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.

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.

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.

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.

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.

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 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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