Half-eigenvalues of periodic Sturm--Liouville problems
type: Item
instance of: scholarly article
title: Half-eigenvalues of periodic Sturm--Liouville problems
zbMATH Open document ID: 1073.34095
author: P. A. Binding
author: Bryan P. Rynne
published in: Journal of Differential Equations
publication date: 2005-01-11T00:00:00Z
review text: Here, the half-eigenvalues and the nodal properties of the eigenfunctions of the nonlinear Sturm-Liouville equation \[ \begin{aligned} &-u''+qu=au^+-bu^-+\lambda u, \;0 < x<2\pi, \tag{E1}\\ &u(0)=u(2\pi),\;u'(0)=u'(2\pi),\tag{E2} \end{aligned} \] are considered, where \(q, a, b \in L_1(0,2\pi),\) \(\lambda\) is a real parameter and \(u^{\pm}(t)=\max \{\pm u(t),0\}.\) Let \[ \begin{aligned} H&=\{u\mid u\in W_1^2(0,2\pi), \; u \text{ satisfies the condition (E2)} \},\\ S_k^1&=\{u\mid u\in H,\; \| u\| =1 \text{ and } u \text{ has exactly } k \text{ simple zeros on }[0,2\pi) \},\\ \Sigma_H&=\{\lambda\in\mathbb R: \text{(E1) has a solution on the unit ball } S^1 \text{ of }H \},\\ \Sigma_H^k&=\{\lambda\in\mathbb R: \text{(E1) has a solution in } S^1_{2k}\},\\ \tilde\Sigma_H^k&=\{(\lambda,u)\in \mathbb R \times S_{2k}^1:(\lambda,u)\text{ satisfies (E1)}\}. \end{aligned} \] The authors show that (1) For each integer \(k\geq 0\), (E1) has two distinct solutions \((\lambda_k^{min} ,\phi_k^{min}),\) \((\lambda_k^{max},\phi_k^{max})\in \mathbb R\times S^1_{2k},\) such that (a) \(\lambda_k^{min}\leq \lambda_k^{max}<\lambda_{k+1}^{min},\) (b) \(\{\lambda_k^{min},\lambda_k^{max}\}\subset \Sigma_H^k\subset [\lambda_k^{min},\lambda_k^{max}],\) and \(\Sigma_H^0=\{\lambda_0^{min},\lambda_0^{max}\}.\) (2) Denote by \((L;a,b)\) the problem (E1), (E2). As functions of \((a,b),\) the eigenvalues \(\lambda_k^{min,max}(L;a,b)\) depend continuously on \(a\) and \(b\) and possess a monotonie property, i.e., if \((a_0,b_0)\leq (a_1,b_1)\) then \[ \lambda_k^{min,max}(L;a_1,b_1)\leq \lambda_k^{min,max}(L;a_0,b_0). \] Moreover, some intrinsic properties for \(\Sigma_H\) are obtained. The results are extended to the more general nonlinear problem \[ -u''+qu={\mathbf f}(u)+h(t). \tag{E3} \]
reviewed by: Chung-Tsun Shieh
Mathematics Subject Classification ID: 34L20
Mathematics Subject Classification ID: 34B24
zbMATH DE Number: 2126458
zbMATH Keywords: periodic Sturm-Liouville problems
zbMATH Keywords: half-eigenvalues
zbMATH Keywords: eigenfunctions
zbMATH Keywords: nodal properties
MaRDI profile type: Publication
full work available at URL: j.jde.2004.05.014
OpenAlex ID: W2057329902
cites work: The functional Fučik spectrum has empty interior
cites work: Structure of the Fučík spectrum and existence of solutions for equations with asymmetric nonlinearities
cites work: On some nonlinear Sturm-Liouville problems
cites work: Sturm–Liouville problems with eigenparameter dependent boundary conditions
cites work: A Prüfer approach to half-linear Sturm-Liouville problems
cites work: On the Nature of the Spectrum of Singular Second Order Linear Differential Equations
cites work: A variational approach to nonresonance with respect to the Fučik spectrum
cites work: On the Dirichlet problem for weakly non-linear elliptic partial differential equations
cites work: Competing species equations with diffusion, large interactions, and jumping nonlinearities
cites work: On the periodic Fučik spectrum and a superlinear Sturm–Liouville equation
cites work: Q5185568
cites work: Existence and multiplicity results for periodic solutions of semilinear Duffing equations
cites work: On the periodic BVP for the forced duffing equation with jumping nonlinearity
cites work: Landesman-Lazer conditions for periodic boundary value problems with asymmetric nonlinearities
cites work: Nonlinear resonance in asymmetric oscillators
cites work: Q3900559
cites work: Q4397001
cites work: Nonresonance with respect to the Fuc̆ik spectrum for periodic solutions of second order ordinary differential equations
cites work: Nonresonance conditions on the potential with respect to the Fučik spectrum for the periodic boundary value problem
cites work: Q3312749
cites work: Q5509059
cites work: Maslov index, Poincaré-Birkhoff theorem and periodic solutions of asymptotically linear planar Hamiltonian systems
cites work: Q3049152
cites work: On the number of \(2\pi\)-periodic solutions for \(u''+g(u)=s(1+h(t))\) using the Poincaré-Birkhoff theorem
cites work: Sturm-Liouville type problems for the \(p\)-Laplacian under asymptotic non-resonance conditions
cites work: Q5613350
cites work: Remarks and generalizations related to a recent multiplicity result of A. Lazer and P. McKenna
cites work: A Nonlinear Fredholm Alternative for Second Order Ordinary Differential Equations
cites work: The Fucik spectrum of general Sturm-Liouville problems
cites work: Non-resonance conditions for semilinear Sturm-Liouville problems with jumping nonlinearities
cites work: Half-eigenvalues of elliptic operators
cites work: Multiplicity of solutions for two-point boundary value problems with asymptotically asymmetric nonlinearities
cites work: Rotation numbers, eigenvalues, and the Poincaré--Birkhoff theorem
cites work: The rotation number approach to the periodic Fučik spectrum
DOI: 10.1016/J.JDE.2004.05.014
description: scientific article; zbMATH DE number 2126458