On the surjectivity conditions of a linear functional differential operator of the second order

Linear second-order functional differential equations without the Volterra condition are studied. Sufficient conditions for the everywhere solvability of the equation (surjectivity of the corresponding functional differential operator) are obtained in terms of norms of the positive and negative parts of the functional operators. These conditions are shown to be sharp in the sense that if they are not satisfied, then there exists an equation with no solution. The obtained solvability conditions are formulated directly for the equation itself, without considering specific boundary value problems.