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

Published online by Cambridge University Press:  20 November 2018

Paul A. Binding ,
Patrick J. Browne  and
Bruce A. Watson
Paul A. Binding
Affiliation:
Department of Mathematics and Statistics, University of Calgary, Calgary, Alberta, T2N 1N4
Patrick J. Browne
Affiliation:
Department of Mathematics and Statistics, University of Saskatchewan, Saskatoon, Saskatchewan, S7N 5E6
Bruce A. Watson
Affiliation:
Department of Mathematics, University of the Witwatersrand, Private Bag 3, P O WITS 2050, South Africa
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Abstract

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The nonlinear Sturm-Liouville equation

$$-{{\left( p{y}' \right)}^{\prime }}\,+\,qy\,=\,\text{ }\!\!\lambda\!\!\text{ }\left( 1-f \right)ry\,on\,[0,\,1]$$

is considered subject to the boundary conditions

$$\left( {{\text{a}}_{j}}\text{ }\lambda \text{ }\text{+}{{\text{b}}_{j}} \right)y\left( j \right)=\left( {{c}_{j}}\text{ }\lambda \text{ }+{{d}_{j}} \right)\left( p{y}' \right)\left( j \right),j=0,1$$
.

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

Keywords

34B2434C2334L30
Type
Research Article
Information
Canadian Journal of Mathematics , Volume 52 , Issue 2 , 01 April 2000 , pp. 248 - 264
Copyright
Copyright © Canadian Mathematical Society 2000

References

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