Conditions for Correct Solvability of a Simplest Singular Boundary Value Problem of General Form. I

Abstract

We consider the singular boundary value problem

where $f\in L_p(\mathbb{R})$, $p\in [1,\infty ]$ ($L_{\infty }(\mathbb{R}):=C(\mathbb{R}))$, $r$ is a continuous positive function on $\mathbb{R}$, $0\le q\in L_1^{\mathrm{loc}}$. A solution of this problem is, by definition, any absolutely continuous function $y$ satisfying the limit condition and almost everywhere the differential equation. This problem is called correctly solvable in a given space $L_p(\mathbb{R})$ if for any function $f\in L_p(\mathbb{R})$ it has a unique solution $y\in L_p(\mathbb{R})$ and if the following inequality holds with an absolute constant $c_p\in (0,\infty ):$

We find minimal requirements for $r$ and $q$ under which the above problem is correctly solvable in $L_p(\mathbb{R})$.