Basis properties of eigenfunctions of nonlinear Sturm-Liouville problems

Zhidkov, P.E.

Electronic Journal of Differential Equations, Tome 2000 (2000).

Résumé

Summary: 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,$\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.

Détail

Classification : 34L10, 34L30, 34L99
Keywords: Riesz basis, nonlinear eigenvalue problem, Sturm-Liouville operator, completeness, basis