Skip to main content
×
×
Home

Spectral Problems for Non-Linear Sturm-Liouville Equations with Eigenparameter Dependent Boundary Conditions

  • Paul A. Binding (a1), Patrick J. Browne (a2) and Bruce A. Watson (a3)
Abstract

The nonlinear Sturm-Liouville equation

−(pyʹ)ʹ + qy = λ(1 − f)ry on [0, 1]

is considered subject to the boundary conditions

(aj λ + bj )y(j) = (cj λ + dj )(pyʹ)(j), j = 0, 1.

Here a 0 = 0 = c 0 and p, r > 0 and q are functions depending on the independent variable x alone, while f depends on x, y and yʹ. Results are given on existence and location of sets of (λ, y) bifurcating from the linearized eigenvalues, and for which y has prescribed oscillation count, and on completeness of the y in an appropriate sense.

    • Send article to Kindle

      To send this article to your Kindle, first ensure no-reply@cambridge.org is added to your Approved Personal Document E-mail List under your Personal Document Settings on the Manage Your Content and Devices page of your Amazon account. Then enter the ‘name’ part of your Kindle email address below. Find out more about sending to your Kindle. Find out more about sending to your Kindle.

      Note you can select to send to either the @free.kindle.com or @kindle.com variations. ‘@free.kindle.com’ emails are free but can only be sent to your device when it is connected to wi-fi. ‘@kindle.com’ emails can be delivered even when you are not connected to wi-fi, but note that service fees apply.

      Find out more about the Kindle Personal Document Service.

      Spectral Problems for Non-Linear Sturm-Liouville Equations with Eigenparameter Dependent Boundary Conditions
      Available formats
      ×
      Send article to Dropbox

      To send this article to your Dropbox account, please select one or more formats and confirm that you agree to abide by our usage policies. If this is the first time you use this feature, you will be asked to authorise Cambridge Core to connect with your <service> account. Find out more about sending content to Dropbox.

      Spectral Problems for Non-Linear Sturm-Liouville Equations with Eigenparameter Dependent Boundary Conditions
      Available formats
      ×
      Send article to Google Drive

      To send this article to your Google Drive account, please select one or more formats and confirm that you agree to abide by our usage policies. If this is the first time you use this feature, you will be asked to authorise Cambridge Core to connect with your <service> account. Find out more about sending content to Google Drive.

      Spectral Problems for Non-Linear Sturm-Liouville Equations with Eigenparameter Dependent Boundary Conditions
      Available formats
      ×
Copyright
References
Hide All
[1] Binding, P. A. and Browne, P. J., Left definite Sturm-Liouville problems with eigenparameter dependent boundary conditions. Differential Integral Equations 12(1999), 167182.
[2] Binding, P. A., Browne, P. J. and Seddighi, K., Sturm-Liouville problems with eigenparameter dependent boundary conditions. Proc. EdinburghMath. Soc. 37(1993), 5772.
[3] Binding, P. A., Browne, P. J. and Watson, B. A., Inverse spectral problems for Sturm-Liouville equations with eigenparameter dependent boundary conditions. J. LondonMath. Soc., to appear.
[4] Brown, K. J., A completeness theorem for a nonlinear problem. Proc. Edinburgh Math. Soc. 19(1974), 169172.
[5] Chow, S.-N. and Hale, J. K., Methods of Bifurcation Theory. Springer, 1982.
[6] Coddington, E. A. and Levinson, N., Theory of Ordinary Differential Equations. Krieger, 1984.
[7] Crandall, M. G. and Rabinowitz, P. H., Nonlinear Sturm-Liouville eigenvalue problems and topological degree. J. Math. Mech. 19(1970), 10831102.
[8] Crandall, M. G. and Rabinowitz, P. H., Bifurcation from simple eigenvalues. J. Funct. Anal. 8(1971), 321340.
[9] Dijksma, A., Eigenfunction expansions for a class of J-selfadjoint ordinary differential operators with boundary conditions containing the eigenparameter. Proc. Roy. Soc. Edinburgh 86A(1980), 127.
[10] Dijksma, A. and Langer, H., Operator theory and ordinary differential operators. Fields Institute Monographs 3, Amer.Math. Soc., 1996, 75139.
[11] Edwards, R. E., Functional Analysis. Dover, 1995.
[12] Fulton, C., Two-point boundary value problems with eigenvalue parameter contained in the boundary conditions. Proc. Roy. Soc. Edinburgh 77A(1977), 293308.
[13] Hörmander, L., Lectures on Nonlinear Hyperbolic Differential Equations. Springer, 1997.
[14] Kato, T., Perturbation Theory for Linear Operators. Springer, 1984.
[15] Naimark, A. M., Linear Differential Operators, Part I. Ungar, 1967.
[16] Rabinowitz, P. H., Nonlinear Sturm-Liouville problems for second order ordinary differential equations. Comm. Pure Appl. Math. 23(1970), 939961.
[17] Russakovskii, E. M., An operator treatment of a boundary value problem with spectral parameter appearing polynomially in the boundary conditions. Funct. Anal. Appl. 9(1975), 9192.
[18] Walter, J., Regular eigenvalue problems with eigenvalue parameter in the boundary conditions. Math. Z. 133(1973), 301312.
Recommend this journal

Email your librarian or administrator to recommend adding this journal to your organisation's collection.

Canadian Journal of Mathematics
  • ISSN: 0008-414X
  • EISSN: 1496-4279
  • URL: /core/journals/canadian-journal-of-mathematics
Please enter your name
Please enter a valid email address
Who would you like to send this to? *
×
MathJax

Keywords

Metrics

Full text views

Total number of HTML views: 0
Total number of PDF views: 0 *
Loading metrics...

Abstract views

Total abstract views: 0 *
Loading metrics...

* Views captured on Cambridge Core between <date>. This data will be updated every 24 hours.

Usage data cannot currently be displayed