The Massera-Schäffer problem for a first order linear differential equation

We consider the Massera-Schäffer problem for the equation (Formula presented.). By a solution of the problem we mean any function y, absolutely continuous and satisfying the above equation almost everywhere in R. Let positive and continuous functions μ(x) and θ(x) for (Formula presented.) be given. Let us introduce the spaces (Formula presented.) We obtain requirements to the functions μ θ and q under which (1) for every function (Formula presented.) there exists a unique solution (Formula presented.) of the above equation; (2) there is an absolute constant c(p) ∈ (0, ∞) such that regardless of the choice of a function (Formula presented.) the solution of the above equation satisfies the inequality (Formula presented.).