Nonlinear discrete Sturm–Liouville problems with global boundary conditions

This paper is devoted to the study of nonlinear difference equations subject to global nonlinear boundary conditions. We provide sufficient conditions for the existence of solutions based on properties of the nonlinearities and the eigenvalues of an associated linear Sturm–Liouville problem.     

On the determination of the impulsive Sturm–Liouville operator with the eigenparameter-dependent boundary conditions

In the present work, we consider the inverse problem for the impulsive Sturm–Liouville equations with eigenparameter-dependent boundary conditions on the whole interval (0,π) from interior spectral data. We prove two uniqueness theorems on the potential q(x) and boundary conditions for the interior inverse problem, and using the Weyl function technique, we show that if coefficients of the first boundary condition, that is, h₁,h₂, are known, then the potential function q(x) and coefficients of the second boundary condition, that is, H₁,H₂, are uniquely determined by information about the eigenfunctions at the midpoint of the interval and one spectrum or partial information on the eigenfunctions at some internal points and some of two spectra.     

On the solvability of inverse Sturm–Liouville problems with self-adjoint boundary conditions

Theorems on the existence and uniqueness of a solution of the inverse Sturm–Liouville problem with self-adjoint nonseparated boundary conditions are proved. As spectral data two spectra and two eigenvalues are used. The theorems generalize the Levitan–Gasymov solvability theorem and Borg’s uniqueness theorem to the case of general boundary conditions.     

Eigenvalue problem of Sturm–Liouville systems with separated boundary conditions

Let \(\lambda _j\) be the jth eigenvalue of Sturm–Liouville systems with separated boundary conditions, we build up the Hill-type formula, which represent \(\prod \nolimits _{j}(1-\lambda _j^{-1})\) as a determinant of finite matrix. Consequently, we get the Krein-type trace formula based on the Hill-type formula, which express \(\sum \nolimits _{j}{1\over \lambda _j^m}\) as trace of finite matrices. The trace formula can be used to estimate the conjugate point alone a geodesic in Riemannian manifold and to get some infinite sum identities.     

Numerical computation of eigenvalues of discontinuous Sturm–Liouville problems with parameter dependent boundary conditions using sinc method

In this paper, we consider a Sturm–Liouville problem which contains an eigenparameter appearing linearly in two boundary conditions, in addition to an internal point of discontinuity. Eigenvalue problems with eigenparameter appearing in the boundary conditions usually have complicated characteristic determinant where zeros cannot be explicitly computed. We apply the sinc method, which is based on the sampling theory to compute approximations of the eigenvalues. An error analysis is exhibited involving rigorous error bounds. Using computable error bounds we obtain eigenvalue enclosures in a simple way. Illustrative examples are included to demonstrate the validity and applicability of the presented technique.     

A Prüfer angle approach to semidefinite Sturm–Liouville problems with coupling boundary conditions

It is shown how to reduce the periodic/antiperiodic Sturm–Liouville problems to analysis of the Prüfer angle. This provides an alternative to the more usual approaches via operator theory or the Hill discriminant in the definite case, and leads to new results in the semidefinite case. An extension to more general coupling boundary conditions is also given.     

Inverse Sturm–Liouville problem with spectral polynomials in nonsplitting boundary conditions

Theorems on the unique reconstruction of a Sturm–Liouville problem with spectral polynomials in nonsplitting boundary conditions are proved. Two spectra and finitely many eigenvalues (one spectrum and finitely many eigenvalues for a symmetric potential) of the problem itself are used as the spectral data. The results generalize the Levinson uniqueness theorem to the case of nonsplitting boundary conditions containing polynomials in the spectral parameter. Algorithms and examples of solving relevant inverse problems are also presented.     

Uniqueness for inverse Sturm–Liouville problems with a finite number of transmission conditions

We establish various uniqueness results for inverse spectral problems of Sturm–Liouville operators with a finite number of discontinuities at interior points at which we impose the usual transmission conditions. We consider both the cases of classical Robin and of eigenparameter dependent boundary conditions.     

Eigenvalues of limit-point Sturm–Liouville problems

The dependence of the eigenvalues of self-adjoint Sturm–Liouville problems on the boundary conditions when each endpoint is regular or in the limit-circle case is now, due to some surprisingly recent results, well understood. Here we study this dependence for singular problems with one endpoint in the limit-point case.     

On a class of singular Sturm–Liouville periodic boundary value problems

Keeping in mind the singular model for the periodic oscillations of the axis of a satellite in the plane of the elliptic orbit around its center of mass, we give sufficient conditions for the solvability of a class of singular Sturm–Liouville equations with periodic boundary value conditions. To this end, under a suitable change of variables, we present a new existence result for problems defined in the real half-line.     

On degenerate boundary conditions in the Sturm–Liouville problem

We describe all degenerate boundary conditions in the homogeneous Sturm–Liouville problem.     

Uniqueness of reconstruction of the Sturm–Liouville problem with spectral polynomials in nonseparated boundary conditions

Uniqueness theorems for solutions of inverse Sturm–Liouville problems with spectral polynomials in nonseparated boundary conditions are proved. As spectral data two spectra and finitely many eigenvalues of the direct problem or, in the case of a symmetric potential, one spectrum and finitely many eigenvalues are used. The obtained results generalize the Levinson uniqueness theorem to the case of nonseparated boundary conditions containing polynomials in the spectral parameter.     

Inequalities among eigenvalues of Sturm–Liouville problems with distribution potentials

This paper deals with the eigenvalue problems for the Sturm–Liouville operators generated by the differential expression

L(y)=−(p(x)y^′)^′+q(x)y$L\left(y\right)=-{\left(p\left(x\right)y^{\prime }\right)}^{\prime }+q\left(x\right)y$

with singular coefficients q(x)$q\left(x\right)$ in the sense of distributions. We obtain the inequalities among the eigenvalues corresponding to different self-adjoint boundary conditions. The continuity region, the differentiability and the monotonicity of the n$n$th eigenvalue corresponding to the separated boundary conditions are given. Oscillation properties of the eigenfunctions of all the coupled Sturm–Liouville problems are characterized. The main results of this paper can also be applied to solve a class of transmission problems.     

Non-real eigenvalues of indefinite Sturm–Liouville problems

The present paper deals with non-real eigenvalues of regular indefinite Sturm–Liouville problems. A priori bounds and sufficient conditions of the existence for non-real eigenvalues are obtained under mild integrable conditions of coefficients.