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