Basic theory of a class of linear functional differential equations with multiplication delay

By introducing a kind of special functions namely exponent-like function, cosine-like function and sine-like function, we obtain explicitly the basic structures of solutions of initial value problem at the original point for this kind of linear pantograph equations. In particular, we get the complete results on the existence, uniqueness and non-uniqueness of the initial value problems at a general point for the kind of linear pantograph equations.