Fractional Telegraph equation with the Caputo derivative

The Cauchy problem for the telegraph equation $(D_t^{\rho }{)}^{2}u(t)+2\alpha D_t^{\rho }u(t)+Au(t)=f(t)$ ($0<t\le T,0<\rho <1$), with the Caputo derivative is considered. Here $A$ is a selfadjoint positive operator, acting in a Hilbert space $H$, $D_t$ is the Caputo fractional derivative. Existence and uniqueness theorems for the solution to the problem under consideration is proved. Inequalities of stability are obtained.