Correct solvability of the Sturm–Liouville equation with delayed argument.

We consider the equation (1) − y ″ ( x ) + q ( x ) y ( x − φ ( x ) ) = f ( x ) , x ∈ R where f ∈ C ( R ) and (2) 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 ) + ‖ q y ‖ C ( R ) ≤ c ‖ f ‖ C ( R ) where the constant c ∈ ( 0 , ∞ ) does not depend on the choice of f ∈ C ( R ) . [ABSTRACT FROM AUTHOR]