Abstract

This paper is concerned with the existence of three solutions to a nonlinear fractional boundary value problem as follows: (𝑑/𝑑𝑡)((1/2)0𝐷𝑡𝛼1(𝐶0𝐷𝛼𝑡𝑢(𝑡))(1/2)𝑡𝐷𝑇𝛼1(𝐶𝑡𝐷𝛼𝑇𝑢(𝑡)))+𝜆𝑎(𝑡)𝑓(𝑢(𝑡))=0,a.e.𝑡[0,𝑇],𝑢(0)=𝑢(𝑇)=0, where 𝛼(1/2,1], and 𝜆 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 [15] and the papers [620] 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 [2125]. 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:𝑑1𝑑𝑡20𝐷𝑡𝛽𝑢+1(𝑡)2𝑡𝐷𝑇𝛽𝑢[],(𝑡)+𝐹(𝑡,𝑢(𝑡))=0,a.e.𝑡0,𝑇𝑢(0)=𝑢(𝑇)=0(1.1) by using the critical point theory, where 0𝐷𝑡𝛽 and 𝑡𝐷𝑇𝛽 are the left and right Riemann-Liouville fractional integrals of order 0𝛽<1, respectively, 𝐹[0,𝑇]×𝐑𝑁𝐑 is a given function and 𝐹(𝑡,𝑥) is the gradient of 𝐹 at 𝑥.

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: 𝑑1𝑑𝑡20𝐷𝑡𝛼1𝐶0𝐷𝑎𝑡1𝑢(𝑡)2𝑡𝐷𝑇𝛼1𝐶𝑡𝐷𝑎𝑇[],𝑢(𝑡)+𝜆𝑎(𝑡)𝑓(𝑢(𝑡))=0,a.e.𝑡0,𝑇𝑢(0)=𝑢(𝑇)=0,(1.2) where 𝛼(1/2,1], 0𝐷𝑡𝛼1 and 𝑡𝐷𝑇𝛼1 are the left and right Riemann-Liouville fractional integrals of order 1𝛼 respectively, 𝑐0𝐷𝛼𝑡 and 𝑐𝑡𝐷𝛼𝑇 are the left and right Caputo fractional derivatives of order 𝛼 respectively, 𝜆 is a positive real parameter, 𝑓𝐑𝐑 is a continuous function, and 𝑎𝐑𝐑 is a nonnegative continuous function with 𝑎(𝑡)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 [𝑎,𝑏]. The left and right Riemann-Liouville fractional integrals of order 𝛼 for function 𝑓 denoted by 𝑎𝐷𝑡𝛼𝑓(𝑡) and 𝑡𝐷𝑏𝛼𝑓(𝑡), respectively, are defined by 𝑎𝐷𝑡𝛼1𝑓(𝑡)=Γ(𝛼)𝑡𝑎(𝑡𝑠)𝛼1[]𝑓(𝑠)𝑑𝑠,𝑡𝑎,𝑏,𝛼>0,𝑡𝐷𝑏𝛼𝑓1(𝑡)=Γ(𝛼)𝑏𝑡(𝑠𝑡)𝛼1𝑓[](𝑠)𝑑𝑠,𝑡𝑎,𝑏,𝛼>0,(2.1) provided the right-hand sides are pointwise defined on [𝑎,𝑏], where Γ(𝛼) is the gamma function.

Definition 2.2 (see [5]). Let 𝛾0 and 𝑛𝐍.
(i) If 𝛾(𝑛1,𝑛) and 𝑓𝐴𝐶𝑛([𝑎,𝑏],𝐑𝑁), then the left and right Caputo fractional derivatives of order 𝛾 for function 𝑓 denoted by 𝐶𝑎𝐷𝛾𝑡𝑓(𝑡) and 𝐶𝑡𝐷𝛾𝑏𝑓(𝑡), respectively, exist almost everywhere on [𝑎,𝑏], 𝐶𝑎𝐷𝛾𝑡𝑓(𝑡) and 𝐶𝑡𝐷𝛾𝑏𝑓(𝑡) are represented by 𝐶𝑎𝐷𝛾𝑡1𝑓(𝑡)=Γ(𝑛𝛾)𝑡𝑎(𝑡𝑠)𝑛𝛾1𝑓(𝑛)([],𝑠)𝑑𝑠,𝑡𝑎,𝑏𝐶𝑡𝐷𝛾𝑏𝑓(𝑡)=(1)𝑛Γ(𝑛𝛾)𝑏𝑡(𝑠𝑡)𝑛𝛾1𝑓(𝑛)[],(𝑠)𝑑𝑠,𝑡𝑎,𝑏(2.2) respectively.
(ii) If 𝛾=𝑛1 and 𝑓𝐴𝐶𝑛1([𝑎,𝑏],𝐑𝑁), then 𝐶𝑎𝐷𝑡𝑛1𝑓(𝑡) and 𝐶𝑡𝐷𝑏𝑛1𝑓(𝑡) are represented by 𝐶𝑎𝐷𝑡𝑛1𝑓(𝑡)=𝑓(𝑛1)(𝑡),𝐶𝑡𝐷𝑏𝑛1𝑓(𝑡)=(1)(𝑛1)𝑓(𝑛1)[].(𝑡),𝑡𝑎,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: 𝑏𝑎𝑎𝐷𝑡𝛾𝑓(𝑡)𝑔(𝑡)𝑑𝑡=𝑏𝑎𝑡𝐷𝑏𝛾𝑔(𝑡)𝑓(𝑡)𝑑𝑡,𝛾>0(2.4) provided that 𝑓𝐿𝑝([𝑎,𝑏],𝐑𝑁), 𝑔𝐿𝑞([𝑎,𝑏],𝐑𝑁), and 𝑝1, 𝑞1, 1/𝑝+1/𝑞1+𝛾 or 𝑝1, 𝑞1, 1/𝑝+1/𝑞=1+𝛾.

Property 2 (see [5]). Let 𝑛𝐍 and 𝑛1<𝛾𝑛. If 𝑓𝐴𝐶𝑛([𝑎,𝑏],𝐑𝑁) or 𝑓𝐶𝑛([𝑎,𝑏],𝐑𝑁), then 𝑎𝐷𝑡𝛾𝐶𝑎𝐷𝛾𝑡𝑓(𝑡)=𝑓(𝑡)𝑛1𝑗=0𝑓(𝑗)(𝑎)𝑗!(𝑡𝑎)𝑗,𝑡𝐷𝑏𝛾𝐶𝑡𝐷𝛾𝑏𝑓(𝑡)=𝑓(𝑡)𝑛1𝑗=0(1)𝑗𝑓(𝑗)(𝑏)𝑗!(𝑏𝑡)𝑗,(2.5) for 𝑡[𝑎,𝑏]. In particular, if 0<𝛾1 and 𝑓𝐴𝐶([𝑎,𝑏],𝐑𝑁) or 𝑓𝐶1([𝑎,𝑏],𝐑𝑁), then 𝑎𝐷𝑡𝛾𝐶𝑎𝐷𝛾𝑡𝑓(𝑡)=𝑓(𝑡)𝑓(𝑎),𝑡𝐷𝑏𝛾𝐶𝑡𝐷𝛾𝑏𝑓(𝑡)=𝑓(𝑡)𝑓(𝑏).(2.6)

Remark 2.3. In view of Property 1 and Definition 2.2, it is obvious that 𝑢𝐴𝐶([0,𝑇]) is a solution of BVP (1.2) if and only if 𝑢 is a solution of the following problem: 𝑑1𝑑𝑡20𝐷𝑡𝛽𝑢+1(𝑡)2𝑡𝐷𝑇𝛽𝑢[],(𝑡)+𝜆𝑎(𝑡)𝑓(𝑢(𝑡))=0,a.e.𝑡0,𝑇𝑢(0)=𝑢(𝑇)=0,(2.7) where 𝛽=2(1𝛼)[0,1).

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

Denote by 𝐶0[0,𝑇] the set of all functions 𝑔𝐶[0,𝑇] with 𝑔(0)=𝑔(𝑇)=0.

Definition 2.4 (see [26]). Let 0<𝛼1. The fractional derivative space 𝐸𝛼0 is defined by the closure of 𝐶0[0,𝑇] with respect to the norm 𝑢𝛼=𝑇0||𝐶0𝐷𝛼𝑡||𝑢(𝑡)2𝑑𝑡+𝑇0||||𝑢(𝑡)2𝑑𝑡1/2,𝑢𝐸𝛼0.(2.8)

Remark 2.5. It is obvious that the fractional derivative space 𝐸𝛼0 is the space of functions 𝑢𝐿2[0,𝑇]having an 𝛼-order Caputo fractional derivative 𝐶0𝐷𝛼𝑡𝑢𝐿2[0,𝑇] and 𝑢(0)=𝑢(𝑇)=0.

Proposition 2.6 (see [26]). Let 0<𝛼1. The fractional derivative space 𝐸𝛼0 is reflexive and separable Banach space.

Lemma 2.7 (see [26]). Let 1/2<𝛼1. For all 𝑢𝐸𝛼0, one has the following:(i) 𝑢𝐿2𝑇𝛼Γ(𝛼+1)𝐶0𝐷𝛼𝑡𝑢𝐿2.(2.9)(ii) 𝑢𝑇𝛼1/2Γ(𝛼)(2(𝛼1)+1)1/2𝐶0𝐷𝛼𝑡𝑢𝐿2.(2.10)

By (2.9), we can consider 𝐸𝛼0 with respect to the norm 𝑢𝛼=𝑇0||𝐶0𝐷𝛼𝑡||𝑢(𝑡)2𝑑𝑡1/2=𝐶0𝐷𝛼𝑡𝑢𝐿2,𝑢𝐸𝛼0(2.11) in the following analysis.

Lemma 2.8 (see [26]). Let 1/2<𝛼1, then for all any 𝑢𝐸𝛼0, one has ||||cos(𝜋𝛼)𝑢2𝛼𝑇0𝐶0𝐷𝛼𝑡𝑢(𝑡)𝐶𝑡𝐷𝛼𝑇1𝑢(𝑡)𝑑𝑡||||cos(𝜋𝛼)𝑢2𝛼.(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; Φ𝑋𝐑 be a nonnegative continuously Gateaux differentiable and sequentially weakly lower semicontinuous functional whose Gateaux derivative admits a continuous inverse on 𝑋; Ψ𝑋𝐑 be a continuously Gateaux differentiable function whose Gateaux derivative is compact. Assume that there exists 𝑥0𝑋 such that Φ(𝑥0)=Ψ(𝑥0)=0, and that(i)lim𝑥+(Φ(𝑥)𝜆Ψ(𝑥))=+, for all 𝜆[0,+]. Further, assume that there are 𝑟>0, 𝑥1𝑋 such that(ii)𝑟<Φ(𝑥1);(iii)sup𝑥Φ1(],𝑟[)𝑤Ψ(𝑥)<(𝑟/(𝑟+Φ(𝑥1)))Ψ(𝑥1). Then, for each 𝜆Λ1=Φ𝑥1Ψ𝑥1sup𝑥Φ1(],𝑟[)𝑤,𝑟Ψ(𝑥)sup𝑥Φ1(],𝑟[)𝑤,Ψ(𝑥)(2.13) the equation Φ(𝑥)𝜆Ψ(𝑥)=0(2.14) has at least three solutions in 𝑋 and, moreover, for each >1, there exists an open interval Λ20,𝑟𝑟Ψ𝑥1𝑥/Φ1sup𝑥Φ1(],𝑟[)𝑤Ψ(𝑥)(2.15) and a positive real number 𝜎 such that, for each 𝜆Λ2, (2.14) has at least three solutions in 𝑋 whose norms are less than 𝜎.

3. Main Result

For given 𝑢𝐸𝛼0, we define functionals Φ,Ψ𝐸𝛼𝐑 as follows: 1Φ(𝑢)=2𝑇0𝐶0𝐷𝛼𝑡𝑢(𝑡)𝐶𝑡𝐷𝛼𝑇𝑢(𝑡)𝑑𝑡,Ψ(𝑢)=𝑇0𝑎(𝑡)𝐹(𝑢(𝑡))𝑑𝑡,(3.1) where 𝐹(𝑢)=𝑢0𝑓(𝑠)𝑑𝑠. Clearly, Φ and Ψ are Gateaux differentiable functional whose Gateaux derivative at the point 𝑢𝐸𝛼0 are given by Φ1(𝑢)𝑣=2𝑇0𝐶0𝐷𝛼𝑡𝑢(𝑡)𝐶𝑡𝐷𝛼𝑇𝑣(𝑡)+𝐶𝑡𝐷𝛼𝑇𝑢(𝑡)𝐶0𝐷𝛼𝑡Ψ𝑣(𝑡)𝑑𝑡,(𝑢)𝑣=𝑇0𝑎(𝑡)𝑓(𝑢(𝑡))𝑣(𝑡)𝑑𝑡=𝑇0𝑡0𝑎(𝑠)𝑓(𝑢(𝑠))𝑑𝑠𝑣(𝑡)𝑑𝑡,(3.2) for every 𝑣𝐸𝛼0. By Definition 2.2 and Property 2, we have Φ(𝑢)𝑣=𝑇0120𝐷𝑡𝛼1𝐶0𝐷𝛼𝑡1𝑢(𝑡)2𝑡𝐷𝑇𝛼1𝐶𝑡𝐷𝛼𝑇𝑢(𝑡)𝑣(𝑡)𝑑𝑡.(3.3) Hence, 𝐼𝜆=Φ𝜆Ψ𝐶1(𝐸𝛼0,𝐑). If 𝑢𝐸𝛼0 is a critical point of 𝐼𝜆, then 0=𝐼𝜆𝑢𝑣=𝑇0120𝐷𝑡𝛼1𝐶0𝐷𝛼𝑡𝑢1(𝑡)2𝑡𝐷𝑇𝛼1𝐶𝑡𝐷𝛼𝑇𝑢(𝑡)+𝜆𝑡0𝑢𝑎(𝑠)𝑓(𝑠)𝑑𝑠𝑣(𝑡)𝑑𝑡,(3.4) for 𝑣𝐸𝛼0. We can choose 𝑣𝐸𝛼0 such that 𝑣(𝑡)=sin2𝑘𝜋𝑡𝑇or𝑣(𝑡)=1cos2𝑘𝜋𝑡𝑇,𝑘=1,2,.(3.5) The theory of Fourier series and (3.4) imply that 120𝐷𝑡𝛼1𝐶0𝐷𝛼𝑡𝑢(1𝑡)2𝑡𝐷𝑇𝛼1𝐶𝑡𝐷𝛼𝑇𝑢(𝑡)+𝜆𝑡0𝑢𝑎(𝑠)𝑓(𝑠)𝑑𝑠=𝐶(3.6) a.e. on [0,𝑇] for some 𝐶𝐑. By (3.6), it is easy to know that 𝑢𝐸𝛼0 is a solution of BVP (1.2).

By Lemma 2.7, if 𝛼>1/2, we have for each 𝑢𝐸𝛼0 that 𝑢Ω𝑇0||𝐶0𝐷𝛼𝑡||𝑢(𝑡)2𝑑𝑡1/2=Ω𝑢𝛼,(3.7) where 𝑇Ω=𝛼1/2Γ(𝛼).2(𝛼1)+1(3.8)

Given two constants 𝑐0 and 𝑑0, with 𝑐(2𝐴(𝛼)/|cos(𝜋𝛼)|)Ω𝑑, where Ω as in (3.8).

For convenience, set 𝐴(𝛼)=8Γ2(2𝛼)𝑇Γ(42𝛼)12𝛼1+332𝛼24𝛼522𝛼3.1(3.9)

Theorem 3.1. Let 𝑓 be a continuous function,  𝑎𝐑𝐑 be a nonnegative continuous function with 𝑎(𝑡)0, and 1/2<𝛼1. Put 𝐹(𝑥)=𝑥0𝑓(𝑠)𝑑𝑠 for every 𝑥, and assume that there exist four positive constants 𝑐,𝑑,𝜇, and 𝑝, with 𝑐<(2𝐴(𝛼)/|cos(𝜋𝛼)|)Ω𝑑 and 𝑝<2, such that(H1)𝐹(𝑥)𝜇(1+|𝑥|𝑝),  for  all 𝑥;(H2)𝐹(𝑥)0 for all 𝑥[0,Γ(2𝛼)𝑑], and ||||𝑐𝐹(𝑥)<cos(𝜋𝛼)2||||𝑐cos(𝜋𝛼)2+2Ω2𝐴(𝛼)𝑑2𝑇0×𝑎(𝑡)𝑑𝑡𝐹(Γ(2𝛼)𝑑)3𝑇/4𝑇/4+𝑇𝑎(𝑡)𝑑𝑡4Γ(2𝛼)𝑑0Γ(2𝛼)𝑑[],𝑏(𝑠)𝐹(𝑠)𝑑𝑠,𝑥𝑐,𝑐(3.10)where 𝑏(𝑠)=𝑎((𝑇/4Γ(2𝛼)𝑑)𝑠)+𝑎(𝑇(𝑇/4Γ(2𝛼)𝑑)𝑠). Then, for each 𝜆Λ1=𝐴(𝛼)𝑑2𝑎+0Γ(2𝛼)𝑑𝑏(𝑥)𝐹(𝑥)𝑑𝑥𝑇0𝑎(𝑡)𝑑𝑡max|𝑥|𝑐,𝑐𝐹(𝑥)2||||cos(𝜋𝛼)2Ω2𝑇0𝑎(𝑡)𝑑𝑡max|𝑥|𝑐,𝐹(𝑥)(3.11)where 𝑎 and denote 𝐹(Γ(2𝛼)𝑑)3𝑇/4𝑇/4𝑎(𝑡)𝑑𝑡 and 𝑇/(4Γ(2𝛼)𝑑) respectively,the problem (1.2) admits at least three solutions in 𝐸𝛼0 and, moreover, for each >1, there exists an open interval Λ20,𝐴(𝛼)𝑑2𝑎+02Γ(2𝛼)𝑏(𝑥)𝐹(𝑥)𝑑𝑥2Ω2𝐴(𝛼)𝑑2/𝑐2||||cos(𝜋𝛼)𝑇0𝑎(𝑡)𝑑𝑡max|𝑥|𝑐𝐹(𝑥)(3.12) such that, for each 𝜆Λ2, the problem (1.2) admits at least three solutions in 𝐸𝛼0 whose norms are less that 𝜎.

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

From (H1) and (2.12), we get lim𝑢𝛼+(Φ(𝑢)𝜆Ψ(𝑢))=+,(3.13) for all 𝜆 [0,+[. Put 𝑢1(𝑡)=4Γ(2𝛼)𝑑𝑇𝑇𝑡,𝑡0,4,𝑇Γ(2𝛼)𝑑,𝑡4,3𝑇4,4Γ(2𝛼)𝑑𝑇𝑇(𝑇𝑡),𝑡4.,𝑇(3.14) It is easy to check that 𝑢1(0)=𝑢1(𝑇)=0 and 𝑢1𝐿2[0,𝑇]. The direct calculation shows 𝐶0𝐷𝛼𝑡𝑢1(𝑡)=4𝑑𝑇𝑡1𝛼𝑇,𝑡0,4,4𝑑𝑇𝑡1𝛼𝑇𝑡41𝛼𝑇,𝑡4,3𝑇4,4𝑑𝑇𝑡1𝛼𝑇𝑡41𝛼𝑡3𝑇41𝛼,𝑡3𝑇4,𝑢,𝑇12𝛼=𝑇0𝐶0𝐷𝛼𝑡𝑢1(𝑡)2𝑑𝑡=0𝑇/4+3𝑇/4𝑇/4+𝑇3𝑇/4𝐶0𝐷𝛼𝑡𝑢1(𝑡)2=𝑑𝑡16𝑑2𝑇2𝑇0𝑡2(1𝛼)𝑑𝑡+𝑇𝑇/4𝑇𝑡42(1𝛼)𝑑𝑡+𝑇3𝑇/4𝑡3𝑇42(1𝛼)𝑑𝑡2𝑇𝑇/4𝑡1𝛼𝑇𝑡41𝛼𝑑𝑡2𝑇3𝑇/4𝑡1𝛼𝑡3𝑇41𝛼𝑑𝑡+2𝑇3𝑇/4𝑇𝑡41𝛼𝑡3𝑇41𝛼=𝑑𝑡16𝑑2𝑇231+432𝛼+1432𝛼𝑇32𝛼32𝛼2𝑇𝑇/4𝑡1𝛼𝑇𝑡41𝛼𝑑𝑡2𝑇3𝑇/4𝑡1𝛼𝑡3𝑇41𝛼𝑑𝑡+2𝑇3𝑇/4𝑇𝑡41𝛼𝑡3𝑇41𝛼𝑑𝑡<.(3.15) That is, 𝐶0𝐷𝛼𝑡𝑢1𝐿2[0,𝑇]. Thus, 𝑢1𝐸𝛼0. Moreover, the direct calculation shows 𝐶𝑡𝐷𝛼𝑇𝑢1(𝑡)=4𝑑𝑇(𝑇𝑡)1𝛼3𝑇4𝑡1𝛼𝑇4𝑡1𝛼𝑇,𝑡0,4,4𝑑𝑇(𝑇𝑡)1𝛼3𝑇4𝑡1𝛼𝑇,𝑡4,3𝑇4,4𝑑𝑇(𝑇𝑡)1𝛼,𝑡3𝑇4,Φ𝑢,𝑇11=2𝑇0𝐶0𝐷𝛼𝑡𝑢1(𝑡)𝐶𝑡𝐷𝛼𝑇𝑢1(𝑡)𝑑𝑡=8𝑑2𝑇20𝑇/4𝑡1𝛼(𝑇𝑡)1𝛼3𝑇4𝑡1𝛼𝑇4𝑡1𝛼+𝑑𝑡3𝑇/4𝑇/4𝑡1𝛼𝑇𝑡41𝛼(𝑇𝑡)1𝛼3𝑇4𝑡1𝛼+𝑑𝑡𝑇3𝑇/4𝑡1𝛼𝑇𝑡41𝛼𝑡3𝑇41𝛼(𝑇𝑡)1𝛼𝑑𝑡=8𝑑2𝑇2𝑇0𝑡1𝛼(𝑇𝑡)1𝛼𝑑𝑡0𝑇/4𝑡1𝛼𝑇4𝑡1𝛼+𝑑𝑡3𝑇/4𝑇/4𝑇𝑡41𝛼3𝑇4𝑡1𝛼𝑑𝑡𝑇3𝑇/4𝑡3𝑇41𝛼(𝑇𝑡)1𝛼𝑑𝑡03𝑇/4𝑡1𝛼3𝑇4𝑡1𝛼𝑇𝑇/4𝑇𝑡41𝛼(𝑇𝑡)1𝛼=𝑑𝑡8Γ2(2𝛼)𝑇Γ(42𝛼)12𝛼𝑑21+332𝛼24𝛼522𝛼31=𝐴(𝛼)𝑑2,Ψ𝑢1=𝑇0𝑎𝑢(𝑡)𝐹1=(𝑡)𝑑𝑡0𝑇/4𝑎(𝑡)𝐹4Γ(2𝛼)𝑑𝑇𝑡𝑑𝑡+3𝑇/4𝑇/4𝑎+(𝑡)𝐹(Γ(2𝛼)𝑑)𝑑𝑡𝑇3𝑇/4𝑎(𝑡)𝐹4Γ(2𝛼)𝑑𝑇(𝑇𝑡)𝑑𝑡=𝐹(Γ(2𝛼)𝑑)3𝑇/4𝑇/4𝑇𝑎(𝑡)𝑑𝑡+4Γ(2𝛼)𝑑0Γ(2𝛼)𝑑𝑏(𝑥)𝐹(𝑥)𝑑𝑥.(3.16)

Let 𝑟=(|cos(𝜋𝛼)|/2Ω2)𝑐2. Since 𝑐<(2𝐴(𝛼)/|cos(𝜋𝛼)|)Ω𝑑, we obtain 𝑟<Φ(𝑢1).

By (2.12) and (3.7), one has Φ(𝑢)𝑟𝑢𝑐. Thus, sup𝑢Φ1(][),𝑟𝑤Ψ(𝑢)=sup𝑢Φ1(]]),𝑟Ψ(𝑢)max|𝑥|𝑐𝐹(𝑥)𝑇0𝑎(𝑡)𝑑𝑡.(3.17)

Moreover, we have 𝑟𝑢𝑟+Φ1Ψ𝑢1=||||cos(𝜋𝛼)/2Ω2𝑐2||||cos(𝜋𝛼)/2Ω2𝑐2+𝐴(𝛼)𝑑2×𝐹(Γ(2𝛼)𝑑)3𝑇/4𝑇/4𝑇𝑎(𝑡)𝑑𝑡+4Γ(2𝛼)𝑑0Γ(2𝛼)𝑑=||||𝑐𝑏(𝑥)𝐹(𝑥)𝑑𝑥cos(𝜋𝛼)2||||𝑐cos(𝜋𝛼)2+2Ω2𝐴(𝛼)𝑑2×𝐹(Γ(2𝛼)𝑑)3𝑇/4𝑇/4𝑇𝑎(𝑡)𝑑𝑡+4Γ(2𝛼)𝑑0Γ(2𝛼)𝑑.𝑏(𝑥)𝐹(𝑥)𝑑𝑥(3.18)

Hence, from (H2) one has sup𝑢Φ1(][),𝑟𝑤Ψ𝑟(𝑢)<𝑢𝑟+Φ1Ψ𝑢1.(3.19)

Now, taking into account that Φ𝑢1Ψ𝑢1sup𝑢Φ1(],𝑟[)𝑤𝐴Ψ(𝑢)(𝛼)𝑑2𝑎+0Γ(2𝛼)𝑑𝑏(𝑥)𝐹(𝑥)𝑑𝑥𝑇0𝑎(𝑡)𝑑𝑡max|𝑥|𝑐,𝑟𝐹(𝑥)sup𝑢Φ1(],𝑟[)𝑤𝑐Ψ(𝑢)2||||cos(𝜋𝛼)2Ω2𝑇0𝑎(𝑡)𝑑𝑡max|𝑥|𝑐𝐹,(𝑥)𝑟𝑟Ψ𝑢1𝑢/Φ1sup𝑢Φ1(],𝑟[)𝑤Ψ(𝑢)𝐴(𝛼)𝑑2𝑎+02Γ(2𝛼)𝑏(𝑥)𝐹(𝑥)𝑑𝑥2Ω2𝐴(𝛼)𝑑2/𝑐2||||cos(𝜋𝛼)𝑇0𝑎(𝑡)𝑑𝑡max|𝑥|𝑐𝐹(𝑥)=𝑚.(3.20) Thus, by Theorem 2.9 it follows that, for each 𝜆Λ1, BVP (1.2) admits at least three solutions, and there exists an open interval Λ2[0,𝑚] and a real positive number 𝜎 such that, for each 𝜆Λ2, 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, 𝑎(𝑡)1, and 𝑓(𝑢)=𝑒𝑢𝑢8(9𝑢)+𝑢. Then BVP (1.2) reduces to the following boundary value problem: 𝑑1𝑑𝑡20𝐷𝑡0.2𝐶0𝐷𝑡0.81𝑢(𝑡)2𝑡𝐷10.2𝐶𝑡𝐷10.8𝑒𝑢(𝑡)+𝜆𝑢𝑢8(9𝑢)+𝑢[],=0,a.e.𝑡0,1𝑢(0)=𝑢(1)=0.(3.21)

Example 3.2. Owing to Theorem 3.1, for each 𝜆]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 Ω=1.1089, 𝐴(0.8)=1.3313, and 2𝐴(0.8)||||cos(0.8𝜋)Ω𝑑=4.0235>1=𝑐.(3.22) Since𝐹(𝑥)=𝑥0𝑓(𝑠)𝑑𝑠=𝑒𝑥𝑥9+23𝑥3/2,(3.23) we have that condition (H1) holds. Moreover, 𝐹(𝑥)0 for each 𝑥[0,2Γ(1.2)], and ||||cos(0.8𝜋)||||cos(0.8𝜋)+2Ω2𝐴(0.8)22121𝐹(2Γ(1.2))+4Γ(1.2)02Γ(1.2)𝐹(𝑠)𝑑𝑠>1.064>1.0345=𝑒1+23𝐹(𝑥),|𝑥|1,(3.24) which implies that condition (H2) holds. Thus, by Theorem 3.1, for each 𝜆]0.291,0.318[, the problem (3.21) admits at least three nontrivial solutions in 𝐸00.8. Moreover, for each >1, there exists an open interval Λ]0,3.4674[ and a real positive number 𝜎 such that, for each 𝜆Λ, 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).