Basis Properties of Eigenfunctions of Nonlinear Sturm-Liouville Problems

Date

2000-04-13

Authors

Zhidkov, Peter E.

Journal Title

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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.

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.

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Attribution 4.0 International

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