criteria for the existence of multiple positive solutions for the following third-order 𝑚-point boundary value problems on time scales (𝜑(𝑢^Δ∇))^∇+𝑎(𝑡)𝑓(𝑢(𝑡))=0,𝑡∈[0,𝑇]_𝐓,𝑢(0)=^𝑚−2_𝑖=1𝑏_𝑖𝑢(𝜉_𝑖),𝑢^Δ(𝑇)=0,𝜑(𝑢^Δ∇(0))=^𝑚−2_𝑖=1𝑐_𝑖𝜑(𝑢^Δ∇(𝜉_𝑖)), (1.6) where 𝜑∶𝑅→𝑅 is an increasing homeomorphism and homomorphism and 𝜑(0)=0, and 𝑏_𝑖,𝑐_𝑖,𝑎,𝑓 satisfy

(𝐻₁)𝑏_𝑖,𝑐_𝑖∈[0,+∞), 0<𝜉₁<⋯<𝜉_𝑚−2<𝜌(𝑇), 0<∑^𝑚−2_𝑖=1𝑏_𝑖<1,0<∑^𝑚−2_𝑖=1𝑐_𝑖<1;(𝐻₂)𝑓∶[0,+∞)→𝑅⁺ is continuous, 𝑎∈𝐶_ld([0,𝑇]_𝐓,𝑅⁺) and there exits 𝑡₀∈[0,𝑇)_𝐓 such that 𝑎(𝑡₀)>0, where 𝑅⁺=[0,+∞).

A projection 𝜑∶𝑅→𝑅 is called an increasing homeomorphism and homomorphism, if the following conditions are satisfied:

(i)if 𝑥≤𝑦, then 𝜑(𝑥)≤𝜑(𝑦),∀𝑥,𝑦∈𝑅;(ii)𝜑 is continuous bijection and its inverse mapping is also continuous;(iii)𝜑(𝑥𝑦)=𝜑(𝑥)𝜑(𝑦),∀𝑥,𝑦∈𝑅.

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, by 𝜎(𝑡)=inf{𝜏∈𝐓∣𝜏>𝑡}∈𝐓,𝜌(𝑟)=sup{𝜏∈𝐓𝜏<𝑟}∈𝐓, (2.1) for all 𝑡,𝑟∈𝐓. If 𝜎(𝑡)>𝑡, 𝑡 is said to be right scattered, and if 𝜌(𝑟)<𝑟, 𝑟 is said to be left scattered; if 𝜎(𝑡)=𝑡, 𝑡 is said to be right dense, and 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 we define the delta integral by ^𝑏_𝑎𝑓(𝑡)Δ𝑡=𝜙(𝑏)−𝜙(𝑎). (2.4) If 𝐹^∇(𝑡)=𝑓(𝑡), then we define the nabla integral by ^𝑏_𝑎𝑓(𝑡)∇𝑡=𝐹(𝑏)−𝐹(𝑎). (2.5)

Definition 2.5. Let 𝐸 be a real Banach space over 𝑅. A nonempty closed set 𝑃⊂𝐸 is said to be a cone provided that
(i)𝑢∈𝑃, 𝑎≥0 implies 𝑎𝑢∈𝑃; (ii)𝑢,−𝑢∈𝑃 implies 𝑢=0.

Definition 2.6. Given a cone 𝑃 in a real Banach space 𝐸, a functional 𝜓∶𝑃→𝑃 is said to be increasing on 𝑃, provided 𝜓(𝑥)≤𝜓(𝑦), for all 𝑥,𝑦∈𝑃 with 𝑥≤𝑦.

Definition 2.7. Given a cone 𝑃 in a real Banach space 𝐸, we define for each 𝑎>0 the set 𝑃_𝑎={𝑥∈𝑃∣‖𝑥‖<𝑎}. (2.6)

Definition 2.8. A map 𝛼 is called nonnegative continuous concave functional on a cone 𝑃 of a real Banach space 𝑋 if 𝛼∶𝑃→[0,+∞) is continuous and 𝛼(𝜆𝑥+(1−𝜆)𝑦)≥𝜆𝛼(𝑥)+(1−𝜆)𝛼(𝑦) (2.7) for all 𝑥,𝑦∈𝑃 and 𝜆∈[0,1]. Similarly we say that the map 𝛽 is called nonnegative continuous concave functional on a cone 𝑃 of a real Banach space 𝑋 if 𝛽∶𝑃→[0,+∞) is continuous and 𝛽(𝜆𝑥+(1−𝜆)𝑦)≤𝜆𝛽(𝑥)+(1−𝜆)𝛽(𝑦) (2.8) for all 𝑥,𝑦∈𝑃 and 𝜆∈[0,1].

Let 𝛾,𝜃 be nonnegative continuous convex functionals on 𝑃, let 𝛼 be a nonnegative continuous concave functional on 𝑃, and let 𝜓 be a nonnegative continuous functional on 𝑃. For nonnegative real numbers 𝑎,𝑏,𝑘, and 𝑐 we define the following convex set: 𝑃(𝛾,𝑐)={𝑥∈𝑃∶𝛾(𝑥)<𝑐},𝑃(𝛼,𝑏;𝛾,𝑐)={𝑥∈𝑃∶𝑏≤𝛼(𝑥),𝛾(𝑥)≤𝑐},𝑅(𝜓,𝑎;𝛾,𝑐)={𝑥∈𝑃∶𝑎≤𝜓(𝑥),𝛾(𝑥)≤𝑐},𝑃(𝛼,𝑏;𝜃,𝑘;𝛾,𝑐)={𝑥∈𝑃∶𝑏≤𝛼(𝑥),𝜃(𝑥)≤𝑘,𝛾(𝑥)≤𝑐}. (2.9)

Theorem 2.9 ([17]). Let 𝑃 be a cone in a real Banach space 𝑋. Let 𝛾 and 𝜃 be nonnegative continuous convex functionals on 𝑃, let 𝛼 be a nonnegative continuous concave functional on 𝑃, and let 𝜓 be a nonnegative continuous functional on 𝑃 satisfying 𝜓(𝜆𝑥)≤𝜆𝜓(𝑥) for 0≤𝜆≤1, such that for some positive numbers 𝑀 and 𝑐, 𝛼(𝑥)≤𝜓(𝑥) and ‖𝑥‖≤𝑀𝛾(𝑥) for all 𝑥∈𝑃(𝛾,𝑐). Suppose that Φ∶𝑃(𝛾,𝑐)→𝑃(𝛾,𝑐) is a completely continuous operator and there exist nonnegative numbers 𝑎,𝑏, and 𝑘 with 0<𝑎<𝑏 such that
(i){𝑥∈𝑃(𝛼,𝑏;𝜃,𝑘;𝛾,𝑐)∶𝛼(𝑥)>𝑏}≠∅ and 𝛼(Φ(𝑥))>𝑏 for 𝑥∈𝑃(𝛼,𝑏;𝜃,𝑘;𝛾,𝑐);(ii)𝛼(Φ(𝑥))>𝑏 for 𝑥∈𝑃(𝛼,𝑏;𝛾,𝑐) with 𝜃(Φ(𝑥))>𝑘;(iii)0∈𝑅(𝜓,𝑎;𝛾,𝑐) and 𝜓(Φ(𝑥))<𝑎 for 𝑥∈𝑅(𝜓,𝑎;𝛾,𝑐) with 𝜓(𝑥)=𝑎.Then Φ has at least three fixed points 𝑥₁,𝑥₂,𝑥₃∈𝑃(𝛾,𝑐) satisfying 𝛾(𝑥_𝑖)≤𝑐,𝑖=1,2,3,𝑏<𝛼(𝑥₁),𝛼(𝑥₂)<𝑏,𝑎<𝜓(𝑥₂),𝜓(𝑥₃)<𝑎. (2.10)

Theorem 2.10 ([18]). Let 𝐴 be a bounded closed convex subset of a Banach space 𝐸. Assume that 𝐴₁,𝐴₂ are disjoint closed convex subsets of 𝐴 and 𝑈₁,𝑈₂ are nonempty open subsets of 𝐴 with 𝑈₁⊂𝐴₁ and 𝑈₂⊂𝐴₂. Suppose that Φ∶𝐴→𝐴 is completely continuous and the following conditions hold:
(i)Φ(𝐴₁)⊂𝐴₁,Φ(𝐴₂)⊂𝐴₂;(ii)Φ has no fixed points in (𝐴₁⧵𝑈₁)∪(𝐴₂⧵𝑈₂).Then Φ has at least three points 𝑥₁,𝑥₂,𝑥₃ such that 𝑥₁∈𝑈₁,𝑥₂∈𝑈₂, and 𝑥₃∈𝐴⧵(𝐴₁∪𝐴₂).

Lemma 2.11. If condition (𝐻₁) holds, then for ℎ∈𝐶_ld[0,𝑇]_𝐓, the boundary value problem (BVP) 𝑢^Δ∇+ℎ(𝑡)=0,𝑡∈(0,𝑇),𝑢(0)=^𝑚−2_𝑖=1𝑏_𝑖𝑢(𝜉_𝑖),𝑢^Δ(𝑇)=0 (2.11) has the unique solution 𝑢(𝑡)=^𝑡₀∑(𝑇−𝑠)ℎ(𝑠)∇𝑠+^{𝑚−2𝑖=1}𝑏_𝑖∫^𝜉𝑖₀(𝑇−𝑠)ℎ(𝑠)∇𝑠1−∑^𝑚−2_𝑖=1𝑏_𝑖. (2.12)

Proof. By caculating, we can easily get (2.12). So we omit it.

Lemma 2.12. If condition (𝐻₁) holds, then for ℎ∈𝐶_ld[0,𝑇]_𝐓, the boundary value problem (BVP) (𝜑(𝑢^Δ∇))^∇+ℎ(𝑡)=0,𝑡∈(0,𝑇),𝑢(0)=^𝑚−2_𝑖=1𝑏_𝑖𝑢(𝜉_𝑖),𝑢^Δ(𝑇)=0,𝜑(𝑢^Δ∇(0))=^𝑚−2_𝑖=1𝑐_𝑖𝜑(𝑢^Δ∇(𝜉_𝑖)) (2.13) has the unique solution 𝑢(𝑡)=^𝑡₀(𝑇−𝑠)𝜑⁻¹(^𝑠₀∑ℎ(𝑟)∇𝑟+𝐶)∇𝑠+^{𝑚−2𝑖=1}𝑏_𝑖∫^𝜉𝑖₀(𝑇−𝑠)𝜑⁻¹(∫^𝑠₀ℎ(𝑟)∇𝑟+𝐶)∇𝑠1−∑^𝑚−2_𝑖=1𝑏_𝑖, (2.14) where 𝐶=∑^𝑚−2_𝑖=1𝑐_𝑖∫^𝜉𝑖₀ℎ(𝑟)∇𝑟/(1−∑^𝑚−2_𝑖=1𝑐_𝑖), 𝜑⁻¹(𝑠) is the inverse function to 𝜑(𝑠).

Proof. Integrating both sides of equation in (2.13) on [0,𝑡], we have 𝜑(𝑢^Δ∇(𝑡))=𝜑(𝑢^Δ∇(0))−^𝑡₀ℎ(𝑟)∇𝑟. (2.15) So, 𝜑(𝑢^Δ∇(𝜉_𝑖))=𝜑(𝑢^Δ∇(0))−^𝜉𝑖₀ℎ(𝑟)∇𝑟. (2.16) By boundary value condition 𝜑(𝑢^Δ∇(0))=∑^𝑚−2_𝑖=1𝑐_𝑖𝜑(𝑢^Δ∇(𝜉_𝑖)), we have 𝜑(𝑢^Δ∇∑(0))=−^{𝑚−2𝑖=1}𝑐_𝑖∫^𝜉𝑖₀ℎ(𝑟)∇𝑟1−∑^𝑚−2_𝑖=1𝑐_𝑖. (2.17) By (2.15) and (2.17) we know 𝑢^Δ∇(𝑡)=−𝜑⁻¹(∑^𝑚−2_𝑖=1𝑐_𝑖∫^𝜉𝑖₀ℎ(𝑟)∇𝑟1−∑^𝑚−2_𝑖=1𝑐_𝑖+^𝑡₀ℎ(𝑟)∇𝑟). (2.18) This together with Lemma 2.11 implies that 𝑢(𝑡)=^𝑡₀(𝑇−𝑠)𝜑⁻¹(^𝑠₀∑ℎ(𝑟)∇𝑟+𝐶)∇𝑠+^{𝑚−2𝑖=1}𝑏_𝑖∫^𝜉𝑖₀(𝑇−𝑠)𝜑⁻¹(∫^𝑠₀ℎ(𝑟)∇𝑟+𝐶)∇𝑠1−∑^𝑚−2_𝑖=1𝑏_𝑖, (2.19) where 𝐶=∑^𝑚−2_𝑖=1𝑐_𝑖∫^𝜉𝑖₀ℎ(𝑟)∇𝑟/(1−∑^𝑚−2_𝑖=1𝑐_𝑖). The proof is complete.

Lemma 2.13. Let condition (𝐻₁) hold. If ℎ∈𝐶_ld[0,𝑇]_𝐓 and ℎ(𝑡)≥0, then the unique solution 𝑢(𝑡) of (2.13) satisfies 𝑢(𝑡)≥0,𝑡∈[0,𝑇]_𝐓. (2.20)

Proof. By 𝑢^Δ∇(𝑡)=−𝜑⁻¹((∑^𝑚−2_𝑖=1𝑐_𝑖∫^𝜉𝑖₀ℎ(𝑟)∇𝑟/(1−∑^𝑚−2_𝑖=1𝑐_𝑖))+∫^𝑡₀ℎ(𝑟)∇𝑟)≤0, we can know that the graph of 𝑢(𝑡) is concave down on (0,𝑇)_𝐓 and 𝑢^Δ(𝑡) is nonincreasing on [0,𝑇]_𝐓. This together with the assumption that the boundary condition is 𝑢^Δ(𝑇)=0 implies that 𝑢^Δ(𝑡)≥0 for 𝑡∈[0,𝑇]_𝐓. This implies that ‖𝑢‖=𝑢(𝑇),min_{𝑡∈[0,𝑇]𝐓}𝑢(𝑡)=𝑢(0). (2.21) So we only prove 𝑢(0)≥0. By condition (𝐻₁) we have ∑𝑢(0)=^{𝑚−2𝑖=1}𝑏_𝑖∫^𝜉𝑖₀(𝑇−𝑠)𝜑⁻¹(∫^𝑠₀ℎ(𝑟)∇𝑟+𝐶)∇𝑠1−∑^𝑚−2_𝑖=1𝑏_𝑖≥0. (2.22) The proof is completed.

3. Triple Positive Solutions

In this section, some existence results of positive solutions to BVP (1.6) are established by imposing some conditions on 𝑓 and defining a suitable Banach space and a cone.

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 by 𝑃={𝑢∶𝑢∈𝐸,𝑢(𝑡)isconcave,nondecreasing,andnonnegativeon[0,𝑇]_𝐓,𝑢^Δ(𝑇)=0}. (3.1) Let 𝑇𝜂=max{𝑡∈𝐓∶𝑡≥2}, (3.2) and fix 𝑙∈𝐓 such that 0<𝜂<𝑙<𝑇, (3.3) and define the nonnegative continuous convex functionals 𝛾 and 𝜃, the nonnegative continuous concave functional 𝛼, and the nonnegative continuous functional 𝜓 on the cone 𝑃 by 𝛾(𝑢)=𝜃(𝑢)=max_{𝑡∈[0,𝑙]𝐓}𝑢(𝑡)=𝑢(𝑙),𝛼(𝑢)=min_{𝑡∈[𝜂,𝑇]𝐓}𝑢(𝑡)=𝑢(𝜂),𝜓(𝑢)=max_{𝑡∈[0,𝜂]𝐓}=𝑢(𝜂). (3.4) For notational convenience, denote ∑𝐶=^{𝑚−2𝑖=1}𝑐_𝑖∫^𝜉𝑖₀𝑎(𝑟)∇𝑟1−∑^𝑚−2_𝑖=1𝑐_𝑖,𝑚_𝜂=^𝜂₀(𝑇−𝑠)𝜑⁻¹(^𝑠₀∑𝑎(𝑟)∇𝑟+𝐶)∇𝑠,𝑚=^{𝑚−2𝑖=1}𝑏_𝑖∫^𝜉𝑖₀(𝑇−𝑠)𝜑⁻¹(∫^𝑠₀𝑎(𝑟)∇𝑟+𝐶)∇𝑠1−∑^𝑚−2_𝑖=1𝑏_𝑖,𝑀_𝑙=^𝑙₀(𝑇−𝑠)𝜑⁻¹(^𝑠₀∑𝑎(𝑟)∇𝑟+𝐶)∇𝑠+^{𝑚−2𝑖=1}𝑏_𝑖∫^𝜉𝑖₀(𝑇−𝑠)𝜑⁻¹(∫^𝑠₀𝑎(𝑟)∇𝑟+𝐶)∇𝑠1−∑^𝑚−2_𝑖=1𝑏_𝑖,𝑀_𝜂=^𝜂₀(𝑇−𝑠)𝜑⁻¹(^𝑠₀∑𝑎(𝑟)∇𝑟+𝐶)∇𝑠+^{𝑚−2𝑖=1}𝑏_𝑖∫^𝜉𝑖₀(𝑇−𝑠)𝜑⁻¹(∫^𝑠₀𝑎(𝑟)∇𝑟+𝐶)∇𝑠1−∑^𝑚−2_𝑖=1𝑏_𝑖,𝑀=^𝑇₀(𝑇−𝑠)𝜑⁻¹(^𝑠₀∑𝑎(𝑟)∇𝑟+𝐶)∇𝑠+^{𝑚−2𝑖=1}𝑏_𝑖∫^𝜉𝑖₀(𝑇−𝑠)𝜑⁻¹(∫^𝑠₀𝑎(𝑟)∇𝑟+𝐶)∇𝑠1−∑^𝑚−2_𝑖=1𝑏_𝑖. (3.5)

Lemma 3.1 ([9]). If 𝑢∈𝑃, then
(1)𝑢(𝑡)≥(𝑡/𝑇)||𝑢|| for all 𝑡∈[0,𝑇]_𝐓;(2)𝑢(𝑠)/𝑠≥𝑢(𝑡)/𝑡 for 𝑡,𝑠∈[0,𝑇]_𝐓 with 𝑠≤𝑡.
Define an operator Φ∶𝑃→𝐸 by (Φ𝑢)(𝑡)=^𝑡₀(𝑇−𝑠)𝜑⁻¹(^𝑠₀+∑𝑎(𝑟)𝑓(𝑢(𝑟))∇𝑟+𝐶)∇𝑠_{𝑚−2𝑖=1}𝑏_𝑖∫^𝜉𝑖₀(𝑇−𝑠)𝜑⁻¹(∫^𝑠₀𝑎(𝑟)𝑓(𝑢(𝑟))∇𝑟+𝐶)∇𝑠1−∑^𝑚−2_𝑖=1𝑏_𝑖, (3.6) where 𝐶=∑^𝑚−2_𝑖=1𝑐_𝑖∫^𝜉i₀𝑎(𝑟)𝑓(𝑢(𝑟))∇𝑟/(1−∑^𝑚−2_𝑖=1𝑐_𝑖). Then, 𝑢 is a solution of boundary value problem (1.6) if and only if 𝑢 is a fixed point of operator Φ. Obviously, for 𝑢∈𝑃 one has (Φ𝑢)(𝑡)≥0 for 𝑡∈[0,𝑇]_𝐓. In addition, (Φ𝑢)^Δ∇(𝑡)≤0 for 𝑡∈[0,𝑇]_𝐓 and (Φ𝑢)^Δ(𝑇)=0, this implies Φ𝑃⊂𝑃. With standard argument one may show that Φ∶𝑃→𝑃 is completely continuous.

Theorem 3.2. Suppose conditions (𝐻₁) and (𝐻₂) hold, and there exist positive numbers 𝑎<(𝜂/𝑇)𝑏<𝑏<(𝑙/𝑇)𝑐,𝑀_𝑙𝑏<𝑚𝑐 such that
(𝐵₁)𝑓(𝑢)≤𝜑(𝑐/𝑀_𝑙),𝑢∈[0,𝑇𝑐/𝑙];(𝐵₂)𝑓(𝑢)>𝜑(𝑏/𝑚_𝜂),𝑢∈[𝑏,𝑇²𝑏/𝑙²];(𝐵₃)𝑓(𝑢)<𝜑(𝑎/𝑀_𝜂),𝑢∈[0,𝑇𝑎/𝜂].Then, the BVP (1.6) has at least three positive solutions 𝑢₁,𝑢₂,𝑢₃∈𝑃(𝛾,𝑐) satisfying 𝛾(𝑢_𝑖)≤𝑐,𝑖=1,2,3,𝑏<𝛼(𝑢₁),𝛼(𝑢₂)<𝑏,𝑎<𝜓(𝑢₂),𝜓(𝑢₃)<𝑎. (3.7)

Proof. Based on Lemma 3.1, it is clear that for 𝑢∈𝑃 and 𝜆∈[0,1], there are 𝛼(𝑢)=𝜓(𝑢),𝜓(𝜆𝑢)=𝜆𝜓(𝑢) and ‖𝑢‖≤(𝑇/𝑙)𝑢(𝑙)=(𝑇/𝑙)𝛾(𝑢). Furthermore, 𝜓(0)=0<𝑎 and therefore 0∈𝑅(𝜓,𝑎;𝛾,𝑐).
Take 𝑢∈𝑃(𝛾,𝑐), then 0≤𝑢≤‖𝑢‖≤(𝑇/𝑙)𝛾(𝑢)≤(𝑇/𝑙)𝑐. By means of (𝐵₁) one derives =𝛾(Φ𝑢)=Φ𝑢(𝑙)_𝑙0(𝑇−𝑠)𝜑⁻¹(^𝑠₀+∑𝑎(𝑟)𝑓(𝑢(𝑟))∇𝑟+𝐶)∇𝑠_{𝑚−2𝑖=1}𝑏_𝑖∫^𝜉𝑖₀(𝑇−𝑠)𝜑⁻¹(∫^𝑠₀𝑎(𝑟)𝑓(𝑢(𝑟))∇𝑟+𝐶)∇𝑠1−∑^𝑚−2_𝑖=1𝑏_𝑖≤𝑐𝑀_𝑙(^𝑙₀(𝑇−𝑠)𝜑⁻¹(^𝑠₀∑𝑎(𝑟)∇𝑟+𝐶)∇𝑠+^{𝑚−2𝑖=1}𝑏_𝑖∫^𝜉𝑖₀(𝑇−𝑠)𝜑⁻¹(∫^𝑠₀𝑎(𝑟)∇𝑟+𝐶)∇𝑠1−∑^𝑚−2_𝑖=1𝑏_𝑖)=𝑐. (3.8) Thus Φ∶𝑃(𝛾,𝑐)→𝑃(𝛾,𝑐).
Set 𝑢≡𝑇𝑏/𝑙 and 𝑘=𝑇𝑏/𝑙, it follows that 𝛼(𝑢)=𝑢(𝜂)=𝑇𝑏𝑙>𝑏,𝜃(𝑢)=𝑢(𝑙)=𝑇𝑏𝑙,𝛾(𝑢)=𝑇𝑏𝑙<𝑐, (3.9) which means {𝑢∈𝑃(𝛼,𝑏;𝜃,𝑇𝑏/𝑙;𝛾,𝑐)∶𝛼(𝑢)>𝑏}≠∅.
For 𝑢∈𝑃(𝛼,𝑏;𝜃,𝑇𝑏/𝑙;𝛾,𝑐), we have 𝑏≤𝑢(𝑡)≤𝑇²𝑏/𝑙² for 𝑡∈[𝜂,𝑇]_𝐓. By condition (𝐵₂) we have =𝛼(Φ𝑢)=Φ𝑢(𝜂)_𝜂0(𝑇−𝑠)𝜑⁻¹(^𝑠₀+∑𝑎(𝑟)𝑓(𝑢(𝑟))∇𝑟+𝐶)∇𝑠_{𝑚−2𝑖=1}𝑏_𝑖∫^𝜉𝑖₀(𝑇−𝑠)𝜑⁻¹(∫^𝑠₀𝑎(𝑟)𝑓(𝑢(𝑟))∇𝑟+𝐶)∇𝑠1−∑^𝑚−2_𝑖=1𝑏_𝑖≥𝑏𝑚_𝜂^𝜂₀(𝑇−𝑠)𝜑⁻¹(^𝑠₀𝑎(𝑟)∇𝑟+𝐶)∇𝑠=𝑏. (3.10) So, (i) of Theorem 2.9 is fulfilled.
If 𝑢∈𝑃(𝛼,𝑏;𝛾,𝑐) and 𝜃(Φ𝑢)>𝑐, then due to (2) of Lemma 3.1𝜂𝛼(Φ𝑢)=Φ𝑢(𝜂)≥𝑙𝜂(Φ𝑢)(𝑙)=𝑙𝜃(Φ𝑢)>𝜂𝑐𝑙>𝑇𝜂𝑏𝑙²>𝑏. (3.11) Therefore, (ii) of Theorem 2.9 is fulfilled.
Take 𝑢∈𝑅(𝜓,𝑎;𝛾,𝑐) and 𝜓(𝑢)=𝑎, then 0≤𝑢≤‖𝑢‖≤(𝑇/𝜂)𝑢(𝜂)=(𝑇/𝜂)𝜓(𝑢)=𝑇𝑎/𝜂, it then follows from (𝐵₃) that =𝜓(Φ𝑢)=Φ𝑢(𝜂)_𝜂0(𝑇−𝑠)𝜑⁻¹(^𝑠₀+∑𝑎(𝑟)𝑓(𝑢(𝑟))∇𝑟+𝐶)∇𝑠_{𝑚−2𝑖=1}𝑏_𝑖∫^𝜉𝑖₀(𝑇−𝑠)𝜑⁻¹(∫^𝑠₀𝑎(𝑟)𝑓(𝑢(𝑟))∇𝑟+𝐶)∇𝑠1−∑^𝑚−2_𝑖=1𝑏_𝑖≤𝑎𝑀_𝜂(^𝜂₀(𝑇−𝑠)𝜑⁻¹(^𝑠₀∑𝑎(𝑟)∇𝑟+𝐶)∇𝑠+^{𝑚−2𝑖=1}𝑏_𝑖∫^𝜉𝑖₀(𝑇−𝑠)𝜑⁻¹(∫^𝑠₀𝑎(𝑟)∇𝑟+𝐶)∇𝑠1−∑^𝑚−2_𝑖=1𝑏_𝑖)=𝑎. (3.12) As a result, all the conditions of Theorem 2.9 are verified. This completes the proof.

Theorem 3.3. Suppose that conditions (𝐻₁) and (𝐻₂) hold. Let 0<𝑎<𝑏<𝑐,𝑀𝑏<𝑚𝑐 and assume that the following conditions are satisfied:
(𝐶₁)𝑓(𝑢)<𝜑(𝑎/𝑀), 𝑢∈[0,𝑎];(𝐶₂) there exists a number 𝑑>𝑐 such that 𝑓(𝑢)<𝜑(𝑑/𝑀), 𝑢∈[0,𝑑];(𝐶₃)𝜑(𝑏/𝑚)<𝑓(𝑢)<𝜑(𝑐/𝑀), 𝑢∈[𝑏,𝑐].Then, the BVP (1.6) has at least three positive solutions 𝑢₁,𝑢₂, and 𝑢₃ such that 𝑏<𝑢₁(𝑡)<𝑐,‖𝑢₂‖<𝑎,‖𝑢₃‖>𝑎. (3.13) Where for real number 𝑏, 𝜙_𝑏∶[0,𝑇]_𝐓→[0,+∞) is continuous, 𝜙_𝑏(𝑡)=𝑏, for 𝑡∈[0,𝑇]_𝐓.

Proof. We first show that 𝐴(𝑃_𝑎)⊆𝑃_𝑎⊂𝑃_𝑎 if condition (𝐶₁) holds. If 𝑢∈𝑃_𝑎, then 0≤𝑢≤‖𝑢‖≤𝑎, which implies 𝑓(𝑢)<𝜑(𝑎/𝑀). We have ≤‖Φ𝑢‖=(Φ𝑢)(𝑇)_𝑇0(𝑇−𝑠)𝜑⁻¹(^𝑠₀+∑𝑎(𝑟)𝑓(𝑢(𝑟))∇𝑟+𝐶)∇𝑠_{𝑚−2𝑖=1}𝑏_𝑖∫^𝜉𝑖₀(𝑇−𝑠)𝜑⁻¹(∫^𝑠₀𝑎(𝑟)𝑓(𝑢(𝑟))∇𝑟+𝐶)∇𝑠1−∑^𝑚−2_𝑖=1𝑏_𝑖<𝑎𝑀(^𝑇₀(𝑇−𝑠)𝜑⁻¹(^𝑠₀∑𝑎(𝑟)∇𝑟+𝐶)∇𝑠+^{𝑚−2𝑖=1}𝑏_𝑖∫^𝜉𝑖₀(𝑇−𝑠)𝜑⁻¹(∫^𝑠₀𝑎(𝑟)∇𝑟+𝐶)∇𝑠1−∑^𝑚−2_𝑖=1𝑏_𝑖)=𝑎. (3.14) This implies that Φ(𝑃_𝑎)⊆𝑃_𝑎⊂𝑃_𝑎.
Next, condition (𝐶₂) indicates that there exists 𝑑>𝑐 such that Φ(𝑃_𝑑)⊂𝑃_𝑑. Now we let 𝐴=𝑃_𝑑,𝐴₁=[𝜑_𝑏,𝜑_𝑐],𝑈₁=int(𝐴₁),𝐴₂=𝑃_𝑎,𝑈₂=𝑃_𝑎, (3.15) where int(𝐴₁) is the interior of 𝐴₁. Then we have Φ(𝐴)⊂𝐴,Φ(𝐴₂)⊂𝐴₂. Moreover, Φ(𝑃_𝑎)⊆𝑃_𝑎⊂𝑃_𝑎 means Φ(𝐴₂)⊆𝑈₂⊂𝐴₂. Thus Φ has no fixed point in (𝐴₂⧵𝑈₂).
To show Φ(𝐴₁)⊂𝐴₁ and Φ has no fixed point in (𝐴₁⧵𝑈₁), set 𝑢∈𝐴₁, following the definition of 𝜑_𝑏, we can know 𝑏≤𝑢(𝑡)≤𝑐, for 𝑡∈[0,𝑇]_𝐓. Condition (𝐶₃) then gives rise to 𝜑(𝑏/𝑚)<𝑓(𝑢)<𝜑(𝑐/𝑀), which in turn produces =∑(Φ𝑢)(𝑡)≥(Φ𝑢)(0)_{𝑚−2𝑖=1}𝑏_𝑖∫^𝜉𝑖₀(𝑇−𝑠)𝜑⁻¹(∫^𝑠₀𝑎(𝑟)𝑓(𝑢(𝑟))∇𝑟+𝐶)∇𝑠1−∑^𝑚−2_𝑖=1𝑏_𝑖>𝑏𝑚∑^𝑚−2_𝑖=1𝑏_𝑖∫^𝜉𝑖₀(𝑇−𝑠)𝜑⁻¹(∫^𝑠₀𝑎(𝑟)∇𝑟+𝐶)∇𝑠1−∑^𝑚−2_𝑖=1𝑏_𝑖≤=𝑏,(Φ𝑢)(𝑡)≤(Φ𝑢)(𝑇)2_𝑇0(𝑇−𝑠)𝜑⁻¹(^𝑠₀+∑𝑎(𝑟)𝑓(𝑢(𝑟))∇𝑟+𝐶)∇𝑠_{𝑚−2𝑖=1}𝑏_𝑖∫^𝜉𝑖₀(𝑇−𝑠)𝜑⁻¹(∫^𝑠₀𝑎(𝑟)𝑓(𝑢(𝑟))∇𝑟+𝐶)∇𝑠1−∑^𝑚−2_𝑖=1𝑏_𝑖<𝑐𝑀(^𝑇₀(𝑇−𝑠)𝜑⁻¹(^𝑠₀∑𝑎(𝑟)∇𝑟+𝐶)∇𝑠+^{𝑚−2𝑖=1}𝑏_𝑖∫^𝜉𝑖₀(𝑇−𝑠)𝜑⁻¹(∫^𝑠₀𝑎(𝑟)∇𝑟+𝐶)∇𝑠1−∑^𝑚−2_𝑖=1𝑏_𝑖)=𝑐. (3.17) Combining the above two inequalities one achieves 𝜙_𝑏(𝑡)=𝑏<(Φ𝑢)(𝑡)<𝑐=𝜙_𝑐(𝑡), for 𝑡∈[0,𝑇]_𝐓. That is, Φ𝑢∈𝑈₁. So Φ(𝐴₁)⊆𝑈₁⊂𝐴₁ and Φ has no fixed point in (𝐴₁⧵𝑈₁). Therefore, all conditions of Theorem 2.10 are fulfilled, and the BVP (1.6) has at least three positive solutions 𝑢₁,𝑢₂, and 𝑢₃ such that 𝑏<𝑢₁(𝑡)<𝑐,‖𝑢₂‖<𝑎,‖𝑢₃‖>𝑎. (3.18)

4. Some Examples

In the 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 third-order three-point boundary value problem: (𝜑(𝑢^′′))^′1+𝑎(𝑡)𝑓(𝑢)=0,0<𝑡<1,𝑢(0)=31𝑢(2),𝑢^′(1)=0,𝜑(𝑢^′′1(0))=4𝜑(𝑢^′′(12)), (4.1) where 𝜑(𝑥)=𝑥,𝑎(𝑡)≡1,𝑏₁=1/3,𝑐₁=1/4,𝜉₁=1/2.

We choose 𝜂=1/2, by computing we can know 𝑚_𝜂=11/24,𝑀_𝑙=351/256,𝑀_𝜂=33/48. Let 𝑎=100,𝑏=245,𝑐=770,𝑙=7/8, then 𝑎<𝜂𝑏<𝑏<𝑙𝑐. Obviously, 𝑀_𝑙𝑏<𝑚𝑐. We define a nonlinearity 𝑓 as follows: 𝑓(𝑢)=140,𝑢∈[0,200],140+410545(𝑢−200),𝑢∈[200,245],550,𝑢∈[245,320],550+560(𝑢−320),𝑢∈[320,+∞). (4.2) Then, by the definition of 𝑓, we have

(i)𝑓(𝑢)≤𝜑(𝑐/𝑀_𝑙)≈557.2,𝑢∈[0,880];(ii)𝑓(𝑢)>𝜑(𝑏/𝑚_𝜂)≈534.5,𝑢∈[245,320];(iii)𝑓(𝑢)<𝜑(𝑎/𝑀_𝜂)≈145.4,𝑢∈[0,200].

By Theorem 3.2, BVP (4.1) has at least three positive solutions.

Example 4.2. Consider the following third-order three-point boundary value problem: (𝜑(𝑢^′′))^′1+𝑎(𝑡)𝑓(𝑢)=0,0<𝑡<1,𝑢(0)=31𝑢(2),𝑢^′(1)=0,𝜑(𝑢^′′1(0))=4𝜑(𝑢^′′(12)), (4.3) where 𝜑(𝑥)=𝑥,𝑎(𝑡)≡1,𝑏₁=1/3,𝑐₁=1/4,𝜉₁=1/2,𝜂=1/2.

By computing, we can know 𝑚=11/48,𝑀=83/48. Let 𝑎=7,𝑏=12,𝑐=336,𝑙=7/8, then 𝑎<𝑏<𝑐. Obviously, 𝑀𝑏<𝑚𝑐. We define a nonlinearity 𝑓 as follows: 𝑓(𝑢)=3,𝑢∈[0,7],3+9725(𝑢−7)²,𝑢∈[7,12],100,𝑢∈[12,336],100+11001764(𝑢−336),𝑢∈[336,+∞). (4.4) Then, by the definition of 𝑓, we have

(i)𝑓(𝑢)<𝜑(𝑎/𝑀)≈4.2,𝑢∈[0,7];(ii)and there exists 𝑑=2100>𝑐 such that 𝑓(𝑢)≤𝜑(𝑑/𝑀)≈1214.4,𝑢∈[0,2100];(iii)𝜑(𝑏/𝑚)≈52.4<𝑓(𝑢)<𝜑(𝑐/𝑀)≈194.3,𝑢∈[12,336].

By Theorem 3.3, BVP (4.3) has at least three positive solutions.

Remark 4.3. Consider following nonlinear m-point boundary value problem: (𝜑(𝑢^Δ∇))^∇+𝑎(𝑡)𝑓(𝑢(𝑡))=0,0<𝑡<𝑇,𝑢(0)=^𝑚−2_𝑖=1𝑏_𝑖𝑢(𝜉_𝑖𝑢),_Δ(𝑇)=0,𝜑(𝑢^Δ∇(0))=^𝑚−2_𝑖=1𝑐_𝑖𝜑(𝑢^Δ∇(𝜉_𝑖)), (4.5) where 𝑢𝜑(𝑢)=³𝑢,𝑢≤0,₂,𝑢>0, (4.6)𝑓 and 𝑎 satisfy the conditions (𝐻₁) and (𝐻₂). It is clear that 𝜑∶𝑅→𝑅 is an increasing homeomorphism and homomorphism and 𝜑(0)=0. Because 𝑝-Laplacian operators are odd, they do not apply to our example. Hence we generalize boundary value problem with 𝑝-Laplacian operator, and the results [8–11, 13–15] do not apply to the example.

Remark 4.4. In a similar way, we can get the corresponding results for the following boundary value problem: (𝜑(𝑢^Δ∇))^∇+𝑎(𝑡)𝑓(𝑢(𝑡))=0,𝑡∈(0,𝑇)_𝐓,𝑢(𝑇)=^𝑚−2_𝑖=1𝑎_𝑖𝑢(𝜉_𝑖),𝑢^Δ(0)=0,𝜑(𝑢^Δ∇(0))=^𝑚−2_𝑖=1𝑐_𝑖𝜑(𝑢^Δ∇(𝜉_𝑖)). (4.7)

Acknowledgment

The project was supported by the National Natural Science Foundation of China (10471075).