  We prove existence of solutions for a nonlinear fractional oscillator
equation with both left Riemann-Liouville and right Caputo fractional
derivatives subject to natural boundary conditions. The proof is based on a
transformation of the problem into an equivalent lower order fractional
boundary value problem followed by the use of an upper and lower solutions
method. To succeed with such approach, we first prove a result on the
monotonicity of the right Caputo derivative.
