Higher-Order Dynamic Delay Differential Equations on Time Scales

Abstract

We study the existence of positive solutions for the nonlinear four-point singular boundary value problem with higher-order 𝑝-Laplacian dynamic delay differential equations on time scales, subject to some boundary conditions. By using the fixed-point index theory, the existence of positive solution and many positive solutions for nonlinear four-point singular boundary value problem with 𝑝-Laplacian operator are obtained.

1. Introduction

The study of dynamic equations on time scales goes back to its founder Stefan Hilger [1] and is a new area of still fairly theoretical exploration in mathematics. Boundary value problems for delay differential equations arise in a variety of areas of applied mathematics, physics, and variational problems of control theory (see [2, 3]). In recent years, many authors have begun to pay attention to the study of boundary value problems or with 𝑝-Laplacian equations or with 𝑝-Laplacian dynamic equations on time scales (see [4–19] and the references therein).

In [7], Sun and Li considered the existence of positive solution of the following dynamic equations on time scales: 𝑢𝛽𝑢^Δ∇((0𝑡))+−𝑎𝛾𝑢(𝑡_Δ)𝑓(𝑡,𝑢(𝑡))=0,𝑡∈(0,𝑇),(0)=0,𝛼𝑢(𝜂)=𝑢(𝑇),(1.1) where 𝛽,𝛾≥0, 𝛽+𝛾>0, 𝜂∈(0,𝜌(𝑇)), 0<𝛼<𝑇/𝜂. They obtained the existence of single and multiple positive solutions of the problem (1.1) by using fixed point theorem and Leggett-Williams fixed point theorem, respectively.

In [10], Avery and Anderson discussed the following dynamic equation on time scales: 𝑢^Δ∇(𝑡𝑢)+(0)𝑎=(𝑡)0𝑓,(𝑢(𝑡𝛼)𝑢)=(𝜂)0=,𝑢𝑡(∈𝑇)(.0,𝑇),(1.2) He obtained some results for the existence of one positive solution of the problem (1.2) based on the limits 𝑓₀=lim_𝑢→0⁺(𝑓(𝑢)/𝑢) and 𝑓_∞=lim_𝑢→∞(𝑓(𝑢)/𝑢).

In [11], Wang et al. discussed the following dynamic equation by using Avery-Peterson fixed theorem (see [10]): 𝜙_𝑝𝑢^^+𝑢𝑢𝑞((𝑡𝑡())𝑡)==𝑓𝜉𝜉((𝑡𝑡𝑡,))𝑢,,(𝑡)−−,𝑢11(≤≤𝑡−𝑡𝑡≤≤1)00,,𝑢,^𝑢𝑢(_𝑡()1)==00,,𝑡∈(0,1),((11..34))(1)=0.(1.5) They obtained some results for the existence, three positive solutions of the problem (1.3), (1.4) and (1.3), (1.5), respectively.

However, there are not many concerning the 𝑝-Laplacian problems on time scales. Especially, for the singular multi point boundary value problems for higher-order 𝑝-Laplacian dynamic delay differential equations on time scales, with the author’s acknowledg, no one has studied the existence of positive solutions in this case.

Recently, in [16], we study the existence of positive solutions for the following nonlinear two-point singular boundary value problem with 𝑝-Laplacian operator 𝛼𝜙_𝑝𝜙_𝑝𝑢^^+𝑎(𝑡)𝑓(𝑢(𝑡))=0,0<𝑡<1,(𝑢(0))−𝛽𝜙_𝑝𝑢^(0)=0,𝛾𝜙_𝑝(𝑢(1))+𝛿𝜙_𝑝𝑢^(1)=0,(1.6) by using the fixed point theorem of cone expansion and compression of norm type, the existence of positive solution and infinitely many positive solutions for nonlinear singular boundary value problem (1.6) with 𝑝-Laplacian operator are obtained.

Now, motivated by the results mentioned above, in this paper, we study the existence of positive solutions for the following nonlinear four-point singular boundary value problem with higher-order 𝑝-Laplacian dynamic delay differential equations operator on time scales (SBVP): 𝜙^𝑝𝑢^Δ𝑛−1(𝑡)^∇+𝑔(𝑡)𝑓𝑢(𝑡),𝑢(𝑡−𝜏),𝑢^Δ(𝑡),…,𝑢^Δ𝑛−2(𝑡)=0,0<𝑡<𝑇,(1.7)𝑢𝑢(𝑡)=𝜁(𝑡),−𝜏≤𝑡≤0,_Δ𝑖𝛼𝜙_𝑝(0)=0,1≤𝑖≤𝑛−3,𝛾𝜙_𝑝𝑢^Δ𝑛−2(0)−𝛽𝜙_𝑝𝑢^Δ𝑛−1(𝜉)=0,𝑢^Δ𝑛−2(𝑇)+𝛿𝜙_𝑝𝑢^Δ𝑛−1(𝜂)=0,𝑛≥3,(1.8) where 𝜙_𝑝(𝑠) is 𝑝-Laplacian operator, that is, 𝜙_𝑝(𝑠)=|𝑠|^𝑝−2𝑠, 𝑝>1, 𝜙_𝑞=𝜙^𝑝−1, 1/𝑝+1/𝑞=1; 𝜉,𝜂∈(0,𝑇), 𝜏∈[0,𝑇] is prescribed and 𝜉<𝜂, 𝑔∶(0,𝑇)→[0,∞), 𝛼>0, 𝛽≥0, 𝛾>0, 𝛿≥0.

In this paper, by constructing one integral equation which is equivalent to the problem (1.7), (1.8), we research the existence of positive solutions for nonlinear singular boundary value problem (1.7), (1.8) when 𝑔 and 𝑓 satisfy some suitable conditions.

Our main tool of this paper is the following fixed point index theory.

Theorem 1.1 (see [18]). Suppose 𝐸 is a real Banach space, 𝐾⊂𝐸 is a cone, let Ω_𝑟={𝑢∈𝐾∶‖𝑢‖≤𝑟}. Let operator 𝑇∶Ω_𝑟→𝐾 be completely continuous and satisfy 𝑇𝑥≠𝑥, for all 𝑥∈𝜕Ω_𝑟. Then(i)if ‖𝑇𝑥‖≤‖𝑥‖, for all 𝑥∈𝜕Ω_𝑟, then 𝑖(𝑇,Ω_𝑟,𝐾)=1; (ii)if ‖𝑇𝑥‖≥‖𝑥‖, for all 𝑥∈𝜕Ω_𝑟, then 𝑖(𝑇,Ω_𝑟,𝐾)=0.

This paper is organized as follows. In Section 2, we present some preliminaries and lemmas that will be used to prove our main results. In Section 3, we study the existence of at least two solutions of the systems (1.7), (1.8). In Section 4, we give an examples as the application.

2. Preliminaries and Lemmas

A time scale 𝐓 is an arbitrary nonempty closed subset of real numbers 𝑅⁺. In [1, 14, 20], we can find some basic definitions about time scale. The operators 𝜎 and 𝜌 from 𝐓 to 𝐓: 𝜎(𝑡)=inf{𝜏∈𝐓∣𝜏>𝑡}∈𝐓,𝜌(𝑡)=sup{𝜏∈𝐓∣𝜏<𝑡}∈𝐓(2.1) are called the forward jump operator and the backward jump operator, respectively.

If 𝐓=𝑅, then 𝑥^Δ(𝑡)=𝑥^∇(𝑡)=𝑥′(𝑡). If 𝐓=𝑍, then 𝑥^Δ(𝑡)=𝑥(𝑡+1)−𝑥(𝑡) is the forward difference operator, while 𝑥^∇(𝑡)=𝑥(𝑡)−𝑥(𝑡−1) is the backward difference operator.

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. It is well known that 𝑓 is 𝑙𝑑-continuous.

If 𝐹^∇(𝑡)=𝑓(𝑡), then we define the nabla integral by ^𝑏𝑎𝑓(𝑡)∇𝑡=𝐹(𝑏)−𝐹(𝑎).(2.2) If 𝐹^Δ(𝑡)=𝑓(𝑡), then we define the delta integral by^𝑏𝑎𝑓(𝑡)Δ𝑡=𝐹(𝑏)−𝐹(𝑎).(2.3)

In the rest of this paper, 𝐓 is closed subset of 𝑅 with 0∈𝐓_𝑘, 𝑇∈𝐓^𝑘. And let𝐵=𝑢∈𝐶[−𝜏,0]∩𝐶^{𝑛−2𝑙𝑑}[0,𝑇]∶𝑢^Δ𝑖(0)=0,0≤𝑖≤𝑛−3.(2.4)

Here,𝐶^{𝑛−2𝑙𝑑}[0,𝑇]={𝑢∶[0,𝑇]→𝑅∣𝑢(𝑡)isleft−dense𝑛−2ordercontinuouslydifferentiable}.(2.5) Then 𝐵 is a Banach space with the norm ‖𝑢‖=max_{𝑡∈[0,𝑇]}|𝑢^Δ𝑛−2(𝑡)|. And let𝐾=𝑢∈𝐵∶𝑢^Δ𝑛−2(𝑡)≥0,𝑢^Δ𝑛−2(𝑡)isconcavefunction,𝑡∈[0,𝑇].(2.6)

Obviously, 𝐾 is a cone in 𝐵. Set 𝐾_𝑟={𝑢∈𝐾∶‖𝑢‖≤𝑟}.

Definition 2.1. 𝑢(𝑡) is called a solution of SBVP (1.7) and (1.8) if it satisfies the following:(1)𝑢∈𝐶[−𝜏,0]∩𝐶^{𝑛−1𝑙𝑑}(0,𝑇);(2)𝑢(𝑡)>0 for all 𝑡∈(0,𝑇) and satisfy conditions (1.8);(3)(𝜙_𝑝(𝑢^Δ𝑛−1(𝑡)))^∇=−𝑔(𝑡)𝑓(𝑢(𝑡),𝑢(𝑡−𝜏),𝑢^Δ(𝑡),…,𝑢^Δ𝑛−2(𝑡)) hold for 𝑡∈(0,𝑇).
In the rest of the paper, we also make the following assumptions:
(𝐻₁)𝑓∈𝐶_𝑙𝑑([0,+∞)^𝑛,[0,+∞));(𝐻₂)𝑔(𝑡)∈𝐶_𝑙𝑑((0,𝑇),[0,+∞)) and there exists 𝑡₀∈(0,𝑇), such that𝑔𝑡₀>0,0<^𝑇0𝑔(𝑠)∇𝑠<+∞;(2.7)(𝐻₃)𝜁(𝑡)∈𝐶([−𝜏,0], 𝜁(𝑡)>0 on [−𝜏,0) and 𝜁(0)=0.It is easy to check that condition (𝐻₂) implies that 0<^𝑇0𝜙_𝑞^𝑠0𝑔𝑠₁∇𝑠₁Δ𝑠<+∞.(2.8)
We can easily get the following lemmas.

Lemma 2.2. Suppose condition (𝐻₂) holds. Then there exists a constant 𝜃∈(0,1/2) satisfing 0<^{𝜃𝑇−𝜃}𝑔(𝑡)∇𝑡<∞.(2.9) Furthermore, the function 𝐴(𝑡)=^𝑡𝜃𝜙_𝑞^𝑡𝑠𝑔𝑠₁∇𝑠₁Δ𝑠+^{𝑡𝑇−𝜃}𝜙_𝑞^𝑠𝑡𝑔𝑠₁∇𝑠₁∇𝑠,𝑡∈[𝜃,𝑇−𝜃],(2.10) is positive continuous functions on [𝜃,𝑇−𝜃]; therefore, 𝐴(𝑡) has minimum on [𝜃,𝑇−𝜃]. Hence we suppose, that there exists 𝐿>0 such that 𝐴(𝑡)≥𝐿, 𝑡∈[𝜃,𝑇−𝜃].

Proof. At first, it is easily seen that 𝐴(𝑡) is continuous on [𝜃,𝑇−𝜃]. Next, let 𝐴₁(𝑡)=^𝑡𝜃𝜙_𝑞^𝑡𝑠𝑔𝑠₁∇𝑠₁Δ𝑠,𝐴²(𝑡)=^{𝑡𝑇−𝜃}𝜙_𝑞^𝑠𝑡𝑔𝑠₁∇𝑠₁Δ𝑠.(2.11) Then, from condition (𝐻₂), we have that the function 𝐴₁(𝑡) is strictly monotone nondecreasing on [𝜃,𝑇−𝜃] and 𝐴₁(𝜃)=0, the function 𝐴₂(𝑡) is strictly monotone nonincreasing on [𝜃,𝑇−𝜃] and 𝐴₂(𝑇−𝜃)=0, which implies 𝐿=min_{𝑡∈[𝜃,𝑇−𝜃]}𝐴(𝑡)>0. The proof is complete.

Lemma 2.3. Let 𝑢∈𝐾 and 𝜃 of Lemma 2.2, then 𝑢(𝑡)≥𝜃‖𝑢‖,𝑡∈[𝜃,𝑇−𝜃].(2.12) The proof of the above lemma is similar to the proof in [17, Lemma 2.2], so we omit it.

Lemma 2.4. Suppose that conditions (𝐻₁),(𝐻₂),(𝐻₃) hold, 𝑢(𝑡)∈𝐵∩𝐶^{𝑛−1𝑙𝑑}(0,1) is a solution of the following boundary value problems: 𝜙^𝑝𝑢^Δ𝑛−1(𝑡)^∇+𝑔(𝑡)𝑓𝑢(𝑡),𝑢(𝑡−𝜏)+ℎ(𝑡−𝜏),𝑢^Δ(𝑡),…,𝑢^Δ𝑛−2(𝑡)=0,0<𝑡<𝑇,(2.13)𝑢𝑢(𝑡)=0,−𝜏≤𝑡≤0,_Δ𝑖𝛼𝜙_𝑝(0)=0,1≤𝑖≤𝑛−3,𝛾𝜙_𝑝𝑢^Δ𝑛−2(0)−𝛽𝜙_𝑝𝑢^Δ𝑛−1(𝜉)=0,𝑢^Δ𝑛−2(𝑇)+𝛿𝜙_𝑝𝑢^Δ𝑛−1(𝜂)=0,𝑛≥3,(2.14) where ℎ(𝑡)=𝜁0,(𝑡),−0𝜏≤≤𝑡≤𝑡≤𝑇.0,(2.15) Then, 𝑢(𝑡)=𝑢(𝑡)+ℎ(𝑡), −𝜏≤𝑡≤𝑇 is a positive solution to the SBVP (1.7) and (1.8).

Proof. It is easy to check that 𝑢(𝑡) satisfies (1.7) and (1.8).

So in the rest of the sections of this paper, we focus on SBVP (2.13) and (2.14).

Lemma 2.5. Suppose that conditions (𝐻₁),(𝐻₂),(𝐻₃) hold, 𝑢(𝑡)∈𝐵∩𝐶^{𝑛−1𝑙𝑑}(0,1) is a solution of boundary value problems (2.13), (2.14) if and only if 𝑢(𝑡)∈𝐵 is a solution of the following integral equation: 𝑢(𝑡)=⎧⎪⎨⎪⎩𝜁(𝑡),−𝜏≤𝑡≤0,^𝑡0^𝑠1₀⋯^𝑠𝑛−3₀𝑤𝑠_𝑛−2Δ𝑠_𝑛−2Δ𝑠_𝑛−3⋯Δ𝑠₁,0≤𝑡≤𝑇,(2.16) where 𝑤(𝑡)=⎧⎪⎪⎪⎪⎪⎨⎪⎪⎪⎪⎪⎩𝜙^𝑞𝛼𝛽^𝜎𝜉+𝑔(𝑠)𝑓𝑢(𝑠),𝑢(𝑠−𝜏)+ℎ(𝑠−𝜏),𝑢^Δ(𝑠),…,𝑢^Δ𝑛−2(𝑠)∇𝑠_𝑡0𝜙_𝑞^𝜎𝑠𝜙𝑔(𝑟)𝑓𝑢(𝑟),𝑢(𝑟−𝜏)+ℎ(𝑟−𝜏),𝑢^Δ(𝑟),…,𝑢^Δ𝑛−2(𝑟)0≤∇𝑟𝑡≤Δ𝜎𝑠,_𝑞,𝛾𝛿^𝜂𝜎+𝑔(𝑠)𝑓𝑢(𝑠),𝑢(𝑠−𝜏)+ℎ(𝑠−𝜏),𝑢^Δ(𝑠),…,𝑢^Δ𝑛−2(𝑠)∇𝑠_𝑇𝑡𝜙_𝑞^𝑠𝜎𝑔(𝑟)𝑓𝑢(𝑟),𝑢(𝑟−𝜏)+ℎ(𝑟−𝜏),𝑢^Δ(𝑟),…,𝑢^Δ𝑛−2(𝑟)𝜎≤∇𝑟𝑡≤Δ𝑇𝑠.,(2.17)

Proof. Necessity. Obviously, for 𝑡∈(−𝜏,0), we have 𝑢(𝑡)=𝜁(𝑡). If 𝑡∈(0,1), by the equation of the boundary condition, we have 𝑢^Δ𝑛−1(𝜉)≥0, 𝑢^Δ𝑛−1(𝜂)≤0, then there exists a constant 𝜎∈[𝜉,𝜂]⊂(0,𝑇) such that 𝑢^Δ𝑛−1(𝜎)=0.
Firstly, by integrating the equation of the problems (2.13) on (𝜎,𝑇), we have𝜙_𝑝𝑢^Δ𝑛−1(𝑡)=𝜙_𝑝𝑢^Δ𝑛−1(𝜎)−^𝑡𝜎𝑔(𝑠)𝑓𝑢(𝑠),𝑢(𝑠−𝜏)+ℎ(𝑠−𝜏),𝑢^Δ(𝑠),…,𝑢^Δ𝑛−2(𝑠)∇𝑠,(2.18) then 𝑢^Δ𝑛−1(𝑡)=−𝜙_𝑞^𝑡𝜎𝑔(𝑠)𝑓𝑢(𝑠),𝑢(𝑠−𝜏)+ℎ(𝑠−𝜏),𝑢^Δ(𝑠),…,𝑢^Δ𝑛−2(𝑠)∇𝑠,(2.19) thus 𝑢^Δ𝑛−2(𝑡)=𝑢^Δ𝑛−2(𝜎)−^𝑡𝜎𝜙_𝑞^𝑠𝜎𝑔(𝑟)𝑓𝑢(𝑟),𝑢(𝑟−𝜏)+ℎ(𝑟−𝜏),𝑢^Δ(𝑟),…,𝑢^Δ𝑛−2(𝑟)∇𝑟Δ𝑠.(2.20)
By 𝑢^Δ𝑛−1(𝜎)=0 and condition (2.18), let 𝑡=𝜂 on (2.18), we have𝜙_𝑝𝑢^Δ𝑛−1(𝜂)=−^𝜂𝜎𝑔(𝑠)𝑓𝑢(𝑠),𝑢(𝑠−𝜏)+ℎ(𝑠−𝜏),𝑢^Δ(𝑠),…,𝑢^Δ𝑛−2(𝑠)∇𝑠.(2.21) By the equation of the boundary condition (2.14), we have 𝜙_𝑝𝑢^Δ𝑛−2(𝑇)=−𝛿𝛾𝜙^𝑝𝑢^Δ𝑛−1(𝜂),(2.22) then 𝑢^Δ𝑛−2(𝑇)=𝜙_𝑞𝛾𝛿^𝜂𝜎𝑔(𝑠)𝑓𝑢(𝑠),𝑢(𝑠−𝜏)+ℎ(𝑠−𝜏),𝑢^Δ(𝑠),…,𝑢^Δ𝑛−2(𝑠)∇𝑠.(2.23) Then, by (2.20) and leting 𝑡=𝑇 on (2.20), we have 𝑢^Δ𝑛−2(𝜎)=𝜙_𝑞𝛾𝛿^𝜂𝜎+𝑔(𝑠)𝑓𝑢(𝑠),𝑢(𝑠−𝜏)+ℎ(𝑠−𝜏),𝑢^Δ(𝑠),…,𝑢^Δ𝑛−2(𝑠)∇𝑠_𝑇𝜎𝜙_𝑞^𝑠𝜎𝑔(𝑟)𝑓𝑢(𝑟),𝑢(𝑟−𝜏)+ℎ(𝑟−𝜏),𝑢^Δ(𝑟),…,𝑢^Δ𝑛−2(𝑟)∇𝑟Δ𝑠.(2.24) Then 𝑢^Δ𝑛−2(𝑡)=𝜙_𝑞𝛾𝛿^𝜂𝜎+𝑔(𝑠)𝑓𝑢(𝑠),𝑢(𝑠−𝜏)+ℎ(𝑠−𝜏),𝑢^Δ(𝑠),…,𝑢^Δ𝑛−2(𝑠)∇𝑠_𝑇𝑡𝜙_𝑞^𝑠𝜎𝑔(𝑟)𝑓𝑢(𝑟),𝑢(𝑟−𝜏)+ℎ(𝑟−𝜏),𝑢^Δ(𝑟),…,𝑢^Δ𝑛−2(𝑟)∇𝑟Δ𝑠.(2.25) Then, by integrating (2.25) for 𝑛−2 times on (0,𝑇), we have 𝑢(𝑡)=^𝑡0^𝑠1₀⋯^𝑠𝑛−3₀𝜙_𝑞𝜎𝛾^𝜂𝛿𝑔(𝑠)𝑓𝑢+(𝑠ℎ)(,𝑠𝑢−(𝑠𝜏−),𝜏𝑢)_Δ(𝑠),…,𝑢^Δ𝑛−2(𝑠)∇𝑠Δ𝑠_𝑠𝑛−2⋯Δ𝑠₂Δ𝑠₁+^𝑡0^𝑠1₀⋯^𝑠𝑛−3₀^𝑇𝑠_𝑛−2𝜙_𝑞^𝑠𝜎×𝑔∇(𝑟𝑟)𝑓Δ𝑠𝑢(𝑟Δ)𝑠,𝑢(_𝑠𝑛−2𝑟−𝜏)+ℎ(𝑟−𝜏),𝑢^Δ(𝑟),…,𝑢^Δ𝑛−2(𝑟)⋯Δ𝑠₂Δ𝑠₁.(2.26) Similarly, for 𝑡∈(0,𝜎), by integrating the equation of problems (2.13) on (0,𝜎), we have 𝑢(𝑡)=^𝑡0^𝑠1₀⋯^𝑠𝑛−3₀𝜙_𝑞𝛼𝛽^𝜎𝜉𝑔(𝑠)𝑓+𝑢ℎ(𝑠(𝑠),−𝑢𝜏(𝑠),−𝑢_Δ𝜏)(𝑠),…,𝑢^Δ𝑛−2(𝑠)∇𝑠Δ𝑠_𝑠𝑛−2⋯Δ𝑠₂Δ𝑠₁+^𝑡0^𝑠1₀⋯^𝑠𝑛−3₀^𝑠𝑛−2₀𝜙_𝑞^𝜎𝑠×𝑔∇(𝑟𝑟)𝑓Δ𝑠𝑢(𝑟Δ)𝑠,𝑢(_𝑠𝑛−2𝑟−𝜏)+ℎ(𝑟−𝜏),𝑢^Δ(𝑟),…,𝑢^Δ𝑛−2(𝑟)⋯Δ𝑠₂Δ𝑠₁.(2.27) Therefore, for any 𝑡∈[0,𝑇], 𝑢(𝑡) can be expressed as equation 𝑢(𝑡)=⎧⎪⎨⎪⎩𝜁(𝑡),−𝜏≤𝑡≤0,^𝑡0^𝑠1₀⋯^𝑠𝑛−3₀𝑤𝑠_𝑛−2Δ𝑠_𝑛−2Δ𝑠_𝑛−3⋯Δ𝑠₁,0≤𝑡≤𝑇,(2.28) where 𝑤(𝑡) is expressed as (2.17). Then the results of Lemma 2.3 hold.
Sufficiency. Suppose that 𝑢(𝑡)=∫^𝑡0∫^𝑠1₀⋯∫^𝑠𝑛−3₀𝑤(𝑠_𝑛−2)Δ𝑠_𝑛−2Δ𝑠_𝑛−3⋯Δ𝑠₁, 0≤𝑡≤𝑇. Then by (2.17), we have𝑢^Δ𝑛−1(𝑡)=⎧⎪⎪⎪⎨⎪⎪⎪⎩𝜙^𝑞^𝜎𝑡−𝜙_𝑞𝑔(𝑠)𝑓𝑢(𝑠),𝑢(𝑠−𝜏)+ℎ(𝑠−𝜏),𝑢^Δ(𝑠),…,𝑢^Δ𝑛−2(𝑠0)≤∇𝑡𝑠≤𝜎≥,0,^𝑡𝜎𝑔(𝑠)𝑓𝑢(𝑠),𝑢(𝑠−𝜏)+ℎ(𝑠−𝜏),𝑢^Δ(𝑠),…,𝑢^Δ𝑛−2(𝑠)𝜎≤∇𝑠𝑡≤≤𝑇.0,(2.29) So, (𝜙_𝑝(𝑢^Δ𝑛−1))^∇+𝑔(𝑡)𝑓(𝑢(𝑡),𝑢(𝑡−𝜏)+ℎ(𝑡−𝜏),𝑢^Δ(𝑡),…,𝑢^Δ𝑛−2(𝑡))=0, 0<𝑡<𝑇. These imply that (2.13) holds. Furthermore, by letting 𝑡=0 and 𝑡=𝑇 on (2.17) and (2.29), we can obtain the boundary value equations of (2.14). The proof is complete.

Now, we define an operator equation 𝑇 given by (𝑇𝑢)(𝑡)=⎧⎪⎨⎪⎩𝜁(𝑡),−𝜏≤𝑡≤0,^𝑡0^𝑠1₀⋯^𝑠𝑛−3₀𝑤𝑠_𝑛−2Δ𝑠_𝑛−2Δ𝑠_𝑛−3⋯Δ𝑠₁,0≤𝑡≤𝑇,(2.30) where 𝑤(𝑡) is given by (2.17).

From the definition of 𝑇 and the previous discussion, we deduce that, for each 𝑢∈𝐾, 𝑇𝑢∈𝐾. Moreover, we have the following lemmas.

Lemma 2.6. 𝑇∶𝐾→𝐾 is completely continuous.

Proof. Because (𝑇𝑢)^Δ𝑛−1(𝑡)=𝑤^Δ(𝑡)=⎧⎪⎪⎪⎨⎪⎪⎪⎩𝜙^𝑞^𝜎𝑡−𝜙_𝑞𝑔(𝑠)𝑓𝑢(𝑠),𝑢(𝑠−𝜏)+ℎ(𝑠−𝜏),𝑢^Δ(𝑠),…,𝑢^Δ𝑛−2(0𝑠)≤∇𝑡≤𝑠𝜎≥,0,^𝑡𝜎𝑔(𝑠)𝑓𝑢(𝑠),𝑢(𝑠−𝜏)+ℎ(𝑠−𝜏),𝑢^Δ(𝑠),…,𝑢^Δ𝑛−2(𝑠𝜎)≤∇𝑡𝑠≤𝑇≤,0,(2.31) is continuous, decreasing on [0,𝑇] and satisfies (𝑇𝑢)^Δ𝑛−1(𝜎)=0, then, 𝑇𝑢∈𝐾 for each 𝑢∈𝐾 and (𝑇𝑢)^Δ𝑛−2(𝜎)=max_{𝑡∈[0,𝑇]}(𝑇𝑢)^Δ𝑛−2(𝑡). This shows that 𝑇𝐾⊂𝐾. Furthermore, it is easy to check by Arzela-ascoli Theorem that 𝑇∶𝐾→𝐾 is completely continuous.

Lemma 2.7. Suppose that conditions (𝐻₁),(𝐻₂),(𝐻₃) hold, the solution 𝑢(𝑡) of problem (2.13), (2.14) satisfies 𝑢(𝑡)≤ma𝑇x𝑢_{0≤𝑡≤𝑇Δ}||𝑢(𝑡−𝜏)+ℎ(𝑡−𝜏)||≤max_{−𝜏≤𝑡≤0}||𝜁(𝑡)||,(𝑡)≤⋯≤𝑇^𝑛−3𝑢^Δ𝑛−3(𝑡),𝑡∈[0,𝑇],(2.32) and for 𝜃∈(0,𝑇/2) in Lemma 2.2, one has 𝑢^Δ𝑛−3(𝑡)≤𝑇𝜃𝑢^Δ𝑛−2(𝑡),𝑡∈[𝜃,𝑇−𝜃].(2.33)

Proof. Firstly, we can have max_{0≤𝑡≤𝑇}||𝑢(𝑡−𝜏)+ℎ(𝑡−𝜏)||≤=mmaxa_{0≤𝑡≤𝑇}x||𝑢_{−𝜏≤𝑡≤𝑇−𝜏}(𝑡−𝜏)||+max_{0≤𝑡≤𝑇}||ℎ(𝑡−𝜏)||=max_{−𝜏≤𝑡≤0}||𝑢(𝑡)||+max_{−𝜏≤𝑡≤𝑇−𝜏}||ℎ(𝑡)||||𝜁(𝑡)||.(2.34)
Next, if 𝑢(𝑡) is the solution of problem (2.13), (2.14), then 𝑢^Δ𝑛−2(𝑡) is concave function, and 𝑢^Δ𝑖(𝑡)≥0(𝑖=0,1,…,𝑛−2), 𝑡∈[0,𝑇]. Thus, we have𝑢^Δ𝑖(𝑡)=^𝑡0𝑢^Δ𝑖+1(𝑠)Δ𝑠≤𝑡𝑢^Δ𝑖+1(𝑡)≤𝑇𝑢^Δ𝑖+1(𝑡),𝑖=0,1,…,𝑛−4,(2.35) that is, 𝑢(𝑡)≤𝑇𝑢^Δ(𝑡)≤⋯≤𝑇^𝑛−3𝑢^Δ𝑛−3(𝑡), 𝑡∈[0,𝑇].
Finally, by Lemma 2.3, for 𝑡∈[𝜃,𝑇−𝜃], we have 𝑢^Δ𝑛−2(𝑡)≥𝜃‖𝑢^Δ𝑛−2‖. By 𝑢^Δ𝑛−3(𝑡)=∫^𝑡0𝑢^Δ𝑛−2(𝑠)Δ𝑠≤𝑇‖𝑢^Δ𝑛−2‖, we have 𝑢^Δ𝑛−3(𝑡)≤𝑇𝜃𝑢^Δ𝑛−2(𝑡),𝑡∈[𝜃,𝑇−𝜃].(2.36) The proof is complete.

For convenience, we set 𝐻=max_{−𝜏≤𝑡≤0}||𝜁(𝑡)||,𝜃^∗=2𝐿,𝜃^∗=1𝑇𝑚+𝜙∈_𝑞(𝛽𝜃/_∗𝛼)𝜙_𝑞∫^𝑇0𝑔(𝑟)∇𝑟,,∞,𝑀∈0,𝜃_∗,(2.37) where 𝐿 is the constant from Lemma 2.2. By Lemma 2.5, we can also set𝑓₀=lim_𝑢𝑛→0max_{(𝑢1,𝑢2,…,𝑢𝑛)∈ℵ}𝑓𝑢₁,𝑢₂,…,𝑢_𝑛𝑢^𝑛𝑝−1𝑓_∞,=lim_{𝑢𝑛→∞}min_{(𝑢1,𝑢2,…,𝑢𝑛)∈ℵ}𝑓𝑢₁,𝑢₂,…,𝑢_𝑛𝑢^𝑛𝑝−1,(2.38) where ℵ={(𝑢₁,𝑢₂,…,𝑢_𝑛)|0≤𝑢₁≤𝑇𝑢₃⋯≤𝑇^𝑛−3𝑢_𝑛−1≤(𝑇^𝑛−2/𝜃)𝑢_𝑛,𝑢₂≤𝐻}.

3. The Existence of Multiple Positive Solutions

In this section, we also make the following conditions:(𝐴₁)𝑓(𝑢₁,𝑢₂,…,𝑢_𝑛)≥(𝑚𝑟)^𝑝−1, for 𝜃𝑟≤𝑢_𝑛≤𝑟, (𝑢₁,𝑢₂,…,𝑢_𝑛)∈ℵ;(𝐴₂)𝑓(𝑢₁,𝑢₂,…,𝑢_𝑛)≤(𝑀𝑅)^𝑝−1, for 0≤𝑢_𝑛≤𝑅, (𝑢₁,𝑢₂,…,𝑢_𝑛)∈ℵ.

Next, we will discuss the existence of multiple positive solutions.

Theorem 3.1. Suppose that conditions (𝐻₁), (𝐻₂), (𝐻₃), and (𝐴₂) hold. Assume that 𝑓 also satisfies(𝐴₃)𝑓₀=+∞;(𝐴₄)𝑓_∞=+∞.Then, the SBVP (2.13), (2.14) hase at last two solutions 𝑢₁, 𝑢₂ such that 0<‖‖𝑢₁‖‖<𝑅<‖‖𝑢₂‖‖.(3.1)

Proof. For any 𝑢∈𝐾, by Lemma 2.3, we have 𝑢^Δ𝑛−2(𝑡)≥𝜃‖𝑢‖,𝑡∈[𝜃,𝑇−𝜃].(3.2) First, by condition (𝐴₃), for any 𝑁>2/𝜃𝐿, there exists a constant 𝜌_∗∈(0,𝑅) such that 𝑓𝑢₁,𝑢₂,…,𝑢_𝑛≥𝑁𝑢_𝑛^𝑝−1,0<𝑢_𝑛≤𝜌_∗,𝑢_𝑛≠0.(3.3) Set Ω_𝜌∗={𝑢∈𝐾∶‖𝑢‖<𝜌_∗}. For any 𝑢∈𝜕Ω_𝜌∗, by (3.2) we have 𝜌_∗=‖𝑢‖≥𝑢^Δ𝑛−2(𝑡)≥𝜃‖𝑢‖=𝜃𝜌_∗,𝑡∈[𝜃,𝑇−𝜃].(3.4) For any 𝑢∈𝜕Ω_𝜌∗, by (3.3) and Lemmas 2.3–2.6, we will discuss it from three perspectives.(i)If 𝜎∈[𝜃,𝑇−𝜃], we have 2‖𝑇𝑢‖=≥2(𝑇𝑢)^Δ𝑛−2(𝜎)_𝜎0𝜙_𝑞^𝜎𝑠+𝑔(𝑟)𝑓𝑢(𝑟),𝑢(𝑟−𝜏)+ℎ(𝑟−𝜏),𝑢^Δ(𝑟),…,𝑢^Δ𝑛−2(𝑟)∇𝑟Δ𝑠_𝑇𝜎𝜙_𝑞^𝑠𝜎≥𝑔(𝑟)𝑓𝑢(𝑟),𝑢(𝑟−𝜏)+ℎ(𝑟−𝜏),𝑢^Δ(𝑟),…,𝑢^Δ𝑛−2(𝑟)∇𝑟Δ𝑠_𝜎𝜃𝜙_𝑞^𝜎𝑠+𝑔(𝑟)𝑓𝑢(𝑟),𝑢(𝑟−𝜏)+ℎ(𝑟−𝜏),𝑢^Δ(𝑟),…,𝑢^Δ𝑛−2(𝑟)∇𝑟Δ𝑠_{𝜎𝑇−𝜃}𝜙_𝑞^𝑠𝜎≥𝑁𝜃𝐴(𝜎)‖𝑢‖≥𝑔(2𝑟)‖𝑓𝑢‖.𝑢(𝑟),𝑢(𝑟−𝜏)+ℎ(𝑟−𝜏),𝑢^Δ(𝑟),…,𝑢^Δ𝑛−2(𝑟)∇𝑟Δ𝑠(3.5)(ii)If 𝜎∈(𝑇−𝜃,𝑇], we have‖𝑇𝑢‖=≥𝜙(𝑇_𝑞𝑢)^Δ𝑛−2(𝜎)𝛼𝛽^𝜎𝜉+𝑔(𝑠)𝑓𝑢(𝑠),𝑢(𝑠−𝜏)+ℎ(𝑠−𝜏),𝑢^Δ(𝑠),…,𝑢^Δ𝑛−2(𝑠)∇𝑠_𝜎0𝜙_𝑞^𝜎𝑠≥𝑔(𝑟)𝑓𝑢(𝑟),𝑢(𝑟−𝜏)+ℎ(𝑟−𝜏),𝑢^Δ(𝑟),…,𝑢^Δ𝑛−2(𝑟)∇𝑟Δ𝑠_{𝜃𝑇−𝜃}𝜙_𝑞^{𝑠𝑇−𝜃}≥𝑁𝜃𝐴(𝑇−𝜃)‖𝑢𝑔‖(𝑟>)𝑓‖𝑢‖𝑢.(𝑟),𝑢(𝑟−𝜏)+ℎ(𝑟−𝜏),𝑢^Δ(𝑟),…,𝑢^Δ𝑛−2(𝑟)∇𝑟Δ𝑠(3.6)(iii)If 𝜎∈(0,𝜃), we have ‖𝑇𝑢‖=≥(𝜙𝑇_𝑞𝑢)^Δ𝑛−2(𝜎)𝛾𝛿^𝜂𝜎+𝑔(𝑠)𝑓𝑢(𝑠),𝑢(𝑠−𝜏)+ℎ(𝑟−𝜏),𝑢^Δ(𝑠),…,𝑢^Δ𝑛−2(𝑠)∇𝑠_𝑇𝜎𝜙_𝑞^𝑠𝜎≥𝑔(𝑟)𝑓𝑢(𝑟),𝑢(𝑟−𝜏)+ℎ(𝑟−𝜏),𝑢^Δ(𝑟),…,𝑢^Δ𝑛−2(𝑟)∇𝑟Δ𝑠_{𝜃𝑇−𝜃}𝜙_𝑞^𝑠𝜃≥𝑁𝜃𝐴(𝜃)‖𝑢‖𝑔>(𝑟‖)𝑢𝑓‖.𝑢(𝑟),𝑢(𝑟−𝜏)+ℎ(𝑟−𝜏),𝑢^Δ(𝑟),…,𝑢^Δ𝑛−2(𝑟)∇𝑟Δ𝑠(3.7) Therefore, no matter under which condition, we all have ‖𝑇𝑢‖≥‖𝑢‖,∀𝑢∈𝜕Ω_𝜌∗.(3.8) Then, by Theorem 1.1, we have 𝑖𝑇,Ω_𝜌∗,𝐾=0.(3.9)Next, by condition (𝐴₄), for any 𝑁>2/𝜃𝐿, there exists a constant 𝜌₀>0 such that 𝑓𝑢₁,𝑢₂,…,𝑢_𝑛≥𝑁𝑢_𝑛^𝑝−1,𝑢_𝑛>𝜌₀.(3.10) We choose a constant 𝜌^∗>max{𝑅,𝜌₀/𝜃}, obviously 𝜌_∗<𝑅<𝜌^∗. Set Ω_𝜌∗={𝑢∈𝐾∶‖𝑢‖<𝜌^∗}. For any 𝑢∈𝜕Ω_𝜌∗, by Lemma 2.3, we have 𝑢(𝑡)≥𝜃‖𝑢‖=𝜃𝜌^∗>𝜌₀,𝑡∈[𝜃,𝑇−𝜃].(3.11) Then, by (3.10), Lemmas 2.3–2.6 and also similar to the previous proof, we can also have from three perspectives that ‖𝑇𝑢‖≥‖𝑢‖,∀𝑢∈𝜕Ω_𝜌∗.(3.12) Then, by Theorem 1.1, we have 𝑖𝑇,Ω_𝜌∗,𝐾=0.(3.13) Finally, set Ω_𝑅={𝑢∈𝐾∶‖𝑢‖<𝑅}. For any 𝑢∈𝜕Ω_𝑅, we have 𝑢(𝑡)≤‖𝑢‖=𝑅, by (𝐴₂) we know ‖𝑇𝑢‖=≤𝜙(𝑇_𝑞𝑢)^Δ𝑛−2(𝜎)𝛼𝛽^𝜎𝜉+𝑔(𝑠)𝑓𝑢(𝑠),𝑢(𝑠−𝜏)+ℎ(𝑠−𝜏),𝑢^Δ(𝑠),…,𝑢^Δ𝑛−2(𝑠)∇𝑠_𝑇0𝜙_𝑞^𝜎𝑠≤𝑇+𝜙_𝑞𝑔(𝑟)𝑓𝑢(𝑟),𝑢(𝑟−𝜏)+ℎ(𝑟−𝜏),𝑢^Δ(𝑟),…,𝑢^Δ𝑛−2(𝑟)∇𝑟Δ𝑠𝛼𝛽𝑀𝑅𝜙^𝑞^𝑇0𝑔(𝑟)∇𝑟≤𝑅=‖𝑢‖.(3.14) Thus, ‖𝑇𝑢‖≤‖𝑢‖,∀𝑢∈𝜕Ω_𝑅.(3.15) Then, by Theorem 1.1, we have 𝑖𝑇,Ω_𝑅,𝐾=1.(3.16) Therefore, by (3.9), (3.13), (3.16), 𝜌_∗<𝑅<𝜌^∗ we have 𝑖𝑇,Ω_𝑅⧵Ω_𝜌∗,𝐾=1,𝑖𝑇,Ω_𝜌∗⧵Ω_𝑅,𝐾=−1.(3.17) Then 𝑇 has fixed point 𝑢₁∈Ω_𝑅⧵Ω_𝜌∗ and fixed point 𝑢₂∈Ω_𝜌∗⧵Ω_𝑅. Obviously, 𝑢₁,𝑢₂ are all positive solutions of problem (2.13), (2.14) and 𝜌_∗<‖𝑢₁‖<𝑅<‖𝑢₂‖<𝜌^∗. Proof of Theorem 3.1 is complete.

Theorem 3.2. Suppose that conditions (𝐻₁), (𝐻₂), (𝐻₃), (𝐴₁) hold. Assume that 𝑓 also satisfies(𝐴₅)𝑓₀=0;(𝐴₆)𝑓_∞=0.Then, the SBVP (2.13), (2.14) has at last two solutions 𝑢₁,𝑢₂ such that 0<‖𝑢₁‖<𝑟<‖𝑢₂‖.

Proof. First, by 𝑓₀=0, for 𝜖₁∈(0,𝜃_∗), there exists a constant 𝜌_∗∈(0,𝑟) such that 𝑓𝑢₁,𝑢₂,…,𝑢_𝑛≤𝜖₁𝑢_𝑛^𝑝−1,0<𝑢_𝑛≤𝜌_∗.(3.18) Set Ω_𝜌∗={𝑢∈𝐾∶‖𝑢‖<𝜌_∗}, for any 𝑢∈𝜕Ω_𝜌∗, by (3.18), we have ‖𝑇𝑢‖=≤𝜙(𝑇_𝑞𝑢)^Δ𝑛−2(𝜎)𝛼𝛽^𝜎𝜉+𝑔(𝑠)𝑓𝑢(𝑠),𝑢(𝑠−𝜏)+ℎ(𝑠−𝜏),𝑢^Δ(𝑠),…,𝑢^Δ𝑛−2(𝑠)∇𝑠_𝑇0𝜙_𝑞^𝜎𝑠≤𝜙_𝑞𝑔(𝑟)𝑓𝑢(𝑟),𝑢(𝑟−𝜏)+ℎ(𝑟−𝜏),𝑢^Δ(𝑟),…,𝑢^Δ𝑛−2(𝑟)∇𝑟Δ𝑠𝛼𝛽^𝜎𝜉+𝑇𝜙_𝑞𝑔(𝑠)𝑓𝑢(𝑠),𝑢(𝑠−𝜏)+ℎ(𝑠−𝜏),𝑢^Δ(𝑠),…,𝑢^Δ𝑛−2(𝑠)∇𝑠^𝑇0≤𝑇+𝜙_𝑞𝑔(𝑟)𝑓𝑢(𝑟),𝑢(𝑟−𝜏)+ℎ(𝑟−𝜏),𝑢^Δ(𝑟),…,𝑢^Δ𝑛−2(𝑟)∇𝑟𝛼𝛽𝜖¹𝜌_∗𝜙_𝑞^𝑇0𝑔(𝑟)∇𝑟≤𝜌_∗=‖𝑢‖,(3.19) that is, ‖𝑇𝑢‖≤‖𝑢‖,∀𝑢∈𝜕Ω_𝜌∗(3.20) Then, by Theorem 1.1, we have 𝑖𝑇,Ω_𝜌∗,𝐾=1.(3.21)
Next, let 𝑓^∗(𝑥)=max_{0≤𝑢𝑛−1≤𝑥}𝑓(𝑢₁,𝑢₂,…,𝑢_𝑛−1); note that 𝑓^∗(𝑥) is monotone increasing with respect to 𝑥≥0. Then, from 𝑓_∞=0, it is easy to see thatlim_𝑥→∞𝑓^∗(𝑥)𝑥^𝑝−1=0.(3.22) Therefore, for any 𝜖₂∈(0,𝜃_∗), there exists a constant 𝜌^∗>𝑟 such that 𝑓^∗(𝑥)≤𝜖₂𝑥^𝑝−1,𝑥≥𝜌^∗.(3.23) Set Ω_𝜌∗={𝑢∈𝐾∶‖𝑢‖<𝜌^∗}, for any 𝑢∈𝜕Ω_𝜌∗, by (3.23), we have ‖𝑇𝑢‖=≤𝜙(𝑇_𝑞𝑢)^Δ𝑛−2(𝜎)𝛼𝛽^𝜎𝜉+𝑔(𝑠)𝑓𝑢(𝑠),𝑢(𝑠−𝜏)+ℎ(𝑠−𝜏),𝑢^Δ(𝑠),…,𝑢^Δ𝑛−2(𝑠)∇𝑠_𝑇0𝜙_𝑞^𝜎𝑠≤𝜙_𝑞𝑔(𝑟)𝑓𝑢(𝑟),𝑢(𝑟−𝜏)+ℎ(𝑟−𝜏),𝑢^Δ(𝑟),…,𝑢^Δ𝑛−2(𝑟)∇𝑟Δ𝑠𝛼𝛽^𝜎𝜉+𝑇𝜙_𝑞𝑔(𝑠)𝑓𝑢(𝑠),𝑢(𝑠−𝜏)+ℎ(𝑠−𝜏),𝑢^Δ(𝑠),…,𝑢^Δ𝑛−2(𝑠)∇𝑠^𝑇0≤𝑇+𝜙_𝑞𝑔(𝑟)𝑓𝑢(𝑟),𝑢(𝑟−𝜏)+ℎ(𝑟−𝜏),𝑢^Δ(𝑟),…,𝑢^Δ𝑛−2(𝑟)∇𝑟𝛼𝛽𝜙^𝑞^𝑇0≤𝑇+𝜙_𝑞𝑔(𝑟)𝑓^∗𝜌^∗∇𝑟𝛼𝛽𝜖²𝜌^∗𝜙_𝑞^𝑇0𝑔(𝑟)∇𝑟≤𝑟^∗=‖𝑢‖,(3.24) that is, ‖𝑇𝑢‖≤‖𝑢‖,∀𝑢∈𝜕Ω_𝜌∗.(3.25) Then, by Theorem 1.1, we have 𝑖𝑇,Ω_𝜌∗,𝐾=1.(3.26)
Finally, set Ω_𝑟={𝑢∈𝐾∶‖𝑢‖<𝑟}. For any 𝑢∈𝜕Ω_𝑟, by (𝐴₁), Lemma 2.3 and also similar to the previous proof of Theorem 3.1, we can also have‖𝑇𝑢‖≥‖𝑢‖,∀𝑢∈𝜕Ω_𝑟.(3.27) Then, by Theorem 1.1, we have 𝑖𝑇,Ω_𝑟,𝐾=0.(3.28) Therefore, by (3.21), (3.28), (3.26), 𝜌_∗<𝑟<𝜌^∗, we have 𝑖𝑇,Ω_𝑟⧵Ω_𝜌∗,𝐾=−1,𝑖𝑇,Ω_𝜌∗⧵Ω_𝑟,𝐾=1.(3.29) Then 𝑇 has fixed point 𝑢₁∈Ω_𝑟⧵Ω_𝜌∗ and fixed point 𝑢₂∈Ω_𝜌∗⧵Ω_𝑟. Obviously, 𝑢₁,𝑢₂ are all positive solutions of problem (2.13), (2.14) and 𝜌_∗<‖𝑢₁‖<𝑟<‖𝑢₂‖<𝜌^∗. The proof of Theorem 3.2 is complete.

Similar to Theorems 3.1 and 3.2, we also obtain the following theorems.

Theorem 3.3. Suppose that conditions (𝐻₁), (𝐻₂), (𝐻₃), and (𝐴₂) hold and(𝐴₇)𝑓_∞=𝜆∈((2𝜃^∗/𝜃)^𝑝−1,∞),(𝐴₈)𝑓₀=𝜑∈((2𝜃^∗/𝜃)^𝑝−1,∞).Then, the SBVP (2.13), (2.14) has at last two solutions 𝑢₁,𝑢₂ such that 0<‖𝑢₁‖<𝑅<‖𝑢₂‖.

Theorem 3.4. Suppose that conditions (𝐻₁), (𝐻₂), (𝐻₃), and (𝐴₁) hold and(𝐴₉)𝑓₀=𝜑∈[0,(𝜃_∗/4)^𝑝−1);(𝐴₁₀)𝑓_∞=𝜆∈[0,(𝜃_∗/4)^𝑝−1).Then, the SBVP (2.13), (2.14) has at last two solutions 𝑢₁,𝑢₂ such that 0<‖𝑢₁‖<𝑟<‖𝑢₂‖.

4. An Example

Example 4.1. Consider the following 3-order singular boundary value problem (SBVP) with 𝑝-Laplacian: 𝜙_𝑝𝑢^ΔΔ^∇(𝑡)+164𝜋⁴𝑡^−1/2(1−𝑡𝑢)(𝑡𝑢)=(𝑡)−+𝑡𝑒𝑢_𝑡(𝑡−1)+𝑢^Δ²(𝑡)+𝑢^Δ⁴(𝑡)=0,0<𝑡<1,2𝜙_𝑝,−1≤𝑡≤0,𝑢^Δ(0)−𝜙_𝑝𝑢^ΔΔ41=0,𝜙^𝑝𝑢^Δ(1)+𝛿𝜙_𝑝𝑢^ΔΔ21=0,(4.1) where 𝛽=𝛾=1,𝛼=2,𝑝=4,𝛿≥0,𝑝=4,𝜉=1𝜂=14,3,𝜃=14,𝜏=𝑇=1.(4.2) So, by Lemma 2.4, we discuss the following SBVP: 𝜙_𝑝𝑢^ΔΔ^∇(𝑡)+164𝜋⁴𝑡^−1/2(1−𝑡+)𝑢𝑢(𝑡)+[𝑢(𝑡−1)+ℎ(𝑡−1)_Δ]2𝜙_𝑝𝑢(𝑡)=0,−1≤𝑡≤²(0𝑡,)+𝑢^Δ⁴(𝑡)=0,0<𝑡<1,𝑢^Δ(0)−𝜙_𝑝𝑢^ΔΔ41=0,𝜙^𝑝𝑢^Δ(1)+𝛿𝜙_𝑝𝑢^ΔΔ21=0,(4.3) where 𝑔(𝑡)=ℎ1(𝑡)=𝜁0(𝑡,),−01≤≤𝑡𝑡≤≤10,,𝜁(𝑡)=−𝑡𝑒^𝑡,64𝜋⁴𝑡^−1/2(1−𝑡),𝑓𝑢₁,𝑢₂,𝑢₃=𝑢₁+𝑢₂+𝑢²³+𝑢⁴³.(4.4) Then, obviously, 𝑞=43,¹⁰𝑔(𝑡)∇𝑡=164𝜋³,𝐻=max_{−1≤𝑡≤0}||𝜁(𝑡)||=𝑒,𝑓_∞=+∞,𝑓₀=+∞,(4.5) so conditions (𝐻₁), (𝐻₂), (𝐻₃), (𝐴₂), and (𝐴₃) hold.
Next,𝜙_𝑞¹⁰𝑎(𝑡)∇𝑡=14𝜋,𝜃^∗=4𝜋1+√³4,(4.6) we choose 𝑅=3, 𝑀=2 and for 𝜃=1/4, because of the monotone increasing of 𝑓(𝑢₁,𝑢₂,𝑢₃) on [0,∞)³, then 𝑓𝑢₁,𝑢₂,𝑢₃≤𝑓34,𝑒,3=43+𝑒+90,0≤𝑢³≤3,0≤𝑢₁≤14𝑢³,0≤𝑢₂≤𝑒.(4.7) Therefore, by 𝑀∈0,𝜃_∗,(𝑀𝑅)^𝑝−1=(6)³=216,(4.8) we know 𝑓𝑢₁,𝑢₂,𝑢₃≤(𝑀𝑅)^𝑝−1,0≤𝑢₃≤3,0≤𝑢₁≤14𝑢³,0≤𝑢₂≤𝑒,(4.9) so condition (𝐴₂) holds. Then, by Theorem 3.1, SBVP (4.3) has at least two positive solutions 𝑣₁,𝑣₂ and 0<‖𝑣₁‖<3<‖𝑣₂‖. Then, by Lemma 2.4, 𝑣₁(𝑡)=𝑣₁(𝑡)+ℎ(𝑡), 𝑣₂(𝑡)=𝑣₂(𝑡)+ℎ(𝑡), 𝑡∈(−1,1) are the positive solutions of the SBVP (4.1).

Acknowldgments

The first and second authors were supported financially by Shandong Province Natural Science Foundation (ZR2009AQ004), and National Natural Science Foundation of China (11071141), and the third author was supported by Shandong Province planning Foundation of Social Science (09BJGJ14).