Well-posedness of a problem with initial conditions for hyperbolic differential-difference equations with shifts in the time argument

We study a problem with initial conditions on the half-line for a differentialdifference equation of the hyperbolic type with deviations of the time argument. We obtain sufficient conditions for the well-posed solvability of the problem in Sobolev spaces with an exponential weight. In terms of the spectrum of the problem operator, we obtain necessary conditions for the well-posed solvability of the problem, sufficient conditions for the absence of solutions, and sufficient conditions for the nonuniqueness of the solution.