Electron. J. Diff. Eqns., Vol. 2000(2000), No. 28, pp. 1-13.

Basis properties of eigenfunctions of nonlinear Sturm-Liouville problems

P. E. Zhidkov

Abstract:
We consider three nonlinear eigenvalue problems that consist of
$-y''+f(y^2)y=\lambda y$
with one of the following boundary conditions:
y(0)=y(1)=0   y'(0)=p,
y'(0)=y(1)=0   y(0)=p,
y'(0)=y'(1)=0   y(0)=p,
where p is a positive constant. Under smoothness and monotonicity conditions on f, we show the existence and uniqueness of a sequence of eigenvalues $\{\lambda _n\}$ and corresponding eigenfunctions $\{y_n\}$ such that $y_n(x)$ has precisely n roots in the interval (0,1), where n=0,1,2,....
For the first boundary condition, we show that $\{y_n\}$ is a basis and that $\{y_n/\|y_n\|\}$ is a Riesz basis in the space $L_2(0,1)$. For the second and third boundary conditions, we show that $\{y_n\}$ is a Riesz basis.

Submitted November 17, 1999. Published April 13, 2000.
Math Subject Classifications: 34L10, 34L30, 34L99.
Key Words: Riesz basis, nonlinear eigenvalue problem, Sturm-Liouville operator, completeness, basis.

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Peter E. Zhidkov
Bogoliubov Laboratory of Theoretical Physics
Joint Institute for Nuclear Research
141980 Dubna (Moscow region), Russia
email: zhidkov@thsun1.jinr.ru

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