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

Binding, Paul A. ; Browne, Patrick J. ; Watson, Bruce A.

Canadian journal of mathematics, Tome 52 (2000) no. 2, pp. 248-264

Résumé

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.

Détail

DOI : 10.4153/CJM-2000-011-1

Mots-clés : 34B24, 34C23, 34L30

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