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

and Applied Analysis 3 ii If γ n − 1 and f ∈ ACn−1 a, b ,R , then CaD t f t and t Dn−1 b f t are represented by C aD n−1 t f t f n−1 t , t D n−1 b f t −1 n−1 f n−1 t , t ∈ a, 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: ∫b a [ aD −γ t f t ] g t dt ∫b a [ tD −γ b g t ] f t dt, γ > 0 2.4 provided that f ∈ L a, b ,R , g ∈ L a, b ,R , and p ≥ 1, q ≥ 1, 1/p 1/q ≤ 1 γ or p / 1, q / 1, 1/p 1/q 1 γ . Property 2 see 5 . Let n ∈ N and n − 1 < γ ≤ n. If f ∈ AC a, b ,R or f ∈ C a, b ,R , then aD −γ t ( C aD γ t f t ) f t − n−1 ∑ j 0 f j a j! t − a j , tD −γ b ( C t D γ bf t ) f t − n−1 ∑ j 0 −1 f j b j! b − t j , 2.5 for t ∈ a, b . In particular, if 0 < γ ≤ 1 and f ∈ AC a, b ,R or f ∈ C1 a, b ,R , then aD −γ t ( C aD γ t f t ) f t − f a , tD b ( C t D γ bf 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 dt ( 1 2 D −β t ( u′ t ) 1 2 t D −β T ( u′ t )) λa t f u t 0, a.e. t ∈ 0, T , u 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 C∞ 0 0, T the set of all functions g ∈ C∞ 0, T with g 0 g T 0. 4 Abstract and Applied Analysis Definition 2.4 see 26 . Let 0 < α ≤ 1. The fractional derivative space E 0 is defined by the closure of C∞ 0 0, T with respect to the norm ‖u‖α (∫T 0 ∣∣ 0Dαt u t ∣∣2dt ∫T 0 |u t |dt )1/2 , ∀u ∈ E 0 . 2.8 Remark 2.5. It is obvious that the fractional derivative space E 0 is the space of functions u ∈ L2 0, T having an α-order Caputo fractional derivative C0D α t u ∈ L2 0, T and u 0 u T 0. Proposition 2.6 see 26 . Let 0 < α ≤ 1. The fractional derivative space E 0 is reflexive and separable Banach space. Lemma 2.7 see 26 . Let 1/2 < α ≤ 1. For all u ∈ E 0 , one has the following: i ‖u‖L2 ≤ T Γ α 1 ∥∥ 0Dαt u ∥∥ L2 . 2.9 ii ‖u‖∞ ≤ Tα−1/2 Γ α 2 α − 1 1 1/2 ∥∥ 0Dαt u ∥∥ L2 . 2.10 By 2.9 , we can consider E 0 with respect to the norm ‖u‖α (∫T 0 ∣∣ 0Dαt u t ∣∣2dt )1/2 ∥∥ 0Dαt u ∥∥ L2 , ∀u ∈ E 0 2.11 in the following analysis. Lemma 2.8 see 26 . Let 1/2 < α ≤ 1, then for all any u ∈ E 0 , one has |cos πα |‖u‖α ≤ − ∫T 0 C 0D α t u t · t D Tu t dt ≤ 1 |cos πα | ‖u‖ 2 α. 2.12 Our main tool is the critical-points theorem 27 which is recalled below. Theorem 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 ∞, forallλ ∈ 0, ∞ . Further, assume that there are r > 0, x1 ∈ X such that Abstract and Applied Analysis 5 ii r < Φ x1 ; iii sup x∈Φ−1 −∞,r w Ψ x < r/ r Φ x1 Ψ x1 . Then, for eachand Applied Analysis 5 ii r < Φ x1 ; iii sup x∈Φ−1 −∞,r w Ψ x < r/ r Φ x1 Ψ x1 . Then, for each λ ∈ Λ1 ⎤ ⎥⎦ Φ x1 Ψ x1 − supx∈Φ−1 −∞,r w Ψ x , r sup x∈Φ−1 −∞,r w Ψ x ⎡ ⎢⎣ , 2.13


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  T are the left and right Riemann-Liouville fractional integrals of order 0 ≤ β < 1, respectively, F : 0, T × R N → 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: are the left and right Riemann-Liouville fractional integrals of order 1 − α respectively, c 0 D α t and c t D α T 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 a : R → R is a nonnegative continuous function with a t / ≡ 0.

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

Abstract and Applied Analysis
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: 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: where In order to establish a variational structure for BVP 1.2 , it is necessary to construct appropriate function spaces.
Denote by C ∞ 0 0, T the set of all functions g ∈ C ∞ 0, T with g 0 g T 0.
Definition 2.4 see 26 .Let 0 < α ≤ 1.The fractional derivative space E α 0 is defined by the closure of C ∞ 0 0, T with respect to the norm Remark 2.5.It is obvious that the fractional derivative space E α 0 is the space of functions u ∈ L 2 0, T having an α-order Caputo fractional derivative C 0 D α t u ∈ L 2 0, T and u 0 u T 0.

2.10
By 2.9 , we can consider E α 0 with respect to the norm in the following analysis.
Lemma 2.8 see 26 .Let 1/2 < α ≤ 1, then for all any u ∈ E α 0 , one has Our main tool is the critical-points theorem 27 which is recalled below.
Theorem 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 x 0 ∈ X such that Φ x 0 Ψ x 0 0, and that Then, for each the equation has at least three solutions in X and, moreover, for each h > 1, there exists an open interval and a positive real number σ such that, for each λ ∈ Λ 2 , 2.14 has at least three solutions in X whose norms are less than σ.

Main Result
For given u ∈ E α 0 , we define functionals Φ, Ψ : E α → R as follows: where F u u 0 f s ds.Clearly, Φ and Ψ are Gateaux differentiable functional whose Gateaux derivative at the point u ∈ E α 0 are given by Abstract and Applied Analysis for every v ∈ E α 0 .By Definition 2.2 and Property 2, we have The theory of Fourier series and 3.4 imply that a.e. on 0, T for some C ∈ R. By 3.6 , it is easy to know that u * ∈ E α 0 is a solution of BVP 1.2 .
For convenience, set 3.9

3.11
where a and denote T/4 a t dt and T/ 4Γ 2 − α d respectively, the problem 1.2 admits at least three solutions in E α 0 and, moreover, for each h > 1, there exists an open interval such that, for each λ ∈ Λ 2 , the problem 1.2 admits at least three solutions in E α 0 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 E α 0 .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 E α 0 is exactly the solution of BVP 1.2 .

3.14
It is easy to check that u 1 0 u 1 T 0 and u 1 ∈ L 2 0, T .The direct calculation shows

3.18
Hence, from H2 one has

3.19
Now, taking into account that Thus, by Theorem 2.9 it follows that, for each λ ∈ Λ 1 , BVP 1.2 admits at least three solutions, and there exists an open interval Λ 2 ⊂ 0, m and a real positive number σ such that, for each λ ∈ Λ 2 , BVP 1.2 admits at least three solutions in E α 0 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 e −u u 8  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 E 0.8 0 .Moreover, for each h > 1, there exists an open interval Λ ⊂ 0, 3.4674h and a real positive number σ such that, for each λ ∈ Λ, the problem 3.21 admits at least three solutions in E 0.8 0 whose norms are less than σ.

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 a D −α t f t and t D −α b f t , respectively, are defined by