JournalszaaVol. 25, No. 2pp. 205–235

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

Abstract

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

\label1\label{1} %$$-r(x)y'(x)+q(x)y(x)=f(x),\quad x\in R$$ %

%

\label2\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 fLp(R),f \in L_p(\mathbb R),\ p[1,\iy]p\in[1,\iy] (L\iy(R):=C(R)),(L_\iy(\mathbb R):=C(\mathbb R)),\ rr is a continuous positive function on R\mathbb R, \ 0qL1\loc.0\le q \in L_1^{\loc}. A solution of this problem is, by definition, any absolutely continuous function yy satisfying the limit condition and almost everywhere the differential equation. This problem is called correctly solvable in a given space Lp(R)L_p(\mathbb R) if for any function fLp(R)f\in L_p(\mathbb R) it has a unique solution yLp(R)y\in L_p(\mathbb R) and if the following inequality holds with an absolute constant cp(0,\iy):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 rr and qq under which the above problem is correctly solvable in L_p(mathbbR).L\_p(\\mathbb R).

Cite this article

L.A. Shuster, N.A. Chernyavskaya, Conditions for Correct Solvability of a Simplest Singular Boundary Value Problem of General Form. I. Z. Anal. Anwend. 25 (2006), no. 2, pp. 205–235

DOI 10.4171/ZAA/1285