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

Open Access article
 Printed: Apr 2000
  • Paul A. Binding
  • Patrick J. Browne
  • Bruce A. Watson
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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.
MSC Classifications: 34B24, 34C23, 34L30 show english descriptions Sturm-Liouville theory [See also 34Lxx]
Bifurcation [See also 37Gxx]
Nonlinear ordinary differential operators
34B24 - Sturm-Liouville theory [See also 34Lxx]
34C23 - Bifurcation [See also 37Gxx]
34L30 - Nonlinear ordinary differential operators

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