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Zeitschrift für Analysis und ihre Anwendungen


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Volume 25, Issue 2, 2006, pp. 205–235
DOI: 10.4171/ZAA/1285

Published online: 2006-06-30

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

N.A. Chernyavskaya[1] and L.A. Shuster[2]

(1) Ben Gurion University of the Negev, Beer-Sheba, Israel
(2) Bar-Ilan University, Ramat-Gan, Israel

We consider the singular boundary value problem %\eqref{1} -- \eqref{2} %\begin{equation}\label{1} %$$-r(x)y'(x)+q(x)y(x)=f(x),\quad x\in R$$ %\end{equation} %\begin{equation}\label{2} %$$\lim_{|x|\to\iy}y(x)=0,$$ %\end{equation} \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 $c_p\in (0,\iy):$ %\begin{equation}\label{3} $$\|y\|_{L_p(\mathbb R)}\le c_p\|f\|_{L_p(\mathbb R)},\quad \ f\in L_p(\mathbb R) . %\end{equation} $$ We find minimal requirements for $r $ and $q$ under which the above problem is correctly solvable in $L_p(\mathbb R).$

Keywords: First order linear differential equation, correct solvability

Chernyavskaya N.A., Shuster L.A.: Conditions for Correct Solvability of a Simplest Singular Boundary Value Problem of General Form. I. Z. Anal. Anwend. 25 (2006), 205-235. doi: 10.4171/ZAA/1285