Open Access

Upper bounds for the eigenvalues of differential equations

Journal of Inequalities and Applications20062006:48606

https://doi.org/10.1155/JIA/2006/48606

Received: 10 February 2004

Accepted: 4 May 2004

Published: 3 January 2006

Abstract

We establish upper bounds for the eigenvalues of second-order and fourth-order differential equations. The inequalities are obtained via rearrangements of higher degree.

[1234567891011]

Authors’ Affiliations

(1)
Department of Mathematics and Statistics, Sultan Qaboos University

References

  1. Bandle C: Isoperimetric Inequalities and Applications, Monographs and Studies in Mathematics. Volume 7. Pitman, Massachusetts; 1980:x+228.Google Scholar
  2. Bandle C: Extremal problems for eigenvalues of the Sturm-Liouville type. In General Inequalities, 5 (Oberwolfach, 1986), Internat. Schriftenreihe Numer. Math.. Volume 80. Birkhäuser, Basel; 1987:319–336.Google Scholar
  3. Banks DO: Bounds for the eigenvalues of nonhomogeneous hinged vibrating rods. Journal of Mathematics Mechanics 1967, 16: 949–966.MATHMathSciNetGoogle Scholar
  4. Barnes DC: Rearrangements of functions and lower bounds for eigenvalues of differential equations. Applicable Analysis. An International Journal 1982,13(4):237–248.MATHMathSciNetView ArticleGoogle Scholar
  5. Barnes DC: Extremal problems for eigenvalues with applications to buckling, vibration and sloshing. SIAM Journal on Mathematical Analysis 1985,16(2):341–357. 10.1137/0516025MATHMathSciNetView ArticleGoogle Scholar
  6. Cochran JA: The Analysis of Linear Integral Equations. McGraw-Hill, New York; 1972:xi+370.MATHGoogle Scholar
  7. Cox SJ, McCarthy CM: The shape of the tallest column. SIAM Journal on Mathematical Analysis 1998,29(3):547–554. 10.1137/S0036141097314537MATHMathSciNetView ArticleGoogle Scholar
  8. Hardy GH, Littlewood JE, Polya G: Inequalities. Cambridge University Press, Cambridge; 1934.MATHGoogle Scholar
  9. Karaa S: Sharp estimates for the eigenvalues of some differential equations. SIAM Journal on Mathematical Analysis 1998,29(5):1279–1300. 10.1137/S0036141096307849MATHMathSciNetView ArticleGoogle Scholar
  10. Karaa S: Inequalities for eigenvalue functionals. Journal of Inequalities and Applications 1999,4(2):175–181. 10.1155/S1025583499000351MATHMathSciNetGoogle Scholar
  11. Schwarz B: On the extrema of the frequencies of nonhomogeneous strings with equimeasurable density. Journal of Mathematics Mechanics 1961, 10: 401–422.MATHMathSciNetGoogle Scholar

Copyright

© Hindawi Publishing Corporation. 2006

This article is published under license to BioMed Central Ltd. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.