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

• ### L.A. Shuster

Bar-Ilan University, Ramat-Gan, Israel
• ### N.A. Chernyavskaya

Ben Gurion University of the Negev, Beer-Sheba, Israel ## Abstract

We consider the singular boundary value problem %\eqref{1} -- \eqref{2} %

$\label{1} %-r(x)y'(x)+q(x)y(x)=f(x),\quad x\in R %$

%

$\label{2} %\lim_{|x|\to\iy}y(x)=0, %$

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

$\label{3} \|y\|_{L_p(\mathbb R)}\le c_p\|f\|_{L_p(\mathbb R)},\quad \ f\in L_p(\mathbb R) . %$

 We find minimal requirements for and under which the above problem is correctly solvable in