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Journal of Applied Mathematics
VolumeΒ 2012Β (2012), Article IDΒ 939162, 19 pages
http://dx.doi.org/10.1155/2012/939162
Research Article

Higher-Order Dynamic Delay Differential Equations on Time Scales

1School of Mathematics and Quantitative Economics, Shandong University of Finance and Economics, Jinan 250014, China
2School of Mathematical Sciences, Qufu Normal University, Qufu 273165, China
3School of Economics, Shandong University, Jinan 250014, China

Received 11 October 2011; Accepted 12 February 2012

Academic Editor: YanshengΒ Liu

Copyright Β© 2012 Hua Su et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

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: π‘’Ξ”βˆ‡(𝑑)+π‘Ž(𝑑)𝑓(𝑑,𝑒(𝑑))=0,π‘‘βˆˆ(0,𝑇),𝛽𝑒(0)βˆ’π›Ύπ‘’Ξ”(0)=0,𝛼𝑒(πœ‚)=𝑒(𝑇),(1.1) where 𝛽,𝛾β‰₯0, 𝛽+𝛾>0, πœ‚βˆˆ(0,𝜌(𝑇)), 0<𝛼<𝑇/πœ‚. 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: π‘’Ξ”βˆ‡(𝑑)+π‘Ž(𝑑)𝑓(𝑒(𝑑))=0,π‘‘βˆˆ(0,𝑇),𝑒(0)=0,𝛼𝑒(πœ‚)=𝑒(𝑇).(1.2) He obtained some results for the existence of one positive solution of the problem (1.2) based on the limits 𝑓0=lim𝑒→0+(𝑓(𝑒)/𝑒) and π‘“βˆž=limπ‘’β†’βˆž(𝑓(𝑒)/𝑒).

In [11], Wang et al. discussed the following dynamic equation by using Avery-Peterson fixed theorem (see [10]): ξ€·πœ™π‘ξ€·π‘’ξ…žξ€Έξ€Έξ…žξ€·+π‘ž(𝑑)𝑓𝑑,𝑒(𝑑),𝑒(π‘‘βˆ’1),π‘’ξ…žξ€Έ(𝑑)=0,π‘‘βˆˆ(0,1),(1.3)𝑒(𝑑)=πœ‰(𝑑),βˆ’1≀𝑑≀0,𝑒(1)=0,(1.4)𝑒(𝑑)=πœ‰(𝑑),βˆ’1≀𝑑≀0,π‘’ξ…ž(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 ξ€·πœ™π‘ξ€·π‘’ξ…žξ€Έξ€Έξ…ž+π‘Ž(𝑑)𝑓(𝑒(𝑑))=0,0<𝑑<1,π›Όπœ™π‘(𝑒(0))βˆ’π›½πœ™π‘ξ€·π‘’ξ…žξ€Έ(0)=0,π›Ύπœ™π‘(𝑒(1))+π›Ώπœ™π‘ξ€·π‘’ξ…žξ€Έ(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): ξ‚€πœ™π‘ξ‚€π‘’Ξ”π‘›βˆ’1(𝑑)ξ‚ξ‚βˆ‡ξ‚€+𝑔(𝑑)𝑓𝑒(𝑑),𝑒(π‘‘βˆ’πœ),𝑒Δ(𝑑),…,π‘’Ξ”π‘›βˆ’2(𝑑)=0,0<𝑑<𝑇,(1.7)𝑒𝑒(𝑑)=𝜁(𝑑),βˆ’πœβ‰€π‘‘β‰€0,Δ𝑖(0)=0,1β‰€π‘–β‰€π‘›βˆ’3,π›Όπœ™π‘ξ‚€π‘’Ξ”π‘›βˆ’2(0)βˆ’π›½πœ™π‘ξ‚€π‘’Ξ”π‘›βˆ’1(πœ‰)=0,π›Ύπœ™π‘ξ‚€π‘’Ξ”π‘›βˆ’2(𝑇)+π›Ώπœ™π‘ξ‚€π‘’Ξ”π‘›βˆ’1(πœ‚)=0,𝑛β‰₯3,(1.8) where πœ™π‘(𝑠) is 𝑝-Laplacian operator, that is, πœ™π‘(𝑠)=|𝑠|π‘βˆ’2𝑠, 𝑝>1, πœ™π‘ž=πœ™π‘βˆ’1, 1/𝑝+1/π‘ž=1; πœ‰,πœ‚βˆˆ(0,𝑇), 𝜏∈[0,𝑇] is prescribed and πœ‰<πœ‚, π‘”βˆΆ(0,𝑇)β†’[0,∞), 𝛼>0, 𝛽β‰₯0, 𝛾>0, 𝛿β‰₯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, πΎβŠ‚πΈ is a cone, let Ξ©π‘Ÿ={π‘’βˆˆπΎβˆΆβ€–π‘’β€–β‰€π‘Ÿ}. Let operator π‘‡βˆΆΞ©π‘Ÿβ†’πΎ be completely continuous and satisfy 𝑇π‘₯β‰ π‘₯, for all π‘₯βˆˆπœ•Ξ©π‘Ÿ. Then(i)if ‖𝑇π‘₯‖≀‖π‘₯β€–, for all π‘₯βˆˆπœ•Ξ©π‘Ÿ, then 𝑖(𝑇,Ξ©π‘Ÿ,𝐾)=1; (ii)if ‖𝑇π‘₯β€–β‰₯β€–π‘₯β€–, for all π‘₯βˆˆπœ•Ξ©π‘Ÿ, then 𝑖(𝑇,Ξ©π‘Ÿ,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(𝑑)=inf{πœβˆˆπ“βˆ£πœ>𝑑}βˆˆπ“,𝜌(𝑑)=sup{πœβˆˆπ“βˆ£πœ<𝑑}βˆˆπ“(2.1) are called the forward jump operator and the backward jump operator, respectively.

If 𝐓=𝑅, then π‘₯Ξ”(𝑑)=π‘₯βˆ‡(𝑑)=π‘₯β€²(𝑑). If 𝐓=𝑍, then π‘₯Ξ”(𝑑)=π‘₯(𝑑+1)βˆ’π‘₯(𝑑) is the forward difference operator, while π‘₯βˆ‡(𝑑)=π‘₯(𝑑)βˆ’π‘₯(π‘‘βˆ’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 πΉβˆ‡(𝑑)=𝑓(𝑑), then we define the nabla integral by ξ€œπ‘π‘Žπ‘“(𝑑)βˆ‡π‘‘=𝐹(𝑏)βˆ’πΉ(π‘Ž).(2.2) If 𝐹Δ(𝑑)=𝑓(𝑑), then we define the delta integral byξ€œπ‘π‘Žπ‘“(𝑑)Δ𝑑=𝐹(𝑏)βˆ’πΉ(π‘Ž).(2.3)

In the rest of this paper, 𝐓 is closed subset of 𝑅 with 0βˆˆπ“π‘˜, π‘‡βˆˆπ“π‘˜. And let[]𝐡=π‘’βˆˆπΆβˆ’πœ,0βˆ©πΆπ‘›βˆ’2𝑙𝑑[]0,π‘‡βˆΆπ‘’Ξ”π‘–ξ‚‡(0)=0,0β‰€π‘–β‰€π‘›βˆ’3.(2.4)

Here,πΆπ‘›βˆ’2𝑙𝑑[][]0,𝑇={π‘’βˆΆ0,π‘‡β†’π‘…βˆ£π‘’(𝑑)isleftβˆ’denseπ‘›βˆ’2ordercontinuouslydifferentiable}.(2.5) Then 𝐡 is a Banach space with the norm ‖𝑒‖=maxπ‘‘βˆˆ[0,𝑇]|π‘’Ξ”π‘›βˆ’2(𝑑)|. And let𝐾=π‘’βˆˆπ΅βˆΆπ‘’Ξ”π‘›βˆ’2(𝑑)β‰₯0,π‘’Ξ”π‘›βˆ’2[](𝑑)isconcavefunction,π‘‘βˆˆ0,𝑇.(2.6)

Obviously, 𝐾 is a cone in 𝐡. Set πΎπ‘Ÿ={π‘’βˆˆπΎβˆΆβ€–π‘’β€–β‰€π‘Ÿ}.

Definition 2.1. 𝑒(𝑑) is called a solution of SBVP (1.7) and (1.8) if it satisfies the following:(1)π‘’βˆˆπΆ[βˆ’πœ,0]βˆ©πΆπ‘›βˆ’1𝑙𝑑(0,𝑇);(2)𝑒(𝑑)>0 for all π‘‘βˆˆ(0,𝑇) and satisfy conditions (1.8);(3)(πœ™π‘(π‘’Ξ”π‘›βˆ’1(𝑑)))βˆ‡=βˆ’π‘”(𝑑)𝑓(𝑒(𝑑),𝑒(π‘‘βˆ’πœ),𝑒Δ(𝑑),…,π‘’Ξ”π‘›βˆ’2(𝑑)) hold for π‘‘βˆˆ(0,𝑇).
In the rest of the paper, we also make the following assumptions:
(𝐻1)π‘“βˆˆπΆπ‘™π‘‘([0,+∞)𝑛,[0,+∞));(𝐻2)𝑔(𝑑)βˆˆπΆπ‘™π‘‘((0,𝑇),[0,+∞)) and there exists 𝑑0∈(0,𝑇), such that𝑔𝑑0ξ€Έξ€œ>0,0<𝑇0𝑔(𝑠)βˆ‡π‘ <+∞;(2.7)(𝐻3)𝜁(𝑑)∈𝐢([βˆ’πœ,0], 𝜁(𝑑)>0 on [βˆ’πœ,0) and 𝜁(0)=0.It is easy to check that condition (𝐻2) implies that ξ€œ0<𝑇0πœ™π‘žξ‚΅ξ€œπ‘ 0𝑔𝑠1ξ€Έβˆ‡π‘ 1Δ𝑠<+∞.(2.8)
We can easily get the following lemmas.

Lemma 2.2. Suppose condition (𝐻2) holds. Then there exists a constant πœƒβˆˆ(0,1/2) satisfing ξ€œ0<πœƒπ‘‡βˆ’πœƒπ‘”(𝑑)βˆ‡π‘‘<∞.(2.9) Furthermore, the function ξ€œπ΄(𝑑)=π‘‘πœƒπœ™π‘žξ‚΅ξ€œπ‘‘π‘ π‘”ξ€·π‘ 1ξ€Έβˆ‡π‘ 1ξ‚Άξ€œΞ”π‘ +π‘‘π‘‡βˆ’πœƒπœ™π‘žξ‚΅ξ€œπ‘ π‘‘π‘”ξ€·π‘ 1ξ€Έβˆ‡π‘ 1ξ‚Ά[]βˆ‡π‘ ,π‘‘βˆˆπœƒ,π‘‡βˆ’πœƒ,(2.10) is positive continuous functions on [πœƒ,π‘‡βˆ’πœƒ]; therefore, 𝐴(𝑑) has minimum on [πœƒ,π‘‡βˆ’πœƒ]. Hence we suppose, that there exists 𝐿>0 such that 𝐴(𝑑)β‰₯𝐿, π‘‘βˆˆ[πœƒ,π‘‡βˆ’πœƒ].

Proof. At first, it is easily seen that 𝐴(𝑑) is continuous on [πœƒ,π‘‡βˆ’πœƒ]. Next, let 𝐴1ξ€œ(𝑑)=π‘‘πœƒπœ™π‘žξ‚΅ξ€œπ‘‘π‘ π‘”ξ€·π‘ 1ξ€Έβˆ‡π‘ 1Δ𝑠,𝐴2ξ€œ(𝑑)=π‘‘π‘‡βˆ’πœƒπœ™π‘žξ‚΅ξ€œπ‘ π‘‘π‘”ξ€·π‘ 1ξ€Έβˆ‡π‘ 1Δ𝑠.(2.11) Then, from condition (𝐻2), we have that the function 𝐴1(𝑑) is strictly monotone nondecreasing on [πœƒ,π‘‡βˆ’πœƒ] and 𝐴1(πœƒ)=0, the function 𝐴2(𝑑) is strictly monotone nonincreasing on [πœƒ,π‘‡βˆ’πœƒ] and 𝐴2(π‘‡βˆ’πœƒ)=0, which implies 𝐿=minπ‘‘βˆˆ[πœƒ,π‘‡βˆ’πœƒ]𝐴(𝑑)>0. The proof is complete.

Lemma 2.3. Let π‘’βˆˆπΎ and πœƒ of Lemma 2.2, then []𝑒(𝑑)β‰₯πœƒβ€–π‘’β€–,π‘‘βˆˆπœƒ,π‘‡βˆ’πœƒ.(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, 𝑒(𝑑)βˆˆπ΅βˆ©πΆπ‘›βˆ’1𝑙𝑑(0,1) is a solution of the following boundary value problems: ξ‚€πœ™π‘ξ‚€π‘’Ξ”π‘›βˆ’1(𝑑)ξ‚ξ‚βˆ‡ξ‚€+𝑔(𝑑)𝑓𝑒(𝑑),𝑒(π‘‘βˆ’πœ)+β„Ž(π‘‘βˆ’πœ),𝑒Δ(𝑑),…,π‘’Ξ”π‘›βˆ’2(𝑑)=0,0<𝑑<𝑇,(2.13)𝑒𝑒(𝑑)=0,βˆ’πœβ‰€π‘‘β‰€0,Δ𝑖(0)=0,1β‰€π‘–β‰€π‘›βˆ’3,π›Όπœ™π‘ξ‚€π‘’Ξ”π‘›βˆ’2(0)βˆ’π›½πœ™π‘ξ‚€π‘’Ξ”π‘›βˆ’1(πœ‰)=0,π›Ύπœ™π‘ξ‚€π‘’Ξ”π‘›βˆ’2(𝑇)+π›Ώπœ™π‘ξ‚€π‘’Ξ”π‘›βˆ’1(πœ‚)=0,𝑛β‰₯3,(2.14) where ξ‚»β„Ž(𝑑)=𝜁(𝑑),βˆ’πœβ‰€π‘‘β‰€0,0,0≀𝑑≀𝑇.(2.15) Then, 𝑒(𝑑)=𝑒(𝑑)+β„Ž(𝑑), βˆ’πœβ‰€π‘‘β‰€π‘‡ is a positive solution to the SBVP (1.7) and (1.8).

Proof. It is easy to check that 𝑒(𝑑) 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, 𝑒(𝑑)βˆˆπ΅βˆ©πΆπ‘›βˆ’1𝑙𝑑(0,1) is a solution of boundary value problems (2.13), (2.14) if and only if 𝑒(𝑑)∈𝐡 is a solution of the following integral equation: ⎧βŽͺ⎨βŽͺβŽ©ξ€œπ‘’(𝑑)=𝜁(𝑑),βˆ’πœβ‰€π‘‘β‰€0,𝑑0ξ€œπ‘ 10β‹―ξ€œπ‘ π‘›βˆ’30π‘€ξ€·π‘ π‘›βˆ’2ξ€ΈΞ”π‘ π‘›βˆ’2Ξ”π‘ π‘›βˆ’3⋯Δ𝑠1,0≀𝑑≀𝑇,(2.16) where ⎧βŽͺβŽͺβŽͺβŽͺβŽͺ⎨βŽͺβŽͺβŽͺβŽͺβŽͺβŽ©πœ™π‘€(𝑑)=π‘žξ‚΅π›½π›Όξ€œπœŽπœ‰ξ‚€π‘”(𝑠)𝑓𝑒(𝑠),𝑒(π‘ βˆ’πœ)+β„Ž(π‘ βˆ’πœ),𝑒Δ(𝑠),…,π‘’Ξ”π‘›βˆ’2+ξ€œ(𝑠)βˆ‡π‘ π‘‘0πœ™π‘žξ‚΅ξ€œπœŽπ‘ ξ‚€π‘”(π‘Ÿ)𝑓𝑒(π‘Ÿ),𝑒(π‘Ÿβˆ’πœ)+β„Ž(π‘Ÿβˆ’πœ),𝑒Δ(π‘Ÿ),…,π‘’Ξ”π‘›βˆ’2ξ‚ξ‚Άπœ™(π‘Ÿ)βˆ‡π‘ŸΞ”π‘ ,0β‰€π‘‘β‰€πœŽ,π‘žξ‚΅π›Ώπ›Ύξ€œπœ‚πœŽπ‘”ξ‚€π‘’(𝑠)𝑓(𝑠),𝑒(π‘ βˆ’πœ)+β„Ž(π‘ βˆ’πœ),𝑒Δ(𝑠),…,π‘’Ξ”π‘›βˆ’2+ξ€œ(𝑠)βˆ‡π‘ π‘‡π‘‘πœ™π‘žξ‚΅ξ€œπ‘ πœŽξ‚€π‘”(π‘Ÿ)𝑓𝑒(π‘Ÿ),𝑒(π‘Ÿβˆ’πœ)+β„Ž(π‘Ÿβˆ’πœ),𝑒Δ(π‘Ÿ),…,π‘’Ξ”π‘›βˆ’2(π‘Ÿ)βˆ‡π‘ŸΞ”π‘ ,πœŽβ‰€π‘‘β‰€π‘‡.(2.17)

Proof. Necessity. Obviously, for π‘‘βˆˆ(βˆ’πœ,0), we have 𝑒(𝑑)=𝜁(𝑑). If π‘‘βˆˆ(0,1), by the equation of the boundary condition, we have π‘’Ξ”π‘›βˆ’1(πœ‰)β‰₯0, π‘’Ξ”π‘›βˆ’1(πœ‚)≀0, then there exists a constant 𝜎∈[πœ‰,πœ‚]βŠ‚(0,𝑇) such that π‘’Ξ”π‘›βˆ’1(𝜎)=0.
Firstly, by integrating the equation of the problems (2.13) on (𝜎,𝑇), we haveπœ™π‘ξ‚€π‘’Ξ”π‘›βˆ’1(𝑑)=πœ™π‘ξ‚€π‘’Ξ”π‘›βˆ’1(ξ‚βˆ’ξ€œπœŽ)π‘‘πœŽξ‚€π‘”(𝑠)𝑓𝑒(𝑠),𝑒(π‘ βˆ’πœ)+β„Ž(π‘ βˆ’πœ),𝑒Δ(𝑠),…,π‘’Ξ”π‘›βˆ’2(𝑠)βˆ‡π‘ ,(2.18) then π‘’Ξ”π‘›βˆ’1(𝑑)=βˆ’πœ™π‘žξ‚΅ξ€œπ‘‘πœŽξ‚€π‘”(𝑠)𝑓𝑒(𝑠),𝑒(π‘ βˆ’πœ)+β„Ž(π‘ βˆ’πœ),𝑒Δ(𝑠),…,π‘’Ξ”π‘›βˆ’2(𝑠)βˆ‡π‘ ,(2.19) thus π‘’Ξ”π‘›βˆ’2(𝑑)=π‘’Ξ”π‘›βˆ’2(ξ€œπœŽ)βˆ’π‘‘πœŽπœ™π‘žξ‚΅ξ€œπ‘ πœŽξ‚€π‘”(π‘Ÿ)𝑓𝑒(π‘Ÿ),𝑒(π‘Ÿβˆ’πœ)+β„Ž(π‘Ÿβˆ’πœ),𝑒Δ(π‘Ÿ),…,π‘’Ξ”π‘›βˆ’2(ξ‚ξ‚Άπ‘Ÿ)βˆ‡π‘ŸΞ”π‘ .(2.20)
By π‘’Ξ”π‘›βˆ’1(𝜎)=0 and condition (2.18), let 𝑑=πœ‚ on (2.18), we haveπœ™π‘ξ‚€π‘’Ξ”π‘›βˆ’1ξ‚ξ€œ(πœ‚)=βˆ’πœ‚πœŽξ‚€π‘”(𝑠)𝑓𝑒(𝑠),𝑒(π‘ βˆ’πœ)+β„Ž(π‘ βˆ’πœ),𝑒Δ(𝑠),…,π‘’Ξ”π‘›βˆ’2(𝑠)βˆ‡π‘ .(2.21) By the equation of the boundary condition (2.14), we have πœ™π‘ξ‚€π‘’Ξ”π‘›βˆ’2𝛿(𝑇)=βˆ’π›Ύπœ™π‘ξ‚€π‘’Ξ”π‘›βˆ’1(πœ‚),(2.22) then π‘’Ξ”π‘›βˆ’2(𝑇)=πœ™π‘žξ‚΅π›Ώπ›Ύξ€œπœ‚πœŽξ‚€π‘”(𝑠)𝑓𝑒(𝑠),𝑒(π‘ βˆ’πœ)+β„Ž(π‘ βˆ’πœ),𝑒Δ(𝑠),…,π‘’Ξ”π‘›βˆ’2(𝑠)βˆ‡π‘ .(2.23) Then, by (2.20) and leting 𝑑=𝑇 on (2.20), we have π‘’Ξ”π‘›βˆ’2(𝜎)=πœ™π‘žξ‚΅π›Ώπ›Ύξ€œπœ‚πœŽξ‚€π‘”(𝑠)𝑓𝑒(𝑠),𝑒(π‘ βˆ’πœ)+β„Ž(π‘ βˆ’πœ),𝑒Δ(𝑠),…,π‘’Ξ”π‘›βˆ’2+ξ€œ(𝑠)βˆ‡π‘ π‘‡πœŽπœ™π‘žξ‚΅ξ€œπ‘ πœŽξ‚€π‘”(π‘Ÿ)𝑓𝑒(π‘Ÿ),𝑒(π‘Ÿβˆ’πœ)+β„Ž(π‘Ÿβˆ’πœ),𝑒Δ(π‘Ÿ),…,π‘’Ξ”π‘›βˆ’2(π‘Ÿ)βˆ‡π‘ŸΞ”π‘ .(2.24) Then π‘’Ξ”π‘›βˆ’2(𝑑)=πœ™π‘žξ‚΅π›Ώπ›Ύξ€œπœ‚πœŽξ‚€π‘”(𝑠)𝑓𝑒(𝑠),𝑒(π‘ βˆ’πœ)+β„Ž(π‘ βˆ’πœ),𝑒Δ(𝑠),…,π‘’Ξ”π‘›βˆ’2+ξ€œ(𝑠)βˆ‡π‘ π‘‡π‘‘πœ™π‘žξ‚΅ξ€œπ‘ πœŽξ‚€π‘”(π‘Ÿ)𝑓𝑒(π‘Ÿ),𝑒(π‘Ÿβˆ’πœ)+β„Ž(π‘Ÿβˆ’πœ),𝑒Δ(π‘Ÿ),…,π‘’Ξ”π‘›βˆ’2(π‘Ÿ)βˆ‡π‘ŸΞ”π‘ .(2.25) Then, by integrating (2.25) for π‘›βˆ’2 times on (0,𝑇), we have ξ€œπ‘’(𝑑)=𝑑0ξ€œπ‘ 10β‹―ξ€œπ‘ π‘›βˆ’30πœ™π‘žξ‚΅πœŽπ›Ύξ€œπœ‚π›Ώξ‚€π‘”(𝑠)𝑓𝑒(𝑠),𝑒(π‘ βˆ’πœ)+β„Ž(π‘ βˆ’πœ),𝑒Δ(𝑠),…,π‘’Ξ”π‘›βˆ’2(𝑠)βˆ‡π‘ Ξ”π‘ π‘ π‘›βˆ’2⋯Δ𝑠2Δ𝑠1+ξ€œπ‘‘0ξ€œπ‘ 10β‹―ξ€œπ‘ π‘›βˆ’30ξ‚΅ξ€œπ‘‡π‘ π‘›βˆ’2πœ™π‘žξ‚΅ξ€œπ‘ πœŽξ‚€π‘”(π‘Ÿ)𝑓𝑒(π‘Ÿ),𝑒(π‘Ÿβˆ’πœ)+β„Ž(π‘Ÿβˆ’πœ),𝑒Δ(π‘Ÿ),…,π‘’Ξ”π‘›βˆ’2(π‘Ÿ)Γ—βˆ‡π‘ŸΞ”π‘ Ξ”π‘ π‘ π‘›βˆ’2⋯Δ𝑠2Δ𝑠1.(2.26) Similarly, for π‘‘βˆˆ(0,𝜎), by integrating the equation of problems (2.13) on (0,𝜎), we have ξ€œπ‘’(𝑑)=𝑑0ξ€œπ‘ 10β‹―ξ€œπ‘ π‘›βˆ’30πœ™π‘žξ‚΅π›½π›Όξ€œπœŽπœ‰ξ‚€π‘”(𝑠)𝑓𝑒(𝑠),𝑒(π‘ βˆ’πœ)+β„Ž(π‘ βˆ’πœ),𝑒Δ(𝑠),…,π‘’Ξ”π‘›βˆ’2(𝑠)βˆ‡π‘ Ξ”π‘ π‘ π‘›βˆ’2⋯Δ𝑠2Δ𝑠1+ξ€œπ‘‘0ξ€œπ‘ 10β‹―ξ€œπ‘ π‘›βˆ’30ξ‚΅ξ€œπ‘ π‘›βˆ’20πœ™π‘žξ‚΅ξ€œπœŽπ‘ π‘”ξ‚€π‘’(π‘Ÿ)𝑓(π‘Ÿ),𝑒(π‘Ÿβˆ’πœ)+β„Ž(π‘Ÿβˆ’πœ),𝑒Δ(π‘Ÿ),…,π‘’Ξ”π‘›βˆ’2(π‘Ÿ)Γ—βˆ‡π‘ŸΞ”π‘ Ξ”π‘ π‘ π‘›βˆ’2⋯Δ𝑠2Δ𝑠1.(2.27) Therefore, for any π‘‘βˆˆ[0,𝑇], 𝑒(𝑑) can be expressed as equation ⎧βŽͺ⎨βŽͺβŽ©ξ€œπ‘’(𝑑)=𝜁(𝑑),βˆ’πœβ‰€π‘‘β‰€0,𝑑0ξ€œπ‘ 10β‹―ξ€œπ‘ π‘›βˆ’30π‘€ξ€·π‘ π‘›βˆ’2ξ€ΈΞ”π‘ π‘›βˆ’2Ξ”π‘ π‘›βˆ’3⋯Δ𝑠1,0≀𝑑≀𝑇,(2.28) where 𝑀(𝑑) is expressed as (2.17). Then the results of Lemma 2.3 hold.
Sufficiency. Suppose that βˆ«π‘’(𝑑)=𝑑0βˆ«π‘ 10β‹―βˆ«π‘ π‘›βˆ’30𝑀(π‘ π‘›βˆ’2)Ξ”π‘ π‘›βˆ’2Ξ”π‘ π‘›βˆ’3⋯Δ𝑠1, 0≀𝑑≀𝑇. Then by (2.17), we haveπ‘’Ξ”π‘›βˆ’1(⎧βŽͺβŽͺβŽͺ⎨βŽͺβŽͺβŽͺβŽ©πœ™π‘‘)=π‘žξ‚΅ξ€œπœŽπ‘‘ξ‚€π‘”(𝑠)𝑓𝑒(𝑠),𝑒(π‘ βˆ’πœ)+β„Ž(π‘ βˆ’πœ),𝑒Δ(𝑠),…,π‘’Ξ”π‘›βˆ’2(𝑠)βˆ‡π‘ β‰₯0,0β‰€π‘‘β‰€πœŽ,βˆ’πœ™π‘žξ‚΅ξ€œπ‘‘πœŽξ‚€π‘”(𝑠)𝑓𝑒(𝑠),𝑒(π‘ βˆ’πœ)+β„Ž(π‘ βˆ’πœ),𝑒Δ(𝑠),…,π‘’Ξ”π‘›βˆ’2(𝑠)βˆ‡π‘ β‰€0,πœŽβ‰€π‘‘β‰€π‘‡.(2.29) So, (πœ™π‘(π‘’Ξ”π‘›βˆ’1))βˆ‡+𝑔(𝑑)𝑓(𝑒(𝑑),𝑒(π‘‘βˆ’πœ)+β„Ž(π‘‘βˆ’πœ),𝑒Δ(𝑑),…,π‘’Ξ”π‘›βˆ’2(𝑑))=0, 0<𝑑<𝑇. These imply that (2.13) holds. Furthermore, by letting 𝑑=0 and 𝑑=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 ⎧βŽͺ⎨βŽͺβŽ©ξ€œ(𝑇𝑒)(𝑑)=𝜁(𝑑),βˆ’πœβ‰€π‘‘β‰€0,𝑑0ξ€œπ‘ 10β‹―ξ€œπ‘ π‘›βˆ’30π‘€ξ€·π‘ π‘›βˆ’2ξ€ΈΞ”π‘ π‘›βˆ’2Ξ”π‘ π‘›βˆ’3⋯Δ𝑠1,0≀𝑑≀𝑇,(2.30) where 𝑀(𝑑) is given by (2.17).

From the definition of 𝑇 and the previous discussion, we deduce that, for each π‘’βˆˆπΎ, π‘‡π‘’βˆˆπΎ. Moreover, we have the following lemmas.

Lemma 2.6. π‘‡βˆΆπΎβ†’πΎ is completely continuous.

Proof. Because (𝑇𝑒)Ξ”π‘›βˆ’1(𝑑)=𝑀Δ(⎧βŽͺβŽͺβŽͺ⎨βŽͺβŽͺβŽͺβŽ©πœ™π‘‘)=π‘žξ‚΅ξ€œπœŽπ‘‘ξ‚€π‘”(𝑠)𝑓𝑒(𝑠),𝑒(π‘ βˆ’πœ)+β„Ž(π‘ βˆ’πœ),𝑒Δ(𝑠),…,π‘’Ξ”π‘›βˆ’2(𝑠)βˆ‡π‘ β‰₯0,0β‰€π‘‘β‰€πœŽ,βˆ’πœ™π‘žξ‚΅ξ€œπ‘‘πœŽξ‚€π‘”(𝑠)𝑓𝑒(𝑠),𝑒(π‘ βˆ’πœ)+β„Ž(π‘ βˆ’πœ),𝑒Δ(𝑠),…,π‘’Ξ”π‘›βˆ’2(𝑠)βˆ‡π‘ β‰€0,πœŽβ‰€π‘‘β‰€π‘‡,(2.31) is continuous, decreasing on [0,𝑇] and satisfies (𝑇𝑒)Ξ”π‘›βˆ’1(𝜎)=0, then, π‘‡π‘’βˆˆπΎ for each π‘’βˆˆπΎ and (𝑇𝑒)Ξ”π‘›βˆ’2(𝜎)=maxπ‘‘βˆˆ[0,𝑇](𝑇𝑒)Ξ”π‘›βˆ’2(𝑑). This shows that π‘‡πΎβŠ‚πΎ. Furthermore, it is easy to check by Arzela-ascoli Theorem that π‘‡βˆΆπΎβ†’πΎ is completely continuous.

Lemma 2.7. Suppose that conditions (𝐻1),(𝐻2),(𝐻3) hold, the solution 𝑒(𝑑) of problem (2.13), (2.14) satisfies max0≀𝑑≀𝑇||||𝑒(π‘‘βˆ’πœ)+β„Ž(π‘‘βˆ’πœ)≀maxβˆ’πœβ‰€π‘‘β‰€0||||,𝜁(𝑑)𝑒(𝑑)≀𝑇𝑒Δ(𝑑)β‰€β‹―β‰€π‘‡π‘›βˆ’3π‘’Ξ”π‘›βˆ’3[],(𝑑),π‘‘βˆˆ0,𝑇(2.32) and for πœƒβˆˆ(0,𝑇/2) in Lemma 2.2, one has π‘’Ξ”π‘›βˆ’3𝑇(𝑑)β‰€πœƒπ‘’Ξ”π‘›βˆ’2[](𝑑),π‘‘βˆˆπœƒ,π‘‡βˆ’πœƒ.(2.33)

Proof. Firstly, we can have max0≀𝑑≀𝑇||||𝑒(π‘‘βˆ’πœ)+β„Ž(π‘‘βˆ’πœ)≀max0≀𝑑≀𝑇||||𝑒(π‘‘βˆ’πœ)+max0≀𝑑≀𝑇||||β„Ž(π‘‘βˆ’πœ)=maxβˆ’πœβ‰€π‘‘β‰€π‘‡βˆ’πœ||||𝑒(𝑑)+maxβˆ’πœβ‰€π‘‘β‰€π‘‡βˆ’πœ||||β„Ž(𝑑)=maxβˆ’πœβ‰€π‘‘β‰€0||||.𝜁(𝑑)(2.34)
Next, if 𝑒(𝑑) is the solution of problem (2.13), (2.14), then π‘’Ξ”π‘›βˆ’2(𝑑) is concave function, and 𝑒Δ𝑖(𝑑)β‰₯0(𝑖=0,1,…,π‘›βˆ’2), π‘‘βˆˆ[0,𝑇]. Thus, we have𝑒Δ𝑖(ξ€œπ‘‘)=𝑑0𝑒Δ𝑖+1(𝑠)Δ𝑠≀𝑑𝑒Δ𝑖+1(𝑑)≀𝑇𝑒Δ𝑖+1(𝑑),𝑖=0,1,…,π‘›βˆ’4,(2.35) that is, 𝑒(𝑑)≀𝑇𝑒Δ(𝑑)β‰€β‹―β‰€π‘‡π‘›βˆ’3π‘’Ξ”π‘›βˆ’3(𝑑), π‘‘βˆˆ[0,𝑇].
Finally, by Lemma 2.3, for π‘‘βˆˆ[πœƒ,π‘‡βˆ’πœƒ], we have π‘’Ξ”π‘›βˆ’2(𝑑)β‰₯πœƒβ€–π‘’Ξ”π‘›βˆ’2β€–. By π‘’Ξ”π‘›βˆ’3∫(𝑑)=𝑑0π‘’Ξ”π‘›βˆ’2(𝑠)Ξ”π‘ β‰€π‘‡β€–π‘’Ξ”π‘›βˆ’2β€–, we have π‘’Ξ”π‘›βˆ’3𝑇(𝑑)β‰€πœƒπ‘’Ξ”π‘›βˆ’2[](𝑑),π‘‘βˆˆπœƒ,π‘‡βˆ’πœƒ.(2.36) The proof is complete.

For convenience, we set 𝐻=maxβˆ’πœβ‰€π‘‘β‰€0||||𝜁(𝑑),πœƒβˆ—=2𝐿,πœƒβˆ—=1𝑇+πœ™π‘žξ€Έπœ™(𝛽/𝛼)π‘žξ‚€βˆ«π‘‡0𝑔,ξ€·πœƒ(π‘Ÿ)βˆ‡π‘Ÿπ‘šβˆˆβˆ—ξ€Έξ€·,∞,π‘€βˆˆ0,πœƒβˆ—ξ€Έ,(2.37) where 𝐿 is the constant from Lemma 2.2. By Lemma 2.5, we can also set𝑓0=lim𝑒𝑛→0max(𝑒1,𝑒2,…,𝑒𝑛)βˆˆβ„΅π‘“ξ€·π‘’1,𝑒2,…,π‘’π‘›ξ€Έπ‘’π‘›π‘βˆ’1,π‘“βˆž=limπ‘’π‘›β†’βˆžmin(𝑒1,𝑒2,…,𝑒𝑛)βˆˆβ„΅π‘“ξ€·π‘’1,𝑒2,…,π‘’π‘›ξ€Έπ‘’π‘›π‘βˆ’1,(2.38) where β„΅={(𝑒1,𝑒2,…,𝑒𝑛)|0≀𝑒1≀𝑇𝑒3β‹―β‰€π‘‡π‘›βˆ’3π‘’π‘›βˆ’1≀(π‘‡π‘›βˆ’2/πœƒ)𝑒𝑛,𝑒2≀𝐻}.

3. The Existence of Multiple Positive Solutions

In this section, we also make the following conditions:(𝐴1)𝑓(𝑒1,𝑒2,…,𝑒𝑛)β‰₯(π‘šπ‘Ÿ)π‘βˆ’1, for πœƒπ‘Ÿβ‰€π‘’π‘›β‰€π‘Ÿ, (𝑒1,𝑒2,…,𝑒𝑛)βˆˆβ„΅;(𝐴2)𝑓(𝑒1,𝑒2,…,𝑒𝑛)≀(𝑀𝑅)π‘βˆ’1, for 0≀𝑒𝑛≀𝑅, (𝑒1,𝑒2,…,𝑒𝑛)βˆˆβ„΅.

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)π‘“βˆž=+∞.Then, the SBVP (2.13), (2.14) hase at last two solutions 𝑒1, 𝑒2 such that ‖‖𝑒0<1‖‖‖‖𝑒<𝑅<2β€–β€–.(3.1)

Proof. For any π‘’βˆˆπΎ, by Lemma 2.3, we have π‘’Ξ”π‘›βˆ’2[](𝑑)β‰₯πœƒβ€–π‘’β€–,π‘‘βˆˆπœƒ,π‘‡βˆ’πœƒ.(3.2) First, by condition (𝐴3), for any 𝑁>2/πœƒπΏ, there exists a constant πœŒβˆ—βˆˆ(0,𝑅) such that 𝑓𝑒1,𝑒2,…,𝑒𝑛β‰₯ξ€·π‘π‘’π‘›ξ€Έπ‘βˆ’1,0<π‘’π‘›β‰€πœŒβˆ—,𝑒𝑛≠0.(3.3) Set Ξ©πœŒβˆ—={π‘’βˆˆπΎβˆΆβ€–π‘’β€–<πœŒβˆ—}. For any π‘’βˆˆπœ•Ξ©πœŒβˆ—, by (3.2) we have πœŒβˆ—=‖𝑒‖β‰₯π‘’Ξ”π‘›βˆ’2(𝑑)β‰₯πœƒβ€–π‘’β€–=πœƒπœŒβˆ—[],π‘‘βˆˆπœƒ,π‘‡βˆ’πœƒ.(3.4) For any π‘’βˆˆπœ•Ξ©πœŒβˆ—, by (3.3) and Lemmas 2.3–2.6, we will discuss it from three perspectives.(i)If 𝜎∈[πœƒ,π‘‡βˆ’πœƒ], we have 2‖𝑇𝑒‖=2(𝑇𝑒)Ξ”π‘›βˆ’2β‰₯ξ€œ(𝜎)𝜎0πœ™π‘žξ‚΅ξ€œπœŽπ‘ ξ‚€π‘”(π‘Ÿ)𝑓𝑒(π‘Ÿ),𝑒(π‘Ÿβˆ’πœ)+β„Ž(π‘Ÿβˆ’πœ),𝑒Δ(π‘Ÿ),…,π‘’Ξ”π‘›βˆ’2+ξ€œ(π‘Ÿ)βˆ‡π‘ŸΞ”π‘ π‘‡πœŽπœ™π‘žξ‚΅ξ€œπ‘ πœŽξ‚€π‘”(π‘Ÿ)𝑓𝑒(π‘Ÿ),𝑒(π‘Ÿβˆ’πœ)+β„Ž(π‘Ÿβˆ’πœ),𝑒Δ(π‘Ÿ),…,π‘’Ξ”π‘›βˆ’2β‰₯ξ€œ(π‘Ÿ)βˆ‡π‘ŸΞ”π‘ πœŽπœƒπœ™π‘žξ‚΅ξ€œπœŽπ‘ π‘”ξ‚€π‘’(π‘Ÿ)𝑓(π‘Ÿ),𝑒(π‘Ÿβˆ’πœ)+β„Ž(π‘Ÿβˆ’πœ),𝑒Δ(π‘Ÿ),…,π‘’Ξ”π‘›βˆ’2+ξ€œ(π‘Ÿ)βˆ‡π‘ŸΞ”π‘ πœŽπ‘‡βˆ’πœƒπœ™π‘žξ‚΅ξ€œπ‘ πœŽξ‚€π‘”(π‘Ÿ)𝑓𝑒(π‘Ÿ),𝑒(π‘Ÿβˆ’πœ)+β„Ž(π‘Ÿβˆ’πœ),𝑒Δ(π‘Ÿ),…,π‘’Ξ”π‘›βˆ’2(ξ‚ξ‚Άπ‘Ÿ)βˆ‡π‘ŸΞ”π‘ β‰₯π‘πœƒπ΄(𝜎)‖𝑒‖β‰₯2‖𝑒‖.(3.5)(ii)If 𝜎∈(π‘‡βˆ’πœƒ,𝑇], we have‖𝑇𝑒‖=(𝑇𝑒)Ξ”π‘›βˆ’2(𝜎)β‰₯πœ™π‘žξ‚΅π›½π›Όξ€œπœŽπœ‰ξ‚€π‘”(𝑠)𝑓𝑒(𝑠),𝑒(π‘ βˆ’πœ)+β„Ž(π‘ βˆ’πœ),𝑒Δ(𝑠),…,π‘’Ξ”π‘›βˆ’2+ξ€œ(𝑠)βˆ‡π‘ πœŽ0πœ™π‘žξ‚΅ξ€œπœŽπ‘ ξ‚€π‘”(π‘Ÿ)𝑓𝑒(π‘Ÿ),𝑒(π‘Ÿβˆ’πœ)+β„Ž(π‘Ÿβˆ’πœ),𝑒Δ(π‘Ÿ),…,π‘’Ξ”π‘›βˆ’2β‰₯ξ€œ(π‘Ÿ)βˆ‡π‘ŸΞ”π‘ πœƒπ‘‡βˆ’πœƒπœ™π‘žξ‚΅ξ€œπ‘ π‘‡βˆ’πœƒξ‚€π‘”(π‘Ÿ)𝑓𝑒(π‘Ÿ),𝑒(π‘Ÿβˆ’πœ)+β„Ž(π‘Ÿβˆ’πœ),𝑒Δ(π‘Ÿ),…,π‘’Ξ”π‘›βˆ’2(π‘Ÿ)βˆ‡π‘ŸΞ”π‘ β‰₯π‘πœƒπ΄(π‘‡βˆ’πœƒ)‖𝑒‖>‖𝑒‖.(3.6)(iii)If 𝜎∈(0,πœƒ), we have ‖𝑇𝑒‖=(𝑇𝑒)Ξ”π‘›βˆ’2(𝜎)β‰₯πœ™π‘žξ‚΅π›Ώπ›Ύξ€œπœ‚πœŽξ‚€π‘”(𝑠)𝑓𝑒(𝑠),𝑒(π‘ βˆ’πœ)+β„Ž(π‘Ÿβˆ’πœ),𝑒Δ(𝑠),…,π‘’Ξ”π‘›βˆ’2+ξ€œ(𝑠)βˆ‡π‘ π‘‡πœŽπœ™π‘žξ‚΅ξ€œπ‘ πœŽξ‚€π‘”(π‘Ÿ)𝑓𝑒(π‘Ÿ),𝑒(π‘Ÿβˆ’πœ)+β„Ž(π‘Ÿβˆ’πœ),𝑒Δ(π‘Ÿ),…,π‘’Ξ”π‘›βˆ’2β‰₯ξ€œ(π‘Ÿ)βˆ‡π‘ŸΞ”π‘ πœƒπ‘‡βˆ’πœƒπœ™π‘žξ‚΅ξ€œπ‘ πœƒπ‘”ξ‚€π‘’(π‘Ÿ)𝑓(π‘Ÿ),𝑒(π‘Ÿβˆ’πœ)+β„Ž(π‘Ÿβˆ’πœ),𝑒Δ(π‘Ÿ),…,π‘’Ξ”π‘›βˆ’2(π‘Ÿ)βˆ‡π‘ŸΞ”π‘ β‰₯π‘πœƒπ΄(πœƒ)‖𝑒‖>‖𝑒‖.(3.7) Therefore, no matter under which condition, we all have ‖𝑇𝑒‖β‰₯‖𝑒‖,βˆ€π‘’βˆˆπœ•Ξ©πœŒβˆ—.(3.8) Then, by Theorem 1.1, we have 𝑖𝑇,Ξ©πœŒβˆ—ξ€Έ,𝐾=0.(3.9)Next, by condition (𝐴4), for any 𝑁>2/πœƒπΏ, there exists a constant 𝜌0>0 such that 𝑓𝑒1,𝑒2,…,𝑒𝑛β‰₯ξ‚€π‘π‘’π‘›ξ‚π‘βˆ’1,𝑒𝑛>𝜌0.(3.10) We choose a constant πœŒβˆ—>max{𝑅,𝜌0/πœƒ}, obviously πœŒβˆ—<𝑅<πœŒβˆ—. Set Ξ©πœŒβˆ—={π‘’βˆˆπΎβˆΆβ€–π‘’β€–<πœŒβˆ—}. For any π‘’βˆˆπœ•Ξ©πœŒβˆ—, by Lemma 2.3, we have 𝑒(𝑑)β‰₯πœƒβ€–π‘’β€–=πœƒπœŒβˆ—>𝜌0[],π‘‘βˆˆπœƒ,π‘‡βˆ’πœƒ.(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 ‖𝑇𝑒‖β‰₯‖𝑒‖,βˆ€π‘’βˆˆπœ•Ξ©πœŒβˆ—.(3.12) Then, by Theorem 1.1, we have 𝑖𝑇,Ξ©πœŒβˆ—ξ€Έ,𝐾=0.(3.13) Finally, set Ω𝑅={π‘’βˆˆπΎβˆΆβ€–π‘’β€–<𝑅}. For any π‘’βˆˆπœ•Ξ©π‘…, we have 𝑒(𝑑)≀‖𝑒‖=𝑅, by (𝐴2) we know ‖𝑇𝑒‖=(𝑇𝑒)Ξ”π‘›βˆ’2(𝜎)β‰€πœ™π‘žξ‚΅π›½π›Όξ€œπœŽπœ‰ξ‚€π‘”(𝑠)𝑓𝑒(𝑠),𝑒(π‘ βˆ’πœ)+β„Ž(π‘ βˆ’πœ),𝑒Δ(𝑠),…,π‘’Ξ”π‘›βˆ’2+ξ€œ(𝑠)βˆ‡π‘ π‘‡0πœ™π‘žξ‚΅ξ€œπœŽπ‘ ξ‚€π‘”(π‘Ÿ)𝑓𝑒(π‘Ÿ),𝑒(π‘Ÿβˆ’πœ)+β„Ž(π‘Ÿβˆ’πœ),𝑒Δ(π‘Ÿ),…,π‘’Ξ”π‘›βˆ’2≀(π‘Ÿ)βˆ‡π‘ŸΞ”π‘ π‘‡+πœ™π‘žξ‚΅π›½π›Όξ‚Άξ‚Άπ‘€π‘…πœ™π‘žξ‚΅ξ€œπ‘‡0𝑔(π‘Ÿ)βˆ‡π‘Ÿβ‰€π‘…=‖𝑒‖.(3.14) Thus, ‖𝑇𝑒‖≀‖𝑒‖,βˆ€π‘’βˆˆπœ•Ξ©π‘….(3.15) Then, by Theorem 1.1, we have 𝑖𝑇,Ω𝑅,𝐾=1.(3.16) Therefore, by (3.9), (3.13), (3.16), πœŒβˆ—<𝑅<πœŒβˆ— we have 𝑖𝑇,Ξ©π‘…β§΅Ξ©πœŒβˆ—ξ‚ξ‚€,𝐾=1,𝑖𝑇,Ξ©πœŒβˆ—β§΅Ξ©π‘…ξ‚,𝐾=βˆ’1.(3.17) Then 𝑇 has fixed point 𝑒1βˆˆΞ©π‘…β§΅Ξ©πœŒβˆ— and fixed point 𝑒2βˆˆΞ©πœŒβˆ—β§΅Ξ©π‘…. Obviously, 𝑒1,𝑒2 are all positive solutions of problem (2.13), (2.14) and πœŒβˆ—<‖𝑒1β€–<𝑅<‖𝑒2β€–<πœŒβˆ—. 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)π‘“βˆž=0.Then, the SBVP (2.13), (2.14) has at last two solutions 𝑒1,𝑒2 such that 0<‖𝑒1β€–<π‘Ÿ<‖𝑒2β€–.

Proof. First, by 𝑓0=0, for πœ–1∈(0,πœƒβˆ—), there exists a constant πœŒβˆ—βˆˆ(0,π‘Ÿ) such that 𝑓𝑒1,𝑒2,…,π‘’π‘›ξ€Έβ‰€ξ€·πœ–1π‘’π‘›ξ€Έπ‘βˆ’1,0<π‘’π‘›β‰€πœŒβˆ—.(3.18) Set Ξ©πœŒβˆ—={π‘’βˆˆπΎβˆΆβ€–π‘’β€–<πœŒβˆ—}, for any π‘’βˆˆπœ•Ξ©πœŒβˆ—, by (3.18), we have ‖𝑇𝑒‖=(𝑇𝑒)Ξ”π‘›βˆ’2(𝜎)β‰€πœ™π‘žξ‚΅π›½π›Όξ€œπœŽπœ‰ξ‚€π‘”(𝑠)𝑓𝑒(𝑠),𝑒(π‘ βˆ’πœ)+β„Ž(π‘ βˆ’πœ),𝑒Δ(𝑠),…,π‘’Ξ”π‘›βˆ’2+ξ€œ(𝑠)βˆ‡π‘ π‘‡0πœ™π‘žξ‚΅ξ€œπœŽπ‘ ξ‚€π‘”(π‘Ÿ)𝑓𝑒(π‘Ÿ),𝑒(π‘Ÿβˆ’πœ)+β„Ž(π‘Ÿβˆ’πœ),𝑒Δ(π‘Ÿ),…,π‘’Ξ”π‘›βˆ’2(π‘Ÿ)βˆ‡π‘ŸΞ”π‘ β‰€πœ™π‘žξ‚΅π›½π›Όξ€œπœŽπœ‰ξ‚€π‘”(𝑠)𝑓𝑒(𝑠),𝑒(π‘ βˆ’πœ)+β„Ž(π‘ βˆ’πœ),𝑒Δ(𝑠),…,π‘’Ξ”π‘›βˆ’2(𝑠)βˆ‡π‘ +π‘‡πœ™π‘žξ‚΅ξ€œπ‘‡0𝑔(π‘Ÿ)𝑓𝑒(π‘Ÿ),𝑒(π‘Ÿβˆ’πœ)+β„Ž(π‘Ÿβˆ’πœ),𝑒Δ(π‘Ÿ),…,π‘’Ξ”π‘›βˆ’2≀(π‘Ÿ)βˆ‡π‘Ÿπ‘‡+πœ™π‘žξ‚΅π›½π›Όπœ–ξ‚Άξ‚Ά1πœŒβˆ—πœ™π‘žξ‚΅ξ€œπ‘‡0𝑔(π‘Ÿ)βˆ‡π‘Ÿβ‰€πœŒβˆ—=‖𝑒‖,(3.19) that is, ‖𝑇𝑒‖≀‖𝑒‖,βˆ€π‘’βˆˆπœ•Ξ©πœŒβˆ—(3.20) Then, by Theorem 1.1, we have 𝑖𝑇,Ξ©πœŒβˆ—ξ€Έ,𝐾=1.(3.21)
Next, let π‘“βˆ—(π‘₯)=max0β‰€π‘’π‘›βˆ’1≀π‘₯𝑓(𝑒1,𝑒2,…,π‘’π‘›βˆ’1); note that π‘“βˆ—(π‘₯) is monotone increasing with respect to π‘₯β‰₯0. Then, from π‘“βˆž=0, it is easy to see thatlimπ‘₯β†’βˆžπ‘“βˆ—(π‘₯)π‘₯π‘βˆ’1=0.(3.22) Therefore, for any πœ–2∈(0,πœƒβˆ—), there exists a constant πœŒβˆ—>π‘Ÿ such that π‘“βˆ—ξ€·πœ–(π‘₯)≀2π‘₯ξ€Έπ‘βˆ’1,π‘₯β‰₯πœŒβˆ—.(3.23) Set Ξ©πœŒβˆ—={π‘’βˆˆπΎβˆΆβ€–π‘’β€–<πœŒβˆ—}, for any π‘’βˆˆπœ•Ξ©πœŒβˆ—, by (3.23), we have ‖𝑇𝑒‖=(𝑇𝑒)Ξ”π‘›βˆ’2(𝜎)β‰€πœ™π‘žξ‚΅π›½π›Όξ€œπœŽπœ‰ξ‚€π‘”(𝑠)𝑓𝑒(𝑠),𝑒(π‘ βˆ’πœ)+β„Ž(π‘ βˆ’πœ),𝑒Δ(𝑠),…,π‘’Ξ”π‘›βˆ’2+ξ€œ(𝑠)βˆ‡π‘ π‘‡0πœ™π‘žξ‚΅ξ€œπœŽπ‘ ξ‚€π‘”(π‘Ÿ)𝑓𝑒(π‘Ÿ),𝑒(π‘Ÿβˆ’πœ)+β„Ž(π‘Ÿβˆ’πœ),𝑒Δ(π‘Ÿ),…,π‘’Ξ”π‘›βˆ’2(π‘Ÿ)βˆ‡π‘ŸΞ”π‘ β‰€πœ™π‘žξ‚΅π›½π›Όξ€œπœŽπœ‰ξ‚€π‘”(𝑠)𝑓𝑒(𝑠),𝑒(π‘ βˆ’πœ)+β„Ž(π‘ βˆ’πœ),𝑒Δ(𝑠),…,π‘’Ξ”π‘›βˆ’2(𝑠)βˆ‡π‘ +π‘‡πœ™π‘žξ‚΅ξ€œπ‘‡0𝑔(π‘Ÿ)𝑓𝑒(π‘Ÿ),𝑒(π‘Ÿβˆ’πœ)+β„Ž(π‘Ÿβˆ’πœ),𝑒Δ(π‘Ÿ),…,π‘’Ξ”π‘›βˆ’2≀(π‘Ÿ)βˆ‡π‘Ÿπ‘‡+πœ™π‘žξ‚΅π›½π›Όπœ™ξ‚Άξ‚Άπ‘žξ‚΅ξ€œπ‘‡0𝑔(π‘Ÿ)π‘“βˆ—ξ€·πœŒβˆ—ξ€Έξ‚Άβ‰€ξ‚΅βˆ‡π‘Ÿπ‘‡+πœ™π‘žξ‚΅π›½π›Όπœ–ξ‚Άξ‚Ά2πœŒβˆ—πœ™π‘žξ‚΅ξ€œπ‘‡0𝑔(π‘Ÿ)βˆ‡π‘Ÿβ‰€π‘Ÿβˆ—=‖𝑒‖,(3.24) that is, ‖𝑇𝑒‖≀‖𝑒‖,βˆ€π‘’βˆˆπœ•Ξ©πœŒβˆ—.(3.25) Then, by Theorem 1.1, we have 𝑖𝑇,Ξ©πœŒβˆ—ξ€Έ,𝐾=1.(3.26)
Finally, set Ξ©π‘Ÿ={π‘’βˆˆπΎβˆΆβ€–π‘’β€–<π‘Ÿ}. For any π‘’βˆˆπœ•Ξ©π‘Ÿ, by (𝐴1), Lemma 2.3 and also similar to the previous proof of Theorem 3.1, we can also have‖𝑇𝑒‖β‰₯‖𝑒‖,βˆ€π‘’βˆˆπœ•Ξ©π‘Ÿ.(3.27) Then, by Theorem 1.1, we have 𝑖𝑇,Ξ©π‘Ÿξ€Έ,𝐾=0.(3.28) Therefore, by (3.21), (3.28), (3.26), πœŒβˆ—<π‘Ÿ<πœŒβˆ—, we have 𝑖𝑇,Ξ©π‘Ÿβ§΅Ξ©πœŒβˆ—ξ‚ξ‚€,𝐾=βˆ’1,𝑖𝑇,Ξ©πœŒβˆ—β§΅Ξ©π‘Ÿξ‚,𝐾=1.(3.29) Then 𝑇 has fixed point 𝑒1βˆˆΞ©π‘Ÿβ§΅Ξ©πœŒβˆ— and fixed point 𝑒2βˆˆΞ©πœŒβˆ—β§΅Ξ©π‘Ÿ. Obviously, 𝑒1,𝑒2 are all positive solutions of problem (2.13), (2.14) and πœŒβˆ—<‖𝑒1β€–<π‘Ÿ<‖𝑒2β€–<πœŒβˆ—. 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)π‘“βˆž=πœ†βˆˆ((2πœƒβˆ—/πœƒ)π‘βˆ’1,∞),(𝐴8)𝑓0=πœ‘βˆˆ((2πœƒβˆ—/πœƒ)π‘βˆ’1,∞).Then, the SBVP (2.13), (2.14) has at last two solutions 𝑒1,𝑒2 such that 0<‖𝑒1β€–<𝑅<‖𝑒2β€–.

Theorem 3.4. Suppose that conditions (𝐻1), (𝐻2), (𝐻3), and (𝐴1) hold and(𝐴9)𝑓0=πœ‘βˆˆ[0,(πœƒβˆ—/4)π‘βˆ’1);(𝐴10)π‘“βˆž=πœ†βˆˆ[0,(πœƒβˆ—/4)π‘βˆ’1).Then, the SBVP (2.13), (2.14) has at last two solutions 𝑒1,𝑒2 such that 0<‖𝑒1β€–<π‘Ÿ<‖𝑒2β€–.

4. An Example

Example 4.1. Consider the following 3-order singular boundary value problem (SBVP) with 𝑝-Laplacian: ξ€·πœ™π‘ξ€·π‘’Ξ”Ξ”ξ€Έξ€Έβˆ‡1(𝑑)+64πœ‹4π‘‘βˆ’1/2𝑒(1βˆ’π‘‘)𝑒(𝑑)+𝑒(π‘‘βˆ’1)+Ξ”ξ€Έ2𝑒(𝑑)+Ξ”ξ€Έ4𝑒(𝑑)=0,0<𝑑<1,(𝑑)=βˆ’π‘‘π‘’π‘‘,βˆ’1≀𝑑≀0,2πœ™π‘ξ€·π‘’Ξ”ξ€Έ(0)βˆ’πœ™π‘ξ‚€π‘’Ξ”Ξ”ξ‚€14=0,πœ™π‘ξ€·π‘’Ξ”ξ€Έ(1)+π›Ώπœ™π‘ξ‚€π‘’Ξ”Ξ”ξ‚€12=0,(4.1) where 1𝛽=𝛾=1,𝛼=2,𝑝=4,𝛿β‰₯0,𝑝=4,πœ‰=4,1πœ‚=31,πœƒ=4,𝜏=𝑇=1.(4.2) So, by Lemma 2.4, we discuss the following SBVP: ξ€·πœ™π‘ξ€·π‘’Ξ”Ξ”ξ€Έξ€Έβˆ‡1(𝑑)+64πœ‹4π‘‘βˆ’1/2[]+𝑒(1βˆ’π‘‘)𝑒(𝑑)+𝑒(π‘‘βˆ’1)+β„Ž(π‘‘βˆ’1)Ξ”ξ€Έ2𝑒(𝑑)+Ξ”ξ€Έ4ξ‚„(𝑑)=0,0<𝑑<1,𝑒(𝑑)=0,βˆ’1≀𝑑≀0,2πœ™π‘ξ€·π‘’Ξ”(ξ€Έ0)βˆ’πœ™π‘ξ‚€π‘’Ξ”Ξ”ξ‚€14=0,πœ™π‘ξ€·π‘’Ξ”(ξ€Έ1)+π›Ώπœ™π‘ξ‚€π‘’Ξ”Ξ”ξ‚€12=0,(4.3) where ξ‚»β„Ž(𝑑)=𝜁(𝑑),βˆ’1≀𝑑≀0,0,0≀𝑑≀1,𝜁(𝑑)=βˆ’π‘‘π‘’π‘‘,1𝑔(𝑑)=64πœ‹4π‘‘βˆ’1/2𝑒(1βˆ’π‘‘),𝑓1,𝑒2,𝑒3ξ€Έ=𝑒1+𝑒2+𝑒23+𝑒43.(4.4) Then, obviously, 4π‘ž=3,ξ€œ101𝑔(𝑑)βˆ‡π‘‘=64πœ‹3,𝐻=maxβˆ’1≀𝑑≀0||||𝜁(𝑑)=𝑒,π‘“βˆž=+∞,𝑓0=+∞,(4.5) so conditions (𝐻1), (𝐻2), (𝐻3), (𝐴2), and (𝐴3) hold.
Next,πœ™π‘žξ‚΅ξ€œ10ξ‚Ά=1π‘Ž(𝑑)βˆ‡π‘‘4πœ‹,πœƒβˆ—=4πœ‹1+3√4,(4.6) we choose 𝑅=3, 𝑀=2 and for πœƒ=1/4, because of the monotone increasing of 𝑓(𝑒1,𝑒2,𝑒3) on [0,∞)3, then 𝑓𝑒1,𝑒2,𝑒3ξ€Έξ‚€3≀𝑓4=3,𝑒,34+𝑒+90,0≀𝑒3≀3,0≀𝑒1≀14𝑒3,0≀𝑒2≀𝑒.(4.7) Therefore, by ξ€·π‘€βˆˆ0,πœƒβˆ—ξ€Έ,(𝑀𝑅)π‘βˆ’1=(6)3=216,(4.8) we know 𝑓𝑒1,𝑒2,𝑒3≀(𝑀𝑅)π‘βˆ’1,0≀𝑒3≀3,0≀𝑒1≀14𝑒3,0≀𝑒2≀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<‖𝑣1β€–<3<‖𝑣2β€–. Then, by Lemma 2.4, 𝑣1(𝑑)=𝑣1(𝑑)+β„Ž(𝑑), 𝑣2(𝑑)=𝑣2(𝑑)+β„Ž(𝑑), π‘‘βˆˆ(βˆ’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).

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