  The Cauchy problem for the telegraph equation $(D_{t}^{\rho
})^{2}u(t)+2\alpha D_{t}^{\rho }u(t)+Au(t)=f(t)$ ($0<t\leq T, \, 0<\rho<1$),
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.
