On boundary value problems for impulsive fractional differential equations

In the present paper, we deal with the existence of solutions for the following impulsive fractional boundary value problem: {D-t(T)alpha((c)(0)D(t)(alpha)u(t)) + a(t)u(t) = f(t, u(t)), t not equal tj, a.e. t is an element of [0, T], Delta(D-t(T)alpha-1((c)(0)D(t)(alpha)u))(t(j)) = l(j)(t(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) < ... < t(n) < t(n+1), T, f: [0, T] x R -> R and I-j : R -> R, j = 1, ... , n, are continuous functions, a is an element of C([0, T]) and Delta(D-t(T)alpha-1((c)(0)D(t)(alpha)u))(t(j)) = D-t(T)alpha-1((c)(0)D(t)(alpha)u)(t(j)(+)) - D-t(T)alpha-1((c)(0)D(t)(alpha)u)(t(j)), D-t(T)alpha-1 ((c)(0)D(t)(alpha)u)(t(j)(+)) = lim(t -> tj+) (D-t(T)alpha-1((c)(0)D(t)(alpha)u)(t)), D-t(T)alpha-1((c)(0)D(t)(alpha)u)(t(j)(-)) = lim(t -> tJ-)(D-t(T)alpha-1((c)(0)D(T)(alpha)u)(t)). By using critical point theory and variational methods, we give some new criteria to guarantee that the above-mentioned impulsive problems have at least one solution or infinitely many solutions. Some examples are also provided. (C) 2015 Elsevier Inc. All rights reserved.