On a Discrete Analogue of Inequalities of Opial and Yang

1968 ◽  
Vol 11 (1) ◽  
pp. 73-77 ◽  
Author(s):  
Cheng-Ming Lee
Keyword(s):  
Integral Inequality ◽  
Discrete Analogue ◽  
Absolutely Continuous ◽  
Negative Numbers ◽  
Image Position

Let be a non-decreasing sequence of non-negative numbers, and let U∘=0. Then we haveYang [3] proved the following integral inequality:If y(x) is absolutely continuous on a≤x≤X, with y(a) = 0, then

A Discrete Analogue of Opial's Inequality

1967 ◽  
Vol 10 (1) ◽  
pp. 115-118 ◽  
Author(s):  
James S. W. Wong
Keyword(s):  
Discrete Analogue ◽  
Absolutely Continuous ◽  
Opial’S Inequality ◽  
Image Position ◽  
Opial's Inequality

In a number of papers [1] - [7], successively simpler proofs were given for the following inequality of Opial [1], in case p=1.Theorem 1. If x(t) is absolutely continuous with x(0)=0, then for any p ≧ 0,(1)Equality holds only if x(t) = Kt for some constant K.

22.—A Critical Class of Examples concerning the Integrable-square Classification of Ordinary Differential Equations

1976 ◽  
Vol 74 ◽  
pp. 285-297 ◽  
Author(s):  
W. N. Everitt ◽  
M. Giertz
Keyword(s):  
Differential Equations ◽  
Ordinary Differential Equations ◽  
Limit Point ◽  
Half Line ◽  
Absolutely Continuous ◽  
Negative Numbers ◽  
Independent Variable ◽  
Image Position ◽  
Critical Class

SynopsisLet the coefficient q be real-valued on the half-line [0, ∞) and let q′ be locally absolutely continuous on [0, ∞). The ordinary symmetric differential expressions M and M2 are determined byIt has been shown in a previous paper by the authors that if for non-negative numbers k and X the coefficient q satisfies the conditionthen M is limit-point and M2 is limit–2 at ∞.This paper is concerned with showing that for powers of the independent variable x the condition (*) is best possible in order that both M and M2 should have the classification at ∞ given above.

On a Class of Nonparametric Tests for Independence—Bivariate Case(1)

1973 ◽  
Vol 16 (3) ◽  
pp. 337-342 ◽  
Author(s):  
M. S. Srivastava
Keyword(s):  
Random Variables ◽  
Nonparametric Tests ◽  
Absolutely Continuous ◽  
Bivariate Case ◽  
Image Position ◽  
Tests For Independence ◽  
Mutually Independent

Let(X1, Y1), (X2, Y2),…, (Xn, Yn) be n mutually independent pairs of random variables with absolutely continuous (hereafter, a.c.) pdf given by(1)where f(ρ) denotes the conditional pdf of X given Y, g(y) the marginal pdf of Y, e(ρ)→ 1 and b(ρ)→0 as ρ→0 and,(2)We wish to test the hypothesis(3)against the alternative(4)For the two-sided alternative we take — ∞< b < ∞. A feature of the model (1) is that it covers both-sided alternatives which have not been considered in the literature so far.

Nonlinear integral inequalities of the Volterra type

1992 ◽  
Vol 111 (3) ◽  
pp. 599-608 ◽  
Author(s):  
Ryszard Szwarc
Keyword(s):  
Power Function ◽  
Integral Inequality ◽  
Integral Inequalities ◽  
Volterra Type ◽  
Identity Function ◽  
Right Hand ◽  
Half Axis ◽  
Image Position ◽  
The Right ◽  
Zero Points

We are studying the integral inequalitywhere all functions appearing are defined and increasing on the right half-axis and take the value zero at zero. We are interested in determining when the inequality admits solutions u(x) which are non-vanishing in a neighbourhood of zero. It is well-known that if ψ(x) is the identity function then no such solution exists. This due to the fact that the operator defined by the integral on the right-hand side of the equation is linear and compact. So if we are interested in non-trivial solutions it is natural to require that ψ(x) > 0 at least for all non-zero points in some neighbourhood of zero. One of the typical examples is the power function ψ(x) = xα, where α < 1. This situation was explored in [2]. The functions a(x) that admit non-zero solutions were characterized by Bushell in [1]. For a general approach to the problem we refer to [2], [3] and [4].

On orthogonality of Riesz products

1974 ◽  
Vol 76 (1) ◽  
pp. 173-181 ◽  
Author(s):  
Gavin Brown ◽  
William Moran
Keyword(s):  
Haar Measure ◽  
Classical Result ◽  
Weak Limit ◽  
Absolutely Continuous ◽  
Riesz Product ◽  
Image Position ◽  
Riesz Products ◽  
Positive Integers

A typical Riesz product on the circle is the weak* limitwhere – 1 ≤ rk ≤ 1, øk ∈ R, λT is Haar measure, and the positive integers nk satisfy nk+1/nk ≥ 3. A classical result of Zygmund (11) implies that either µ is absolutely continuous with respect to λT (when ) or µ is purely singular (when ).

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.

scholarly journals ASYMPTOTICALLY LINEAR ELLIPTIC SYSTEMS WITH PARAMETERS

2010 ◽  
Vol 52 (2) ◽  
pp. 383-389 ◽  
Author(s):  
CHAOQUAN PENG
Keyword(s):  
Bounded Domain ◽  
Trivial Solution ◽  
Elliptic Systems ◽  
Continuous Functions ◽  
Asymptotically Linear ◽  
Smooth Bounded Domain ◽  
Negative Numbers ◽  
Solution Pair ◽  
Image Position

AbstractIn this paper, we show that the semi-linear elliptic systems of the form (0.1) possess at least one non-trivial solution pair (u, v) ∈ H01(Ω) × H01(Ω), where Ω is a smooth bounded domain in ℝN, λ and μ are non-negative numbers, f(x, t) and g(x, t) are continuous functions on Ω × ℝ and asymptotically linear at infinity.

A dichotomy for infinite convolutions of discrete measures

1973 ◽  
Vol 73 (2) ◽  
pp. 307-316 ◽  
Author(s):  
G. Brown ◽  
W. Moran
Keyword(s):  
Number Theory ◽  
Basic Property ◽  
Discrete Measure ◽  
Absolutely Continuous ◽  
Image Position ◽  
Discrete Measures ◽  
Infinite Convolution

Measures, μ which can be realized as an infinite convolutionwhere each measure μn is a discrete measure, arise naturally in many parts of analysis and number theory (see (15)). The basic property of these measures is ‘purity’; i.e. such a measure μ 1must be absolutely continuous, continuous and singular, or discrete.

Asymptotic Expansions

1956 ◽  
Vol 8 ◽  
pp. 225-233 ◽  
Author(s):  
Leo Moser ◽  
Max Wyman
Keyword(s):  
Asymptotic Expansions ◽  
Combinatorial Problems ◽  
Negative Numbers ◽  
Image Position

1. Introduction. Let a1 a2, …, am be a set of real non-negative numbers and let1.1 P(x) = a1x + a2x2 + … + amxm (am ≠ 0).Many combinatorial problems can be reduced to the study of numbers Bn generated by1.2.Some problems of this type were treated by Touchard (7), Jacobsthal (3), Chowla, Herstein, Moore and Scott (1; 2), and the present authors (4).

Differential Operators with Abstract Boundary Conditions

1978 ◽  
Vol 30 (02) ◽  
pp. 262-288 ◽  
Author(s):  
R. C. Brown
Keyword(s):  
Boundary Value Problem ◽  
Boundary Conditions ◽  
Vector Space ◽  
Differential Operators ◽  
Dimensional Vector ◽  
Absolutely Continuous ◽  
Abstract Boundary ◽  
Image Position ◽  
Vector Valued Functions ◽  
Vector Valued

Suppose F is a topological vector space. Let ACm ≡ ACm[a, b] be the absolutely continuous m-dimensional vector valued functions y on the compact interval [a, b] with essentially bounded components. Consider the boundary value problem where A0, A are respectively... operator with range in F.

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