scholarly journals Oscillation of Half-Linear Differential Equations with Delay

2013 ◽  
Vol 2013 ◽  
pp. 1-6 ◽  
Author(s):  
Simona Fišnarová ◽  
Robert Mařík
Keyword(s):  
Differential Equation ◽  
A Priori ◽  
Nonoscillatory Solution ◽  
Neutral Equations ◽  
Oscillation Criteria ◽  
A Priori Bound ◽  
Linear Delay ◽  
Linear Differential ◽  
Delay Differential ◽  
Equations With Delay

We study the half-linear delay differential equation , , We establish a new a priori bound for the nonoscillatory solution of this equation and utilize this bound to derive new oscillation criteria for this equation in terms of oscillation criteria for an ordinary half-linear differential equation. The presented results extend and improve previous results of other authors. An extension to neutral equations is also provided.

scholarly journals New Criteria for Sharp Oscillation of Second-OrderNeutral Delay Differential Equations

Mathematics ◽  
2021 ◽  
Vol 9 (17) ◽  
pp. 2089
Author(s):  
Irena Jadlovská
Keyword(s):  
Differential Equations ◽  
Delay Differential Equations ◽  
A Priori ◽  
Nonoscillatory Solution ◽  
Neutral Delay ◽  
Oscillation Criteria ◽  
A Priori Bound ◽  
Neutral Delay Differential Equations ◽  
Delay Differential ◽  
New Criteria

In this paper, new oscillation criteria for second-order half-linear neutral delay differential equations are established, using a recently developed method of iteratively improved monotonicity properties of a nonoscillatory solution. Our approach allows removing several disadvantages which were commonly associated with the method based on a priori bound for the nonoscillatory solution, and deriving new results which are optimal in a nonneutral case. It is shown that the newly obtained results significantly improve a large number of existing ones.

scholarly journals On Sharp Oscillation Criteria for General Third-Order Delay Differential Equations

Mathematics ◽  
2021 ◽  
Vol 9 (14) ◽  
pp. 1675
Author(s):  
Irena Jadlovská ◽  
George E. Chatzarakis ◽  
Jozef Džurina ◽  
Said R. Grace
Keyword(s):  
Differential Equation ◽  
Differential Equations ◽  
Delay Differential Equation ◽  
Delay Differential Equations ◽  
Nonoscillatory Solution ◽  
Third Order ◽  
Oscillation Criteria ◽  
Delay Differential

In this paper, effective oscillation criteria for third-order delay differential equations of the form, r2r1y′′′(t)+q(t)y(τ(t))=0 ensuring that any nonoscillatory solution tends to zero asymptotically, are established. The results become sharp when applied to a Euler-type delay differential equation and, to the best of our knowledge, improve all existing results from the literature. Examples are provided to illustrate the importance of the main results.

scholarly journals Robustness of solutions of linear differential equations with infinite delay

2006 ◽  
Vol 2006 ◽  
pp. 1-13
Author(s):  
Anwar A. Al-Badarneh (Al-Nayef) ◽  
Rabaa K. Maaitah
Keyword(s):  
Differential Equation ◽  
Differential Equations ◽  
Delay Differential Equation ◽  
Infinite Delay ◽  
Nonlinear Perturbation ◽  
Linear Differential Equations ◽  
Solution Operator ◽  
Linear Delay ◽  
Linear Differential ◽  
Delay Differential

We use some consequences of the concept of semihyperbolicity of the solution operator to show robustness of solutions of the linear delay differential equation x′(t)=Ax(t)+Bx(t−r) with infinite delay with respect to a small nonlinear perturbation.

scholarly journals Oscillation tests for first-order linear differential equations with non-monotone delays

2021 ◽  
Vol 2021 (1) ◽  
Author(s):  
Emad R. Attia
Keyword(s):  
Differential Equation ◽  
Differential Equations ◽  
Delay Differential Equation ◽  
Linear Differential Equations ◽  
Order Linear ◽  
First Order ◽  
Linear Delay ◽  
Linear Differential ◽  
A New Technique ◽  
Delay Differential

AbstractWe study the oscillation of a first-order linear delay differential equation. A new technique is developed and used to obtain new oscillatory criteria for differential equation with non-monotone delay. Some of these results can improve many previous works. An example is introduced to illustrate the effectiveness and applicability of our results.

scholarly journals Lp-Solution to the Random Linear Delay Differential Equation with a Stochastic Forcing Term

Mathematics ◽  
2020 ◽  
Vol 8 (6) ◽  
pp. 1013
Author(s):  
Juan Carlos Cortés ◽  
Marc Jornet
Keyword(s):  
Differential Equation ◽  
Solution Process ◽  
Random Variables ◽  
Initial Condition ◽  
Forcing Term ◽  
Random Forcing ◽  
Linear Delay ◽  
Method Of Steps ◽  
Linear Differential ◽  
Delay Differential

This paper aims at extending a previous contribution dealing with the random autonomous-homogeneous linear differential equation with discrete delay τ > 0 , by adding a random forcing term f ( t ) that varies with time: x ′ ( t ) = a x ( t ) + b x ( t − τ ) + f ( t ) , t ≥ 0 , with initial condition x ( t ) = g ( t ) , − τ ≤ t ≤ 0 . The coefficients a and b are assumed to be random variables, while the forcing term f ( t ) and the initial condition g ( t ) are stochastic processes on their respective time domains. The equation is regarded in the Lebesgue space L p of random variables with finite p-th moment. The deterministic solution constructed with the method of steps and the method of variation of constants, which involves the delayed exponential function, is proved to be an L p -solution, under certain assumptions on the random data. This proof requires the extension of the deterministic Leibniz’s integral rule for differentiation to the random scenario. Finally, we also prove that, when the delay τ tends to 0, the random delay equation tends in L p to a random equation with no delay. Numerical experiments illustrate how our methodology permits determining the main statistics of the solution process, thereby allowing for uncertainty quantification.

scholarly journals New Stability Conditions for Linear Differential Equations with Several Delays

2011 ◽  
Vol 2011 ◽  
pp. 1-19 ◽  
Author(s):  
Leonid Berezansky ◽  
Elena Braverman
Keyword(s):  
Differential Equation ◽  
Differential Equations ◽  
Exponential Stability ◽  
Delay Differential Equation ◽  
Linear Differential Equations ◽  
Stability Conditions ◽  
Linear Delay ◽  
Linear Differential ◽  
Several Delays ◽  
Delay Differential

New explicit conditions of asymptotic and exponential stability are obtained for the scalar nonautonomous linear delay differential equationx˙(t)+∑k=1mak(t)x(hk(t))=0with measurable delays and coefficients. These results are compared to known stability tests.

scholarly journals Oscillation Criteria for Second Order Neutral Differential Equations

1996 ◽  
Vol 48 (4) ◽  
pp. 871-886 ◽  
Author(s):  
Horng-Jaan Li ◽  
Wei-Ling Liu
Keyword(s):  
Differential Equation ◽  
Differential Equations ◽  
Delay Differential Equations ◽  
Second Order ◽  
Neutral Delay ◽  
Oscillation Criteria ◽  
Neutral Differential Equations ◽  
Neutral Delay Differential Equation ◽  
Image Position ◽  
Delay Differential

AbstractSome oscillation criteria are given for the second order neutral delay differential equationwhere τ and σ are nonnegative constants, . These results generalize and improve some known results about both neutral and delay differential equations.

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