Basis Properties of Eigenfunctions of Nonlinear Sturm-Liouville Problems
Date
2000-04-13
Authors
Zhidkov, Peter E.
Journal Title
Journal ISSN
Volume Title
Publisher
Southwest Texas State University, Department of Mathematics
Abstract
We consider three nonlinear eigenvalue problems that consist of
-y'' + ƒ(y2)y = λy
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 ƒ, we show the existence and uniqueness of a sequence of eigen-values {λn} and corresponding eigenfunctions {yn} such that yn(x) has precisely n roots in the interval (0,1), where n = 0, 1, 2,.... For the first boundary condition, we show that {yn} is a basis and that {yn/
yn
} is a Riesz basis in the space L2(0, 1). For the second and third boundary conditions, we show that {yn} is a Riesz basis.
yn
} is a Riesz basis in the space L2(0, 1). For the second and third boundary conditions, we show that {yn} is a Riesz basis.
Description
Keywords
Riesz basis, Nonlinear eigenvalue problem, Sturm-Liouville operator, Completeness, Basis
Citation
Zhidkov, P. E. (2000). Basis properties of eigenfunctions of nonlinear Sturm-Liouville problems. <i>Electronic Journal of Differential Equations, 2000</i>(28), pp. 1-13.
Rights
Attribution 4.0 International