Abstract
In this paper, we prove the existence and multiplicity of (weak) solutions for the following fractional boundary value problem:
{-d/dt (p(t) (1/2 0D(t)(-zeta)(u'(t)) + 1/2(t)D(T)(-zeta) (u'(t))) +r(t) (1/2 0D(t)(-zeta)(u'(t)) + 1/2(t)D(T)(-zeta) (u'(t))) + q(t)u(t) = f(t, u(t)), a.e. t is an element of [0, T],
where D-0(t)-zeta and D-t(T)-zeta are the left and right Riemann-Liouville fractional integrals of order 0 <= zeta < 1 respectively, L(t) := f(0)(t)(r(s)/p(s))ds, 0 < m <= e(-L(t))p(t) <= M and q(t) - p(t) >= 0 where t is an element of [0, T], f is an element of C([0, T] x R, R). Our approach is based on variational methods.