Abstract

We study the existence of solutions to a nonlinear fractional differential equation in Hilbert spaces associated with three-point boundary conditions at resonance x(0)=θ,Dα−1x(1)=ADα−1x(η) by using Mawhin’s continuation theorem. We propose a new technique to improve the conditions on A which have been used previously. In addition, a necessary and sufficient condition for that the fractional differential operator is Fredholm with zero-index is established, especially for the first time when the fractional differential operator takes values in an arbitrary Hilbert space.