scholarly journals Differential inequality and non-oscillation of fourth order differential equation

2021 ◽  
Vol 104 (4) ◽  
pp. 103-109
Author(s):  
A.A. Kalybay ◽  
◽  
A.O. Baiarystanov ◽  
Keyword(s):  
Differential Equations ◽  
Variational Method ◽  
Fourth Order ◽  
Differential Inequality ◽  
Order Differential Equation ◽  
Boundary Behavior ◽  
Power Functions ◽  
Hardy Type ◽  
Fourth Order Differential Equation ◽  
Oscillatory Properties

The oscillatory theory of fourth order differential equations has not yet been developed well enough. The results are known only for the case when the coefficients of differential equations are power functions. This fact can be explained by the absence of simple effective methods for studying such higher order equations. In this paper, the authors investigate the oscillatory properties of a class of fourth order differential equations by the variational method. The presented variational method allows to consider any arbitrary functions as coefficients, and our main results depend on their boundary behavior in neighborhoods of zero and infinity. Moreover, this variational method is based on the validity of a certain weighted differential inequality of Hardy type, which is of independent interest. The authors of the article also find two-sided estimates of the least constant for this inequality, which are especially important for their applications to the main results on the oscillatory properties of these differential equations.

scholarly journals Weighted differential inequality and oscillatory properties of fourth order differential equations

2021 ◽  
Vol 2021 (1) ◽  
Author(s):  
Aigerim Kalybay ◽  
Ryskul Oinarov ◽  
Yaudat Sultanaev
Keyword(s):  
Differential Equations ◽  
Fourth Order ◽  
Differential Inequality ◽  
Boundary Behavior ◽  
Oscillatory Properties

AbstractIn this paper, we investigate the oscillatory properties of two fourth order differential equations in dependence on boundary behavior of its coefficients at infinity. These properties are established based on two-sided estimates of the least constant of a certain weighted differential inequality.

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 Oscillation Theorems for Nonlinear Differential Equations of Fourth-Order

Mathematics ◽  
2020 ◽  
Vol 8 (4) ◽  
pp. 520 ◽  
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.

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.

scholarly journals Lyapunov's Type Inequalities for Fourth-Order Differential Equations

2012 ◽  
Vol 2012 ◽  
pp. 1-25 ◽  
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.

Instability results for fourth order differential equations

1980 ◽  
Vol 85 (3-4) ◽  
pp. 247-250 ◽  
Author(s):  
W. A. Skrapek
Keyword(s):  
Differential Equation ◽  
Differential Equations ◽  
Fourth Order ◽  
Order Differential Equation ◽  
Sufficient Conditions ◽  
Fourth Order Differential Equation

SYNOPSISIn this paper we give sufficient conditions (Theorems 1 and 2) for the instability of the fourth order differential equation

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 Emden–Fowler-type neutral differential equations: oscillatory properties of solutions

2021 ◽  
Vol 2021 (1) ◽  
Author(s):  
Omar Bazighifan ◽  
Alanoud Almutairi
Keyword(s):  
Differential Equations ◽  
Fourth Order ◽  
Delay Differential Equations ◽  
Comparison Method ◽  
Riccati Transformation ◽  
Neutral Differential Equations ◽  
Neutral Term ◽  
Delay Differential ◽  
Oscillatory Properties

AbstractIn this paper, we study the oscillation of a class of fourth-order Emden–Fowler delay differential equations with neutral term. Using the Riccati transformation and comparison method, we establish several new oscillation conditions. These new conditions complement a number of results in the literature. We give examples to illustrate our main results.

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