scholarly journals Criteria for the Oscillation of Solutions to Linear Second-Order Delay Differential Equation with a Damping Term

Axioms ◽  
2021 ◽  
Vol 10 (4) ◽  
pp. 246
Author(s):  
Osama Moaaz ◽  
Elmetwally M. E. Elabbasy ◽  
Jan Awrejcewicz ◽  
Aml Abdelnaser
Keyword(s):  
Differential Equation ◽  
Delay Differential Equation ◽  
Delay Differential Equations ◽  
Asymptotic Properties ◽  
Second Order ◽  
Iterative Technique ◽  
Damping Term ◽  
Delay Differential ◽  
New Criterion ◽  
Oscillation Of Solutions

The aim of this work is to present new oscillation results for a class of second-order delay differential equations with damping term. The new criterion of oscillation depends on improving the asymptotic properties of the positive solutions of the studied equation by using an iterative technique. Our results extend some of the results recently published in the literature.

Exponential stability of a second order delay differential equation without damping term

2015 ◽  
Vol 258 ◽  
pp. 483-488 ◽  
Author(s):  
Leonid Berezansky ◽  
Alexander Domoshnitsky ◽  
Mikhail Gitman ◽  
Valery Stolbov
Keyword(s):  
Differential Equation ◽  
Exponential Stability ◽  
Delay Differential Equation ◽  
Second Order ◽  
Damping Term ◽  
Delay Differential

scholarly journals Oscillatory results for second-order noncanonical delay differential equations

2019 ◽  
Vol 39 (4) ◽  
pp. 483-495 ◽  
Author(s):  
Jozef Džurina ◽  
Irena Jadlovská ◽  
Ioannis P. Stavroulakis
Keyword(s):  
Differential Equation ◽  
Differential Equations ◽  
Positive Solutions ◽  
Delay Differential Equation ◽  
Delay Differential Equations ◽  
Second Order ◽  
Linear Delay ◽  
Delay Differential

The main purpose of this paper is to improve recent oscillation results for the second-order half-linear delay differential equation \[\left(r(t)\left(y'(t)\right)^\gamma\right)'+q(t)y^\gamma(\tau(t))= 0, \quad t\geq t_0,\] under the condition \[\int_{t_0}^{\infty}\frac{\text{d} t}{r^{1/\gamma}(t)} \lt \infty.\] Our approach is essentially based on establishing sharper estimates for positive solutions of the studied equation than those used in known works. Two examples illustrating the results are given.

scholarly journals Necessary and Sufficient Conditions for Oscillation of Second-Order Delay Differential Equations

2020 ◽  
Vol 75 (1) ◽  
pp. 135-146
Author(s):  
Shyam Sundar Santra
Keyword(s):  
Differential Equation ◽  
Delay Differential Equation ◽  
Delay Differential Equations ◽  
Sufficient Conditions ◽  
Second Order ◽  
Necessary And Sufficient Conditions ◽  
All Solutions ◽  
Linear Delay ◽  
Delay Differential ◽  
Necessary And Sufficient

AbstractIn this work, we obtain necessary and sufficient conditions for the oscillation of all solutions of second-order half-linear delay differential equation of the form {\left( {r{{\left( {x'} \right)}^\gamma }} \right)^\prime }\left( t \right) + q\left( t \right){x^\alpha }\left( {\tau \left( t \right)} \right) = 0Under the assumption ∫∞(r(n))−1/γdη=∞, we consider the two cases when γ > α and γ < α. Further, some illustrative examples showing applicability of the new results are included, and state an open problem.

scholarly journals Asymptotic properties of trinomial delay differential equations

2009 ◽  
Vol 43 (1) ◽  
pp. 71-79
Author(s):  
Jozef Džurina ◽  
Renáta Kotorová
Keyword(s):  
Differential Equation ◽  
Differential Equations ◽  
Canonical Form ◽  
Delay Differential Equation ◽  
Delay Differential Equations ◽  
Asymptotic Properties ◽  
Third Order ◽  
The Third ◽  
Delay Differential ◽  
New Criteria

AbstractNew criteria for asymptotic properties of the solutions of the third order delay differential equation, by transforming this equation to its binomial canonical form are presented

scholarly journals Asymptotic and Oscillatory Properties of Noncanonical Delay Differential Equations

2021 ◽  
Vol 5 (4) ◽  
pp. 259
Author(s):  
Osama Moaaz ◽  
Clemente Cesarano ◽  
Sameh Askar
Keyword(s):  
Differential Equation ◽  
Differential Equations ◽  
Delay Differential Equation ◽  
Delay Differential Equations ◽  
Asymptotic Properties ◽  
Oscillatory Solutions ◽  
Even Order ◽  
Delay Differential ◽  
New Criteria ◽  
Oscillatory Properties

In this work, by establishing new asymptotic properties of non-oscillatory solutions of the even-order delay differential equation, we obtain new criteria for oscillation. The new criteria provide better results when determining the values of coefficients that correspond to oscillatory solutions. To explain the significance of our results, we apply them to delay differential equation of Euler-type.

scholarly journals New Asymptotic Properties of Positive Solutions of Delay Differential Equations and Their Application

Mathematics ◽  
2021 ◽  
Vol 9 (16) ◽  
pp. 1971
Author(s):  
Osama Moaaz ◽  
Clemente Cesarano
Keyword(s):  
Differential Equation ◽  
Differential Equations ◽  
Positive Solutions ◽  
Delay Differential Equation ◽  
Delay Differential Equations ◽  
Asymptotic Properties ◽  
Even Order ◽  
Delay Differential ◽  
New Criteria ◽  
Noncanonical Operator

In this study, new asymptotic properties of positive solutions of the even-order delay differential equation with the noncanonical operator are established. The new properties are of an iterative nature, which allows it to be applied several times. Moreover, we use these properties to obtain new criteria for the oscillation of the solutions of the studied equation using the principles of comparison.

scholarly journals Asymptotic Properties of Solutions of Fourth-Order Delay Differential Equations

Symmetry ◽  
2019 ◽  
Vol 11 (5) ◽  
pp. 628 ◽  
Author(s):  
Clemente Cesarano ◽  
Sandra Pinelas ◽  
Faisal Al-Showaikh ◽  
Omar Bazighifan
Keyword(s):  
Differential Equation ◽  
Differential Equations ◽  
Fourth Order ◽  
Delay Differential Equations ◽  
Order Differential Equation ◽  
Asymptotic Properties ◽  
Riccati Transformation ◽  
Delay Differential ◽  
Oscillation Of Solutions ◽  
Comparison Technique

In the paper, we study the oscillation of fourth-order delay differential equations, the present authors used a Riccati transformation and the comparison technique for the fourth order delay differential equation, and that was compared with the oscillation of the certain second order differential equation. Our results extend and improve many well-known results for oscillation of solutions to a class of fourth-order delay differential equations. Some examples are also presented to test the strength and applicability of the results obtained.

scholarly journals Oscillation results for second order half-linear neutral delay differential equations with "maxima"

2017 ◽  
Vol 48 (3) ◽  
pp. 289-299 ◽  
Author(s):  
Selvarangam Srinivasan ◽  
Rani Bose ◽  
Ethiraju Thandapani
Keyword(s):  
Differential Equation ◽  
Differential Equations ◽  
Delay Differential Equation ◽  
Delay Differential Equations ◽  
Second Order ◽  
Neutral Delay ◽  
Oscillation Criteria ◽  
Neutral Delay Differential Equation ◽  
Neutral Delay Differential Equations ◽  
Delay Differential

In this paper, we present some oscillation criteria for the second order half-linear neutral delay differential equation with ``maxima" of the from\begin{equation*}\left(r(t)((x(t)+p(t)x(\tau(t)))')^{\alpha}\right)'+q(t) \max_{[\sigma(t),\;t]}x^{\alpha}(s)=0\end{equation*}under the condition $\int_{t_0}^{\infty}\frac{1}{r^{1/ \alpha}(t)}dt<\infty.$ The results obtained here extend and complement to some known results in the literature. Examples are provided in support of our results.

ON STABILITY OF THE SECOND ORDER DELAY DIFFERENTIAL EQUATION: THREE METHODS

2021 ◽  
Vol 28 (1-2) ◽  
pp. 3-17
Author(s):  
LEONID BEREZANSKY
Keyword(s):  
Differential Equation ◽  
Differential Equations ◽  
Exponential Stability ◽  
Delay Differential Equation ◽  
Delay Differential Equations ◽  
Second Order ◽  
Operator Equations ◽  
Linear Delay ◽  
Delay Differential

The aim of the paper is a review of some methods on exponential stability for linear delay differential equations of the second order. All these methods are based on Bohl-Perron theorem which reduces stability investi-gations to study the properties of operator equations in some functional spaces. As an example of application of these methods we consider the following equation x¨(t)+ a(t)˙x(g(t)) + b(t)x(h(t)) = 0.

scholarly journals On the oscillation of third-order quasi-linear delay differential equations

2011 ◽  
Vol 48 (1) ◽  
pp. 117-123 ◽  
Author(s):  
Tongxing Li ◽  
Chenghui Zhang ◽  
Blanka Baculíková ◽  
Jozef Džurina
Keyword(s):  
Differential Equation ◽  
Differential Equations ◽  
Delay Differential Equation ◽  
Delay Differential Equations ◽  
Asymptotic Properties ◽  
Third Order ◽  
The Third ◽  
Linear Delay ◽  
Delay Differential

Abstract The aim of this work is to study asymptotic properties of the third-order quasi-linear delay differential equation , (E) where and τ(t) ≤ t. We establish a new condition which guarantees that every solution of (E) is either oscillatory or converges to zero. These results improve some known results in the literature. An example is given to illustrate the main results.

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