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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
Abstract
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.
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- Research Article
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- Copyright © Canadian Mathematical Society 2000
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