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.

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.

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.

Maximum and anti-maximum principles for singular Sturm–Liouville problems

1998 ◽  
Vol 128 (3) ◽  
pp. 525-547 ◽  
Author(s):  
M. Duhoux
Keyword(s):  
Boundary Conditions ◽  
Weight Function ◽  
Maximum Principles ◽  
Constant Sign ◽  
Bounded Interval ◽  
Dirichlet Type ◽  
Sturm Liouville ◽  
Image Position

The maximum and anti-maximum principles are extended to the case of eigenvalue Sturm–Liouville problemswith boundary conditions of Dirichlet type (if possible) on a bounded interval [a, b]. The function r is assumed to be continuous and > 0 on ]a, b[, but the function 1/r is not necessarily integrable on [a, b]. The conditions on the functions p, m and h depend on the integrability or nonintegrability of 1/r on [a, c] and/or [c, b], for some c ∈ ]a, b[. The weight function m is not necessarily of constant sign.

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 A note on the characterisation of eigencurves for certain two parameter eigenvalue problems in ordinary differential equations

1993 ◽  
Vol 35 (1) ◽  
pp. 63-67 ◽  
Author(s):  
Patrick J. Browne ◽  
B. D. Sleeman
Keyword(s):  
Differential Equations ◽  
Boundary Conditions ◽  
Weight Function ◽  
Analytic Functions ◽  
Dirichlet Boundary ◽  
Eigenvalue Problems ◽  
Dirichlet Boundary Conditions ◽  
Survey Paper ◽  
Image Position ◽  
Two Parameter

We are interested in two parameter eigenvalue problems of the formsubject to Dirichlet boundary conditionsThe weight function 5 and the potential q will both be assumed to lie in L2[0,1]. The problem (1.1), (1.2) generates eigencurvesin the sense that for any fixed λ, ν(λ) is the nth eigenvalue ν, (according to oscillation indexing) of (1.1), (1.2). These curves are in fact analytic functions of λ and have been the object of considerable study in recent years. The survey paper [1] provides background in this area and itemises properties of eigencurves.

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

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.

Resolvent Means and Inverting Generalized Fourier Transforms

1986 ◽  
Vol 38 (4) ◽  
pp. 861-877 ◽  
Author(s):  
Louise A. Raphael
Keyword(s):  
Boundary Conditions ◽  
Weight Function ◽  
Eigenfunction Expansion ◽  
Resolvent Operator ◽  
Fourier Transforms ◽  
Half Line ◽  
Positive Spectrum ◽  
Sturm Liouville ◽  
Image Position ◽  
The Resolvent Operator

Let S-L denote a singular Sturm-Liouville system on the half line with homogeneous boundary conditions, possessing a discrete negative and continuous positive spectrum. Let A be the S-L operator and Sα(f; x) the S-L eigenfunction expansion associated with the resolvent operator (z – A)–1, z complex. That is, Sα(f; x) denotes the resolvent summability means with weight function z(z – λ)–1 (or (1 + tλ)–1 where t = – 1/z).We first study the problem of determining when(1)where is the Green's function associated with a certain perturbation of our system.

A Note on Klein’s Oscillation Theorem for Periodic Boundary Conditions

1975 ◽  
Vol 17 (5) ◽  
pp. 749-755 ◽  
Author(s):  
M. Faierman
Keyword(s):  
Eigenvalue Problem ◽  
Boundary Conditions ◽  
Periodic Boundary ◽  
Periodic Boundary Conditions ◽  
Oscillation Theory ◽  
Continuous Functions ◽  
Oscillation Theorem ◽  
Image Position ◽  
Two Parameter

Recently Howe [4] has considered the oscillation theory for the two-parameter eigenvalue problem1a1bsubjected to the boundary conditions2a2bwhere for i = 1, 2, — ∞<ai<bi<∞, and qi are real-valued, continuous functions in [ai, bi], pi is positive in [ai, biz], and pi(ai)=pi(bi). Furthermore, it is also assumed that (A1B2—A2B1)≠0 for all values of x1 and x2 in their respective intervals.

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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