Spectral asymptotics for nonlinear Sturm-Liouville problems.

Tetsutaro Shibata

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Spectral Theory of Singular Hahn Difference Equation of the Sturm-Liouville Type

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In this work, we consider the singular Hahn difference equation of the Sturm-Liouville type. We prove the existence of the spectral function for this equation. We establish Parseval equality and an expansion formula for this equation on a semi-unbounded interval.

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We study a Sturm-Liouville problem containing a spectral parameter in the boundary conditions. We associate to this problem a self-adjoint operator in a Pontryagin space Π₁. Using this operator-theoretic formulation and analytic methods, we study the asymptotic behavior of the eigenvalues under the variation of a large physical parameter in the boundary conditions. The spectral analysis is applied to investigate the well-posedness and stability of the wave equation of a string. ...

Some global results for nonlinear Sturm-Liouville problems with spectral parameter in the boundary condition

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We consider nonlinear Sturm-Liouville problems with spectral parameter in the boundary condition. We investigate the structure of the set of bifurcation points, and study the behavior of two families of continua of nontrivial solutions of this problem contained in the classes of functions having oscillation properties of the eigenfunctions of the corresponding linear problem, and bifurcating from the points and intervals of the line of trivial solutions.

A simple proof of the spectral continuity of the Sturm-Liouville problem

PrzemysŁaw Kosowski (1997)

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The aim of this article is to present a simple proof of the theorem about perturbation of the Sturm-Liouville operator in Liouville normal form.

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A Note on Pairs of Solutions of a Nonlinear Sturm-Liouville Problem.

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On the Green function of Sturm-Liouville differential equation with normal operator coefficient.

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A note on a generalization of the Sturm-Picone theorem

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Spectral singularities of Sturm-Liouville problems with eigenvalue-dependent boundary conditions.

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The nonlinear limit-point/limit-circle problem for higher order equations

Miroslav Bartušek, Zuzana Došlá, John R. Graef (1998)

Archivum Mathematicum

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We describe the nonlinear limit-point/limit-circle problem for the $n$ -th order differential equation $y^{(n)} + r (t) f (y, y^{'}, \dots, y^{(n - 1)}) = 0 .$ The results are then applied to higher order linear and nonlinear equations. A discussion of fourth order equations is included, and some directions for further research are indicated.