Generalized Antiperiodic Boundary Value Problems for the Fractional Differential Equation with p-Laplacian Operator

We discuss the existence of solutions about generalized antiperiodic boundary value problems for the fractional differential equation with p-Laplacian operator phi(p)((c)D(0+)(alpha)u(t)) = f(t,u(t),u'(t)), 0 < t < T, 1 < alpha <= 2, u( 0) + (-1)(theta)au(T) = 0, (c)D(0+)(beta)u(0) + (-1)(theta)b (c)D(0+)(beta)u(T) = lambda, 0 < beta < 1, where D-c(0+)alpha is the Caputo fractional derivative, theta = 0, 1, a > 0, a not equal 1, b > 0 and theta(p)(s) = vertical bar s vertical bar(p-2)s, p > 1, phi(-1)(p) = phi(q), 1/p + 1q = 1. Our results are based on fixed point theorem and contraction mapping principle. Furthermore, three examples are also given to illustrate the results.