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.

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).

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.

Spectral properties of two-parameter eigenvalue problems

1981 ◽  
Vol 89 (1-2) ◽  
pp. 157-173 ◽  
Author(s):  
Paul Binding ◽  
Patrick J. Browne
Keyword(s):  
Spectral Properties ◽  
Eigenvalue Problems ◽  
Sufficient Conditions ◽  
Asymptotic Results ◽  
Solution Sets ◽  
Image Position ◽  
Two Parameters ◽  
Necessary And Sufficient ◽  
Bounded Below ◽  
Two Parameter

SynopsisWe study the self-adjoint eigenvalue problem W(λ)x = 0, (*), in Hilbert space for one equation in two parameters. Hereis bounded below with compact resolvent for each λ = (λ1, λ2). We give necessary and sufficient conditions for the existence of λ so that (*) holds with W(λ)= ≧0 and we investigate the geometry of the set Z0 of such λ. We also discuss higher order solution sets Zi where the ith eigenvalue of W(λ) vanishes, deriving various asymptotic results in a unified fashion.

Application of two parameter eigencurves to Sturm–Liouville problems with eigenparameter-dependent boundary conditions

1995 ◽  
Vol 125 (6) ◽  
pp. 1205-1218 ◽  
Author(s):  
P. A. Binding ◽  
Patrick J. Browne
Keyword(s):  
Boundary Conditions ◽  
Asymptotic Theory ◽  
Liouville Problem ◽  
Prüfer Angle ◽  
Sturm Liouville ◽  
Sign Conditions ◽  
Image Position ◽  
Eigenvalue Dependent Boundary Conditions ◽  
Parameter Problem ◽  
Two Parameter

Oscillation, comparison and asymptotic theory for the Sturm-Liouville problemwith 1/p, q, r ε L1 ([0, 1]), p, r > 0, are studied subject to eigenvalue-dependent boundary conditionsThis continues previous work on cases with (− 1)j δj ≦ 0 where δj = ajdj − bjcj. We now consider the remaining sign conditions for δj, exploiting the interplay between the graph of cot θ− (λ, 1), for a modified Prüfer angle θ−, and the eigencurves of a related two-parameter problem.

Two-parameter right definite Sturm—Liouville problems with eigenparameter-dependent boundary conditions

2001 ◽  
Vol 131 (1) ◽  
pp. 45-58 ◽  
Author(s):  
T. Bhattacharyya ◽  
P. A. Binding ◽  
K. Seddighi
Keyword(s):  
Boundary Conditions ◽  
Sturm Liouville ◽  
Image Position ◽  
Two Parameter

Linked equations are studied on [0,1] subject to boundary conditions of the form Results are given on existence, location, asymptotics and perturbation of the eigenvalues λj and oscillation of the eigenfunctions yi.

scholarly journals Fundamental Spectral Theory of Fractional Singular Sturm-Liouville Operator

2013 ◽  
Vol 2013 ◽  
pp. 1-7 ◽  
Author(s):  
Erdal Bas
Keyword(s):  
Spectral Theory ◽  
Liouville Operator ◽  
Spectral Properties ◽  
Liouville Problem ◽  
Eigenvalues And Eigenfunctions ◽  
Sturm Liouville

We give the theory of spectral properties for eigenvalues and eigenfunctions of Bessel type of fractional singular Sturm-Liouville problem. We show that the eigenvalues and eigenfunctions of the problem are real and orthogonal, respectively. Furthermore, we prove new approximations about the topic.

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

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 → ∞.

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.

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