A new approach to minimum and comparison principles for nonlinear ordinary differential operators of second order

1995 ◽  
Vol 25 (9-10) ◽  
pp. 1071-1078 ◽  
Author(s):  
Wolfgang Walter
Keyword(s):  
Differential Operators ◽  
Second Order ◽  
New Approach ◽  
Comparison Principles ◽  
Ordinary Differential Operators

V.— On the Spectrum of Second Order Partial Differential Operators.

1970 ◽  
Vol 69 (1) ◽  
pp. 77-84
Author(s):  
K. J. Brown ◽  
I. M. Michael
Keyword(s):  
Differential Equations ◽  
Spectral Properties ◽  
Differential Operators ◽  
Second Order ◽  
Partial Differential ◽  
Partial Differential Operators ◽  
Ordinary Differential Operators

SynopsisIn a recent paper, Jyoti Chaudhuri and W. N. Everitt linked the spectral properties of certain second order ordinary differential operators with the analytic properties of the solutions of the corresponding differential equations. This paper considers similar properties of the spectrum of the corresponding partial differential operators.

Oscillation criteria of Nehari-type for Sturm–Liouville operators and elliptic differential operators of second order and the lower spectrum

1988 ◽  
Vol 109 (1-2) ◽  
pp. 127-144 ◽  
Author(s):  
F. Fiedler
Keyword(s):  
Differential Equation ◽  
Differential Operators ◽  
Second Order ◽  
Oscillation Criteria ◽  
Elliptic Differential Equations ◽  
Ordinary Differential Operators ◽  
Elliptic Differential Operators ◽  
Sturm Liouville ◽  
Lower Spectrum ◽  
Liouville Operators

SynopsisSufficient oscillation criteria of Nehari-type are established for the differential equation −uʺ(t) + q(t)u(t) = 0, 0<t<∞, with and without sign restrictions on q(t), respectively. These results are extended to Sturm-Liouville equations and elliptic differential equations of second order.In Section 7 we present conclusions for the lower spectrum of elliptic differential operators and also for the discreteness of the spectrum of certain ordinary differential operators of second order.

Inverse-monotone nonlinear differential operators of the second order

1978 ◽  
Vol 79 (3-4) ◽  
pp. 193-226 ◽  
Author(s):  
Johann Schröder
Keyword(s):  
Differential Operators ◽  
Sufficient Conditions ◽  
Monotone Operators ◽  
Second Order ◽  
Differential Inequalities ◽  
Partial Differential ◽  
Nonlinear Differential Operators ◽  
Partial Differential Operators ◽  
Ordinary Differential Operators

SynopsisThis paper provides a survey on a class of methods to obtain sufficient conditions for the inversemonotonicity of second-order differential operators. Pointwise differential inequalities as well as weak differential inequalities are treated. In particular, the theory yields results on the relation between inverse-mo no tone operators and monotone definite operators, i.e. monotone operators in the Browder–Minty sense. This presentation is restricted to ordinary differential operators. Most methods explained here can also be applied to elliptic-parabolic partial differential operators in essentially the same way.

The distribution of eigenvalues of ordinary differential operators

1969 ◽  
Vol 310 (1503) ◽  
pp. 565-577 ◽  
Keyword(s):  
Qualitative Analysis ◽  
Differential Operators ◽  
Quantitative Information ◽  
Second Order ◽  
Distribution Of Eigenvalues ◽  
Ordinary Differential Operators ◽  
Remarkable Agreement

K. O. Friedrichs devised a technique for the qualitative analysis of the spectra of second order differential operators. It is shown that a modification of this technique can be used to obtain quantitative information regarding the distribution of eigenvalues of such operators. These calculations are easy to carry out and explicit examples demonstrate remarkable agreement with precise calculations.

Resolvent Estimate in L_p for Discretisations of Second Order Ordinary Differential Operators on Variable Grids

2003 ◽  
Vol 3 (2) ◽  
pp. 235-252
Author(s):  
R. D. Grigorieff
Keyword(s):  
Differential Operators ◽  
A Priori ◽  
A Priori Estimate ◽  
Second Order ◽  
Dirichlet Boundary Conditions ◽  
Resolvent Estimate ◽  
Uniform Grid ◽  
Fully Discrete ◽  
Ordinary Differential Operators ◽  
Discrete Galerkin Method

Abstract In this paper, a resolvent estimate is proved for discretisations of second order ordinary differential operators subject to Dirichlet boundary conditions on a finite or infinite interval. As a discretisation method, a fully discrete Galerkin method using continuous splines of order r > 2 on a locally quasi-uniform grid is considered. As a byproduct, an a priori estimate for the discretised differential operator is obtained.

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