Multiplicity results for Sturm–Liouville boundary value problems

Multiplicity results for Sturm–Liouville boundary value problems are obtained. Proofs are based on variational methods.     

Corrected finite difference eigenvalues of periodic Sturm–Liouville problems

Computation of eigenvalues of regular Sturm–Liouville problems with periodic boundary conditions is considered. We show that a proof similar to that given by Andrew (1989) can be used to prove that a correction technique applied to a finite difference scheme given by Vanden Berghe et al. (1995) reduces the error in the kth eigenvalue estimate from O(k⁴h²) to O(kh²), where h is the uniform mesh length. We also provide a significantly shorter proof of a slightly weaker result.     

Asymptotic profiles of variational eigenvalues of two-parameter non-linear Sturm–Liouville problems

We consider the two-parameter non-linear Sturm–Liouville problems. By using the variational method on general level sets, the variational eigenvalues are obtained. The purpose of this paper is to study the properties of these variational eigenvalues with respect to the parameter of general level sets. © 1998 B. G. Teubner Stuttgart—John Wiley & Sons, Ltd.     

A collocation/quadrature-based Sturm–Liouville problem solver

We present a computational method for solving a class of boundary-value problems in Sturm–Liouville form. The algorithms are based on global polynomial collocation methods and produce discrete representations of the eigenfunctions. Error control is performed by evaluating the eigenvalue problem residuals generated when the eigenfunctions are interpolated to a finer discretization grid; eigenfunctions that produce residuals exceeding an infinity-norm bound are discarded. Because the computational approach involves the generation of quadrature weights and arrays for discrete differentiation operations, our computational methods provide a convenient framework for solving boundary-value problems by eigenfunction expansion and other projection methods.     

Eigenvalues of regular fourth‐order Sturm–Liouville problems with transmission conditions

In this paper, a class of fourth‐order Sturm‐Liouville problems with transmission conditions is considered. The eigenvalues depend not only continuously but also smoothly on the problem. An expression for the derivative of the eigenvalues with respect to a given parameter: an endpoint, a boundary condition, a transmission condition, a coefficient, or the weight function, is found. Copyright © 2016 John Wiley & Sons, Ltd.     

Discrete Sturm–Liouville problems, Jacobi matrices and Lagrange interpolation series

The close relationship between discrete Sturm–Liouville problems belonging to the so-called limit-circle case, the indeterminate Hamburger moment problem and the search of self-adjoint extensions of the associated semi-infinite Jacobi matrix is well known. In this paper, all these important topics are also related with associated sampling expansions involving analytic Lagrange-type interpolation series.     

Self‐adjoint Sturm–Liouville problems with an infinite number of boundary conditions

We study Sturm–Liouville (SL) problems on an infinite number of intervals, adjacent endpoints are linked by means of boundary conditions, and characterize the conditions which determine self‐adjoint operators in a Hilbert space which is the direct sum of the spaces for each interval. These conditions can be regular or singular, separated or coupled. Furthermore, the inner products of the summand spaces may be multiples of the usual inner products with different spaces having different multiples. We also extend the GKN Theorem to cover the infinite number of intervals theory with modified inner products and discuss the connection between our characterization and the classical one with the usual inner products. Our results include the finite number of intervals case.     

A Sturm–Liouville problem depending rationally on the eigenvalue parameter

We consider the Sturm–Liouville problem (1.1) and (1.2) with a potential depending rationally on the eigenvalue parameter. With these equations a λ ‐linear eigenvalue problem is associated in such a way that L₂‐solutions of (1.1), (1.2) correspond to eigenvectors of a linear operator. If the functions q and u are real and satisfy some additional conditions, the corresponding linear operator is a definitizable self‐adjoint operator in some Krein space. Moreover we consider the problem (1.1) and (1.3) on the positive half‐axis. Here we use results on the absense of positive eigenvalues for Sturm–Liouville operators to exclude critical points of the associated definitizable operator. (© 2007 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)     

Second order problems with functional conditions including Sturm–Liouville and multipoint conditions

In this paper we study the solvability of equations of the form ‐(d/dt)φ (t, u, u (t), u ′(t)) = f (t, u, u (t), u ′(t)) for a.e. t ∈ I = [a, b ], together with functional‐boundary conditions which cover, amongst others, Sturm–Liouville and multipoint boundary data as particular cases. Our approach uses upper and lower solutions together with growth restrictions of Nagumo's type. An example is presented to show the applicability of the obtained results. (© 2008 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)     

Uniqueness of solutions to inverse Sturm–Liouville problems with potential using three spectra

The uniqueness of solutions to two inverse Sturm–Liouville problems using three spectra is proven, based on the uniqueness of the solution-pair to an overdetermined Goursat–Cauchy boundary value problem. We discuss the uniqueness of the potential for a Dirichlet boundary condition at an arbitrary interior node, and for a Robin boundary condition at an arbitrary interior node, whereas at the exterior nodes we have Dirichlet boundary conditions in both situations. Here we are particularly concerned with potential functions that are L²(0,a).     

Linear Sturm–Liouville problems with general homogeneous linear multi‐point boundary conditions

In this paper, we study second order linear Sturm–Liouville problems involving one or two homogeneous linear multi‐point boundary conditions in the most general form. We obtain conditions for the existence of a sequence of positive eigenvalues with consecutive zero counts of the eigenfunctions. Furthermore, we reveal the interlacing relations between the eigenvalues of such Sturm–Liouville problems and those of Sturm–Liouville problems with certain two‐point separated boundary conditions.     

Sturm–Liouville properties for Atkinson's semi‐definite p‐Laplacian eigenvalue problems

We define Atkinson's semi‐definite p‐Laplacian eigenvalue problems, which include the regular p‐Laplacian eigenvalue problems with L ¹ coefficient functions. Then we show that the Sturm oscillation theorem also holds for this eigenvalue problem.     

The essential spectrum of a singular Sturm–Liouville operator

In this paper we study the essential spectrum of the operator where is a positive absolutely continuous function on (0, 1) that resembles for some . We prove that the essential spectrum of coincides with the essential spectrum of the operator .     

Half‐inverse problems for the quadratic pencil of the Sturm–Liouville equations with impulse

In this paper, we consider the inverse spectral problem for the impulsive Sturm–Liouville differential pencils on [0, π] with the Robin boundary conditions and the jump conditions at the point . We prove that two potentials functions on the whole interval and the parameters in the boundary and jump conditions can be determined from a set of eigenvalues for two cases: (i) the potentials given on and (ii) the potentials given on , where 0 < α < 1 , respectively. Inverse spectral problems, Sturm–Liouville operator, spectrum, uniqueness.     

On a Sturm–Liouville type problem with retarded argument

In this work, a Sturm–Liouville‐type problem with retarded argument, which contains spectral parameter in the boundary conditions and with transmission conditions at the point of discontinuity are investigated. We obtained asymptotic formulas for the eigenvalues and eigenfunctions. Copyright © 2012 John Wiley & Sons, Ltd.     

Inverse problems for Sturm–Liouville operators on a compact equilateral graph by partial nodal data

Partial inverse nodal problems for Sturm–Liouville operators on a compact equilateral star graph are investigated in this paper. Uniqueness theorems from partial twin‐dense nodal subsets in interior subintervals or arbitrary interior subintervals having the central vertex are proved. In particular, we posed and solved a new type partial inverse nodal problems for the Sturm–Liouville operator on the compact equilateral star graph.     

Paley–Wiener and Boas theorems for singular Sturm–Liouville integral transforms

This paper deals with a class of integral transforms arising from a singular Sturm–Liouville problem y″−q(x)y=−λy, x(a,b), in the limit-point case at one end or both ends of the interval (a,b). The paper completely solves the problem of characterization of the image of a function that has compact support (Paley–Wiener theorem) and also of a function that vanishes on some interval (Boas problem) under this class of transforms. The characterizations are obtained with no restriction on q(x) other than being locally integrable.