Solvability of differential operator equations with deviating argument and construction of their solutions in the class of distributions

The unique solvability of the initial value problem with the initial function for a differential operator equation with a perturbed argument is studied. Researches are carried out by methods of the theory of the Sobolev-Schwartz generalized functions with values in Banach space. The concept of fundamental solution of differential operator with delay is used. This approach is used to prove the existence and uniqueness of the solution of the considered problem in the class of distributions with a left-bounded support and to construct its generalized solution. We have obtained conditions under which generalized solution are equal to the classical solution of the considered problem