Eigenvalue asymptotics for Sturm-Liouville operators with singular potentials

We derive eigenvalue asymptotics for Sturm-Liouville operators with singular complex-valued potentials from the space , α∈[0,1], and Dirichlet or Neumann-Dirichlet boundary conditions. We also give application of the obtained results to the inverse spectral problem of recovering the potential from these two spectra.     

Inverse spectral theory for Sturm-Liouville problems with finite spectrum

For any positive integer and any given distinct real numbers we construct a Sturm-Liouville problem whose spectrum is precisely the given set of numbers. Such problems are of Atkinson type in the sense that the weight function or the reciprocal of the leading coefficient is identically zero on at least one subinterval.

     

Inverse spectral problems for the Sturm-Liouville operator on a d-star graph

In this work, we consider inverse spectral problems for the Sturm-Liouville differential operator on a d-star-type graph with standard matching conditions in the internal vertex, where the integer d?2. By using the Yurko's method (Yurko (2008) [27], Yurko (2009) [28]) we show that

(1)

if the potential function qj(x) on a fixed edge ej is prescribed on the interval , then the reciprocal of d of the spectrum suffices to determine qj(x) on the whole interval [0,π];

(2)

the 2 over d of the spectrum suffices to determine qj(x) on a fixed edge ej.

     

Inverse spectral problems for Sturm-Liouville operators with singular potentials. Part III: Reconstruction by three spectra

We solve the inverse spectral problem of recovering the singular potential from W⁻¹₂(0,1) of a Sturm-Liouville operator by its spectra on the three intervals [0,1], [0,a], and [a,1] for some a∈(0,1). Necessary and sufficient conditions on the spectral data are derived, and uniqueness of the solution is analyzed.     

On inverse spectral problem for non-selfadjoint Sturm-Liouville operator on a finite interval

An inverse spectral problem is studied for a non-selfadjoint Sturm-Liouville operator on a finite interval with an arbitrary behavior of the spectrum. The spectral data introduced generalize the classical discrete spectral data corresponding to the specification of the spectral function in the selfadjoint case. The connection with other types of spectral characteristics is investigated and a uniqueness theorem is proved. A constructive procedure for solving the inverse problem is given.     

On the existence and nonexistence of positive solutions for nonlinear Sturm-Liouville boundary value problems

In this paper the existence and nonexistence results of positive solutions are obtained for Sturm-Liouville boundary value problem

     

N维向量Sturm-Liouville算子的正则迹及其在反谱问题中的应用

研究了赋予一般分离型边界条件的N维向量Sturm-Liouville方程的特征值问题,获得了该系统的一个正则迹公式.迹公式不仅形式美观,而且它在反谱理论中具有重要的作用.     

Discrete Sturm-Liouville problems with nonlinear parameter in the boundary conditions

This paper deals with discrete second order Sturm-Liouville problems where the parameter that is part of the Sturm-Liouville difference equation appears nonlinearly in the boundary conditions. The operator is extended to a self-adjoint operator and an expansion theorem is proved.     

On existence of positive solutions of Sturm-Liouville boundary value problems for a nonlinear singular differential system

By employing the fixed point theorem of cone expansion and compression of norm type, we investigate the existence of positive solutions of Sturm-Liouville boundary value problems for a nonlinear singular differential system. Some well-known results in the literature are generalized and improved. An example is presented to illustrate the application of our main result.     

Uniqueness of positive solutions for Sturm-Liouville boundary value problems

Sufficient conditions for the uniqueness of positive solutions of singular Sturm-Liouville boundary value problems

where and , are established.

     

EXISTENCE OF SOLUTIONS OF STURM-LIOUVILLE BOUNDARY VALUE PROBLEMS FOR NONLINEAR SECOND ORDER IMPULSIVE DIFFERENTIAL EQUATIONS IN BANACH SPACES

In this paper, the fixed point theorem is applied to investigate the existence of solutions of Sturm-Liouville boundary value problems for nonlinear second order impulsive differential equations in Banaeh spaces.     

Inversion of Trace Formulas for a Sturm-Liouville Operator

This paper revisits the classical problem “Can we hear the density of a string?”, which can be formulated as an inverse spectral problem for a Sturm-Liouville operator. Instead of inverting the map from density to spectral data directly, we propose a novel method to reconstruct the density based on inverting a sequence of trace formulas which bridge the density and its spectral data clearly in terms of a series of nonlinear integral equations. Numerical experiments are presented to verify the validity and effectiveness of the proposed numerical algorithm. The impact of different parameters involved in the algorithm is also discussed.     

An irregular extension of the oscillation theory of the Sturm-Liouville spectral problem

Oscillation spectral properties (such as the number of zeros, the alternation of zeros for eigenfunctions, the simplicity of the spectrum, etc.) for the Sturm-Liouville problem with generalized coefficients are described.     

A sturm-liouville problem with physical and spectral parameters in boundary conditions

We study the spectral probleml(u)=−u″+q(x)u(x)=λu(x),u′(0)=0, u′(π)=mλu(π), where λ andm are a spectral and a physical parameter. Form<0, we associate with the problem a self-adjoint operator in Pontryagin space II₁. Using this fact and developing analytic methods of the theory of Sturm-Liouville operators, we study the dynamics of eigenvalues and eigenfunctions of the problems asm→−0. Translated fromMatematicheskie Zametki, Vol. 66, No. 2, pp. 163–172, August, 1999.     

Inverse nodal and inverse spectral problems for discontinuous boundary value problems

Inverse nodal and inverse spectral problems are studied for second-order differential operators on a finite interval with discontinuity conditions inside the interval. Uniqueness theorems are proved, and a constructive procedure for the solution is provided.     

Existence and multiplicity of weak solutions for a class of fractional Sturm-Liouville boundary value problems with impulsive conditions

In this paper, we consider the existence and multiplicity of weak solutions for a class of fractional differential equations with non-homogeneous Sturm-Liouville conditions and impulsive conditions by using the critical point theory. In addition, at the end of this paper, we also give the existence results of infinite weak solutions of fractional differential equations under homogeneous Sturm-Liouville boundary value conditions. Finally, several examples are given to illustrate our main results.     

EXISTENCE OF SOLUTIONS OF NONLINEAR TWO AND THREE-POINT BOUNDARY VALUE PROBLEMS FOR FOURTH ORDER NONLINEAR DIFFERENTIAL EQUATIONS

In this paper, the two and three-point boundsry problems (with nonlinear boundary conditions)for the genaral noniinear equations of fourth order are discussed.We have set some grups of the assurnpion coditions and proved the existence of solutins for corresponding boundary value problems under these conditons.     

一类四阶非线性奇摄动方程的边值问题

利用匹配渐近展开法,讨论了一类四阶非线性方程的具有两个边界层的奇摄动边值问题．引进伸长变量,根据边界条件与匹配原则,在一定的可解性条件下,给出了外部解和左右边界层附近的内层解,得到了该问题的二阶渐近解,并举例说明了这类非线性问题渐近解的存在性．     

A method for solving nonlinear spectral problems for a class of systems of differential algebraic equations

A method for calculating eigenvalues of a nonlinear spectral problem for one class of linear differential algebraic equations is proposed under the assumption of an analytical dependence on spectral parameter of the matrices appearing in the system of equations and the matrices determining boundary conditions.