Abstract and Applied Analysis

Abstract and Applied Analysis / 2012 / Article
Special Issue

Advanced Theoretical and Applied Studies of Fractional Differential Equations

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Research Article | Open Access

Volume 2012 |Article ID 963105 | 13 pages | https://doi.org/10.1155/2012/963105

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

Academic Editor: Bashir Ahmad
Received13 May 2012
Revised08 Jul 2012
Accepted09 Jul 2012
Published06 Sep 2012

Abstract

This paper is concerned with the existence of three solutions to a nonlinear fractional boundary value problem as follows: (𝑑/𝑑𝑡)((1/2)0𝐷𝑡𝛼−1(𝐶0𝐷𝛼𝑡𝑢(𝑡))−(1/2)𝑡𝐷𝑇𝛼−1(𝐶𝑡𝐷𝛼𝑇𝑢(𝑡)))+ğœ†ğ‘Ž(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 [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1𝑑𝑡20ğ·ğ‘¡âˆ’ğ›½î€·ğ‘¢î…žî€¸+1(𝑡)2ğ‘¡ğ·ğ‘‡âˆ’ğ›½î€·ğ‘¢î…žî€¸î‚[],(𝑡)+∇𝐹(𝑡,𝑢(𝑡))=0,a.e.𝑡∈0,𝑇𝑢(0)=𝑢(𝑇)=0(1.1) by using the critical point theory, where 0𝐷𝑡−𝛽 and 𝑡𝐷𝑇−𝛽 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 ∇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: 𝑑1𝑑𝑡20𝐷𝑡𝛼−1𝐶0ğ·ğ‘Žğ‘¡î€¸âˆ’1𝑢(𝑡)2𝑡𝐷𝑇𝛼−1î€·ğ¶ğ‘¡ğ·ğ‘Žğ‘‡î€¸î‚[],𝑢(𝑡)+ğœ†ğ‘Ž(𝑡)𝑓(𝑢(𝑡))=0,a.e.𝑡∈0,𝑇𝑢(0)=𝑢(𝑇)=0,(1.2) where 𝛼∈(1/2,1], 0𝐷𝑡𝛼−1 and 𝑡𝐷𝑇𝛼−1 are the left and right Riemann-Liouville fractional integrals of order 1−𝛼 respectively, 𝑐0𝐷𝛼𝑡 and 𝑐𝑡𝐷𝛼𝑇 are the left and right Caputo fractional derivatives of order 𝛼 respectively, 𝜆 is a positive real parameter, 𝑓∶𝐑→𝐑 is a continuous function, and ğ‘Žâˆ¶ğ‘â†’ğ‘ is a nonnegative continuous function with ğ‘Ž(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 [ğ‘Ž,𝑏]. The left and right Riemann-Liouville fractional integrals of order 𝛼 for function 𝑓 denoted by ğ‘Žğ·ğ‘¡âˆ’ğ›¼ğ‘“(𝑡) and 𝑡𝐷𝑏−𝛼𝑓(𝑡), respectively, are defined by ğ‘Žğ·ğ‘¡âˆ’ğ›¼1𝑓(𝑡)=Γ(𝛼)ğ‘¡ğ‘Ž(𝑡−𝑠)𝛼−1[]𝑓(𝑠)𝑑𝑠,ğ‘¡âˆˆğ‘Ž,𝑏,𝛼>0,𝑡𝐷𝑏−𝛼𝑓1(𝑡)=Γ(𝛼)𝑏𝑡(𝑠−𝑡)𝛼−1𝑓[](𝑠)𝑑𝑠,ğ‘¡âˆˆğ‘Ž,𝑏,𝛼>0,(2.1) provided the right-hand sides are pointwise defined on [ğ‘Ž,𝑏], where Γ(𝛼) is the gamma function.

Definition 2.2 (see [5]). Let 𝛾≥0 and 𝑛∈𝐍.
(i) If 𝛾∈(𝑛−1,𝑛) and 𝑓∈𝐴𝐶𝑛([ğ‘Ž,𝑏],𝐑𝑁), then the left and right Caputo fractional derivatives of order 𝛾 for function 𝑓 denoted by ğ¶ğ‘Žğ·ğ›¾ğ‘¡ğ‘“(𝑡) and 𝐶𝑡𝐷𝛾𝑏𝑓(𝑡), respectively, exist almost everywhere on [ğ‘Ž,𝑏], ğ¶ğ‘Žğ·ğ›¾ğ‘¡ğ‘“(𝑡) and 𝐶𝑡𝐷𝛾𝑏𝑓(𝑡) are represented by ğ¶ğ‘Žğ·ğ›¾ğ‘¡1𝑓(𝑡)=Γ(𝑛−𝛾)ğ‘¡ğ‘Ž(𝑡−𝑠)𝑛−𝛾−1𝑓(𝑛)([],𝑠)𝑑𝑠,ğ‘¡âˆˆğ‘Ž,𝑏𝐶𝑡𝐷𝛾𝑏𝑓(𝑡)=(−1)𝑛Γ(𝑛−𝛾)𝑏𝑡(𝑠−𝑡)𝑛−𝛾−1𝑓(𝑛)[],(𝑠)𝑑𝑠,ğ‘¡âˆˆğ‘Ž,𝑏(2.2) respectively.
(ii) If 𝛾=𝑛−1 and 𝑓∈𝐴𝐶𝑛−1([ğ‘Ž,𝑏],𝐑𝑁), then ğ¶ğ‘Žğ·ğ‘¡ğ‘›âˆ’1𝑓(𝑡) and 𝐶𝑡𝐷𝑏𝑛−1𝑓(𝑡) are represented by ğ¶ğ‘Žğ·ğ‘¡ğ‘›âˆ’1𝑓(𝑡)=𝑓(𝑛−1)(𝑡),𝐶𝑡𝐷𝑏𝑛−1𝑓(𝑡)=(−1)(𝑛−1)𝑓(𝑛−1)[].(𝑡),ğ‘¡âˆˆğ‘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: î€œğ‘ğ‘Žî€ºğ‘Žğ·ğ‘¡âˆ’ğ›¾î€»î€œğ‘“(𝑡)𝑔(𝑡)𝑑𝑡=ğ‘ğ‘Žî€ºğ‘¡ğ·ğ‘âˆ’ğ›¾î€»ğ‘”(𝑡)𝑓(𝑡)𝑑𝑡,𝛾>0(2.4) provided that 𝑓∈𝐿𝑝([ğ‘Ž,𝑏],𝐑𝑁), ğ‘”âˆˆğ¿ğ‘ž([ğ‘Ž,𝑏],𝐑𝑁), and 𝑝≥1, ğ‘žâ‰¥1, 1/𝑝+1/ğ‘žâ‰¤1+𝛾 or 𝑝≠1, ğ‘žâ‰ 1, 1/𝑝+1/ğ‘ž=1+𝛾.

Property 2 (see [5]). Let 𝑛∈𝐍 and 𝑛−1<𝛾≤𝑛. If 𝑓∈𝐴𝐶𝑛([ğ‘Ž,𝑏],𝐑𝑁) or 𝑓∈𝐶𝑛([ğ‘Ž,𝑏],𝐑𝑁), then ğ‘Žğ·ğ‘¡âˆ’ğ›¾î€·ğ¶ğ‘Žğ·ğ›¾ğ‘¡î€¸ğ‘“(𝑡)=𝑓(𝑡)−𝑛−1𝑗=0𝑓(𝑗)(ğ‘Ž)𝑗!(ğ‘¡âˆ’ğ‘Ž)𝑗,𝑡𝐷𝑏−𝛾𝐶𝑡𝐷𝛾𝑏𝑓(𝑡)=𝑓(𝑡)−𝑛−1𝑗=0(−1)𝑗𝑓(𝑗)(𝑏)𝑗!(𝑏−𝑡)𝑗,(2.5) for 𝑡∈[ğ‘Ž,𝑏]. In particular, if 0<𝛾≤1 and 𝑓∈𝐴𝐶([ğ‘Ž,𝑏],𝐑𝑁) or 𝑓∈𝐶1([ğ‘Ž,𝑏],𝐑𝑁), then ğ‘Žğ·ğ‘¡âˆ’ğ›¾î€·ğ¶ğ‘Žğ·ğ›¾ğ‘¡î€¸ğ‘“(𝑡)=𝑓(𝑡)−𝑓(ğ‘Ž),𝑡𝐷𝑏−𝛾𝐶𝑡𝐷𝛾𝑏𝑓(𝑡)=𝑓(𝑡)−𝑓(𝑏).(2.6)

Remark 2.3. In view of Property 1 and Definition 2.2, it is obvious that 𝑢∈𝐴𝐶([0,𝑇]) is a solution of BVP (1.2) if and only if 𝑢 is a solution of the following problem: 𝑑1𝑑𝑡20ğ·ğ‘¡âˆ’ğ›½î€·ğ‘¢î…žî€¸+1(𝑡)2ğ‘¡ğ·ğ‘‡âˆ’ğ›½î€·ğ‘¢î…žî€¸î‚[],(𝑡)+ğœ†ğ‘Ž(𝑡)𝑓(𝑢(𝑡))=0,a.e.𝑡∈0,𝑇𝑢(0)=𝑢(𝑇)=0,(2.7) where 𝛽=2(1−𝛼)∈[0,1).

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

Denote by ğ¶âˆž0[0,𝑇] the set of all functions ğ‘”âˆˆğ¶âˆž[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 ğ¶âˆž0[0,𝑇] with respect to the norm ‖𝑢‖𝛼=𝑇0||𝐶0𝐷𝛼𝑡||𝑢(𝑡)2𝑑𝑡+𝑇0||||𝑢(𝑡)2𝑑𝑡1/2,∀𝑢∈𝐸𝛼0.(2.8)

Remark 2.5. It is obvious that the fractional derivative space 𝐸𝛼0 is the space of functions 𝑢∈𝐿2[0,𝑇]having an 𝛼-order Caputo fractional derivative 𝐶0𝐷𝛼𝑡𝑢∈𝐿2[0,𝑇] and 𝑢(0)=𝑢(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 𝑢∈𝐸𝛼0, one has the following:(i) ‖𝑢‖𝐿2≤𝑇𝛼Γ‖‖(𝛼+1)𝐶0𝐷𝛼𝑡𝑢‖‖𝐿2.(2.9)(ii) â€–ğ‘¢â€–âˆžâ‰¤ğ‘‡ğ›¼âˆ’1/2Γ(𝛼)(2(𝛼−1)+1)1/2‖‖𝐶0𝐷𝛼𝑡𝑢‖‖𝐿2.(2.10)

By (2.9), we can consider 𝐸𝛼0 with respect to the norm ‖𝑢‖𝛼=𝑇0||𝐶0𝐷𝛼𝑡||𝑢(𝑡)2𝑑𝑡1/2=‖‖𝐶0𝐷𝛼𝑡𝑢‖‖𝐿2,∀𝑢∈𝐸𝛼0(2.11) in the following analysis.

Lemma 2.8 (see [26]). Let 1/2<𝛼≤1, then for all any 𝑢∈𝐸𝛼0, one has ||||cos(𝜋𝛼)‖𝑢‖2𝛼≤−𝑇0𝐶0𝐷𝛼𝑡𝑢(𝑡)⋅𝐶𝑡𝐷𝛼𝑇1𝑢(𝑡)𝑑𝑡≤||||cos(𝜋𝛼)‖𝑢‖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; Φ∶𝑋→𝐑 be a nonnegative continuously Gateaux differentiable and sequentially weakly lower semicontinuous functional whose Gateaux derivative admits a continuous inverse on 𝑋∗; Ψ∶𝑋→𝐑 be a continuously Gateaux differentiable function whose Gateaux derivative is compact. Assume that there exists 𝑥0∈𝑋 such that Φ(𝑥0)=Ψ(𝑥0)=0, and that(i)lim‖𝑥‖→+∞(Φ(𝑥)−𝜆Ψ(𝑥))=+∞, for all 𝜆∈[0,+∞]. Further, assume that there are 𝑟>0, 𝑥1∈𝑋 such that(ii)𝑟<Φ(𝑥1);(iii)sup𝑥∈Φ−1(]−∞,𝑟[)𝑤Ψ(𝑥)<(𝑟/(𝑟+Φ(𝑥1)))Ψ(𝑥1). Then, for each 𝜆∈Λ1=âŽ¤âŽ¥âŽ¥âŽ¥âŽ¦Î¦î€·ğ‘¥1Ψ𝑥1−sup𝑥∈Φ−1(]−∞,𝑟[)𝑤,𝑟Ψ(𝑥)sup𝑥∈Φ−1(]−∞,𝑟[)ğ‘¤âŽ¡âŽ¢âŽ¢âŽ¢âŽ£,Ψ(𝑥)(2.13) the equation Φ(𝑥)âˆ’ğœ†Î¨î…ž(𝑥)=0(2.14) has at least three solutions in 𝑋 and, moreover, for each ℎ>1, there exists an open interval Λ2⊂⎡⎢⎢⎣0,â„Žğ‘Ÿî€·ğ‘Ÿî€·Î¨î€·ğ‘¥1𝑥/Φ1−sup𝑥∈Φ−1(]−∞,𝑟[)ğ‘¤Î¨âŽ¤âŽ¥âŽ¥âŽ¦(𝑥)(2.15) and a positive real number ğœŽ such that, for each 𝜆∈Λ2, (2.14) has at least three solutions in 𝑋 whose norms are less than ğœŽ.

3. Main Result

For given 𝑢∈𝐸𝛼0, we define functionals Φ,Ψ∶𝐸𝛼→𝐑 as follows: 1Φ(𝑢)∶=−2𝑇0𝐶0𝐷𝛼𝑡𝑢(𝑡)⋅𝐶𝑡𝐷𝛼𝑇𝑢(𝑡)𝑑𝑡,Ψ(𝑢)∶=𝑇0ğ‘Ž(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 𝑢∈𝐸𝛼0 are given by Φ1(𝑢)𝑣=−2𝑇0𝐶0𝐷𝛼𝑡𝑢(𝑡)⋅𝐶𝑡𝐷𝛼𝑇𝑣(𝑡)+𝐶𝑡𝐷𝛼𝑇𝑢(𝑡)⋅𝐶0𝐷𝛼𝑡Ψ𝑣(𝑡)𝑑𝑡,(𝑢)𝑣=𝑇0î€œğ‘Ž(𝑡)𝑓(𝑢(𝑡))𝑣(𝑡)𝑑𝑡=−𝑇0𝑡0ğ‘Ž(𝑠)𝑓(𝑢(𝑠))ğ‘‘ğ‘ â‹…ğ‘£î…ž(𝑡)𝑑𝑡,(3.2) for every 𝑣∈𝐸𝛼0. By Definition 2.2 and Property 2, we have Φ(𝑢)𝑣=𝑇0120𝐷𝑡𝛼−1𝐶0𝐷𝛼𝑡−1𝑢(𝑡)2𝑡𝐷𝑇𝛼−1𝐶𝑡𝐷𝛼𝑇𝑢(𝑡)â‹…ğ‘£î…ž(𝑡)𝑑𝑡.(3.3) Hence, 𝐼𝜆=Φ−𝜆Ψ∈𝐶1(𝐸𝛼0,𝐑). If 𝑢∗∈𝐸𝛼0 is a critical point of 𝐼𝜆, then 0=ğ¼î…žğœ†î€·ğ‘¢âˆ—î€¸ğ‘£=𝑇0120𝐷𝑡𝛼−1𝐶0𝐷𝛼𝑡𝑢∗−1(𝑡)2𝑡𝐷𝑇𝛼−1𝐶𝑡𝐷𝛼𝑇𝑢∗(𝑡)+𝜆𝑡0î€·ğ‘¢ğ‘Ž(𝑠)𝑓∗(𝑠)ğ‘‘ğ‘ â‹…ğ‘£î…ž(𝑡)𝑑𝑡,(3.4) for 𝑣∈𝐸𝛼0. We can choose 𝑣∈𝐸𝛼0 such that 𝑣(𝑡)=sin2𝑘𝜋𝑡𝑇or𝑣(𝑡)=1−cos2𝑘𝜋𝑡𝑇,𝑘=1,2,….(3.5) The theory of Fourier series and (3.4) imply that 120𝐷𝑡𝛼−1𝐶0𝐷𝛼𝑡𝑢∗(−1𝑡)2𝑡𝐷𝑇𝛼−1𝐶𝑡𝐷𝛼𝑇𝑢∗(𝑡)+𝜆𝑡0î€·ğ‘¢ğ‘Ž(𝑠)𝑓∗(𝑠)𝑑𝑠=𝐶(3.6) a.e. on [0,𝑇] for some 𝐶∈𝐑. By (3.6), it is easy to know that 𝑢∗∈𝐸𝛼0 is a solution of BVP (1.2).

By Lemma 2.7, if 𝛼>1/2, we have for each 𝑢∈𝐸𝛼0 that â€–ğ‘¢â€–âˆžî‚µî€œâ‰¤Î©ğ‘‡0||𝐶0𝐷𝛼𝑡||𝑢(𝑡)2𝑑𝑡1/2=Ω‖𝑢‖𝛼,(3.7) where 𝑇Ω=𝛼−1/2√Γ(𝛼).2(𝛼−1)+1(3.8)

Given two constants 𝑐≥0 and 𝑑≠0, with √𝑐≠(2𝐴(𝛼)/|cos(𝜋𝛼)|)Ω⋅𝑑, where Ω as in (3.8).

For convenience, set 𝐴(𝛼)∶=8Γ2(2−𝛼)𝑇Γ(4−2𝛼)1−2𝛼1+33−2𝛼24𝛼−5−22𝛼−3.−1(3.9)

Theorem 3.1. Let 𝑓∶ℝ→ℝ be a continuous function,â€‰â€‰ğ‘Žâˆ¶ğ‘â†’ğ‘ be a nonnegative continuous function with ğ‘Ž(𝑡)≢0, and 1/2<𝛼≤1. Put ∫𝐹(𝑥)=𝑥0𝑓(𝑠)𝑑𝑠 for every 𝑥∈ℝ, and assume that there exist four positive constants 𝑐,𝑑,𝜇, and 𝑝, with √𝑐<(2𝐴(𝛼)/|cos(𝜋𝛼)|)Ω⋅𝑑 and 𝑝<2, such that(H1)𝐹(𝑥)≤𝜇(1+|𝑥|𝑝),  for  all 𝑥∈ℝ;(H2)𝐹(𝑥)≥0 for all 𝑥∈[0,Γ(2−𝛼)𝑑], and ||||𝑐𝐹(𝑥)<cos(𝜋𝛼)2||||𝑐cos(𝜋𝛼)2+2Ω2𝐴(𝛼)𝑑2∫𝑇0Ã—î‚¸î€œğ‘Ž(𝑡)𝑑𝑡𝐹(Γ(2−𝛼)𝑑)3𝑇/4𝑇/4+ğ‘‡ğ‘Ž(𝑡)𝑑𝑡4Γ(2−𝛼)𝑑0Γ(2−𝛼)𝑑[],𝑏(𝑠)𝐹(𝑠)𝑑𝑠,∀𝑥∈−𝑐,𝑐(3.10)where 𝑏(𝑠)=ğ‘Ž((𝑇/4Γ(2−𝛼)𝑑)𝑠)+ğ‘Ž(𝑇−(𝑇/4Γ(2−𝛼)𝑑)𝑠). Then, for each 𝜆∈Λ1=𝐴(𝛼)𝑑2â„œğ‘Žâˆ«+ℜ0Γ(2−𝛼)𝑑∫𝑏(𝑥)𝐹(𝑥)𝑑𝑥−𝑇0ğ‘Ž(𝑡)𝑑𝑡⋅max|𝑥|≤𝑐,𝑐𝐹(𝑥)2||||cos(𝜋𝛼)2Ω2∫𝑇0ğ‘Ž(𝑡)𝑑𝑡⋅max|𝑥|≤𝑐,𝐹(𝑥)(3.11)where â„œğ‘Ž and ℜ denote ∫𝐹(Γ(2−𝛼)𝑑)3𝑇/4𝑇/4ğ‘Ž(𝑡)𝑑𝑡 and 𝑇/(4Γ(2−𝛼)𝑑) respectively,the problem (1.2) admits at least three solutions in 𝐸𝛼0 and, moreover, for each ℎ>1, there exists an open interval Λ2⊂0,â„Žğ´(𝛼)𝑑2â„œğ‘Žâˆ«+ℜ02Γ(2−𝛼)𝑏(𝑥)𝐹(𝑥)𝑑𝑥−2Ω2𝐴(𝛼)𝑑2/𝑐2||||∫cos(𝜋𝛼)𝑇0ğ‘Ž(𝑡)𝑑𝑡⋅max|𝑥|≤𝑐𝐹(𝑥)(3.12) such that, for each 𝜆∈Λ2, the problem (1.2) admits at least three solutions in 𝐸𝛼0 whose norms are less that ğœŽ.

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 Φ−𝜆Ψ in 𝐸𝛼0 is exactly the solution of BVP (1.2).

From (H1) and (2.12), we get lim‖𝑢‖𝛼→+∞(Φ(𝑢)−𝜆Ψ(𝑢))=+∞,(3.13) for all 𝜆∈ [0,+∞[. Put 𝑢1⎧⎪⎪⎪⎨⎪⎪⎪⎩(𝑡)=4Γ(2−𝛼)𝑑𝑇𝑇𝑡,𝑡∈0,4,𝑇Γ(2−𝛼)𝑑,𝑡∈4,3𝑇4,4Γ(2−𝛼)𝑑𝑇𝑇(𝑇−𝑡),𝑡∈4.,𝑇(3.14) It is easy to check that 𝑢1(0)=𝑢1(𝑇)=0 and 𝑢1∈𝐿2[0,𝑇]. The direct calculation shows 𝐶0𝐷𝛼𝑡𝑢1⎧⎪⎪⎨⎪⎪⎩(𝑡)=4𝑑𝑇𝑡1−𝛼𝑇,𝑡∈0,4,4𝑑𝑇𝑡1−𝛼−𝑇𝑡−41−𝛼𝑇,𝑡∈4,3𝑇4,4𝑑𝑇𝑡1−𝛼−𝑇𝑡−41−𝛼−𝑡−3𝑇41−𝛼,𝑡∈3𝑇4,‖‖𝑢,𝑇1‖‖2𝛼=𝑇0𝐶0𝐷𝛼𝑡𝑢1(𝑡)2𝑑𝑡=0𝑇/4+3𝑇/4𝑇/4+𝑇3𝑇/4𝐶0𝐷𝛼𝑡𝑢1(𝑡)2=𝑑𝑡16𝑑2𝑇2𝑇0𝑡2(1−𝛼)𝑑𝑡+𝑇𝑇/4𝑇𝑡−42(1−𝛼)𝑑𝑡+𝑇3𝑇/4𝑡−3𝑇42(1−𝛼)𝑑𝑡−2𝑇𝑇/4𝑡1−𝛼𝑇𝑡−41−𝛼𝑑𝑡−2𝑇3𝑇/4𝑡1−𝛼𝑡−3𝑇41−𝛼𝑑𝑡+2𝑇3𝑇/4𝑇𝑡−41−𝛼𝑡−3𝑇41−𝛼=𝑑𝑡16𝑑2𝑇231+43−2𝛼+143−2𝛼𝑇3−2𝛼3−2𝛼−2𝑇𝑇/4𝑡1−𝛼𝑇𝑡−41−𝛼𝑑𝑡−2𝑇3𝑇/4𝑡1−𝛼𝑡−3𝑇41−𝛼𝑑𝑡+2𝑇3𝑇/4𝑇𝑡−41−𝛼𝑡−3𝑇41−𝛼𝑑𝑡<∞.(3.15) That is, 𝐶0𝐷𝛼𝑡𝑢1∈𝐿2[0,𝑇]. Thus, 𝑢1∈𝐸𝛼0. Moreover, the direct calculation shows 𝐶𝑡𝐷𝛼𝑇𝑢1⎧⎪⎪⎨⎪⎪⎩(𝑡)=4𝑑𝑇(𝑇−𝑡)1−𝛼−3𝑇4−𝑡1−𝛼−𝑇4−𝑡1−𝛼𝑇,𝑡∈0,4,4𝑑𝑇(𝑇−𝑡)1−𝛼−3𝑇4−𝑡1−𝛼𝑇,𝑡∈4,3𝑇4,4𝑑𝑇(𝑇−𝑡)1−𝛼,𝑡∈3𝑇4,Φ𝑢,𝑇11=−2𝑇0𝐶0𝐷𝛼𝑡𝑢1(𝑡)⋅𝐶𝑡𝐷𝛼𝑇𝑢1(𝑡)𝑑𝑡=−8𝑑2𝑇20𝑇/4𝑡1−𝛼(𝑇−𝑡)1−𝛼−3𝑇4−𝑡1−𝛼−𝑇4−𝑡1−𝛼+𝑑𝑡3𝑇/4𝑇/4𝑡1−𝛼−𝑇𝑡−41−𝛼(𝑇−𝑡)1−𝛼−3𝑇4−𝑡1−𝛼+𝑑𝑡𝑇3𝑇/4𝑡1−𝛼−𝑇𝑡−41−𝛼−𝑡−3𝑇41−𝛼(𝑇−𝑡)1−𝛼𝑑𝑡=−8𝑑2𝑇2𝑇0𝑡1−𝛼(𝑇−𝑡)1−𝛼𝑑𝑡−0𝑇/4𝑡1−𝛼𝑇4−𝑡1−𝛼+𝑑𝑡3𝑇/4𝑇/4𝑇𝑡−41−𝛼3𝑇4−𝑡1−𝛼𝑑𝑡−𝑇3𝑇/4𝑡−3𝑇41−𝛼(𝑇−𝑡)1−𝛼−𝑑𝑡03𝑇/4𝑡1−𝛼3𝑇4−𝑡1−𝛼−𝑇𝑇/4𝑇𝑡−41−𝛼(𝑇−𝑡)1−𝛼=𝑑𝑡8Γ2(2−𝛼)𝑇Γ(4−2𝛼)1−2𝛼𝑑21+33−2𝛼24𝛼−5−22𝛼−3−1=𝐴(𝛼)𝑑2,Ψ𝑢1=𝑇0ğ‘Žî€·ğ‘¢(𝑡)𝐹1=(𝑡)𝑑𝑡0𝑇/4ğ‘Žî‚µ(𝑡)𝐹4Γ(2−𝛼)𝑑𝑇𝑡𝑑𝑡+3𝑇/4𝑇/4ğ‘Ž+(𝑡)𝐹(Γ(2−𝛼)𝑑)𝑑𝑡𝑇3𝑇/4î‚µğ‘Ž(𝑡)𝐹4Γ(2−𝛼)𝑑𝑇(𝑇−𝑡)𝑑𝑡=𝐹(Γ(2−𝛼)𝑑)3𝑇/4𝑇/4ğ‘‡ğ‘Ž(𝑡)𝑑𝑡+4Γ(2−𝛼)𝑑0Γ(2−𝛼)𝑑𝑏(𝑥)𝐹(𝑥)𝑑𝑥.(3.16)

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

By (2.12) and (3.7), one has Φ(𝑢)â‰¤ğ‘Ÿâ‡’â€–ğ‘¢â€–âˆžâ‰¤ğ‘. Thus, sup𝑢∈Φ−1(][)−∞,𝑟𝑤Ψ(𝑢)=sup𝑢∈Φ−1(]])−∞,𝑟Ψ(𝑢)≤max|𝑥|≤𝑐𝐹(𝑥)𝑇0ğ‘Ž(𝑡)𝑑𝑡.(3.17)

Moreover, we have 𝑟𝑢𝑟+Φ1Ψ𝑢1=||||cos(𝜋𝛼)/2Ω2𝑐2||||cos(𝜋𝛼)/2Ω2𝑐2+𝐴(𝛼)𝑑2×𝐹(Γ(2−𝛼)𝑑)3𝑇/4𝑇/4ğ‘‡ğ‘Ž(𝑡)𝑑𝑡+4Γ(2−𝛼)𝑑0Γ(2−𝛼)𝑑=||||𝑐𝑏(𝑥)𝐹(𝑥)𝑑𝑥cos(𝜋𝛼)2||||𝑐cos(𝜋𝛼)2+2Ω2𝐴(𝛼)𝑑2×𝐹(Γ(2−𝛼)𝑑)3𝑇/4𝑇/4ğ‘‡ğ‘Ž(𝑡)𝑑𝑡+4Γ(2−𝛼)𝑑0Γ(2−𝛼)𝑑.𝑏(𝑥)𝐹(𝑥)𝑑𝑥(3.18)

Hence, from (H2) one has sup𝑢∈Φ−1(][)−∞,𝑟𝑤Ψ𝑟(𝑢)<𝑢𝑟+Φ1Ψ𝑢1.(3.19)

Now, taking into account that Φ𝑢1Ψ𝑢1−sup𝑢∈Φ−1(]−∞,𝑟[)𝑤≤𝐴Ψ(𝑢)(𝛼)𝑑2â„œğ‘Žâˆ«+ℜ0Γ(2−𝛼)𝑑∫𝑏(𝑥)𝐹(𝑥)𝑑𝑥−𝑇0ğ‘Ž(𝑡)𝑑𝑡⋅max|𝑥|≤𝑐,𝑟𝐹(𝑥)sup𝑢∈Φ−1(]−∞,𝑟[)𝑤≥𝑐Ψ(𝑢)2||||cos(𝜋𝛼)2Ω2∫𝑇0ğ‘Ž(𝑡)𝑑𝑡⋅max|𝑥|≤𝑐𝐹,(𝑥)â„Žğ‘Ÿğ‘Ÿî€·Î¨î€·ğ‘¢1𝑢/Φ1−sup𝑢∈Φ−1(]−∞,𝑟[)𝑤Ψ≤(𝑢)â„Žğ´(𝛼)𝑑2â„œğ‘Žâˆ«+ℜ02Γ(2−𝛼)𝑏(𝑥)𝐹(𝑥)𝑑𝑥−2Ω2𝐴(𝛼)𝑑2/𝑐2||||∫cos(𝜋𝛼)𝑇0ğ‘Ž(𝑡)𝑑𝑡⋅max|𝑥|≤𝑐𝐹(𝑥)=𝑚.(3.20) Thus, by Theorem 2.9 it follows that, for each 𝜆∈Λ1, BVP (1.2) admits at least three solutions, and there exists an open interval Λ2⊂[0,𝑚] and a real positive number ğœŽ such that, for each 𝜆∈Λ2, BVP (1.2) admits at least three solutions in 𝐸𝛼0 whose norms are less than ğœŽ.

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

Let 𝛼=0.8, 𝑇=1, ğ‘Ž(𝑡)≡1, and 𝑓(𝑢)=𝑒−𝑢𝑢8√(9−𝑢)+𝑢. Then BVP (1.2) reduces to the following boundary value problem: 𝑑1𝑑𝑡20𝐷𝑡−0.2𝐶0𝐷𝑡0.8−1𝑢(𝑡)2𝑡𝐷1−0.2𝐶𝑡𝐷10.8𝑒𝑢(𝑡)+𝜆−𝑢𝑢8√(9−𝑢)+𝑢[],=0,a.e.𝑡∈0,1𝑢(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)Ω⋅𝑑=4.0235>1=𝑐.(3.22) Since𝐹(𝑥)=𝑥0𝑓(𝑠)𝑑𝑠=𝑒−𝑥𝑥9+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)⋅22121𝐹(2Γ(1.2))+4Γ(1.2)02Γ(1.2)𝐹(𝑠)𝑑𝑠>1.064>1.0345=𝑒−1+23≥𝐹(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 ℎ>1, there exists an open interval Λ⊂]0,3.4674ℎ[ and a real positive number ğœŽ such that, for each 𝜆∈Λ, the problem (3.21) admits at least three solutions in 𝐸00.8 whose norms are less than ğœŽ.

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).

References

  1. K. S. Miller and B. Ross, An Introduction to the Fractional Calculus and Fractional Differential Equations, John Wiley & Sons, New York, NY, USA, 1993. View at: Zentralblatt MATH
  2. S. G. Samko, A. A. Kilbas, and O. I. Marichev, Fractional Integrals and Derivatives: Theory and Applications, Gordon and Breach Science Publishers, Longhorne, Pa, USA, 1993. View at: Zentralblatt MATH
  3. I. Podlubny, Fractional Differential Equations, Academic Press, San Diego, Calif, USA, 1999. View at: Zentralblatt MATH
  4. R. Hilfer, Applications of Fractional Calculus in Physics, World Scientic, Singapore, Singapore, 2000.
  5. A. A. Kilbas, H. M. Srivastava, and J. J. Trujillo, Theory and Applications of Fractional Dierential Equations, Elsevier, Amsterdam, The Netherlands, 2006.
  6. V. Lakshmikantham and A. S. Vatsala, “Basic theory of fractional differential equations,” Nonlinear Analysis. Theory, Methods & Applications, vol. 69, no. 8, pp. 2677–2682, 2008. View at: Publisher Site | Google Scholar | Zentralblatt MATH
  7. R. P. Agarwal, M. Benchohra, and S. Hamani, “A survey on existence results for boundary value problems of nonlinear fractional differential equations and inclusions,” Acta Applicandae Mathematicae, vol. 109, no. 3, pp. 973–1033, 2010. View at: Publisher Site | Google Scholar
  8. B. Ahmad and J. J. Nieto, “Existence results for a coupled system of nonlinear fractional differential equations with three-point boundary conditions,” Computers & Mathematics with Applications, vol. 58, no. 9, pp. 1838–1843, 2009. View at: Publisher Site | Google Scholar | Zentralblatt MATH
  9. Z. Bai and H. Lü, “Positive solutions for boundary value problem of nonlinear fractional differential equation,” Journal of Mathematical Analysis and Applications, vol. 311, no. 2, pp. 495–505, 2005. View at: Publisher Site | Google Scholar | Zentralblatt MATH
  10. C. Bai, “Impulsive periodic boundary value problems for fractional differential equation involving Riemann-Liouville sequential fractional derivative,” Journal of Mathematical Analysis and Applications, vol. 384, no. 2, pp. 211–231, 2011. View at: Publisher Site | Google Scholar | Zentralblatt MATH
  11. C. Bai, “Solvability of multi-point boundary value problem of nonlinear impulsive fractional differential equation at resonance,” Electronic Journal of Qualitative Theory of Differential Equations, vol. 89, pp. 1–19, 2011. View at: Google Scholar
  12. M. Benchohra, S. Hamani, and S. K. Ntouyas, “Boundary value problems for differential equations with fractional order and nonlocal conditions,” Nonlinear Analysis. Theory, Methods & Applications, vol. 71, no. 7-8, pp. 2391–2396, 2009. View at: Publisher Site | Google Scholar | Zentralblatt MATH
  13. N. Kosmatov, “Integral equations and initial value problems for nonlinear differential equations of fractional order,” Nonlinear Analysis. Theory, Methods & Applications, vol. 70, no. 7, pp. 2521–2529, 2009. View at: Publisher Site | Google Scholar | Zentralblatt MATH
  14. J. Wang and Y. Zhou, “A class of fractional evolution equations and optimal controls,” Nonlinear Analysis. Real World Applications, vol. 12, no. 1, pp. 262–272, 2011. View at: Publisher Site | Google Scholar | Zentralblatt MATH
  15. J. Wang and Y. Zhou, “Existence and controllability results for fractional semilinear differential inclusions,” Nonlinear Analysis. Real World Applications, vol. 12, no. 6, pp. 3642–3653, 2011. View at: Publisher Site | Google Scholar | Zentralblatt MATH
  16. J. Wang, Y. Zhou, and W. Wei, “Optimal feedback control for semilinear fractional evolution equations in Banach spaces,” Systems and Control Letters, vol. 61, no. 4, pp. 472–476, 2012. View at: Publisher Site | Google Scholar
  17. J. Wang, Y. Zhou, and M. Medved, “On the solvability and optimal controls of fractional integrodifferential evolution systems with infinite delay,” Journal of Optimization Theory and Applications, vol. 152, no. 1, pp. 31–50, 2012. View at: Publisher Site | Google Scholar
  18. J. Wang, M. Fěčkan, and Y. Zhou, “On the new concept of solutions and existence results for impulsive fractional evolution equations,” Dynamics of Partial Differential Equations, vol. 8, no. 4, pp. 345–361, 2011. View at: Google Scholar
  19. Z. Wei, W. Dong, and J. Che, “Periodic boundary value problems for fractional differential equations involving a Riemann-Liouville fractional derivative,” Nonlinear Analysis. Theory, Methods & Applications, vol. 73, no. 10, pp. 3232–3238, 2010. View at: Publisher Site | Google Scholar | Zentralblatt MATH
  20. S. Zhang, “Positive solutions to singular boundary value problem for nonlinear fractional differential equation,” Computers & Mathematics with Applications, vol. 59, no. 3, pp. 1300–1309, 2010. View at: Publisher Site | Google Scholar | Zentralblatt MATH
  21. P. H. Rabinowitz, “Minimax methods in critical point theory with applications to differential equations,” in Proceedings of the Conference Board of the Mathematical Sciences (CBMS '86), vol. 65, American Mathematical Society, 1986. View at: Google Scholar
  22. J. Mawhin and M. Willem, Critical Point Theory and Hamiltonian Systems, Springer-Verlag, New York, NY, USA, 1989.
  23. F. Li, Z. Liang, and Q. Zhang, “Existence of solutions to a class of nonlinear second order two-point boundary value problems,” Journal of Mathematical Analysis and Applications, vol. 312, no. 1, pp. 357–373, 2005. View at: Publisher Site | Google Scholar | Zentralblatt MATH
  24. C.-L. Tang and X.-P. Wu, “Some critical point theorems and their applications to periodic solution for second order Hamiltonian systems,” Journal of Differential Equations, vol. 248, no. 4, pp. 660–692, 2010. View at: Publisher Site | Google Scholar | Zentralblatt MATH
  25. J.-N. Corvellec, V. V. Motreanu, and C. Saccon, “Doubly resonant semilinear elliptic problems via nonsmooth critical point theory,” Journal of Differential Equations, vol. 248, no. 8, pp. 2064–2091, 2010. View at: Publisher Site | Google Scholar | Zentralblatt MATH
  26. F. Jiao and Y. Zhou, “Existence of solutions for a class of fractional boundary value problems via critical point theory,” Computers & Mathematics with Applications, vol. 62, no. 3, pp. 1181–1199, 2011. View at: Publisher Site | Google Scholar | Zentralblatt MATH
  27. G. Bonanno, “A critical points theorem and nonlinear differential problems,” Journal of Global Optimization, vol. 28, no. 3-4, pp. 249–258, 2004. View at: Publisher Site | Google Scholar | Zentralblatt MATH

Copyright © 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.

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