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A criterion for correct solvability of the Sturm-Liouville equation in the space $L_{p}(R)$


Authors: N. Chernyavskaya and L. Shuster
Journal: Proc. Amer. Math. Soc. 130 (2002), 1043-1054
MSC (2000): Primary 34C11, 34B40, 47E05
DOI: https://doi.org/10.1090/S0002-9939-01-06145-7
Published electronically: September 14, 2001
MathSciNet review: 1873778
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Abstract: We consider an equation

\begin{displaymath}{(1)}\quad\qquad\qquad\qquad -y''(x) + q(x) y(x) = f(x),\qquad x\in R, \qquad\qquad\qquad\qquad\end{displaymath}

where $f(x) \in L_{p}(R), p\in [1,\infty ] \left (\Vert f \Vert _{\infty } := C (R) \right )$, and $0 \le q(x)\in L_{1}^{\operatorname{loc}} (R).$By a solution of equation (1), we mean any function $y(x)$ such that $y(x), y'(x) \in AC^{\operatorname{loc}} (R),$and equality (1) holds almost everywhere on $R.$In this paper, we obtain a criterion for the correct solvability of (1) in $L_{p} (R)$, $p \in [1,\infty ].$


References [Enhancements On Off] (What's this?)

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Additional Information

N. Chernyavskaya
Affiliation: Department of Mathematics and Computer Science, Ben-Gurion University of the Negev, P.O.B. 653, Beer-Sheva, 84105, Israel

L. Shuster
Affiliation: Department of Mathematics and Computer Science, Bar-Ilan University, Ramat-Gan, 52900, Israel

DOI: https://doi.org/10.1090/S0002-9939-01-06145-7
Keywords: Correct solvability, Sturm-Liouville equation
Received by editor(s): April 6, 2000
Received by editor(s) in revised form: October 4, 2000
Published electronically: September 14, 2001
Additional Notes: This research was supported by the Israel Academy of Sciences under Grant 431/95
Communicated by: Carmen C. Chicone
Article copyright: © Copyright 2001 American Mathematical Society

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