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Minimum Properties of Eigenvalues — Elementary Proofs

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General Inequalities 2

Abstract

The purpose of this paper is to use an elementary integral inequality and some simple linear algebra to give a completely elementary proof of the minimum properties of all eigenvalues of Sturm-Liouville problems. The results are a simplification of work published in [1], where singular cases were considered but general boundary conditions were not.

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References

  1. P. R. Beesack, Elementary proofs of the extremal properties of the eigenvalues of the Sturm-Liouville equation, Can. Math. Bull. 3 (1960), 59–77.

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  2. E. Coddington and N. Levinson, Theory of Ordinary Differential Equations, McGraw-Hill Book Company, Inc., New York, 1955.

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  3. R. Courant and D. Hilbert, Methods of Mathematical Physics, V. 1, Interscience Publishers, Inc., New York, 1953.

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  4. P. Hartman, Ordinary Differential Equations, S. M. Hartman, Baltimore, 1973.

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  5. J. Hersch, Propriétés de convexité du type de Weyl pour des problèmes de vibration ou d’équilibre, Jour. Math. et de phys. appl. (ZAMP) 12 (1961), 298–322.

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  6. E. Kamke, Differentialgleichungen reeller Funktionen, Akademische Verlagsgesellschaft, Berlin, 1930, reprinted by Chelsea Publishing Company, N. Y., 1947.

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  7. W. T. Reid, Ordinary Differential Equations, John Wiley and Sons, Inc., New York, 1971.

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Author information

Authors and Affiliations

  1. Department of Mathematics, Carleton University, Ottawa, Ontario, KLS 5B6, Canada

    Paul R. Beesack

Authors
  1. Paul R. Beesack
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Editor information

Editors and Affiliations

  1. University of California, Los Angeles, USA

    E. F. Beckenbach

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© 1980 Springer Basel AG

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Beesack, P.R. (1980). Minimum Properties of Eigenvalues — Elementary Proofs. In: Beckenbach, E.F. (eds) General Inequalities 2. International Series of Numerical Mathematics / Internationale Schriftenreihe zur Numerischen Mathematik / Série Internationale d’Analyse Numérique, vol 47. Birkhäuser, Basel. https://doi.org/10.1007/978-3-0348-6324-7_10

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  • DOI: https://doi.org/10.1007/978-3-0348-6324-7_10

  • Publisher Name: Birkhäuser, Basel

  • Print ISBN: 978-3-7643-1056-1

  • Online ISBN: 978-3-0348-6324-7

  • eBook Packages: Springer Book Archive

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