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.

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

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.

Asymptotics for third-order nonlinear differential equations: Non-oscillatory and oscillatory cases

2021 ◽  
pp. 1-19
Author(s):  
Calogero Vetro ◽  
Dariusz Wardowski
Keyword(s):  
Differential Equation ◽  
Differential Equations ◽  
Order Differential Equation ◽  
Nonlinear Differential Equations ◽  
Oscillatory Behavior ◽  
Third Order ◽  
First Order ◽  
Third Order Differential Equation ◽  
Comparison Technique ◽  
First Order Differential Equations

We discuss a third-order differential equation, involving a general form of nonlinearity. We obtain results describing how suitable coefficient functions determine the asymptotic and (non-)oscillatory behavior of solutions. We use comparison technique with first-order differential equations together with the Kusano–Naito’s and Philos’ approaches.

Oscillation criteria for third-order nonlinear differential equations

2008 ◽  
Vol 58 (2) ◽  
Author(s):  
B. Baculíková ◽  
E. Elabbasy ◽  
S. Saker ◽  
J. Džurina
Keyword(s):  
Differential Equation ◽  
Differential Equations ◽  
Order Differential Equation ◽  
Nonlinear Differential Equations ◽  
Sufficient Conditions ◽  
Third Order ◽  
Oscillation Criteria ◽  
The Third ◽  
Third Order Differential Equation ◽  
Oscillation Properties

AbstractIn this paper, we are concerned with the oscillation properties of the third order differential equation $$ \left( {b(t) \left( {[a(t)x'(t)'} \right)^\gamma } \right)^\prime + q(t)x^\gamma (t) = 0, \gamma > 0 $$. Some new sufficient conditions which insure that every solution oscillates or converges to zero are established. The obtained results extend the results known in the literature for γ = 1. Some examples are considered to illustrate our main results.

scholarly journals Kamenev-Type Asymptotic Criterion of Fourth-Order Delay Differential Equation

2020 ◽  
Vol 4 (1) ◽  
pp. 7 ◽  
Author(s):  
Omar Bazighifan
Keyword(s):  
Differential Equation ◽  
Fourth Order ◽  
Delay Differential Equation ◽  
Order Differential Equation ◽  
Sufficient Conditions ◽  
Oscillation Criterion ◽  
Delay Differential ◽  
Necessary And Sufficient ◽  
Type Oscillation ◽  
Fourth Order Differential Equation

In this paper, we obtain necessary and sufficient conditions for a Kamenev-type oscillation criterion of a fourth order differential equation of the form r 3 t r 2 t r 1 t y ′ t ′ ′ ′ + q t f y σ t = 0 , where t ≥ t 0 . The results presented here complement some of the known results reported in the literature. Moreover, the importance of the obtained conditions is illustrated via some examples.

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 Second-Order Differential Equation: Oscillation Theorems and Applications

2020 ◽  
Vol 2020 ◽  
pp. 1-6 ◽  
Author(s):  
Shyam S. Santra ◽  
Omar Bazighifan ◽  
Hijaz Ahmad ◽  
Yu-Ming Chu
Keyword(s):  
Differential Equation ◽  
Differential Equations ◽  
Delay Differential Equations ◽  
Order Differential Equation ◽  
Sufficient Conditions ◽  
Second Order ◽  
Linear Delay ◽  
Delay Differential ◽  
Necessary And Sufficient ◽  
Oscillation Theorems

Differential equations of second order appear in a wide variety of applications in physics, mathematics, and engineering. In this paper, necessary and sufficient conditions are established for oscillations of solutions to second-order half-linear delay differential equations of the form ς y u ′ y a ′ + p y u c ϑ y = 0 ,  for  y ≥ y 0 , under the assumption ∫ ∞ ς η − 1 / a = ∞ . Two cases are considered for a < c and a > c , where a and c are the quotients of two positive odd integers. Two examples are given to show the effectiveness and applicability of the result.

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.

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 ON SPECIFIC ASYMPTOTIC STABILITY SOLUTIONS OF A LINEAR HOMOGENEOUS WELTERER INTEGRO-DIFFERENTIAL EQUATION OF THE FOURTH ORDER

2021 ◽  
pp. 110-117
Author(s):  
Z. A. Japarova
Keyword(s):  
Differential Equation ◽  
Asymptotic Stability ◽  
Fourth Order ◽  
Order Differential Equation ◽  
Sufficient Conditions ◽  
Third Order ◽  
Volterra Type ◽  
Integro Differential Equation ◽  
The Third ◽  
Fourth Order Differential Equation

Specific sufficient conditions for the asymptotic stability of a linear homogeneous fourthorder integro-differential equation of the Volterra type are established in the case when all nonzero solutions of the corresponding fourth-order differential equation do not have the property of asymptotic stability of the solutions. In this paper, we obtain estimates on the semiaxis of the solution and the derivative up to the third order.

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