Electronic Journal of Differential Equations,
Vol. 2000(2000), No. 28, pp. 1-13.
Title: Basis properties of eigenfunctions of nonlinear Sturm-Liouville problems
Author: P. E. Zhidkov (Joint Inst. for Nuclear Research, Dubna, Russia)
Abstract:
We consider three nonlinear eigenvalue problems that consist of
$$-y''+f(y^2)y=\lambda y$$
with one of the following boundary conditions:
$$\displaylines{ y(0)=y(1)=0 \quad y'(0)=p \,,\cr y'(0)=y(1)=0
\quad y(0)=p\,, \cr y'(0)=y'(1)=0 \quad y(0)=p\,, }$$
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 $n=0,1,2,\dots$.
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.