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

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 $f\in L_p(\mathbb{R}),$\ \( p\in[1,\iy] \) \( (L_\iy(\mathbb R):=C(\mathbb R)), \)\ $r$ is a continuous positive function on $\mathbb{R}$, \ \( 0\le q \in L_1^{\loc}. \) 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 \( c_p\in (0,\iy): \) %

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

$$ We find minimal requirements for $r$ and $q$ under which the above problem is correctly solvable in $$\begin{gathered} L\_p( \\ mathbbR). \end{gathered}$$