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Proceedings of the American Mathematical Society
Proceedings of the American Mathematical Society
ISSN 1088-6826(online) ISSN 0002-9939(print)

 

A limit-point criterion for a class of Sturm-Liouville operators defined in ${L^p}$ spaces


Author: R. C. Brown
Journal: Proc. Amer. Math. Soc. 132 (2004), 2273-2280
MSC (2000): Primary 47E05, 34C11, 34B24; Secondary 34C10
Published electronically: March 25, 2004
MathSciNet review: 2052403
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Abstract: Using a recent result of Chernyavskaya and Shuster we show that the maximal operator determined by $M[y]=-y''+qy$ on $[a,\infty)$, $a>-\infty$, where $q\ge 0$ and the mean value of $q$ computed over all subintervals of $\mathbb{R} $ of a fixed length is bounded away from zero, shares several standard ``limit-point at $\infty$" properties of the $L^2$ case. We also show that there is a unique solution of $M[y]=0$ that is in all $L^p[a, \infty)$, $p=[1,\infty]$.


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Additional Information

R. C. Brown
Affiliation: Department of Mathematics, University of Alabama, Tuscaloosa, Alabama 35487-0350
Email: dbrown@gp.as.ua.edu

DOI: http://dx.doi.org/10.1090/S0002-9939-04-07471-4
PII: S 0002-9939(04)07471-4
Keywords: Second-order differential operators of symmetric form in $L^p$ spaces, correct solvability, limit-point, $L^p$ solutions
Received by editor(s): December 18, 2002
Published electronically: March 25, 2004
Communicated by: Carmen C. Chicone
Article copyright: © Copyright 2004 American Mathematical Society