On the strong limit-point and Dirichlet properties of second order differential expressions

1985 ◽  
Vol 101 (3-4) ◽  
pp. 283-296 ◽  
Author(s):  
D. Race
Keyword(s):  
Weight Function ◽  
Limit Point ◽  
Second Order ◽  
Strong Limit ◽  
Positive Weight ◽  
End Point

SynopsisSecond order differential expressions of the form w−1(−(pf′)′ + qf) are considered at a singular end-point. Some of the known relationships between properties such as Dirichlet, weak Dirichlet and strong limit-point, are extended to incorporate an arbitrary, positive weight function and complexvalued coefficients.

HELP inequalities for limit-circle and regular problems

1991 ◽  
Vol 432 (1886) ◽  
pp. 367-390 ◽  
Keyword(s):  
Differential Equation ◽  
Limit Point ◽  
Strong Limit ◽  
End Points ◽  
Limit Circle ◽  
End Point ◽  
Limit Circle Case ◽  
Limit Point Case ◽  
Regular Problems ◽  
Circle Case

Everitt’s criterion for the validity of the generalized Hardy-Littlewood inequality presupposes that the associated differential equation is singular at one end-point of the interval of definition and is in the strong-limit-point case at the end-point. In this paper we investigate the cases when the differential equation is in the limit-circle case and non-oscillatory at the singular end-point and when both end-points of the interval are regular.

On bounds for Titchmarsh–Weyl m-coefficients and for spectral functions for second-order differential operators

1984 ◽  
Vol 97 ◽  
pp. 1-7 ◽  
Author(s):  
F. V. Atkinson
Keyword(s):  
Weight Function ◽  
Differential Operators ◽  
Order Term ◽  
Weyl Function ◽  
Second Order ◽  
Spectral Functions ◽  
Positive Weight ◽  
Nevanlinna Function ◽  
Fixed Sign ◽  
Order Of Magnitude

SynopsisOrder-of-magnitude results are extended to the case of general second-order term, with coefficient not necessarily of fixed sign, with general positive weight-function. The bounds are used to establish the expression for the Titchmarsh–Weyl function m(λ) as a Nevanlinna function in terms of the spectral function.

20.—On the Limit-point and Limit-circle Theory of Second-order Differential Equations

1975 ◽  
Vol 72 (3) ◽  
pp. 245-256
Author(s):  
K. S. Ong
Keyword(s):  
Differential Equations ◽  
Weight Function ◽  
Limit Point ◽  
Circle Method ◽  
Second Order ◽  
Limit Circle ◽  
Second Order Differential Equations

SynopsisIn this paper the Weyl limit-point and limit-circle theory of second-order differential equations is extended to the case that the weight function is allowed to take on both positive and negative values—the polar case. This extension is achieved using Weyl's limit circle method.

Theory of Boundary Eigensolutions in Engineering Mechanics

2000 ◽  
Vol 68 (1) ◽  
pp. 101-108 ◽  
Author(s):  
A. R. Hadjesfandiari ◽  
G. F. Dargush
Keyword(s):  
Weight Function ◽  
Mixed Boundary ◽  
Traditional Approach ◽  
Mixed Boundary Conditions ◽  
Boundary Element Methods ◽  
Positive Weight ◽  
Nonsmooth Problems ◽  
Integral Equation Methods ◽  
Potential Problems ◽  
Engineering Mechanics

A theory of boundary eigensolutions is presented for boundary value problems in engineering mechanics. While the theory is quite general, the presentation here is restricted to potential problems. Contrary to the traditional approach, the eigenproblem is formed by inserting the eigenparameter, along with a positive weight function, into the boundary condition. The resulting spectra are real and the eigenfunctions are mutually orthogonal on the boundary, thus providing a basis for solutions. The weight function permits effective treatment of nonsmooth problems associated with cracks, notches and mixed boundary conditions. Several ideas related to the convergence characteristics are also introduced. Furthermore, the connection is made to integral equation methods and variational methods. This paves the way toward the development of new computational formulations for finite element and boundary element methods. Two numerical examples are included to illustrate the applicability.

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