Global behavior and nontrivial solutions for discrete Sturm–Liouville problems with eigenparameter-dependent boundary condition

2021 ◽  
Vol 13 (1) ◽  
Author(s):  
Fumei Ye ◽  
Xiaoling Han ◽  
Chenghua Gao
Keyword(s):  
Boundary Condition ◽  
Global Behavior ◽  
Nontrivial Solutions ◽  
Sturm Liouville

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.

scholarly journals On some classes of generalized Schrödinger equations

2020 ◽  
Vol 10 (1) ◽  
pp. 522-533
Author(s):  
Amanda S. S. Correa Leão ◽  
Joelma Morbach ◽  
Andrelino V. Santos ◽  
João R. Santos Júnior
Keyword(s):  
Boundary Condition ◽  
Dirichlet Boundary Condition ◽  
Dirichlet Boundary ◽  
Laplacian Operator ◽  
Schrodinger Equations ◽  
Schrödinger Equations ◽  
Nontrivial Solutions ◽  
Homogeneous Dirichlet Boundary Condition ◽  
Stationary Problems

Abstract Some classes of generalized Schrödinger stationary problems are studied. Under appropriated conditions is proved the existence of at least 1 + $\begin{array}{} \sum_{i=2}^{m} \end{array}$ dim Vλi pairs of nontrivial solutions if a parameter involved in the equation is large enough, where Vλi denotes the eigenspace associated to the i-th eigenvalue λi of laplacian operator with homogeneous Dirichlet boundary condition.

scholarly journals Asymptotic Approximations of Eigen-Functions for Regular Sturm-Liouville Problems with Eigenvalue Parameter in the Boundary Condition for Integrable Potential

2013 ◽  
Vol 113 (1) ◽  
pp. 143 ◽  
Author(s):  
Haskiz Coşkun ◽  
Ayşe Kabataş
Keyword(s):  
Boundary Condition ◽  
Asymptotic Approximations ◽  
Asymptotic Estimates ◽  
Integrable Potential ◽  
Eigenvalue Parameter ◽  
Sturm Liouville

In this paper we obtain asymptotic estimates of eigenfunctions for regular Sturm-Liouville problems having the eigenparameter in the boundary condition without smoothness conditions on the potential.

scholarly journals On a class of semilinear elliptic equations with boundary conditions and potentials which change sign

2005 ◽  
Vol 2005 (2) ◽  
pp. 95-104
Author(s):  
M. Ouanan ◽  
A. Touzani
Keyword(s):  
Boundary Condition ◽  
Boundary Conditions ◽  
Elliptic Equations ◽  
First Eigenvalue ◽  
Nontrivial Solutions ◽  
Bounded Smooth Domain ◽  
The First Eigenvalue ◽  
Semilinear Elliptic ◽  
Superlinear Growth ◽  
Bounded Smooth

We study the existence of nontrivial solutions for the problemΔu=u, in a bounded smooth domainΩ⊂ℝℕ, with a semilinear boundary condition given by∂u/∂ν=λu−W(x)g(u), on the boundary of the domain, whereWis a potential changing sign,ghas a superlinear growth condition, and the parameterλ∈]0,λ1];λ1is the first eigenvalue of the Steklov problem. The proofs are based on the variational and min-max methods.

A priori bounds and existence of non-real eigenvalues for singular indefinite Sturm–Liouville problems with limit-circle type endpoints

2019 ◽  
Vol 150 (5) ◽  
pp. 2607-2619 ◽  
Author(s):  
Fu Sun ◽  
Jiangang Qi
Keyword(s):  
Boundary Condition ◽  
A Priori ◽  
A Priori Bounds ◽  
Limit Circle ◽  
Sturm Liouville

AbstractThe present paper deals with non-real eigenvalues of singular indefinite Sturm–Liouville problems with limit-circle type endpoints. A priori bounds and the existence of non-real eigenvalues of the problem associated with a special separated boundary condition are obtained.

Bounds for distance between eigenvalues of boundary value problems with retarded argument

2019 ◽  
Vol 69 (2) ◽  
pp. 399-408
Author(s):  
Erdoğan Şen
Keyword(s):  
Boundary Condition ◽  
Weight Function ◽  
Boundary Value Problems ◽  
Liouville Operator ◽  
Boundary Value ◽  
Physical Parameters ◽  
Retarded Argument ◽  
Transmission Coefficients ◽  
Sturm Liouville ◽  
Special Case

Abstract In this study we are concerned with spectrum of boundary value problems with retarded argument with discontinuous weight function, two supplementary transmission conditions at the point of discontinuity, spectral and physical parameters in the boundary condition and we obtain bounds for the distance between eigenvalues. We extend and generalize some approaches and results of the classical regular and discontinuous Sturm-Liouville problems. In the special case that ω (x) ≡ 1, the transmission coefficients γ1 = δ1, γ2 = δ2 and retarded argument Δ ≡ 0 in the results obtained in this work coincide with corresponding results in the classical Sturm-Liouville operator.

Solutions of second order multi-point boundary value problems

2008 ◽  
Vol 145 (2) ◽  
pp. 489-510 ◽  
Author(s):  
JOHN R. GRAEF ◽  
LINGJU KONG
Keyword(s):  
Boundary Condition ◽  
Boundary Value Problems ◽  
Fixed Point Index ◽  
Boundary Value ◽  
Second Order ◽  
Point Boundary ◽  
Nontrivial Solutions ◽  
Point Index ◽  
Linear Integral ◽  
The Relationship

AbstractWe consider classes of second order boundary value problems with a nonlinearity f(t, x) in the equations and subject to a multi-point boundary condition. Criteria are established for the existence of nontrivial solutions, positive solutions, and negative solutions of the problems under consideration. The symmetry of solutions is also studied. Conditions are determined by the relationship between the behavior of the quotient f(t, x)/x for x near 0 and ∞ and the largest positive eigenvalue of a related linear integral operator. Our analysis mainly relies on the topological degree and fixed point index theories.

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