Asymptotic theory for a critical class of fourth-order differential equations

1982 ◽  
Vol 383 (1785) ◽  
pp. 465-476 ◽  
Keyword(s):  
Differential Equations ◽  
Fourth Order ◽  
Critical Case ◽  
Asymptotic Theory ◽  
All Solutions ◽  
Exponential Character ◽  
Independent Variable ◽  
Fourth Order Equations ◽  
Critical Class ◽  
Asymptotic Forms

A recently formulated asymptotic theory for differential equations with large coefficients that are like powers of the independent variable is developed here for fourth-order equations with general coefficients. In this general approach a critical case is identified that forms a borderline between situations where all solutions have asymptotically a certain exponential character in terms of the coefficients and where only two solutions have this character. The asymptotic forms of a fundamental set of solutions are obtained in the critical case.

scholarly journals Asymptotic theory for a critical case for a general fourth-order differential equation

1998 ◽  
Vol 21 (3) ◽  
pp. 479-488
Author(s):  
A. S. A. Al-Hammadi
Keyword(s):  
Differential Equation ◽  
Fourth Order ◽  
Critical Case ◽  
Asymptotic Theory ◽  
Order Differential Equation ◽  
Fourth Order Equations ◽  
Fourth Order Differential Equation

In this paper we identify a relation between the coefficients that represents a critical case for general fourth-order equations. We obtained the forms of solutions under this critical case.

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.

scholarly journals Simplified and improved criteria for oscillation of delay differential equations of fourth order

2021 ◽  
Vol 2021 (1) ◽  
Author(s):  
O. Moaaz ◽  
A. Muhib ◽  
D. Baleanu ◽  
W. Alharbi ◽  
E. E. Mahmoud
Keyword(s):  
Differential Equations ◽  
Fourth Order ◽  
Delay Differential Equations ◽  
Oscillatory Behavior ◽  
Interesting Point ◽  
All Solutions ◽  
Special Cases ◽  
Delay Differential ◽  
Noncanonical Operator

AbstractAn interesting point in studying the oscillatory behavior of solutions of delay differential equations is the abbreviation of the conditions that ensure the oscillation of all solutions, especially when studying the noncanonical case. Therefore, this study aims to reduce the oscillation conditions of the fourth-order delay differential equations with a noncanonical operator. Moreover, the approach used gives more accurate results when applied to some special cases, as we explained in the examples.

scholarly journals Comparison theorems for fourth order differential equations

1986 ◽  
Vol 9 (1) ◽  
pp. 105-109
Author(s):  
Garret J. Etgen ◽  
Willie E. Taylor
Keyword(s):  
Continuous Function ◽  
Differential Equations ◽  
Fourth Order ◽  
Linear Equations ◽  
Oscillation Criterion ◽  
Comparison Theorems ◽  
Positive Continuous Function ◽  
Order Linear ◽  
All Solutions

This paper establishes an apparently overlooked relationship between the pair of fourth order linear equationsyiv−p(x)y=0andyiv+p(x)y=0, wherepis a positive, continuous function defined on[0,∞). It is shown that if all solutions of the first equation are nonoscillatory, then all solutions of the second equation must be nonoscillatory as well. An oscillation criterion for these equations is also given.

On oscillatory fourth order nonlinear neutral differential equations – III

2018 ◽  
Vol 68 (6) ◽  
pp. 1385-1396 ◽  
Author(s):  
Arun Kumar Tripathy ◽  
Rashmi Rekha Mohanta
Keyword(s):  
Differential Equations ◽  
Fourth Order ◽  
Functional Differential Equations ◽  
Neutral Type ◽  
Sufficient Conditions ◽  
Neutral Differential Equations ◽  
All Solutions ◽  
Functional Differential ◽  
T Tau

Abstract In this paper, several sufficient conditions for oscillation of all solutions of fourth order functional differential equations of neutral type of the form $$\begin{array}{} \displaystyle \bigl(r(t)(y(t)+p(t)y(t-\tau))''\bigr)''+q(t)G\bigl(y(t-\sigma)\bigr)=0 \end{array}$$ are studied under the assumption $$\begin{array}{} \displaystyle \int\limits^{\infty}_{0}\frac{t}{r(t)}{\rm d} t =\infty \end{array}$$

scholarly journals Oscillation of Fourth-Order Functional Differential Equations with Distributed Delay

Axioms ◽  
2019 ◽  
Vol 8 (2) ◽  
pp. 61 ◽  
Author(s):  
Clemente Cesarano ◽  
Omar Bazighifan
Keyword(s):  
Differential Equations ◽  
Fourth Order ◽  
Delay Differential Equations ◽  
Functional Differential Equations ◽  
Sufficient Conditions ◽  
Distributed Delay ◽  
All Solutions ◽  
Functional Differential ◽  
Delay Differential ◽  
Comparison Technique

In this paper, the authors obtain some new sufficient conditions for the oscillation of all solutions of the fourth order delay differential equations. Some new oscillatory criteria are obtained by using the generalized Riccati transformations and comparison technique with first order delay differential equation. Our results extend and improve many well-known results for oscillation of solutions to a class of fourth-order delay differential equations. The effectiveness of the obtained criteria is illustrated via examples.

Oscillation criteria for fourth-order functional differential equations

2013 ◽  
Vol 63 (6) ◽  
Author(s):  
Said Grace ◽  
Martin Bohner ◽  
Ailian Liu
Keyword(s):  
Differential Equations ◽  
Fourth Order ◽  
Functional Differential Equations ◽  
Oscillation Criteria ◽  
All Solutions ◽  
Functional Differential ◽  
New Criteria

AbstractSome new criteria for the oscillation of all solutions of certain fourth-order functional differential equations are established.

scholarly journals Some new oscillation criteria of fourth-order quasi-linear differential equations with neutral term

2021 ◽  
Vol 2021 (1) ◽  
Author(s):  
Saeed Althubiti ◽  
Fahad Alsharari ◽  
Omar Bazighifan ◽  
George E. Chatzarakis
Keyword(s):  
Differential Equations ◽  
Fourth Order ◽  
Delay Differential Equations ◽  
Oscillatory Behavior ◽  
Average Technique ◽  
Neutral Differential Equations ◽  
Linear Differential ◽  
Fourth Order Equations ◽  
Delay Differential ◽  
Integral Average

AbstractIn this article, we are interested in studying the asymptotic behavior of fourth-order neutral differential equations. Despite the growing interest in studying the oscillatory behavior of delay differential equations of second-order, fourth-order equations have received less attention. We get more than one criterion to check the oscillation by the generalized Riccati method and the integral average technique. Our results are an extension and complement to some results published in the literature. Examples are given to prove the significance of new theorems.

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