Functional-differential equations in the space of functions of bounded variation

For nonlinear systems of functional-differential equations with a generalized action on the right-hand side, the notion of a solution is formalized based on the closure of smooth solutions in the space of functions of bounded variation. Sufficient conditions are obtained for the existence and uniqueness of the solution. For linear systems with distributed delay and generalized action in the system matrix, conditions for the existence of a solution and the Cauchy formula for representing the solution are derived.