Existence of solutions for a nonlinear fractional boundary value problem via a local minimum theorem

This article concerns the existence of solutions to the nonlinear fractional boundary-value problem $$displaylines{ frac{d}{dt} Big({}_0 D_t^{alpha-1}({}_0^c D_t^{alpha} u(t)) -{}_t D_T^{alpha-1}({}_t^c D_T^{alpha} u(t))Big) +lambda f(u(t)) = 0, quadhbox{a.e. } t in [0, T], cr u(0) = u(T) = 0, }$$ where $alpha in (1/2, 1]$, and $lambda$ is a positive real parameter. The approach is based on a local minimum theorem established by Bonanno.