On Estimates of the First Eigenvalue for the Sturm–Liouville Problem with Symmetric Boundary Conditions and Integral Condition

2013 ◽  
pp. 457-464
Author(s):  
Elena Karulina
Keyword(s):  
Boundary Conditions ◽  
Liouville Problem ◽  
Integral Condition ◽  
First Eigenvalue ◽  
The First Eigenvalue ◽  
Sturm Liouville

scholarly journals ON THE UPPER ESTIMATES FOR THE FIRST EIGENVALUE OF A STURM - LIOUVILLE PROBLEM WITH A WEIGHTED INTEGRAL CONDITION

2017 ◽  
Vol 21 (6) ◽  
pp. 124-129
Author(s):  
M.Yu. Telnova
Keyword(s):  
Weight Function ◽  
Eigenvalue Problems ◽  
Liouville Problem ◽  
Integral Condition ◽  
First Eigenvalue ◽  
Second Order Differential Equations ◽  
The First Eigenvalue ◽  
Lagrange Problem ◽  
Sturm Liouville ◽  
The Given

In this paper a problem for which the origin problem was a problem known as the Lagrange problem or the problem on finding the form of the firmest column of the given volume is viewed. The Lagrange problem was the source for different extremal eigenvalue problems, among them for eigenvalue problems for the second-order differential equations, with an integral condition on the potential. In this paper the problem of that kind is considered under the con- dition that the integral condition contains a weight function. The method of finding the sharp upper estimates for the first eigenvalue of a Sturm - Liouville problem with Dirichlet conditions for some values of parameters in the integral condition was found and attainability of those estimates was proved. In this paper a problem for which the origin problem was a problem known as the Lagrange problem or the problem on finding the form of the firmest column of the given volume is viewed. The Lagrange problem was the source for different extremal eigenvalue problems, among them for eigenvalue problems for the second-order differential equations, with an integral condition on the potential. In this paper the problem of that kind is considered under the con- dition that the integral condition contains a weight function. The method of finding the sharp upper estimates for the first eigenvalue of a Sturm - Liouville problem with Dirichlet conditions for some values of parameters in the integral condition was found and attainability of those estimates was proved.

scholarly journals The Sturm-Liouville Problem with Singular Potential and the Extrema of the First Eigenvalue

2013 ◽  
Vol 54 (1) ◽  
pp. 101-118
Author(s):  
Elena S. Karulina ◽  
Anton A. Vladimirov
Keyword(s):  
Singular Potential ◽  
Liouville Problem ◽  
First Eigenvalue ◽  
Constant Sign ◽  
The First Eigenvalue ◽  
Sturm Liouville

Abstract We get the infima and suprema of the first eigenvalue of the problem where q belongs to the set of constant-sign summable functions on [0,1] such that

scholarly journals The dual eigenvalue problems of the conformable fractional Sturm–Liouville problems

2021 ◽  
Vol 2021 (1) ◽  
Author(s):  
Yan-Hsiou Cheng
Keyword(s):  
Eigenvalue Problems ◽  
Liouville Problem ◽  
First Eigenvalue ◽  
Liouville Property ◽  
The First Eigenvalue ◽  
Eigenvalue Gap ◽  
Single Well ◽  
Eigenvalue Ratio ◽  
Sturm Liouville ◽  
Prüfer Substitution

AbstractIn this paper, we are concerned with the eigenvalue gap and eigenvalue ratio of the Dirichlet conformable fractional Sturm–Liouville problems. We show that this kind of differential equation satisfies the Sturm–Liouville property by the Prüfer substitution. That is, the nth eigenfunction has $n-1$ n − 1 zero in $( 0,\pi ) $ ( 0 , π ) for $n\in \mathbb{N}$ n ∈ N . Then, using the homotopy argument, we find the minimum of the first eigenvalue gap under the class of single-well potential functions and the first eigenvalue ratio under the class of single-barrier density functions. The result of the eigenvalue gap is different from the classical Sturm–Liouville problem.

scholarly journals Fundamental Results of Conformable Sturm-Liouville Eigenvalue Problems

Complexity ◽  
2017 ◽  
Vol 2017 ◽  
pp. 1-7 ◽  
Author(s):  
Mohammed Al-Refai ◽  
Thabet Abdeljawad
Keyword(s):  
Boundary Conditions ◽  
Eigenvalue Problems ◽  
Liouville Problem ◽  
Rayleigh Quotient ◽  
First Eigenvalue ◽  
Suggested Model ◽  
Conformable Derivative ◽  
Arbitrary Boundary ◽  
Sturm Liouville ◽  
Natural Boundary Conditions

We suggest a regular fractional generalization of the well-known Sturm-Liouville eigenvalue problems. The suggested model consists of a fractional generalization of the Sturm-Liouville operator using conformable derivative and with natural boundary conditions on bounded domains. We establish fundamental results of the suggested model. We prove that the eigenvalues are real and simple and the eigenfunctions corresponding to distinct eigenvalues are orthogonal and we establish a fractional Rayleigh Quotient result that can be used to estimate the first eigenvalue. Despite the fact that the properties of the fractional Sturm-Liouville problem with conformable derivative are very similar to the ones with the classical derivative, we find that the fractional problem does not display an infinite number of eigenfunctions for arbitrary boundary conditions. This interesting result will lead to studying the problem of completeness of eigenfunctions for fractional systems.

scholarly journals Ambarzumyan-type theorems on a time scale

2018 ◽  
Vol 26 (5) ◽  
pp. 633-637 ◽  
Author(s):  
Ahmet Sinan Ozkan
Keyword(s):  
Boundary Conditions ◽  
Dynamic Equation ◽  
Time Scale ◽  
Robin Boundary Conditions ◽  
First Eigenvalue ◽  
The First Eigenvalue ◽  
Sturm Liouville ◽  
Robin Boundary

Abstract In this paper, we give Ambarzumyan-type theorems for a Sturm–Liouville dynamic equation with Robin boundary conditions on a time scale. Under certain conditions, we prove that the potential can be specified from only the first eigenvalue.

Sturm-Liouville Problem for Stationary Differential Operator with Nonlocal Two-Point Boundary Conditions

2006 ◽  
Vol 11 (1) ◽  
pp. 47-78 ◽  
Author(s):  
S. Pečiulytė ◽  
A. Štikonas
Keyword(s):  
Boundary Condition ◽  
Differential Operator ◽  
Boundary Conditions ◽  
Nonlocal Boundary Condition ◽  
Nonlocal Boundary ◽  
Liouville Problem ◽  
Complex Case ◽  
Qualitative Behavior ◽  
Point Boundary ◽  
Sturm Liouville

The Sturm-Liouville problem with various types of two-point boundary conditions is considered in this paper. In the first part of the paper, we investigate the Sturm-Liouville problem in three cases of nonlocal two-point boundary conditions. We prove general properties of the eigenfunctions and eigenvalues for such a problem in the complex case. In the second part, we investigate the case of real eigenvalues. It is analyzed how the spectrum of these problems depends on the boundary condition parameters. Qualitative behavior of all eigenvalues subject to the nonlocal boundary condition parameters is described.

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