Asymptotic methods for fourth-order differential equations

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.

Download Full-text

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

Proceedings of the Royal Society of Edinburgh Section A Mathematics ◽

10.1017/s0308210500016814 ◽

1977 ◽

Vol 79 (1-2) ◽

pp. 51-59 ◽

Cited By ~ 8

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.

Download Full-text

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

Proceedings of the Royal Society of Edinburgh Section A Mathematical and Physical Sciences ◽

10.1017/s0080454100008797 ◽

1972 ◽

Vol 69 (4) ◽

pp. 295-333 ◽

Cited By ~ 3

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.

Download Full-text

Euler case for a general fourth-order differential equation

International Journal of Mathematics and Mathematical Sciences ◽

10.1155/s0161171204401367 ◽

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.

Download Full-text

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

Proceedings of the Royal Society of Edinburgh Section A Mathematics ◽

10.1017/s0308210500031887 ◽

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.

Download Full-text

Oscillation Theorems for Nonlinear Differential Equations of Fourth-Order

Mathematics ◽

10.3390/math8040520 ◽

2020 ◽

Vol 8 (4) ◽

pp. 520 ◽

Cited By ~ 16

Author(s):

Osama Moaaz ◽

Ioannis Dassios ◽

Omar Bazighifan ◽

Ali Muhib

Keyword(s):

Differential Equation ◽

Differential Equations ◽

Fourth Order ◽

Order Differential Equation ◽

Nonlinear Differential Equations ◽

Sufficient Conditions ◽

Oscillatory Behavior ◽

Middle Term ◽

Oscillation Theorems ◽

Fourth Order Differential Equation

We study the oscillatory behavior of a class of fourth-order differential equations and establish sufficient conditions for oscillation of a fourth-order differential equation with middle term. Our theorems extend and complement a number of related results reported in the literature. One example is provided to illustrate the main results.

Download Full-text

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

Proceedings of the Royal Society of Edinburgh Section A Mathematics ◽

10.1017/s030821051800001x ◽

2018 ◽

Vol 148 (3) ◽

pp. 605-617 ◽

Cited By ~ 2

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.

Download Full-text

Lyapunov's Type Inequalities for Fourth-Order Differential Equations

Abstract and Applied Analysis ◽

10.1155/2012/795825 ◽

2012 ◽

Vol 2012 ◽

pp. 1-25 ◽

Cited By ~ 2

Author(s):

Samir H. Saker

Keyword(s):

Differential Equation ◽

Differential Equations ◽

Nontrivial Solution ◽

Lower Bounds ◽

Fourth Order ◽

Order Differential Equation ◽

Higher Order ◽

Hardy’S Inequality ◽

Hardy's Inequality ◽

Fourth Order Differential Equation

For a fourth-order differential equation, we will establish some new Lyapunov-type inequalities, which give lower bounds of the distance between zeros of a nontrivial solution and also lower bounds of the distance between zeros of a solution and/or its derivatives. The main results will be proved by making use of Hardy’s inequality and some generalizations of Opial-Wirtinger-type inequalities involving higher-order derivatives. Some examples are considered to illustrate the main results.

Download Full-text