Sturm–Liouville Problems on Time Scales with Separated Boundary Conditions

2008 ◽  
Vol 52 (1-2) ◽  
pp. 111-121 ◽  
Author(s):  
Qingkai Kong
Keyword(s):  
Boundary Conditions ◽  
Time Scales ◽  
Separated Boundary Conditions ◽  
Sturm Liouville

Sturm–Liouville problems with non-separated eigenvalue dependent boundary conditions

2000 ◽  
Vol 130 (2) ◽  
pp. 239-247 ◽  
Author(s):  
P. A. Binding ◽  
P. J. Browne
Keyword(s):  
Differential Equations ◽  
Boundary Conditions ◽  
Separated Boundary Conditions ◽  
Sturm Liouville ◽  
Eigenvalue Dependent Boundary Conditions

Sturm–Liouville differential equations are studied under non-separated boundary conditions whose coefficients are first degree polynomials in the eigenparameter. Situations are examined where there are at most finitely many non-real eigenvalues and also where there are only finitely many real ones.

scholarly journals A solution to the inverse problem for the Sturm-Liouville-type equation with a delay

Filomat ◽  
2013 ◽  
Vol 27 (7) ◽  
pp. 1237-1245 ◽  
Author(s):  
Milenko Pikula ◽  
Vladimir Vladicic ◽  
Olivera Markovic
Keyword(s):  
Inverse Problem ◽  
Boundary Conditions ◽  
Fourier Coefficients ◽  
Liouville Problem ◽  
Type Equation ◽  
Liouville Type ◽  
Separated Boundary Conditions ◽  
Sturm Liouville ◽  
Delay Factor ◽  
Spectral Assignment

The paper is devoted to study of the inverse problem of the boundary spectral assignment of the Sturm-Liouville with a delay. -y'(x) + q(x)y(? ? x) = ?y(x), q ? AS[0, ?], ? ? (0,1] (1) with separated boundary conditions: y(0) = y(?) = 0 (2) y(0) = y'(?) = 0 (3) It is argued that if the sequence of eigenvalues is given ?n(1) and ?n(2) tasks (1-2) and (1-3) respectively, then the delay factor ? ? (0,1) and the potential q ? AS[0, ?] are unambiguous. The potential q is composed by means of trigonometric Fourier coefficients. The method can be easily transferred to the case of ? = 1 i.e. to the classical Sturm-Liouville problem.

Geometric aspects of Sturm—Liouville problems I. Structures on spaces of boundary conditions

2000 ◽  
Vol 130 (3) ◽  
pp. 561-589 ◽  
Author(s):  
Q. Kong ◽  
H. Wu ◽  
A. Zettl
Keyword(s):  
Differential Equation ◽  
Boundary Condition ◽  
Boundary Conditions ◽  
Real Number ◽  
Geometric Structure ◽  
Separated Boundary Conditions ◽  
Coupled Boundary Conditions ◽  
Sturm Liouville

We consider some geometric aspects of regular Sturm—Liouville problems. First, we clarify a natural geometric structure on the space of boundary conditions. This structure is the base for studying the dependence of Sturm—Liouville eigenvalues on the boundary condition, and reveals many new properties of these eigenvalues. In particular, the eigenvalues for separated boundary conditions and those for coupled boundary conditions, or the eigenvalues for self-adjoint boundary conditions and those for non-self-adjoint boundary conditions, are closely related under this structure. Then we give complete characterizations of several subsets of boundary conditions such as the set of self-adjoint boundary conditions that have a given real number as an eigenvalue, and determine their shapes. The shapes are shown to be independent of the differential equation in question. Moreover, we investigate the differentiability of continuous eigenvalue branches under this structure, and discuss the relationships between the algebraic and geometric multiplicities of an eigenvalue.

An inverse problem for Sturm–Liouville operators with non-separated boundary conditions containing the spectral parameter

2016 ◽  
Vol 24 (4) ◽  
Author(s):  
Chinare G. Ibadzadeh ◽  
Ibrahim M. Nabiev
Keyword(s):  
Inverse Problem ◽  
Boundary Value Problem ◽  
Boundary Conditions ◽  
Boundary Value Problems ◽  
Spectral Data ◽  
Spectral Parameter ◽  
Boundary Value ◽  
Separated Boundary Conditions ◽  
Sturm Liouville ◽  
Liouville Operators

AbstractIn this paper a boundary value problem is considered generated by the Sturm–Liouville equation and non-separated boundary conditions, one of which contains a spectral parameter. We give a uniqueness theorem, develop an algorithm for solving the inverse problem of reconstruction of boundary value problems with spectral data. We use the spectra of two boundary value problems and some sequence of signs as a spectral data.

scholarly journals Inverse problems for differential operators with nonseparated boundary conditions in the central symmetric case

2017 ◽  
Vol 48 (4) ◽  
pp. 377-387 ◽  
Author(s):  
Vjacheslav Yurko
Keyword(s):  
Inverse Problems ◽  
Boundary Conditions ◽  
Differential Operators ◽  
Sufficient Conditions ◽  
Symmetric Case ◽  
Nonseparated Boundary Conditions ◽  
Separated Boundary Conditions ◽  
Sturm Liouville ◽  
Necessary And Sufficient ◽  
Liouville Operators

Inverse spectral problems for Sturm-Liouville operators on a finite interval with non-separated boundary conditions are studied in the central symmetric case, when the potential is symmetric with respect to the middle of the interval. We discuss statements of the problems, provide algorithms for their solutions along with necessary and sufficient conditions for the solvability of the inverse problems considered.

scholarly journals Finite spectrum of Sturm-Liouville problems with eigenparameter-dependent boundary conditions on time scales

Filomat ◽  
2019 ◽  
Vol 33 (6) ◽  
pp. 1747-1757
Author(s):  
Ji-Jun Ao ◽  
Juan Wang
Keyword(s):  
Spectral Analysis ◽  
Boundary Conditions ◽  
Time Scales ◽  
Liouville Equation ◽  
Time Scale ◽  
The Self ◽  
Finite Spectrum ◽  
Sturm Liouville

The spectral analysis of a class of Sturm-Liouville problems with eigenparameter-dependent boundary conditions on bounded time scales is investigated. By partitioning the bounded time scale such that the coefficients of Sturm-Liouville equation satisfy certain conditions on the adjacent subintervals, the finite eigenvalue results are obtained. The results show that the number of eigenvalues not only depend on the partition of the bounded time scale, but also depend on the eigenparameter-dependent boundary conditions. Both of the self-adjoint and non-self-adjoint cases are considered in this paper.

scholarly journals A Prüfer approach to half-linear Sturm-Liouville problems

1998 ◽  
Vol 41 (3) ◽  
pp. 573-583 ◽  
Author(s):  
Patrick J. Browne
Keyword(s):  
Boundary Conditions ◽  
Liouville Problem ◽  
Oscillation Theory ◽  
Separated Boundary Conditions ◽  
Sturm Liouville ◽  
Image Position

We consider the half linear Sturm-Liouville problemon the interval [0,1] subject to separated boundary conditions (which may be eigenparameter dependent at x = 1) and use Prüfer techniques to produce an oscillation theory for this problem. Both right definite (r > 0) and left definite (r of both signs) cases are discussed.

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