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.

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].

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

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.

scholarly journals On a Differential Equation and the Construction of Milner's Lamp

1886 ◽  
Vol 5 ◽  
pp. 99-101 ◽  
Author(s):  
Professor Cayley
Keyword(s):  
Differential Equation ◽  
Differential Equations ◽  
Ordinary Differential Equations ◽  
Image Position

What sort of an equation isWriteand start with the equationsThis last givesand the system thus isviz., this is a system of ordinary differential equations between the five variables θ, r, X, Y, Z: the system can therefore be integrated with 4 arbitrary constants, and these may be so determined that for the value β of θ, X, Y, Z shall be each = 0; and r shall have the value r0.

Factorization Ladders and Eigenfunctions

1949 ◽  
Vol 1 (4) ◽  
pp. 379-396 ◽  
Author(s):  
G. F. D. Duff
Keyword(s):  
Differential Equations ◽  
Boundary Value Problem ◽  
Boundary Conditions ◽  
Ordinary Differential Equations ◽  
Integral Equations ◽  
Manufacturing Process ◽  
Boundary Value ◽  
Factorization Method ◽  
Sturm Liouville ◽  
The Subject

The eigenfunctions of a boundary value problem are characterized by two quite distinct properties. They are solutions of ordinary differential equations, and they satisfy prescribed boundary conditions. It is a definite advantage to combine these two requirements into a single problem expressed by a unified formula. The use of integral equations is an example in point. The subject of this paper, namely the Schrödinger-Infeld Factorization Method, which is applicable to certain restricted. Sturm-Liouville problems, is based upon another combination of the two properties. The Factorization Method prescribes a manufacturing process.

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.

Solutions of ordinary differential equations as limits of pure jump markov processes

1970 ◽  
Vol 7 (1) ◽  
pp. 49-58 ◽  
Author(s):  
Thomas G. Kurtz
Keyword(s):  
Differential Equation ◽  
Differential Equations ◽  
Population Growth ◽  
Ordinary Differential Equations ◽  
Markov Processes ◽  
Radioactive Decay ◽  
Dynamic Processes ◽  
Deterministic Models ◽  
Image Position ◽  
Jump Markov Processes

In a great variety of fields, e.g., biology, epidemic theory, physics, and chemistry, ordinary differential equations are used to give continuous deterministic models for dynamic processes which are actually discrete and random in their development. Perhaps the simplest example is the differential equation used to describe a number of processes including radioactive decay and population growth.

Qualitative properties of nonlinear ordinary differential equations

1977 ◽  
Vol 79 (1-2) ◽  
pp. 79-85 ◽  
Author(s):  
Nelson Onuchic ◽  
Plácido Z. Táboas
Keyword(s):  
Differential Equation ◽  
Banach Space ◽  
Ordinary Differential Equation ◽  
Differential Equations ◽  
Ordinary Differential Equations ◽  
Linear Ordinary Differential Equation ◽  
Point Of View ◽  
Nonlinear Ordinary Differential Equations ◽  
Qualitative Properties ◽  
Image Position

SynopsisThe perturbed linear ordinary differential equationis considered. Adopting the same approach of Massera and Schäffer [6], Corduneanu states in [2] the existence of a set of solutions of (1) contained in a given Banach space. In this paper we investigate some topological aspects of the set and analyze some of the implications from a point of view ofstability theory.

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