Electron. J. Diff. Eqns., Vol. 2000(2000), No. 28, pp. 113.
Basis properties of eigenfunctions of nonlinear SturmLiouville problems
P. E. Zhidkov
Abstract:
We consider three nonlinear eigenvalue problems that consist of
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
and corresponding eigenfunctions
such that
has precisely n roots in the interval (0,1),
where n=0,1,2,....
For the first boundary condition, we show that
is a basis and that
is a Riesz basis in the space
.
For the second and third boundary conditions, we show that
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,
SturmLiouville 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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