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

Published:2000-04-01
Printed: Apr 2000
• Paul A. Binding
• Patrick J. Browne
• Bruce A. Watson
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Abstract

The nonlinear Sturm-Liouville equation $$-(py')' + qy = \lambda(1 - f)ry \text{ on } [0,1]$$ is considered subject to the boundary conditions $$(a_j\lambda + b_j) y(j) = (c_j\lambda + d_j) (py') (j), \quad j = 0,1.$$ 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 $y'$. 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.