On an inequality of the Kolmogorov type for a second-order differential expression

1993 ◽  
Vol 442 (1916) ◽  
pp. 555-569 ◽  
Keyword(s):  
Differential Expression ◽  
Differential Inequality ◽  
Second Order ◽  
Numerical Treatment ◽  
Kolmogorov Type ◽  
Best Constant

In this paper we discuss an integro-differential inequality formed from the square of a second-order differential expression. A connection between the existence of the inequality and the Titchmarsh–Weyl m -function is established and it is shown that the best constant in the inequality is determined by the behaviour of the m -function. Analytic results are established for specific classes of problems. A numerical treatment of the inequality is also discussed and estimates of the best constant in specific examples are given.

XXIII.—On an Extension to an Integro-differential Inequality of Hardy, Littlewood and Polya

1972 ◽  
Vol 69 (4) ◽  
pp. 295-333 ◽  
Author(s):  
W. N. Everitt
Keyword(s):  
Differential Equation ◽  
Fourth Order ◽  
Differential Inequality ◽  
Order Differential Equation ◽  
Order Equation ◽  
Second Order ◽  
Image Position ◽  
Fourth Order Differential Equation

SynopsisThis paper considers an extension of the following inequality given in the book Inequalities by Hardy, Littlewood and Polya; let f be real-valued, twice differentiable on [0, ∞) and such that f and f are both in the space fn, ∞), then f′ is in L,2(0, ∞) andThe extension consists in replacing f′ by M[f] wherechoosing f so that f and M[f] are in L2(0, ∞) and then seeking to determine if there is an inequality of the formwhere K is a positive number independent of f.The analysis involves a fourth-order differential equation and the second-order equation associated with M.A number of examples are discussed to illustrate the theorems obtained and to show that the extended inequality (*) may or may not hold.

scholarly journals Spatial Behavior of a Coupled System of Wave-Plate Type

2014 ◽  
Vol 2014 ◽  
pp. 1-13 ◽  
Author(s):  
Gusheng Tang ◽  
Yan Liu ◽  
Wenhui Liao
Keyword(s):  
Differential Inequality ◽  
Coupled System ◽  
Second Order ◽  
Spatial Behavior ◽  
Decay Estimates ◽  
Spatial Decay ◽  
Area Measure ◽  
First Order ◽  
Wave Plate ◽  
Plate Type

The spatial behavior of a coupled system of wave-plate type is studied. We get the alternative results of Phragmén-Lindelöf type in terms of an area measure of the amplitude in question based on a first-order differential inequality. We also get the spatial decay estimates based on a second-order differential inequality.

Generalizations of the Landau–Hadamard inequality and inequalities for quadratic polynomials of operators

1986 ◽  
Vol 102 (1-2) ◽  
pp. 123-129
Author(s):  
Khr. N. Boyadzhiev
Keyword(s):  
Differential Expression ◽  
Second Order ◽  
Functional Inequalities ◽  
Hadamard Inequality ◽  
Quadratic Polynomials

SynopsisWe give generalizations of the Landau–Hadamard inequality ‖u′‖2 ≦ K ‖u‖ ‖u″‖ replacing u” by the second-order differential expression u″ − (α + β)u′ + αβu (α, β ∈ ℂ). The new functional inequalities are then used to obtain similar inequalities for dissipative and skew-Hermitian operators.

On a fourth-order singular integral inequality

1978 ◽  
Vol 80 (3-4) ◽  
pp. 249-260 ◽  
Author(s):  
A. Russell
Keyword(s):  
Differential Expression ◽  
Fourth Order ◽  
Singular Integral ◽  
Integral Inequality ◽  
Second Order ◽  
New Order ◽  
Alternative Account ◽  
The Real ◽  
Image Position ◽  
Complete Solutions

SynopsisThe inequality considered in this paper iswhereNis the real-valued symmetric differential expression defined byGeneral properties of this inequality are considered which result in giving an alternative account of a previously considered inequalityto which (*) reduces in the casep=q= 0,r= 1.Inequality (*) is also an extension of the inequalityas given by Hardy and Littlewood in 1932. This last inequality has been extended by Everitt to second-order differential expressions and the methods in this paper extend it to fourth-order differential expressions. As with many studies of symmetric differential expressions the jump from the second-order to the fourth-order introduces difficulties beyond the extension of technicalities: problems of a new order appear for which complete solutions are not available.

Full- and partial-range eigenfunction expansions for Sturm-Liouville problems with indefinite weights

1984 ◽  
Vol 98 (1-2) ◽  
pp. 69-88 ◽  
Author(s):  
Hans G. Kaper ◽  
Man Kam Kwong ◽  
C. G. Lekkerkerker ◽  
A. Zettl
Keyword(s):  
Hilbert Space ◽  
Differential Expression ◽  
Positive Operator ◽  
Eigenvalue Problems ◽  
Full Range ◽  
Second Order ◽  
Eigenfunction Expansions ◽  
Multiplicative Operator ◽  
Indefinite Weights ◽  
Sturm Liouville

SynopsisThis article is concerned with eigenvalue problems of the form Au = λTu in a Hilbert space H, where Ais a selfadjoint positive operator generated by a second-order Sturm-Liouville differential expression and T a selfadjoint indefinite multiplicative operator which is one-to-one. Emphasis is on the full-range and partial-range expansionproperties of the eigenfunctions.

The deficiency indices of powers of second order expressions with large leading coefficient

1985 ◽  
Vol 101 (3-4) ◽  
pp. 227-235
Author(s):  
Thomas T. Read
Keyword(s):  
Large Class ◽  
Differential Expression ◽  
Limit Point ◽  
Second Order ◽  
Simple Function ◽  
Deficiency Index ◽  
Deficiency Indices

SynopsisThe deficiency index of each power of the differential expression M[y] = w−1(−(py′)′ + qy), defined on [a, ∞), is calculated exactly in terms of the behaviour of a simple function of p and w for a large class of expressions satisfying a hypothesis which requires that p be large compared with w and q. In general, not all powers of M are limit-point.

scholarly journals Numerical Treatment of Interfaces for Second-Order Wave Equations

2014 ◽  
Vol 62 (3) ◽  
pp. 875-897
Author(s):  
Florencia Parisi ◽  
Mariana Cécere ◽  
Mirta Iriondo ◽  
Oscar Reula
Keyword(s):  
Wave Equations ◽  
Second Order ◽  
Numerical Treatment
Sign in / Sign up

Export Citation Format

Share Document