This article discusses linear functional differential equations that can serve as the basis for modern modeling in various fields of science, technology, and economics, including the study of neural networks and machine learning. These equations describe a wide class of processes where the rate of change of a certain quantity depends not only on the values at the current time, but also on the values in the past and future.The aim of the work is to obtain precise conditions on the parameters of the equation, under which the equation has a solution for any integrable right-hand side, which reflects the existence of the modeled object for a reasonably large class of external influences.It is shown that to establish the fact of everywhere solvability of a first-order functional differential equation, it is sufficient to study only three boundary value problems: a periodic boundary value problem, a Cauchy problem, and a problem with a boundary condition at the right end.In terms of the values of the norms of the positive and negative parts of the functional operator, necessary and sufficient conditions are obtained for a linear functional differential equation of the first order to be solvable everywhere. If these conditions on the norms are not satisfied, then there exists an operator with the given norms of the positive and negative parts such that the equation will have no solutions for some integrable right-hand sides.The developed research methods are based on the apparatus of the theory of functional differential equations and can be applied to study other classes of functional equations, in particular, higher-order equations.The results obtained can be used to analyze and model various dynamic systems with delays and (or) advances. These delays and advances can be described by the most general functional operators, including both positive and negative parts, which corresponds to the consideration of systems with both positive and negative feedback. This allows for a more accurate description and prediction of the behavior of such systems.

functional differential equations, conditions of everywhere solvability, positive operators, periodic boundary value problem, modeling of dynamic systems.