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Abstract and Applied Analysis
VolumeΒ 2012Β (2012), Article IDΒ 963105, 13 pages
http://dx.doi.org/10.1155/2012/963105
Research Article

Existence of Three Solutions for a Nonlinear Fractional Boundary Value Problem via a Critical Points Theorem

Department of Mathematics, Huaiyin Normal University, Jiangsu, Huaian 223300, China

Received 13 May 2012; Revised 8 July 2012; Accepted 9 July 2012

Academic Editor: BashirΒ Ahmad

Copyright Β© 2012 Chuanzhi Bai. 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

This paper is concerned with the existence of three solutions to a nonlinear fractional boundary value problem as follows: (𝑑/𝑑𝑑)((1/2)0π·π‘‘π›Όβˆ’1(𝐢0𝐷𝛼𝑑𝑒(𝑑))βˆ’(1/2)π‘‘π·π‘‡π›Όβˆ’1(𝐢𝑑𝐷𝛼𝑇𝑒(𝑑)))+πœ†π‘Ž(𝑑)𝑓(𝑒(𝑑))=0,a.e.π‘‘βˆˆ[0,𝑇],𝑒(0)=𝑒(𝑇)=0, where π›Όβˆˆ(1/2,1], and πœ† is a positive real parameter. The approach is based on a critical-points theorem established by G. Bonanno.

1. Introduction

Differential equations with fractional order have recently proved to be strong tools in the modeling of many physical phenomena in various fields of physical, chemical, biology, engineering, and economics. There has been significant development in fractional differential equations, one can see the monographs [1–5] and the papers [6–20] and the references therein.

Critical-point theory, which proved to be very useful in determining the existence of solution for integer-order differential equation with some boundary conditions, for example, one can refer to [21–25]. But till now, there are few results on the solution to fractional boundary value problem which were established by the critical-point theory, since it is often very difficult to establish a suitable space and variational functional for fractional boundary value problem. Recently, Jiao and Zhou [26] investigated the following fractional boundary value problem:𝑑1𝑑𝑑20π·π‘‘βˆ’π›½ξ€·π‘’ξ…žξ€Έ+1(𝑑)2π‘‘π·π‘‡βˆ’π›½ξ€·π‘’ξ…žξ€Έξ‚[],(𝑑)+βˆ‡πΉ(𝑑,𝑒(𝑑))=0,a.e.π‘‘βˆˆ0,𝑇𝑒(0)=𝑒(𝑇)=0(1.1) by using the critical point theory, where 0π·π‘‘βˆ’π›½ and π‘‘π·π‘‡βˆ’π›½ are the left and right Riemann-Liouville fractional integrals of order 0≀𝛽<1, respectively, 𝐹∢[0,𝑇]×𝐑𝑁→𝐑 is a given function and βˆ‡πΉ(𝑑,π‘₯) is the gradient of 𝐹 at π‘₯.

In this paper, by using the critical-points theorem established by Bonanno in [27], a new approach is provided to investigate the existence of three solutions to the following fractional boundary value problems: 𝑑1𝑑𝑑20π·π‘‘π›Όβˆ’1𝐢0π·π‘Žπ‘‘ξ€Έβˆ’1𝑒(𝑑)2π‘‘π·π‘‡π›Όβˆ’1ξ€·πΆπ‘‘π·π‘Žπ‘‡ξ€Έξ‚[],𝑒(𝑑)+πœ†π‘Ž(𝑑)𝑓(𝑒(𝑑))=0,a.e.π‘‘βˆˆ0,𝑇𝑒(0)=𝑒(𝑇)=0,(1.2) where π›Όβˆˆ(1/2,1], 0π·π‘‘π›Όβˆ’1 and π‘‘π·π‘‡π›Όβˆ’1 are the left and right Riemann-Liouville fractional integrals of order 1βˆ’π›Ό respectively, 𝑐0𝐷𝛼𝑑 and 𝑐𝑑𝐷𝛼𝑇 are the left and right Caputo fractional derivatives of order 𝛼 respectively, πœ† is a positive real parameter, π‘“βˆΆπ‘β†’π‘ is a continuous function, and π‘ŽβˆΆπ‘β†’π‘ is a nonnegative continuous function with π‘Ž(𝑑)β‰’0.

2. Preliminaries

In this section, we first introduce some necessary definitions and properties of the fractional calculus which are used in this paper.

Definition 2.1 (see [5]). Let 𝑓 be a function defined on [π‘Ž,𝑏]. The left and right Riemann-Liouville fractional integrals of order 𝛼 for function 𝑓 denoted by π‘Žπ·π‘‘βˆ’π›Όπ‘“(𝑑) and π‘‘π·π‘βˆ’π›Όπ‘“(𝑑), respectively, are defined by π‘Žπ·π‘‘βˆ’π›Ό1𝑓(𝑑)=ξ€œΞ“(𝛼)π‘‘π‘Ž(π‘‘βˆ’π‘ )π›Όβˆ’1[]𝑓(𝑠)𝑑𝑠,π‘‘βˆˆπ‘Ž,𝑏,𝛼>0,π‘‘π·π‘βˆ’π›Όπ‘“1(𝑑)=ξ€œΞ“(𝛼)𝑏𝑑(π‘ βˆ’π‘‘)π›Όβˆ’1𝑓[](𝑠)𝑑𝑠,π‘‘βˆˆπ‘Ž,𝑏,𝛼>0,(2.1) provided the right-hand sides are pointwise defined on [π‘Ž,𝑏], where Ξ“(𝛼) is the gamma function.

Definition 2.2 (see [5]). Let 𝛾β‰₯0 and π‘›βˆˆπ.
(i) If π›Ύβˆˆ(π‘›βˆ’1,𝑛) and π‘“βˆˆπ΄πΆπ‘›([π‘Ž,𝑏],𝐑𝑁), then the left and right Caputo fractional derivatives of order 𝛾 for function 𝑓 denoted by πΆπ‘Žπ·π›Ύπ‘‘π‘“(𝑑) and 𝐢𝑑𝐷𝛾𝑏𝑓(𝑑), respectively, exist almost everywhere on [π‘Ž,𝑏], πΆπ‘Žπ·π›Ύπ‘‘π‘“(𝑑) and 𝐢𝑑𝐷𝛾𝑏𝑓(𝑑) are represented by πΆπ‘Žπ·π›Ύπ‘‘1𝑓(𝑑)=ξ€œΞ“(π‘›βˆ’π›Ύ)π‘‘π‘Ž(π‘‘βˆ’π‘ )π‘›βˆ’π›Ύβˆ’1𝑓(𝑛)([],𝑠)𝑑𝑠,π‘‘βˆˆπ‘Ž,𝑏𝐢𝑑𝐷𝛾𝑏𝑓(𝑑)=(βˆ’1)π‘›ξ€œΞ“(π‘›βˆ’π›Ύ)𝑏𝑑(π‘ βˆ’π‘‘)π‘›βˆ’π›Ύβˆ’1𝑓(𝑛)[],(𝑠)𝑑𝑠,π‘‘βˆˆπ‘Ž,𝑏(2.2) respectively.
(ii) If 𝛾=π‘›βˆ’1 and π‘“βˆˆπ΄πΆπ‘›βˆ’1([π‘Ž,𝑏],𝐑𝑁), then πΆπ‘Žπ·π‘‘π‘›βˆ’1𝑓(𝑑) and πΆπ‘‘π·π‘π‘›βˆ’1𝑓(𝑑) are represented by πΆπ‘Žπ·π‘‘π‘›βˆ’1𝑓(𝑑)=𝑓(π‘›βˆ’1)(𝑑),πΆπ‘‘π·π‘π‘›βˆ’1𝑓(𝑑)=(βˆ’1)(π‘›βˆ’1)𝑓(π‘›βˆ’1)[].(𝑑),π‘‘βˆˆπ‘Ž,𝑏(2.3)
With these definitions, we have the rule for fractional integration by parts, and the composition of the Riemann-Liouville fractional integration operator with the Caputo fractional differentiation operator, which were proved in [2, 5].

Property 1 (see [2, 5]). We have the following property of fractional integration: ξ€œπ‘π‘Žξ€Ίπ‘Žπ·π‘‘βˆ’π›Ύξ€»ξ€œπ‘“(𝑑)𝑔(𝑑)𝑑𝑑=π‘π‘Žξ€Ίπ‘‘π·π‘βˆ’π›Ύξ€»π‘”(𝑑)𝑓(𝑑)𝑑𝑑,𝛾>0(2.4) provided that π‘“βˆˆπΏπ‘([π‘Ž,𝑏],𝐑𝑁), π‘”βˆˆπΏπ‘ž([π‘Ž,𝑏],𝐑𝑁), and 𝑝β‰₯1, π‘žβ‰₯1, 1/𝑝+1/π‘žβ‰€1+𝛾 or 𝑝≠1, π‘žβ‰ 1, 1/𝑝+1/π‘ž=1+𝛾.

Property 2 (see [5]). Let π‘›βˆˆπ and π‘›βˆ’1<𝛾≀𝑛. If π‘“βˆˆπ΄πΆπ‘›([π‘Ž,𝑏],𝐑𝑁) or π‘“βˆˆπΆπ‘›([π‘Ž,𝑏],𝐑𝑁), then π‘Žπ·π‘‘βˆ’π›Ύξ€·πΆπ‘Žπ·π›Ύπ‘‘ξ€Έπ‘“(𝑑)=𝑓(𝑑)βˆ’π‘›βˆ’1𝑗=0𝑓(𝑗)(π‘Ž)𝑗!(π‘‘βˆ’π‘Ž)𝑗,π‘‘π·π‘βˆ’π›Ύξ€·πΆπ‘‘π·π›Ύπ‘π‘“ξ€Έ(𝑑)=𝑓(𝑑)βˆ’π‘›βˆ’1𝑗=0(βˆ’1)𝑗𝑓(𝑗)(𝑏)𝑗!(π‘βˆ’π‘‘)𝑗,(2.5) for π‘‘βˆˆ[π‘Ž,𝑏]. In particular, if 0<𝛾≀1 and π‘“βˆˆπ΄πΆ([π‘Ž,𝑏],𝐑𝑁) or π‘“βˆˆπΆ1([π‘Ž,𝑏],𝐑𝑁), then π‘Žπ·π‘‘βˆ’π›Ύξ€·πΆπ‘Žπ·π›Ύπ‘‘ξ€Έπ‘“(𝑑)=𝑓(𝑑)βˆ’π‘“(π‘Ž),π‘‘π·π‘βˆ’π›Ύξ€·πΆπ‘‘π·π›Ύπ‘ξ€Έπ‘“(𝑑)=𝑓(𝑑)βˆ’π‘“(𝑏).(2.6)

Remark 2.3. In view of Property 1 and Definition 2.2, it is obvious that π‘’βˆˆπ΄πΆ([0,𝑇]) is a solution of BVP (1.2) if and only if 𝑒 is a solution of the following problem: 𝑑1𝑑𝑑20π·π‘‘βˆ’π›½ξ€·π‘’ξ…žξ€Έ+1(𝑑)2π‘‘π·π‘‡βˆ’π›½ξ€·π‘’ξ…žξ€Έξ‚[],(𝑑)+πœ†π‘Ž(𝑑)𝑓(𝑒(𝑑))=0,a.e.π‘‘βˆˆ0,𝑇𝑒(0)=𝑒(𝑇)=0,(2.7) where 𝛽=2(1βˆ’π›Ό)∈[0,1).

In order to establish a variational structure for BVP (1.2), it is necessary to construct appropriate function spaces.

Denote by 𝐢∞0[0,𝑇] the set of all functions π‘”βˆˆπΆβˆž[0,𝑇] with 𝑔(0)=𝑔(𝑇)=0.

Definition 2.4 (see [26]). Let 0<𝛼≀1. The fractional derivative space 𝐸𝛼0 is defined by the closure of 𝐢∞0[0,𝑇] with respect to the norm ‖𝑒‖𝛼=ξ‚΅ξ€œπ‘‡0||𝐢0𝐷𝛼𝑑||𝑒(𝑑)2ξ€œπ‘‘π‘‘+𝑇0||||𝑒(𝑑)2𝑑𝑑1/2,βˆ€π‘’βˆˆπΈπ›Ό0.(2.8)

Remark 2.5. It is obvious that the fractional derivative space 𝐸𝛼0 is the space of functions π‘’βˆˆπΏ2[0,𝑇]having an 𝛼-order Caputo fractional derivative 𝐢0π·π›Όπ‘‘π‘’βˆˆπΏ2[0,𝑇] and 𝑒(0)=𝑒(𝑇)=0.

Proposition 2.6 (see [26]). Let 0<𝛼≀1. The fractional derivative space 𝐸𝛼0 is reflexive and separable Banach space.

Lemma 2.7 (see [26]). Let 1/2<𝛼≀1. For all π‘’βˆˆπΈπ›Ό0, one has the following:(i) ‖𝑒‖𝐿2≀𝑇𝛼Γ‖‖(𝛼+1)𝐢0𝐷𝛼𝑑𝑒‖‖𝐿2.(2.9)(ii) β€–π‘’β€–βˆžβ‰€π‘‡π›Όβˆ’1/2Ξ“(𝛼)(2(π›Όβˆ’1)+1)1/2‖‖𝐢0𝐷𝛼𝑑𝑒‖‖𝐿2.(2.10)

By (2.9), we can consider 𝐸𝛼0 with respect to the norm ‖𝑒‖𝛼=ξ‚΅ξ€œπ‘‡0||𝐢0𝐷𝛼𝑑||𝑒(𝑑)2𝑑𝑑1/2=‖‖𝐢0𝐷𝛼𝑑𝑒‖‖𝐿2,βˆ€π‘’βˆˆπΈπ›Ό0(2.11) in the following analysis.

Lemma 2.8 (see [26]). Let 1/2<𝛼≀1, then for all any π‘’βˆˆπΈπ›Ό0, one has ||||cos(πœ‹π›Ό)‖𝑒‖2π›Όξ€œβ‰€βˆ’π‘‡0𝐢0𝐷𝛼𝑑𝑒(𝑑)⋅𝐢𝑑𝐷𝛼𝑇1𝑒(𝑑)𝑑𝑑≀||||cos(πœ‹π›Ό)‖𝑒‖2𝛼.(2.12)

Our main tool is the critical-points theorem [27] which is recalled below.

Theorem 2.9 2.9(see [27]). Let 𝑋 be a separable and reflexive real Banach space; Ξ¦βˆΆπ‘‹β†’π‘ be a nonnegative continuously Gateaux differentiable and sequentially weakly lower semicontinuous functional whose Gateaux derivative admits a continuous inverse on π‘‹βˆ—; Ξ¨βˆΆπ‘‹β†’π‘ be a continuously Gateaux differentiable function whose Gateaux derivative is compact. Assume that there exists π‘₯0βˆˆπ‘‹ such that Ξ¦(π‘₯0)=Ξ¨(π‘₯0)=0, and that(i)limβ€–π‘₯β€–β†’+∞(Ξ¦(π‘₯)βˆ’πœ†Ξ¨(π‘₯))=+∞, for all πœ†βˆˆ[0,+∞]. Further, assume that there are π‘Ÿ>0, π‘₯1βˆˆπ‘‹ such that(ii)π‘Ÿ<Ξ¦(π‘₯1);(iii)supπ‘₯βˆˆΞ¦βˆ’1(]βˆ’βˆž,π‘Ÿ[)𝑀Ψ(π‘₯)<(π‘Ÿ/(π‘Ÿ+Ξ¦(π‘₯1)))Ξ¨(π‘₯1). Then, for each πœ†βˆˆΞ›1=⎀βŽ₯βŽ₯βŽ₯βŽ¦Ξ¦ξ€·π‘₯1ξ€ΈΞ¨ξ€·π‘₯1ξ€Έβˆ’supπ‘₯βˆˆΞ¦βˆ’1(]βˆ’βˆž,π‘Ÿ[)𝑀,π‘ŸΞ¨(π‘₯)supπ‘₯βˆˆΞ¦βˆ’1(]βˆ’βˆž,π‘Ÿ[)π‘€βŽ‘βŽ’βŽ’βŽ’βŽ£,Ξ¨(π‘₯)(2.13) the equation Ξ¦ξ…ž(π‘₯)βˆ’πœ†Ξ¨ξ…ž(π‘₯)=0(2.14) has at least three solutions in 𝑋 and, moreover, for each β„Ž>1, there exists an open interval Ξ›2βŠ‚βŽ‘βŽ’βŽ’βŽ£0,β„Žπ‘Ÿξ€·π‘Ÿξ€·Ξ¨ξ€·π‘₯1ξ€Έξ€·π‘₯/Ξ¦1ξ€Έξ€Έξ€Έβˆ’supπ‘₯βˆˆΞ¦βˆ’1(]βˆ’βˆž,π‘Ÿ[)π‘€Ξ¨βŽ€βŽ₯βŽ₯⎦(π‘₯)(2.15) and a positive real number 𝜎 such that, for each πœ†βˆˆΞ›2, (2.14) has at least three solutions in 𝑋 whose norms are less than 𝜎.

3. Main Result

For given π‘’βˆˆπΈπ›Ό0, we define functionals Ξ¦,Ξ¨βˆΆπΈπ›Όβ†’π‘ as follows: 1Ξ¦(𝑒)∢=βˆ’2ξ€œπ‘‡0𝐢0𝐷𝛼𝑑𝑒(𝑑)β‹…πΆπ‘‘π·π›Όπ‘‡ξ€œπ‘’(𝑑)𝑑𝑑,Ξ¨(𝑒)∢=𝑇0π‘Ž(𝑑)𝐹(𝑒(𝑑))𝑑𝑑,(3.1) where ∫𝐹(𝑒)=𝑒0𝑓(𝑠)𝑑𝑠. Clearly, Ξ¦ and Ξ¨ are Gateaux differentiable functional whose Gateaux derivative at the point π‘’βˆˆπΈπ›Ό0 are given by Ξ¦ξ…ž1(𝑒)𝑣=βˆ’2ξ€œπ‘‡0𝐢0𝐷𝛼𝑑𝑒(𝑑)⋅𝐢𝑑𝐷𝛼𝑇𝑣(𝑑)+𝐢𝑑𝐷𝛼𝑇𝑒(𝑑)⋅𝐢0𝐷𝛼𝑑Ψ𝑣(𝑑)𝑑𝑑,ξ…žξ€œ(𝑒)𝑣=𝑇0ξ€œπ‘Ž(𝑑)𝑓(𝑒(𝑑))𝑣(𝑑)𝑑𝑑=βˆ’π‘‡0ξ€œπ‘‘0π‘Ž(𝑠)𝑓(𝑒(𝑠))π‘‘π‘ β‹…π‘£ξ…ž(𝑑)𝑑𝑑,(3.2) for every π‘£βˆˆπΈπ›Ό0. By Definition 2.2 and Property 2, we have Ξ¦ξ…žξ€œ(𝑒)𝑣=𝑇0ξ‚€120π·π‘‘π›Όβˆ’1𝐢0π·π›Όπ‘‘ξ€Έβˆ’1𝑒(𝑑)2π‘‘π·π‘‡π›Όβˆ’1𝐢𝑑𝐷𝛼𝑇𝑒(𝑑)β‹…π‘£ξ…ž(𝑑)𝑑𝑑.(3.3) Hence, πΌπœ†=Ξ¦βˆ’πœ†Ξ¨βˆˆπΆ1(𝐸𝛼0,𝐑). If π‘’βˆ—βˆˆπΈπ›Ό0 is a critical point of πΌπœ†, then 0=πΌξ…žπœ†ξ€·π‘’βˆ—ξ€Έπ‘£=ξ€œπ‘‡0ξ‚΅120π·π‘‘π›Όβˆ’1𝐢0π·π›Όπ‘‘π‘’βˆ—ξ€Έβˆ’1(𝑑)2π‘‘π·π‘‡π›Όβˆ’1ξ€·πΆπ‘‘π·π›Όπ‘‡π‘’βˆ—ξ€Έξ€œ(𝑑)+πœ†π‘‘0ξ€·π‘’π‘Ž(𝑠)π‘“βˆ—ξ€Έξ‚Ά(𝑠)π‘‘π‘ β‹…π‘£ξ…ž(𝑑)𝑑𝑑,(3.4) for π‘£βˆˆπΈπ›Ό0. We can choose π‘£βˆˆπΈπ›Ό0 such that 𝑣(𝑑)=sin2π‘˜πœ‹π‘‘π‘‡or𝑣(𝑑)=1βˆ’cos2π‘˜πœ‹π‘‘π‘‡,π‘˜=1,2,….(3.5) The theory of Fourier series and (3.4) imply that 120π·π‘‘π›Όβˆ’1𝐢0π·π›Όπ‘‘π‘’βˆ—(ξ€Έβˆ’1𝑑)2π‘‘π·π‘‡π›Όβˆ’1ξ€·πΆπ‘‘π·π›Όπ‘‡π‘’βˆ—(ξ€Έξ€œπ‘‘)+πœ†π‘‘0ξ€·π‘’π‘Ž(𝑠)π‘“βˆ—(𝑠)𝑑𝑠=𝐢(3.6) a.e. on [0,𝑇] for some πΆβˆˆπ‘. By (3.6), it is easy to know that π‘’βˆ—βˆˆπΈπ›Ό0 is a solution of BVP (1.2).

By Lemma 2.7, if 𝛼>1/2, we have for each π‘’βˆˆπΈπ›Ό0 that β€–π‘’β€–βˆžξ‚΅ξ€œβ‰€Ξ©π‘‡0||𝐢0𝐷𝛼𝑑||𝑒(𝑑)2𝑑𝑑1/2=Ω‖𝑒‖𝛼,(3.7) where 𝑇Ω=π›Όβˆ’1/2βˆšΞ“(𝛼).2(π›Όβˆ’1)+1(3.8)

Given two constants 𝑐β‰₯0 and 𝑑≠0, with βˆšπ‘β‰ (2𝐴(𝛼)/|cos(πœ‹π›Ό)|)Ω⋅𝑑, where Ξ© as in (3.8).

For convenience, set 𝐴(𝛼)∢=8Ξ“2(2βˆ’π›Ό)𝑇Γ(4βˆ’2𝛼)1βˆ’2𝛼1+33βˆ’2𝛼24π›Όβˆ’5βˆ’22π›Όβˆ’3ξ€Έ.βˆ’1(3.9)

Theorem 3.1. Let π‘“βˆΆβ„β†’β„ be a continuous function,β€‰β€‰π‘ŽβˆΆπ‘β†’π‘ be a nonnegative continuous function with π‘Ž(𝑑)β‰’0, and 1/2<𝛼≀1. Put ∫𝐹(π‘₯)=π‘₯0𝑓(𝑠)𝑑𝑠 for every π‘₯βˆˆβ„, and assume that there exist four positive constants 𝑐,𝑑,πœ‡, and 𝑝, with βˆšπ‘<(2𝐴(𝛼)/|cos(πœ‹π›Ό)|)Ω⋅𝑑 and 𝑝<2, such that(H1)𝐹(π‘₯)β‰€πœ‡(1+|π‘₯|𝑝),  for  all π‘₯βˆˆβ„;(H2)𝐹(π‘₯)β‰₯0 for all π‘₯∈[0,Ξ“(2βˆ’π›Ό)𝑑], and ||||𝑐𝐹(π‘₯)<cos(πœ‹π›Ό)2ξ€·||||𝑐cos(πœ‹π›Ό)2+2Ξ©2𝐴(𝛼)𝑑2ξ€Έβˆ«π‘‡0Γ—ξ‚Έξ€œπ‘Ž(𝑑)𝑑𝑑𝐹(Ξ“(2βˆ’π›Ό)𝑑)3𝑇/4𝑇/4+π‘‡π‘Ž(𝑑)π‘‘π‘‘ξ€œ4Ξ“(2βˆ’π›Ό)𝑑0Ξ“(2βˆ’π›Ό)𝑑[],𝑏(𝑠)𝐹(𝑠)𝑑𝑠,βˆ€π‘₯βˆˆβˆ’π‘,𝑐(3.10)where 𝑏(𝑠)=π‘Ž((𝑇/4Ξ“(2βˆ’π›Ό)𝑑)𝑠)+π‘Ž(π‘‡βˆ’(𝑇/4Ξ“(2βˆ’π›Ό)𝑑)𝑠). Then, for each πœ†βˆˆΞ›1=𝐴(𝛼)𝑑2β„œπ‘Žβˆ«+β„œ0Ξ“(2βˆ’π›Ό)π‘‘βˆ«π‘(π‘₯)𝐹(π‘₯)𝑑π‘₯βˆ’π‘‡0π‘Ž(𝑑)𝑑𝑑⋅max|π‘₯|≀𝑐,𝑐𝐹(π‘₯)2||||cos(πœ‹π›Ό)2Ξ©2βˆ«π‘‡0π‘Ž(𝑑)𝑑𝑑⋅max|π‘₯|≀𝑐,𝐹(π‘₯)(3.11)where β„œπ‘Ž and β„œ denote ∫𝐹(Ξ“(2βˆ’π›Ό)𝑑)3𝑇/4𝑇/4π‘Ž(𝑑)𝑑𝑑 and 𝑇/(4Ξ“(2βˆ’π›Ό)𝑑) respectively,the problem (1.2) admits at least three solutions in 𝐸𝛼0 and, moreover, for each β„Ž>1, there exists an open interval Ξ›2βŠ‚ξƒ¬0,β„Žπ΄(𝛼)𝑑2β„œπ‘Žβˆ«+β„œ02Ξ“(2βˆ’π›Ό)𝑏(π‘₯)𝐹(π‘₯)𝑑π‘₯βˆ’2Ξ©2𝐴(𝛼)𝑑2/𝑐2||||ξ€Έβˆ«cos(πœ‹π›Ό)𝑇0π‘Ž(𝑑)𝑑𝑑⋅max|π‘₯|≀𝑐𝐹(π‘₯)(3.12) such that, for each πœ†βˆˆΞ›2, the problem (1.2) admits at least three solutions in 𝐸𝛼0 whose norms are less that 𝜎.

Proof. Let Ξ¦,Ξ¨ be the functionals defined in the above. By the Lemma 5.1 in [26], Ξ¦ is continuous and convex, hence it is weakly sequentially lower semicontinuous. Moreover, Ξ¦ is coercive, continuously Gateaux differentiable functional whose Gateaux derivative admits a continuous inverse on 𝐸𝛼0. The functional Ξ¨ is well defined, continuously Gateaux differentiable and with compact derivative. It is well known that the critical point of the functional Ξ¦βˆ’πœ†Ξ¨ in 𝐸𝛼0 is exactly the solution of BVP (1.2).

From (H1) and (2.12), we get lim‖𝑒‖𝛼→+∞(Ξ¦(𝑒)βˆ’πœ†Ξ¨(𝑒))=+∞,(3.13) for all πœ†βˆˆ [0,+∞[. Put 𝑒1⎧βŽͺβŽͺβŽͺ⎨βŽͺβŽͺβŽͺ⎩(𝑑)=4Ξ“(2βˆ’π›Ό)𝑑𝑇𝑇𝑑,π‘‘βˆˆ0,4,𝑇Γ(2βˆ’π›Ό)𝑑,π‘‘βˆˆ4,3𝑇4ξ‚„,4Ξ“(2βˆ’π›Ό)𝑑𝑇𝑇(π‘‡βˆ’π‘‘),π‘‘βˆˆ4ξ‚„.,𝑇(3.14) It is easy to check that 𝑒1(0)=𝑒1(𝑇)=0 and 𝑒1∈𝐿2[0,𝑇]. The direct calculation shows 𝐢0𝐷𝛼𝑑𝑒1⎧βŽͺβŽͺ⎨βŽͺβŽͺ⎩(𝑑)=4𝑑𝑇𝑑1βˆ’π›Όξ‚ƒπ‘‡,π‘‘βˆˆ0,4,4𝑑𝑇𝑑1βˆ’π›Όβˆ’ξ‚€π‘‡π‘‘βˆ’41βˆ’π›Όξ‚Άξ‚ƒπ‘‡,π‘‘βˆˆ4,3𝑇4ξ‚„,4𝑑𝑇𝑑1βˆ’π›Όβˆ’ξ‚€π‘‡π‘‘βˆ’41βˆ’π›Όβˆ’ξ‚€π‘‘βˆ’3𝑇41βˆ’π›Όξ‚Άξ‚„,π‘‘βˆˆ3𝑇4ξ‚„,‖‖𝑒,𝑇1β€–β€–2𝛼=ξ€œπ‘‡0𝐢0𝐷𝛼𝑑𝑒1ξ€Έ(𝑑)2ξ€œπ‘‘π‘‘=0𝑇/4+ξ€œ3𝑇/4𝑇/4+ξ€œπ‘‡3𝑇/4𝐢0𝐷𝛼𝑑𝑒1ξ€Έ(𝑑)2=𝑑𝑑16𝑑2𝑇2ξ‚Έξ€œπ‘‡0𝑑2(1βˆ’π›Ό)ξ€œπ‘‘π‘‘+𝑇𝑇/4ξ‚€π‘‡π‘‘βˆ’42(1βˆ’π›Ό)ξ€œπ‘‘π‘‘+𝑇3𝑇/4ξ‚€π‘‘βˆ’3𝑇42(1βˆ’π›Ό)ξ€œπ‘‘π‘‘βˆ’2𝑇𝑇/4𝑑1βˆ’π›Όξ‚€π‘‡π‘‘βˆ’41βˆ’π›Όξ€œπ‘‘π‘‘βˆ’2𝑇3𝑇/4𝑑1βˆ’π›Όξ‚€π‘‘βˆ’3𝑇41βˆ’π›Όξ€œπ‘‘π‘‘+2𝑇3𝑇/4ξ‚€π‘‡π‘‘βˆ’41βˆ’π›Όξ‚€π‘‘βˆ’3𝑇41βˆ’π›Όξ‚Ή=𝑑𝑑16𝑑2𝑇2ξ‚€3ξ‚Έξ‚΅1+43βˆ’2𝛼+ξ‚€143βˆ’2𝛼𝑇3βˆ’2π›Όξ€œ3βˆ’2π›Όβˆ’2𝑇𝑇/4𝑑1βˆ’π›Όξ‚€π‘‡π‘‘βˆ’41βˆ’π›Όξ€œπ‘‘π‘‘βˆ’2𝑇3𝑇/4𝑑1βˆ’π›Όξ‚€π‘‘βˆ’3𝑇41βˆ’π›Όξ€œπ‘‘π‘‘+2𝑇3𝑇/4ξ‚€π‘‡π‘‘βˆ’41βˆ’π›Όξ‚€π‘‘βˆ’3𝑇41βˆ’π›Όξ‚Ήπ‘‘π‘‘<∞.(3.15) That is, 𝐢0𝐷𝛼𝑑𝑒1∈𝐿2[0,𝑇]. Thus, 𝑒1βˆˆπΈπ›Ό0. Moreover, the direct calculation shows 𝐢𝑑𝐷𝛼𝑇𝑒1⎧βŽͺβŽͺ⎨βŽͺβŽͺ⎩(𝑑)=4𝑑𝑇(π‘‡βˆ’π‘‘)1βˆ’π›Όβˆ’ξ‚€3𝑇4ξ‚βˆ’π‘‘1βˆ’π›Όβˆ’ξ‚€π‘‡4ξ‚βˆ’π‘‘1βˆ’π›Όξ‚Άξ‚ƒπ‘‡,π‘‘βˆˆ0,4,4𝑑𝑇(π‘‡βˆ’π‘‘)1βˆ’π›Όβˆ’ξ‚€3𝑇4ξ‚βˆ’π‘‘1βˆ’π›Όξ‚Άξ‚ƒπ‘‡,π‘‘βˆˆ4,3𝑇4ξ‚„,4𝑑𝑇(π‘‡βˆ’π‘‘)1βˆ’π›Όξ‚„,π‘‘βˆˆ3𝑇4ξ‚„,Φ𝑒,𝑇1ξ€Έ1=βˆ’2ξ€œπ‘‡0𝐢0𝐷𝛼𝑑𝑒1(𝑑)⋅𝐢𝑑𝐷𝛼𝑇𝑒1(𝑑)𝑑𝑑=βˆ’8𝑑2𝑇2ξ‚Έξ€œ0𝑇/4𝑑1βˆ’π›Όξ‚΅(π‘‡βˆ’π‘‘)1βˆ’π›Όβˆ’ξ‚€3𝑇4ξ‚βˆ’π‘‘1βˆ’π›Όβˆ’ξ‚€π‘‡4ξ‚βˆ’π‘‘1βˆ’π›Όξ‚Ά+ξ€œπ‘‘π‘‘3𝑇/4𝑇/4𝑑1βˆ’π›Όβˆ’ξ‚€π‘‡π‘‘βˆ’41βˆ’π›Όξ‚Άξ‚΅(π‘‡βˆ’π‘‘)1βˆ’π›Όβˆ’ξ‚€3𝑇4ξ‚βˆ’π‘‘1βˆ’π›Όξ‚Ά+ξ€œπ‘‘π‘‘π‘‡3𝑇/4𝑑1βˆ’π›Όβˆ’ξ‚€π‘‡π‘‘βˆ’41βˆ’π›Όβˆ’ξ‚€π‘‘βˆ’3𝑇41βˆ’π›Όξ‚Ά(π‘‡βˆ’π‘‘)1βˆ’π›Όξ‚Ήπ‘‘π‘‘=βˆ’8𝑑2𝑇2ξ‚Έξ€œπ‘‡0𝑑1βˆ’π›Ό(π‘‡βˆ’π‘‘)1βˆ’π›Όξ€œπ‘‘π‘‘βˆ’0𝑇/4𝑑1βˆ’π›Όξ‚€π‘‡4ξ‚βˆ’π‘‘1βˆ’π›Ό+ξ€œπ‘‘π‘‘3𝑇/4𝑇/4ξ‚€π‘‡π‘‘βˆ’41βˆ’π›Όξ‚€3𝑇4ξ‚βˆ’π‘‘1βˆ’π›Όξ€œπ‘‘π‘‘βˆ’π‘‡3𝑇/4ξ‚€π‘‘βˆ’3𝑇41βˆ’π›Ό(π‘‡βˆ’π‘‘)1βˆ’π›Όβˆ’ξ€œπ‘‘π‘‘03𝑇/4𝑑1βˆ’π›Όξ‚€3𝑇4ξ‚βˆ’π‘‘1βˆ’π›Όβˆ’ξ€œπ‘‡π‘‡/4ξ‚€π‘‡π‘‘βˆ’41βˆ’π›Ό(π‘‡βˆ’π‘‘)1βˆ’π›Όξ‚Ή=𝑑𝑑8Ξ“2(2βˆ’π›Ό)𝑇Γ(4βˆ’2𝛼)1βˆ’2𝛼𝑑2ξ€·ξ€·1+33βˆ’2𝛼24π›Όβˆ’5βˆ’22π›Όβˆ’3ξ€Έβˆ’1=𝐴(𝛼)𝑑2,Ψ𝑒1ξ€Έ=ξ€œπ‘‡0π‘Žξ€·π‘’(𝑑)𝐹1ξ€Έ=ξ€œ(𝑑)𝑑𝑑0𝑇/4π‘Žξ‚΅(𝑑)𝐹4Ξ“(2βˆ’π›Ό)π‘‘π‘‡π‘‘ξ‚Άξ€œπ‘‘π‘‘+3𝑇/4𝑇/4π‘Ž+ξ€œ(𝑑)𝐹(Ξ“(2βˆ’π›Ό)𝑑)𝑑𝑑𝑇3𝑇/4ξ‚΅π‘Ž(𝑑)𝐹4Ξ“(2βˆ’π›Ό)π‘‘π‘‡ξ‚Άξ€œ(π‘‡βˆ’π‘‘)𝑑𝑑=𝐹(Ξ“(2βˆ’π›Ό)𝑑)3𝑇/4𝑇/4π‘‡π‘Ž(𝑑)𝑑𝑑+ξ€œ4Ξ“(2βˆ’π›Ό)𝑑0Ξ“(2βˆ’π›Ό)𝑑𝑏(π‘₯)𝐹(π‘₯)𝑑π‘₯.(3.16)

Let π‘Ÿ=(|cos(πœ‹π›Ό)|/2Ξ©2)𝑐2. Since βˆšπ‘<(2𝐴(𝛼)/|cos(πœ‹π›Ό)|)Ω⋅𝑑, we obtain π‘Ÿ<Ξ¦(𝑒1).

By (2.12) and (3.7), one has Ξ¦(𝑒)β‰€π‘Ÿβ‡’β€–π‘’β€–βˆžβ‰€π‘. Thus, supπ‘’βˆˆΞ¦βˆ’1(][)βˆ’βˆž,π‘Ÿπ‘€Ξ¨(𝑒)=supπ‘’βˆˆΞ¦βˆ’1(]])βˆ’βˆž,π‘ŸΞ¨(𝑒)≀max|π‘₯|β‰€π‘ξ€œπΉ(π‘₯)𝑇0π‘Ž(𝑑)𝑑𝑑.(3.17)

Moreover, we have π‘Ÿξ€·π‘’π‘Ÿ+Ξ¦1Ψ𝑒1ξ€Έ=ξ€·||||cos(πœ‹π›Ό)/2Ξ©2𝑐2ξ€·||||cos(πœ‹π›Ό)/2Ξ©2𝑐2+𝐴(𝛼)𝑑2Γ—ξ‚Έξ€œπΉ(Ξ“(2βˆ’π›Ό)𝑑)3𝑇/4𝑇/4π‘‡π‘Ž(𝑑)𝑑𝑑+ξ€œ4Ξ“(2βˆ’π›Ό)𝑑0Ξ“(2βˆ’π›Ό)𝑑=||||𝑐𝑏(π‘₯)𝐹(π‘₯)𝑑π‘₯cos(πœ‹π›Ό)2||||𝑐cos(πœ‹π›Ό)2+2Ξ©2𝐴(𝛼)𝑑2Γ—ξ‚Έξ€œπΉ(Ξ“(2βˆ’π›Ό)𝑑)3𝑇/4𝑇/4π‘‡π‘Ž(𝑑)𝑑𝑑+ξ€œ4Ξ“(2βˆ’π›Ό)𝑑0Ξ“(2βˆ’π›Ό)𝑑.𝑏(π‘₯)𝐹(π‘₯)𝑑π‘₯(3.18)

Hence, from (H2) one has supπ‘’βˆˆΞ¦βˆ’1(][)βˆ’βˆž,π‘Ÿπ‘€Ξ¨π‘Ÿ(𝑒)<ξ€·π‘’π‘Ÿ+Ξ¦1Ψ𝑒1ξ€Έ.(3.19)

Now, taking into account that Φ𝑒1Ψ𝑒1ξ€Έβˆ’supπ‘’βˆˆΞ¦βˆ’1(]βˆ’βˆž,π‘Ÿ[)𝑀≀𝐴Ψ(𝑒)(𝛼)𝑑2β„œπ‘Žβˆ«+β„œ0Ξ“(2βˆ’π›Ό)π‘‘βˆ«π‘(π‘₯)𝐹(π‘₯)𝑑π‘₯βˆ’π‘‡0π‘Ž(𝑑)𝑑𝑑⋅max|π‘₯|≀𝑐,π‘ŸπΉ(π‘₯)supπ‘’βˆˆΞ¦βˆ’1(]βˆ’βˆž,π‘Ÿ[)𝑀β‰₯𝑐Ψ(𝑒)2||||cos(πœ‹π›Ό)2Ξ©2βˆ«π‘‡0π‘Ž(𝑑)𝑑𝑑⋅max|π‘₯|≀𝑐𝐹,(π‘₯)β„Žπ‘Ÿπ‘Ÿξ€·Ξ¨ξ€·π‘’1𝑒/Ξ¦1ξ€Έξ€Έβˆ’supπ‘’βˆˆΞ¦βˆ’1(]βˆ’βˆž,π‘Ÿ[)𝑀Ψ≀(𝑒)β„Žπ΄(𝛼)𝑑2β„œπ‘Žβˆ«+β„œ02Ξ“(2βˆ’π›Ό)𝑏(π‘₯)𝐹(π‘₯)𝑑π‘₯βˆ’2Ξ©2𝐴(𝛼)𝑑2/𝑐2||||ξ€Έβˆ«cos(πœ‹π›Ό)𝑇0π‘Ž(𝑑)𝑑𝑑⋅max|π‘₯|≀𝑐𝐹(π‘₯)=π‘š.(3.20) Thus, by Theorem 2.9 it follows that, for each πœ†βˆˆΞ›1, BVP (1.2) admits at least three solutions, and there exists an open interval Ξ›2βŠ‚[0,π‘š] and a real positive number 𝜎 such that, for each πœ†βˆˆΞ›2, BVP (1.2) admits at least three solutions in 𝐸𝛼0 whose norms are less than 𝜎.

Finally, we give an example to show the effectiveness of the results obtained here.

Let 𝛼=0.8, 𝑇=1, π‘Ž(𝑑)≑1, and 𝑓(𝑒)=π‘’βˆ’π‘’π‘’8√(9βˆ’π‘’)+𝑒. Then BVP (1.2) reduces to the following boundary value problem: 𝑑1𝑑𝑑20π·π‘‘βˆ’0.2𝐢0𝐷𝑑0.8ξ‚βˆ’1𝑒(𝑑)2𝑑𝐷1βˆ’0.2𝐢𝑑𝐷10.8𝑒𝑒(𝑑)+πœ†βˆ’π‘’π‘’8√(9βˆ’π‘’)+𝑒[],=0,a.e.π‘‘βˆˆ0,1𝑒(0)=𝑒(1)=0.(3.21)

Example 3.2. Owing to Theorem 3.1, for each πœ†βˆˆ]0.291,0.318[, BVP (3.21) admits at least three solutions. In fact, put 𝑐=1 and 𝑑=2, it is easy to calculate that Ξ©=1.1089, 𝐴(0.8)=1.3313, and ξƒŽ2𝐴(0.8)||||cos(0.8πœ‹)Ω⋅𝑑=4.0235>1=𝑐.(3.22) Sinceξ€œπΉ(π‘₯)=π‘₯0𝑓(𝑠)𝑑𝑠=π‘’βˆ’π‘₯π‘₯9+23π‘₯3/2,(3.23) we have that condition (H1) holds. Moreover, 𝐹(π‘₯)β‰₯0 for each π‘₯∈[0,2Ξ“(1.2)], and ||||cos(0.8πœ‹)||||cos(0.8πœ‹)+2Ξ©2𝐴(0.8)β‹…22ξ‚Έ121𝐹(2Ξ“(1.2))+ξ€œ4Ξ“(1.2)02Ξ“(1.2)𝐹(𝑠)𝑑𝑠>1.064>1.0345=π‘’βˆ’1+23β‰₯𝐹(π‘₯),|π‘₯|≀1,(3.24) which implies that condition (H2) holds. Thus, by Theorem 3.1, for each πœ†βˆˆ]0.291,0.318[, the problem (3.21) admits at least three nontrivial solutions in 𝐸00.8. Moreover, for each β„Ž>1, there exists an open interval Ξ›βŠ‚]0,3.4674β„Ž[ and a real positive number 𝜎 such that, for each πœ†βˆˆΞ›, the problem (3.21) admits at least three solutions in 𝐸00.8 whose norms are less than 𝜎.

Acknowledgments

The author thanks the reviewers for their suggestions and comments which improved the presentation of this paper. This work is supported by Natural Science Foundation of Jiangsu Province (BK2011407) and Natural Science Foundation of China (10771212).

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