Existence and Multiplicity of Solutions for Impulsive Fractional Differential Equations

In this paper, we study the existence of solutions for the following impulsive fractional boundary-value problem: {-d/dt (1/2 D-t(alpha-1) (D-c(0)t(alpha) u(t)) - 1/2(t) D-T(alpha-1)((c)(t) D-T(alpha) u(t))) = lambda u(t) + f(t, u(t)), t not equal t, a.e, t is an element of [0, T], Delta(1/2 D-t(alpha-1)(D-c(0)t(alpha) u(t(j))) - 1/2(t) D-T(alpha-1) ((c)(t) D-T(alpha) u(t(j))) = I-j (u(t(j))), j = 1,2, ..., n, u(0) = u (T) = 0, where alpha is an element of (1/2, 1], 0 = t(0) < t(1) < t(2) < center dot center dot center dot < t(n) < t(n+1) = T, lambda is a parameter and f : [ 0, T] x R -> R and I-j : R -> R, j = 1, ..., n are continuous functions and Delta(1/2D(t)(alpha-1) (D-c(0)t(alpha) u(t(j))) - 1/2t D-T(alpha-1)((c)(t) D(T)(alpha)u(t(j)))) =1/2D(t)(alpha-1) ((c)(0) D-t(alpha) u(t(j)(+))) - 1/2t D-T(alpha-1) ((c)(t)D(T)(alpha)u(t)), -1/2D(t)(alpha-1) ((c)(0) D-t(alpha) u(t(j)(-))) - 1/2t D-T(alpha-1) ((c)(t) D(T)(alpha)u(t(j)(+))) =(t -> t)lim(+j) (1/2 D-t(alpha-1) ((c)(0) D-t(alpha) u(t)) - 1/2(t)D(T)(alpha-1) ((c)(t) D(T)(alpha)u(t)), 1/2D(t)(alpha-1) ((c)(0) D(t)(alpha)u(t(j)(-))) - 1/2t D-T(alpha-1) ((c)(t) D(T)(alpha)u(t(j)(-))) =t -> t(j)(-) (1/2 D-t(alpha-1) ((c)(0) D(t)(alpha)u(t)) - 1/2tD(T)(alpha-1) ((c)(t) D(T)(alpha)u (t)). By using critical point theory and variational methods, we give some new criteria to guarantee that the impulsive problems have at least one solution and infinitely many solutions.