Non-standard oscillation theory for multiparameter eigenvalue problems

2012 ◽  
Vol 142 (4) ◽  
pp. 673-690
Author(s):  
P. A. Binding ◽  
H. Volkmer
Keyword(s):  
Eigenvalue Problem ◽  
Eigenvalue Problems ◽  
Oscillation Theory ◽  
Second Order ◽  
Order Derivative ◽  
Standard Case ◽  
Sturm Liouville ◽  
Two Parameters

An eigenvalue problem for k Sturm–Liouville equations coupled by k parameters λ1,…,λk is considered. In contrast to the standard case, for each r, the second-order derivative in the rth equation is multiplied by λr. This problem presents various interesting features. For example, the existence of eigenvalues with oscillation counts beyond a certain (computable) value is obtained without any of the restrictive definiteness conditions known from the standard case. Uniqueness is also analysed, and the results are given greater precision via eigencurve methods in the case of two equations coupled by two parameters.

Full- and partial-range eigenfunction expansions for Sturm-Liouville problems with indefinite weights

1984 ◽  
Vol 98 (1-2) ◽  
pp. 69-88 ◽  
Author(s):  
Hans G. Kaper ◽  
Man Kam Kwong ◽  
C. G. Lekkerkerker ◽  
A. Zettl
Keyword(s):  
Hilbert Space ◽  
Differential Expression ◽  
Positive Operator ◽  
Eigenvalue Problems ◽  
Full Range ◽  
Second Order ◽  
Eigenfunction Expansions ◽  
Multiplicative Operator ◽  
Indefinite Weights ◽  
Sturm Liouville

SynopsisThis article is concerned with eigenvalue problems of the form Au = λTu in a Hilbert space H, where Ais a selfadjoint positive operator generated by a second-order Sturm-Liouville differential expression and T a selfadjoint indefinite multiplicative operator which is one-to-one. Emphasis is on the full-range and partial-range expansionproperties of the eigenfunctions.

scholarly journals Asymptotic spectrum of multiparameter eigenvalue problems

1996 ◽  
Vol 39 (1) ◽  
pp. 119-132 ◽  
Author(s):  
Hans Volkmer
Keyword(s):  
Hilbert Space ◽  
Maximum Principle ◽  
Eigenvalue Problem ◽  
Eigenvalue Problems ◽  
Liouville Problem ◽  
Asymptotic Spectrum ◽  
Multiparameter Eigenvalue Problem ◽  
Sturm Liouville

Results are given for the asymptotic spectrum of a multiparameter eigenvalue problem in Hilbert space. They are based on estimates for eigenvalues derived from the minim un-maximum principle. As an application, a multiparameter Sturm-Liouville problem is considered.

scholarly journals Positive Solutions for Second-Order Three-Point Eigenvalue Problems

2010 ◽  
Vol 2010 ◽  
pp. 1-8 ◽  
Author(s):  
Chuanzhi Bai
Keyword(s):  
Fixed Point ◽  
Eigenvalue Problem ◽  
Positive Solution ◽  
Existence Theorem ◽  
Fixed Point Index ◽  
Eigenvalue Problems ◽  
Index Theorem ◽  
Second Order ◽  
Point Index ◽  
Fixed Point Index Theorem

With the help of the fixed point index theorem in cones, we get an existence theorem concerning the existence of positive solution for a second-order three-point eigenvalue problemx′′(t)+λf(t,x(t))=0,  0≤t≤1,  x(0)=0,  x(1)=x(η), whereλis a parameter. An illustrative example is given to demonstrate the effectiveness of the obtained result.

The eigenvalue problem with interaction conditions at one interior singular point

Filomat ◽  
2017 ◽  
Vol 31 (17) ◽  
pp. 5411-5420 ◽  
Author(s):  
Oktay Mukhtarov ◽  
Kadriye Aydemir
Keyword(s):  
Singular Point ◽  
Eigenvalue Problem ◽  
Quantum Physics ◽  
Eigenvalue Problems ◽  
Physical Processes ◽  
Liouville Type ◽  
Vibrating String ◽  
Sturm Liouville ◽  
Transfer Problems ◽  
String Problems

Some physical processes, both classical physics and quantum physics reduced to eigenvalue problems for Sturm-Liouville equations. In the recent years there has been an increasing interest in discontinuous eigenvalue problems for various Sturm-Liouville type equations. Such problems are connected with heat transfer problems, vibrating string problems, diffraction problems and etc. In this study we shall investigate a class of two order eigenvalue problem with supplementary transmission conditions at one interior singular point. We give an operator-theoretic interpretation in suitable Hilbert space.

Homogeneous robin boundary conditions and discrete spectrum of fractional eigenvalue problem

2019 ◽  
Vol 22 (1) ◽  
pp. 78-94 ◽  
Author(s):  
Malgorzata Klimek
Keyword(s):  
Eigenvalue Problem ◽  
Boundary Conditions ◽  
Eigenvalue Problems ◽  
Liouville Problem ◽  
Continuous Functions ◽  
Real Spectrum ◽  
Robin Boundary Conditions ◽  
Schmidt Operator ◽  
Sturm Liouville ◽  
Robin Boundary

Abstract We discuss a fractional eigenvalue problem with the fractional Sturm-Liouville operator mixing the left and right derivatives of order in the range (1/2, 1], subject to a variant of Robin boundary conditions. The considered differential fractional Sturm-Liouville problem (FSLP) is equivalent to an integral eigenvalue problem on the respective subspace of continuous functions. By applying the properties of the explicitly calculated integral Hilbert-Schmidt operator, we prove the existence of a purely atomic real spectrum for both eigenvalue problems. The orthogonal eigenfunctions’ systems coincide and constitute a basis in the corresponding weighted Hilbert space. An analogous result is obtained for the reflected fractional Sturm-Liouville problem.

scholarly journals Computation of Some Second Order Sturm-Liouville Bvps using Chebyshev-Legendre Collocation Method

2016 ◽  
Vol 35 ◽  
pp. 95-112
Author(s):  
Humaira Farzana ◽  
Md Shafiqul Islam
Keyword(s):  
Collocation Method ◽  
Eigenvalue Problems ◽  
Second Order ◽  
Type Problem ◽  
Nonlinear Eigenvalue Problems ◽  
Legendre Collocation Method ◽  
Derivative Matrix ◽  
Sturm Liouville ◽  
Type Boundary ◽  
Good Agreement

We propose Chebyshev-Legendre spectral collocation method for solving second order linear and nonlinear eigenvalue problems exploiting Legendre derivative matrix. The Sturm-Liouville (SLP) problems are formulated utilizing Chebyshev-Gauss-Lobatto (CGL) nodes instead of Legendre Gauss-Lobatto (LGL) nodes and Legendre polynomials are taken as basis function. We discuss, in details, the formulations of the present method for the Sturm-Liouville problems (SLP) with Dirichlet and mixed type boundary conditions. The accuracy of this method is demonstrated by computing eigenvalues of three regular and two singular SLP's. Nonlinear Bratu type problem is also tested in this article. The numerical results are in good agreement with the other available relevant studies.GANIT J. Bangladesh Math. Soc.Vol. 35 (2015) 95-112

scholarly journals Inverse Eigenvalue Problems for Singular Rank One Perturbations of a Sturm-Liouville Operator

2021 ◽  
Vol 2021 ◽  
pp. 1-5
Author(s):  
Xuewen Wu
Keyword(s):  
Eigenvalue Problem ◽  
Potential Function ◽  
Liouville Operator ◽  
Eigenvalue Problems ◽  
Inverse Eigenvalue Problem ◽  
Inverse Eigenvalue Problems ◽  
Rank One ◽  
Inverse Eigenvalue ◽  
Sturm Liouville

This paper is concerned with the inverse eigenvalue problem for singular rank one perturbations of a Sturm-Liouville operator. We determine uniquely the potential function from the spectra of the Sturm-Liouville operator and its rank one perturbations.

Eigenvalue Problem for the Second Order Differential Equation with Nonlocal

2006 ◽  
Vol 11 (1) ◽  
pp. 13-32 ◽  
Author(s):  
B. Bandyrskii ◽  
I. Lazurchak ◽  
V. Makarov ◽  
M. Sapagovas
Keyword(s):  
Differential Equation ◽  
Eigenvalue Problem ◽  
Variable Coefficient ◽  
Order Differential Equation ◽  
Integral Condition ◽  
Second Order ◽  
Computational Results ◽  
Discrete Method ◽  
Complex Eigenvalues ◽  
Nonlocal Integral Condition

The paper deals with numerical methods for eigenvalue problem for the second order ordinary differential operator with variable coefficient subject to nonlocal integral condition. FD-method (functional-discrete method) is derived and analyzed for calculating of eigenvalues, particulary complex eigenvalues. The convergence of FD-method is proved. Finally numerical procedures are suggested and computational results are schown.

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