Hostname: page-component-78c5997874-t5tsf Total loading time: 0 Render date: 2024-11-19T07:34:01.797Z Has data issue: false hasContentIssue false
  • English
  • Français

Sturm–Liouville problems with indefinite weights and Everitt's inequality

Published online by Cambridge University Press:  14 November 2011

Hans Volkmer
Hans Volkmer
Affiliation:
Department of Mathematical Sciences, University of Wisconsin-Milwaukee, P.O. Box 413, Milwaukee, WI 53201, U.S.A.
Get access Rights & Permissions [Opens in a new window]

Abstract

It is shown that spectral properties of Sturm–Liouville eigenvalue problems with indefinite weights are related to integral inequalities studied by Everitt. A result of Beals on indefinite problems leads to a sufficient condition for the validity of such an inequality. A Baire category argument is used to show that, in general, the inequality under consideration does not hold.

Type
Research Article
Information
Proceedings of the Royal Society of Edinburgh Section A: Mathematics , Volume 126 , Issue 5 , 1996 , pp. 1097 - 1112
Copyright
Copyright © Royal Society of Edinburgh 1996

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

1Akhiezer, N. I. and Glazman, I. M.. Theory of Linear Operators in Hilbert Space (New York: Dover reprint, 1993).Google Scholar
2Beals, R.. Indefinite Sturm–Liouville problems and half-range completeness. J. Differential Equations 56 (1985), 391–407.CrossRefGoogle Scholar
3Binding, P. and Volkmer, H.. Eigencurves for two-parameter Sturm–Liouville equations. SIAM Rev. 38 (1996), 27–48.CrossRefGoogle Scholar
4Curgus, B.. On the regularity of the critical point infinity of definitizable operators. J. Integral Equations Operator Theory 8 (1985), 462–88.CrossRefGoogle Scholar
5Curgus, B. and Langer, H.. A Krein space approach to symmetric ordinary differential operators with an indefinite weight function. J. Differential Equations 79 (1989), 31–61.CrossRefGoogle Scholar
6Evans, W. D. and Everitt, W. N.. A return to the Hardy-Littlewood integral inequality. Proc. Roy. Soc. London Ser. A 380 (1982), 447–86.Google Scholar
7Everitt, W. N.. On an extension to an integro-differential inequality of Hardy, Littlewood and Pólya. Proc. Roy. Soc. Edinburgh Sect. A 69 (1971/1972), 295–333.Google Scholar
8Everitt, W. N.. Hardy-Littlewood integral inequalities. In Inequalities: Fifty Years on from Hardy, Littlewood and Pólya, ed. Everitt, W. N., 29–51 (New York: Dekker, 1991).Google Scholar
9Faierman, M.. Elliptic problems involving an indefinite weight. Trans. Amer. Math. Soc. 320 (1990), 253–79.CrossRefGoogle Scholar
10Fleige, A.. The 'Turning Point Condition' of Beals for indefinite Sturm–Liouville problems. Math. Nachr. 172 (1995), 109–12.CrossRefGoogle Scholar
11Fleige, A.. A counterexample to completeness properties for indefinite Sturm–Liouville problems (submitted).Google Scholar
12Hardy, G. H., Littlewood, J. E. and Pólya, G.. Inequalities (London: Cambridge University Press, 1934).Google Scholar
13Ince, E. L.. Ordinary Differential Equations (New York: Dover reprint, 1956).Google Scholar
14Kaper, H. G., Kwong, M. K., Lekkerkerker, C. G. and Zettl, A.. Full- and half-range completeness theory of Sturm–Liouville problems with indefinite weights. Proc. Roy. Soc. Edinburgh Sect. A 98 (1984), 69–88.CrossRefGoogle Scholar
15Kato, T.. Perturbation Theory for Linear Operators (New York: Springer, 1976).Google Scholar
16Kwong, M. K. and Zettl, A.. Norm Inequalities for Derivatives and Differences, Lecture Notes in Mathematics 1536 (New York: Springer, 1992).CrossRefGoogle Scholar
17Pyatkov, S. G.. Some properties of eigenfunctions of linear pencils. Siberian Math. J. 30 (1989), 587–97.CrossRefGoogle Scholar
18Pyatkov, S. G.. Certain properties of eigenfunctions of linear pencils. Math. Notes 51 (1992), 90–5.CrossRefGoogle Scholar
19Renardy, M. and Rogers, R. C.. An Introduction to Partial Differential Equations (New York: Springer, 1993).Google Scholar
20Young, R. M.. An Introduction to Nonharmonic Fourier Series (New York: Academic Press, 1980).Google Scholar