The following arose in a physical problem:

Solve the two equations \begin{eqnarray} p(t)+D^\alpha x(t)+\omega x(t)=0\\ _T^- D^\alpha p(t)-\omega p(t)+k x(t)=0 \end{eqnarray} subject to the conditions: $x(0)=x_i$ and $x(T)=x_f$. Given: $0<\alpha\leq 1$ and $\omega$ and $k$ are positive constants. In the above $D^\alpha$ is the Caputo fractional derivative $$D^\alpha f(t)=\frac{1}{\Gamma(1-\alpha)}\int_0^tds \frac{ds}{(t-s)^{1-\alpha}}\frac{df(s)}{ds}$$ and $$_T^-D^\alpha f(t)=\frac{1}{\Gamma(1-\alpha)}\frac{d}{dt}\int_t^T ds \frac{ds}{(s-t)^{1-\alpha}}f(s).$$