Dilna N. Solvability of the initial-values problem for positive systems of functional-differential equations.

We obtain new and optimal, in a sense, unique solvability conditions for a wide range of functional-differential equations. We prove a general theorem on the linear functional-differential unequalities. By using it, we establish new theorems on the unique solvability of the Cauchy problem involving properties of certain onestep and multistep iteration sequenses determined by the corresponding homogeneous initial-value problem. We show that, under the conditions obtained, the fulfilment of a certain positivity property of the right-hand side of the equation gurantees the monotonous dependence of the solution of initial-value problem on the additive perturbations of the equation and initial data.