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} %

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\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 $c_p\in (0,\iy):$ %

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