Large eigenvalues of a Sturm-Liouville problem

1967 ◽  
Vol 63 (2) ◽  
pp. 473-475 ◽  
Author(s):  
J. H. E. Cohn
Keyword(s):  
Eigenvalue Problem ◽  
Bounded Function ◽  
Liouville Problem ◽  
Sturm Liouville ◽  
Image Position

Consider the eigenvalue problemwhere q(x) is a (bounded) function in the class L(a, b). We may suppose without loss of generality that 0 ≤ α < π and 0 ≤ β < π. Then as is well known there are infinitely many eigenvalues λr (r = 0, 1, 2, …) and λn ∽ n2π2 (b − a)−2 as n → ∞.

X.—The Two Parameter Sturm-Liouville Problem for Ordinary Differential Equations

1971 ◽  
Vol 69 (2) ◽  
pp. 139-148
Author(s):  
B. D. Sleeman
Keyword(s):  
Differential Equation ◽  
Eigenvalue Problem ◽  
Differential Equations ◽  
Boundary Conditions ◽  
Ordinary Differential Equations ◽  
Liouville Problem ◽  
Sturm Liouville ◽  
Image Position ◽  
Two Parameter ◽  
General Conditions

SynopsisThis paper discusses the existence, under fairly general conditions, of solutions of the two-parameter eigenvalue problem denned by the differential equation,and three point Sturm-Liouville boundary conditions.

Expansions in terms of sets of functions with complex eigenvalues

1948 ◽  
Vol 44 (2) ◽  
pp. 242-250 ◽  
Author(s):  
R. Peierls
Keyword(s):  
Boundary Condition ◽  
Eigenvalue Problem ◽  
Boundary Conditions ◽  
Real Function ◽  
Liouville Problem ◽  
Nuclear Collisions ◽  
Complex Eigenvalues ◽  
Sturm Liouville ◽  
Image Position ◽  
Unusual Type

In the following I discuss the properties, in particular the completeness of the set of eigenfunctions, of an eigenvalue problem which differs from the well-known Sturm-Liouville problem by the boundary condition being of a rather unusual type.The problem arises in the theory of nuclear collisions, and for our present purpose we take it in the simplified formwhere 0 ≤ x ≤ 1. V(x) is a given real function, which we assume to be integrable and to remain between the bounds ± M, and W is an eigenvalue. The eigenfunction ψ(x) is subject to the boundary conditionsand

11.— A Left Definite Multiparameter Eigenvalue Problem in Ordinary Differential Equations

1976 ◽  
Vol 74 ◽  
pp. 145-155 ◽  
Author(s):  
A. Källström ◽  
B. D. Sleeman
Keyword(s):  
Eigenvalue Problem ◽  
Differential Equations ◽  
Boundary Conditions ◽  
Ordinary Differential Equations ◽  
Ellipticity Condition ◽  
Spectral Parameters ◽  
Multiparameter Eigenvalue Problem ◽  
Sturm Liouville ◽  
Image Position

SynopsisThe main result of this paper is to establish the completeness of the eigenfunctions for the multiparameter eigenvalue problem defined by the system of ordinary differential equations0 ≤ x, ≤ 1, r = 1, …, k, subject to the Sturm-Liouville boundary conditionsr = 1, …, k. In addition it is assumed that the coefficients ars of the spectral parameters λs, satisfy the ellipticity condition , s = 1, …, k, for all xrɛ[0, 1], r = 1, …, k, and some real k-tuple μ1, …, μk and where is the co-factor of asr in the determinant . The theory developed here contrasts with the results known when …k is assumed non-vanishing for all xrɛ[0,1].

scholarly journals Sturm–Liouville problems with eigenparameter dependent boundary conditions

1994 ◽  
Vol 37 (1) ◽  
pp. 57-72 ◽  
Author(s):  
P. A. Binding ◽  
P. J. Browne ◽  
K. Seddighi
Keyword(s):  
Boundary Conditions ◽  
Liouville Problem ◽  
Asymptotic Estimates ◽  
Comparison Results ◽  
Sturm Liouville ◽  
Image Position ◽  
Sturm Theory

Sturm theory is extended to the equationfor 1/p, q, r∈L1 [0, 1] with p, r > 0, subject to boundary conditionsandOscillation and comparison results are given, and asymptotic estimates are developed. Interlacing of eigenvalues with those of a standard Sturm–Liouville problem where the boundary conditions are ajy(j) = cj(py′)(j), j=0, 1, forms a key tool.

Sturm-Liouville problems for the p-Laplacian on a half-line

2010 ◽  
Vol 53 (2) ◽  
pp. 271-291 ◽  
Author(s):  
Paul Binding ◽  
Patrick J. Browne ◽  
Illya M. Karabash
Keyword(s):  
Eigenvalue Problem ◽  
Nonlinear Eigenvalue Problem ◽  
Half Line ◽  
Sturm Liouville ◽  
Image Position ◽  
Nonlinear Eigenvalue

AbstractThe nonlinear eigenvalue problemfor 0 ≤ x < ∞, fixed p ∈ (1, ∞), and with y′(0)/y(0) specified is studied under various conditions on the coefficients s and q, leading to either oscillatory or non-oscillatory situations.

Consequences of the connection formulae for Sturm-Liouville spectral functions

2002 ◽  
Vol 132 (2) ◽  
pp. 387-393 ◽  
Author(s):  
S. M. Riehl
Keyword(s):  
Explicit Formula ◽  
Liouville Equation ◽  
Liouville Problem ◽  
Initial Condition ◽  
Spectral Functions ◽  
Sturm Liouville ◽  
Image Position ◽  
Connection Formulae ◽  
Special Case

For a special case of the Sturm-Liouville equation, −(py′)′ + qy = λwy on [0, ∞) with the initial condition y(0) cos α + p(0)y′(0) sin α = 0, α ∈ [0, π), it is shown that, given the spectral derivative for two values of α ∈ [0, π) at a fixed μ = Re{λ} ≥ Λ0, it is possible to uniquely determine . An explicit formula is derived to accomplish this. Further, in a more general case of the Sturm-Liouville problem for μ with both finite and positive, then the following inequality holds

Multiparameter Variational Eigenvalue Problems With Indefinite Nonlinearity

1997 ◽  
Vol 49 (5) ◽  
pp. 1066-1088 ◽  
Author(s):  
Tetsutaro Shibata
Keyword(s):  
Asymptotic Behavior ◽  
Asymptotic Formula ◽  
Optimal Condition ◽  
Level Set ◽  
Eigenvalue Problems ◽  
Liouville Problem ◽  
General Level ◽  
Sturm Liouville ◽  
Image Position

AbstractWe consider the multiparameter nonlinear Sturm-Liouville problemwhere are parameters. We assume that1 ≤ q ≤ p1 < p2 < ... ≤ pn < 2q + 3.We shall establish an asymptotic formula of variational eigenvalue λ = λ(μ, α) obtained by using Ljusternik-Schnirelman theory on general level set Nμ, α(α < 0 : parameter of level set). Furthermore,we shall give the optimal condition of {(μ, α)}, under which μi(m + 1 ≤ i ≤ n : fixed) dominates the asymptotic behavior of λ(μ, α).

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.

Spectral properties of a two parameter nonlinear Sturm-Liouville problem

1993 ◽  
Vol 123 (6) ◽  
pp. 1041-1058 ◽  
Author(s):  
Tetsutaro Shibata
Keyword(s):  
Level Set ◽  
Spectral Properties ◽  
Liouville Problem ◽  
General Level ◽  
Asymptotic Formulae ◽  
Sturm Liouville ◽  
Image Position ◽  
Two Parameters ◽  
Two Parameter

SynopsisWe consider the nonlinear Sturm–Liouville problem with two parameters on the general level setWe establish asymptotic formulae of the n-th variational eigenvalue λ = λn(μ, α) as α→∞ and α↓(nπ)2.

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