Multiple Solutions for a Fractional Difference Boundary Value Problem via Variational Approach

and Applied Analysis 3 Definition 2.2. Let f be any real-valued function and ν ∈ 0, 1 . The left discrete fractional difference and the right discrete fractional difference operators are, respectively, defined as tΔaf t ΔtΔ − 1−ν a f t 1 Γ 1 − ν Δ t ν−1 ∑ s a t − s − 1 −ν f s , t ≡ a − ν 1 mod1 , bΔt f t −ΔbΔ 1−ν t f t 1 Γ 1 − ν −Δ b ∑ s t 1−ν s − t − 1 −ν f s , t ≡ b ν − 1 mod1 . 2.2 Definition 2.3. For I ∈ C1 E,R , we say I satisfies the Palais-Smale condition henceforth denoted by PS condition if any sequence {xn} ⊂ E for which I xn is bounded and I ′ xn → 0 as n → ∞ possesses a convergent subsequence. Lemma 2.4 see 18 . A real symmetric matrixA is positive definite if there exists a real nonsingular matrixM such that A M†M, whereM† is the transpose. Lemma 2.5 see 9 : linking theorem . Let E be a real Banach space, and I ∈ C1 E,R satisfies (PS) condition and is bounded from below. Suppose I has a local linking at the origin θ, namely, there is a decomposition E Y ⊕W and a positive number ρ such that k dimY < ∞, I y < I θ for y ∈ Y with 0 < ‖y‖ ≤ ρ; I y ≥ I θ for y ∈ W with ‖y‖ ≤ ρ. Then I has at least three critical points. Lemma 2.6 see 6 . Let E be a real reflexive Banach space, and let the functional I : E → R be weakly lower (upper) semicontinuous and coercive, that is, lim||x||→∞I x ∞ (resp., anticoercive, i.e., lim||x||→∞I x −∞). Then there exists x0 ∈ E such that I x0 infEI x (resp., I x0 sup E I x ). Moreover, if I ∈ C1 E,R , then x0 is a critical point of functional I. Recall that, in the finite dimensional setting, it is well known that a coercive functional satisfies the (PS) condition. Let Br denote the open ball in a real Banach space of radius r about 0, and let ∂Br denote its boundary. Now some critical point theorems needed later can be stated. Lemma 2.7 mountain pass theorem 8 . Let E be a real Banach space and I ∈ C1 E,R , satisfying (PS) condition. Suppose I θ 0 and I1 there are constants ρ, α > 0 such that I|∂Bρ ≥ α, I2 there is e ∈ E \ Bρ such that I e ≤ 0. Then I possesses a critical value c ≥ α. Moreover c can be characterized as c inf g∈Γ max u∈g 0,1 I u , 2.3