Correct solvability of the Sturm–Liouville equation with delayed argument

We consider the equation −y^″(x)+q(x)y(x−φ(x))=f(x),x∈R where f∈C(R) and 0≤φ∈C^loc(R),1≤q∈C^loc(R). Here C^loc(R) is the set of functions continuous in every point of the number axis. By a solution of (1), we mean any function y, doubly continuously differentiable everywhere in R, which satisfies (1). We show that under certain additional conditions on the functions φ and q to (2), (1) has a unique solution y, satisfying the inequality ‖y^″‖_C(R)+‖y^′‖_C(R)+‖qy‖_C(R)≤c‖f‖_C(R) where the constant c∈(0,∞) does not depend on the choice of f∈C(R).