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 Euler case for a general fourth-order differential equation

2004 ◽  
Vol 2004 (51) ◽  
pp. 2705-2717
Author(s):  
A. S. A. Al-Hammadi
Keyword(s):  
Differential Equation ◽  
General Formula ◽  
Fourth Order ◽  
Asymptotic Form ◽  
Order Differential Equation ◽  
Order Equation ◽  
Fourth Order Equation ◽  
Fourth Order Differential Equation

We deal with an Euler case for a general fourth-order equation and under this case, we obtain the general formula for the asymptotic form of the solutions.

The asymptotic form of the Titchmarsh–Weyl coefficient for a fourth order differential equation

1984 ◽  
Vol 97 ◽  
pp. 109-123
Author(s):  
D. B. Hinton ◽  
J. K. Shaw
Keyword(s):  
Differential Equation ◽  
Asymptotic Formula ◽  
Real Axis ◽  
Fourth Order ◽  
Asymptotic Form ◽  
Order Differential Equation ◽  
Order Equation ◽  
Fourth Order Equation ◽  
The Matrix ◽  
Fourth Order Differential Equation

SynopsisThis paper considers the asymptotic form, as λ tends to infinity in sectors omitting the real axis, of the matrix Titchmarsh-Weyl coefficient M(λ) for the fourth order equation y(4) + q(x)y = λy, where q(x) is real and locally absolutely integrable. By letting M0(λ) denote the m-coefficient for the Fourier case y(4) = λy, the asymptotic formula M(λ) = M0(λ) + 0(1) is established.

Nonclassical Orthogonal Polynomials as Solutions to Second Order Differential Equations

1982 ◽  
Vol 25 (3) ◽  
pp. 291-295 ◽  
Author(s):  
Lance L. Littlejohn ◽  
Samuel D. Shore
Keyword(s):  
Differential Equation ◽  
Differential Equations ◽  
Orthogonal Polynomial ◽  
Orthogonal Polynomials ◽  
Order Differential Equation ◽  
Order Equation ◽  
Second Order ◽  
Second Order Differential Equations ◽  
Image Position

AbstractOne of the more popular problems today in the area of orthogonal polynomials is the classification of all orthogonal polynomial solutions to the second order differential equation:In this paper, we show that the Laguerre type and Jacobi type polynomials satisfy such a second order equation.

The Krall Polynomials as Solutions to a Second Order Differential Equation

1983 ◽  
Vol 26 (4) ◽  
pp. 410-417 ◽  
Author(s):  
Lance L. Littlejohn
Keyword(s):  
Differential Equation ◽  
Differential Equations ◽  
Orthogonal Polynomial ◽  
Orthogonal Polynomials ◽  
Order Differential Equation ◽  
Order Equation ◽  
Second Order ◽  
Polynomial Solutions ◽  
Second Order Differential Equations ◽  
Image Position

AbstractA popular problem today in orthogonal polynomials is that of classifying all second order differential equations which have orthogonal polynomial solutions. We show that the Krall polynomials satisfy a second order equation of the form1.1

Positive periodic solutions for singular fourth-order differential equations with a deviating argument

2018 ◽  
Vol 148 (3) ◽  
pp. 605-617 ◽  
Author(s):  
Fanchao Kong ◽  
Zaitao Liang
Keyword(s):  
Differential Equation ◽  
Periodic Solutions ◽  
Fourth Order ◽  
Order Differential Equation ◽  
Positive Periodic Solutions ◽  
Mawhin’S Continuation Theorem ◽  
Deviating Argument ◽  
Small Force ◽  
Image Position ◽  
Fourth Order Differential Equation

In this paper, we study the singular fourth-order differential equation with a deviating argument:By using Mawhin's continuation theorem and some analytic techniques, we establish some criteria to guarantee the existence of positive periodic solutions. The significance of this paper is that g has a strong singularity at x = 0 and satisfies a small force condition at x = ∞, which is different from the known ones in the literature.

Asymptotic methods for fourth-order differential equations

1980 ◽  
Vol 87 (1-2) ◽  
pp. 53-63 ◽  
Author(s):  
C. G. M. Grudniewicz
Keyword(s):  
Differential Equation ◽  
Differential Equations ◽  
Fourth Order ◽  
Asymptotic Form ◽  
Order Differential Equation ◽  
Asymptotic Methods ◽  
Index Theory ◽  
New Method ◽  
Image Position ◽  
Fourth Order Differential Equation

SynopsisA new method is developed for obtaining the asymptotic form of solutions of the fourth-order differential equationwherem, nare integers and 1 ≦m,n≦ 2. The method gives new, shorter proofs of the well-known results of Walker in deficiency index theory and covers the cases not considered by Walker.

Stagnation Point Flow of a Non-Newtonian Second Order Fluid

1988 ◽  
Vol 12 (2) ◽  
pp. 57-61 ◽  
Author(s):  
I. Teipel
Keyword(s):  
Differential Equation ◽  
Reynolds Number ◽  
Stagnation Point ◽  
Fourth Order ◽  
Order Differential Equation ◽  
Equation Of Motion ◽  
Weissenberg Number ◽  
Second Order ◽  
Two Dimensional ◽  
Fourth Order Differential Equation

In this paper the flow near a two-dimensional stagnation point for a particular non-Newtonian fluid has been studied. For a second order fluid the equation of motion for the stream function has been solved by using a similarity approach. A new parameter which is a combination of the Weissenberg number and the Reynolds number characterizes the visco-elastic effects. A fourth order differential equation has to be solved numerically. Only three boundary conditions are necessary. Results for various cases will be shown. In addition an approxamation theory has been derived in order to recognize the influence of the new parameter.

scholarly journals Fourth-Order Differential Equation with Deviating Argument

2012 ◽  
Vol 2012 ◽  
pp. 1-17 ◽  
Author(s):  
M. Bartušek ◽  
M. Cecchi ◽  
Z. Došlá ◽  
M. Marini
Keyword(s):  
Differential Equation ◽  
Fourth Order ◽  
Order Differential Equation ◽  
Sufficient Conditions ◽  
Order Equation ◽  
Asymptotically Linear ◽  
Middle Term ◽  
Deviating Argument ◽  
Necessary And Sufficient ◽  
Fourth Order Differential Equation

We consider the fourth-order differential equation with middle-term and deviating argumentx(4)(t)+q(t)x(2)(t)+r(t)f(x(φ(t)))=0, in case when the corresponding second-order equationh″+q(t)h=0is oscillatory. Necessary and sufficient conditions for the existence of bounded and unbounded asymptotically linear solutions are given. The roles of the deviating argument and the nonlinearity are explained, too.

The limit-4 case of fourth-order self-adjoint differential equations

1977 ◽  
Vol 79 (1-2) ◽  
pp. 51-59 ◽  
Author(s):  
M. S. P. Eastham
Keyword(s):  
Differential Equation ◽  
Differential Equations ◽  
Fourth Order ◽  
Asymptotic Theory ◽  
Order Differential Equation ◽  
New Method ◽  
All Solutions ◽  
Image Position ◽  
Fourth Order Differential Equation

SynopsisA new method is developed for identifying real-valued coefficientsr(x),p(x), andq(x)for which all solutions of the fourth-order differential equationareL2(0, ∞). The results are compared with those derived from the asymptotic theory of Devinatz, Walker, Kogan and Rofe-Beketov.

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