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].

scholarly journals On refinements of some integral inequalities for differentiable prequasiinvex functions

Filomat ◽  
2019 ◽  
Vol 33 (14) ◽  
pp. 4377-4385
Author(s):  
Serap Özcan
Keyword(s):  
Integral Inequality ◽  
Type Inequality ◽  
Integral Inequalities ◽  
Right Hand ◽  
The Right ◽  
New Inequalities ◽  
Better Than

In this paper, using the new and improved form of H?lder?s integral inequality called H?lder-??can integral inequality, some new inequalities of the right-hand side of Hermite-Hadamard type inequality for prequasiinvex functions are established. The results obtained are compared with the known results. It is shown that the results obtained in this paper are better than those known ones.

scholarly journals A Fourth Century Head in Central Museum, Athens

1895 ◽  
Vol 15 ◽  
pp. 194-201 ◽  
Author(s):  
E. F. Benson
Keyword(s):  
Fourth Century ◽  
Present Condition ◽  
Left Hand ◽  
Right Hand ◽  
Image Position ◽  
The Right

There is among the fourth century works in the Central Museum at Athens a head found at Laurium. It is made of Parian marble but it has been completely discoloured by slag or refuse from the lead mines, and is now quite black. In its present condition it is quite impossible to obtain a satisfactory photograph of it, and the reproduction given of it in the figure is from a cast.It has been published, as far as I am aware, only in M. Kavvadias' catalogue. There it is described as a head of the Lykeian Apollo. This identification rests solely on a passage of Lucian, who mentions a statue of the Lykeian Apollo in the gymnasium at Athens.He says of it ( 7)—It will be seen from a glance at the photograph that the grounds for this identification are very slender. The left hand with the bow does not exist, and the only reason for supposing therefore that this is a head of the Lykeian Apollo consists in the fact that the right hand of the statue rests on the head. This in itself seems insufficient and, among other reasons, it is I think rendered impossible by the phrase For the hand is not idly resting, it is not a tired hand; the posture of the fingers is firm and energetic.

An inequality relating means

1944 ◽  
Vol 40 (3) ◽  
pp. 253-255
Author(s):  
J. Bronowski
Keyword(s):  
Arithmetic Mean ◽  
Arithmetic Means ◽  
Geometric Means ◽  
Left Hand ◽  
Right Hand ◽  
Image Position ◽  
The Right

1. Let a, b be positive constants; and let y1, y2, …, yn be real exponents, not all equal, having arithmetic mean y defined by(here, and in what follows, the summation ∑ extends over the values i = 1, 2, …, n). Then it is clear thatsince the right-hand sides are the geometric means of the positive numbers whose arithmetic means stand on the left-hand sides. I know of no results, however, which relate the ratios and and I have had occasion recently to require such results. This note gives an inequality between these ratios, subject to certain restrictions on a and b.

The extremal values of Legendre polynomials and of certain related functions

1950 ◽  
Vol 46 (4) ◽  
pp. 549-554 ◽  
Author(s):  
R. Cooper
Keyword(s):  
Legendre Polynomials ◽  
Extreme Value ◽  
Positive Zero ◽  
Right Hand ◽  
Extremal Values ◽  
Image Position ◽  
The Right

1. The tabulated values of the Legendre polynomials suggest that the right-hand minimum of Pn(x) changes monotonically as n increases. Let xr, n be the value of x which gives the rth extreme value to the left of 1 of Pn(x). Then we can show thatwhere jr is the rth pösitive zero of J1(z), and that after some term the sequence Pn(xr, n) is monotonic with the moduli of the terms decreasing. We cannot, however, show that the sequence is monotonic from the place at which its terms become significant.

scholarly journals An Expansion in Terms of Associated Legendre Functions

1952 ◽  
Vol 1 (1) ◽  
pp. 13-15
Author(s):  
T. M. Macrobert
Keyword(s):  
Legendre Functions ◽  
Associated Legendre Functions ◽  
Right Hand ◽  
Image Position ◽  
The Right

If the right-hand side of the expansionis integrated m times from 1 to µ, it becomes

Almost-conservative second-order differential equations

1975 ◽  
Vol 77 (1) ◽  
pp. 159-169 ◽  
Author(s):  
H. P. F. Swinnerton-Dyer
Keyword(s):  
Differential Equations ◽  
Rate Of Convergence ◽  
Second Order ◽  
Good Function ◽  
Second Order Differential Equations ◽  
Right Hand ◽  
Image Position ◽  
The Right

During the last thirty years an immense amount of research has been done on differential equations of the formwhere ε > 0 is small. It is usually assumed that the perturbing term on the right-hand side is a ‘good’ function of its arguments and that its dependence on t is purely trigonometric; this means that there is an expansion of the formwhere the ωn are constants, and that one may impose any conditions on the rate of convergence of the series which turn out to be convenient. Without loss of generality we can assumeand for convenience we shall sometimes write ω0 = 0. Often f is assumed to be periodic in t, in which case it is implicit that the period is independent of x and ẋ (We can also allow f to depend on ε, provided it does so in a sensible manner.)

scholarly journals An extension of a Hardy-Littlewood-Pólya inequality

1978 ◽  
Vol 21 (1) ◽  
pp. 11-15 ◽  
Author(s):  
A. Erdélyi
Keyword(s):  
Integrable Function ◽  
Right Hand ◽  
Usual Norm ◽  
Locally Integrable Function ◽  
Image Position ◽  
The Right

The Hardy-Littlewood-Pólya inequality in question can be written in the formHere and throughout, all functions are assumed to be locally integrable on ]0,∞[, 1≤p≤∞,p-1+(p′)-1=1 (with similar conventions for q,r,s), is the usual norm on Lp(0,∞), and if the right hand side is finite, then (1.1) is understood to mean thatdefines a locally integrable function Kf for which (1.1) holds.

On the asymptotic behaviour of the solutions of a class of non-linear differential equations

1950 ◽  
Vol 46 (3) ◽  
pp. 406-418
Author(s):  
F. G. Friedlander
Keyword(s):  
Existence And Uniqueness ◽  
Lipschitz Constant ◽  
Asymptotic Properties ◽  
Physical Significance ◽  
Right Hand ◽  
Non Linear ◽  
Linear Differential ◽  
Non Linear Mechanics ◽  
Image Position ◽  
The Right

1. This paper is concerned with certain asymptotic properties of the solutions of the differential equationwhere dots indicate differentiation with respect to t, k is a small parameter, and f(x, ẋ, t) satisfies certain conditions which will be formulated below. Equations of this type occur frequently in non-linear mechanics; for k = 0 a system satisfying (1·1) behaves as a harmonic oscillator. To ensure the existence and uniqueness of the solutions of (1·1) it must be assumed that the right-hand side is bounded and satisfies a Lipschitz condition, at least for finite x, ẋ and say all t ≥ 0. The parameter k may be considered as a measure of the ‘smallness’ of the upper bound, and of the Lipschitz constant, of the right-hand side, and need not have any intrinsic physical significance.

On some theorems of Hurwitz and Sheil-Small

1986 ◽  
Vol 100 (2) ◽  
pp. 365-370
Author(s):  
R. R. Hall
Keyword(s):  
Trigonometric Polynomial ◽  
Sign Changes ◽  
Right Hand ◽  
Image Position ◽  
The Right

We recall a theorem of Hurwitz [5]: let f(θ) be real-valued and periodic, f(θ) ≢ 0, andwhere an = bn = 0, 0 ≤ n < N. Then f(θ) has at least 2N sign changes in each period. A similar result with a trigonometric polynomial on the right-hand side of (1) had been obtained earlier by Sturm [7].

A Medallion of Antoninus Pius

1911 ◽  
Vol 1 ◽  
pp. 187-195
Author(s):  
A. W. Van Buren
Keyword(s):  
Right Hand ◽  
Casual Observer ◽  
Image Position ◽  
The Right

There are published several examples of a bronze medallion of Antoninus Pius, dated A.D. 140–144, with the following reverse (fig 33):AESCVLAPIVS in exergue.Ship to right under arched structure; one man is visible on ship; at prow, serpent springing to right; to right, river-god facing left with right hand extended in greeting; above him, indications of rocks; above them, three buildings and a tree.The casual observer would naturally interpret the scene as follows : “From under a bridge comes a ship in motion, from the prow of which a serpent leaps to the land; a river-god greets him; the inscription AESCVLAPIVS identifies the scene as the arrival of the sacred snake of Aesculapius at the Insula Tiberina, the Insula Tiberina being represented to the right, and a bridge across the Tiber (Pons Aemilius?) to the left; the river-god being the Tiber.”

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