We consider the Massera-Schäffer problem for the equation −y′(x) + q(x)y(x) = f(x), where (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 x ∈ R 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 f ∈ Lp(R, θ) there exists a unique solution y ∈ Lp(R, μ) of the above equation; (2) there is an absolute constant c(p) ∈ (0, ∞) such that regardless of the choice of a function f ∈ Lp(R, θ) the solution of the above equation satisfies the inequality (Formula presented.).
- admissible space
- first order linear differential equation
ASJC Scopus subject areas
- Mathematics (all)