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 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.

An Oscillation Criterion for First Order Linear Delay Differential Equations

1998 ◽  
Vol 41 (2) ◽  
pp. 207-213 ◽  
Author(s):  
CH. G. Philos ◽  
Y. G. Sficas
Keyword(s):  
Differential Equation ◽  
Differential Equations ◽  
Delay Differential Equation ◽  
Delay Differential Equations ◽  
Oscillation Criterion ◽  
Order Linear ◽  
First Order ◽  
Linear Delay ◽  
Delay Differential

AbstractA new oscillation criterion is given for the delay differential equation , where and the function T defined by is increasing and such that . This criterion concerns the case where .

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 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.

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 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.

scholarly journals Oscillation Results for Third-Order Semi-Canonical Quasi-Linear Delay Differential Equations

2021 ◽  
Vol 8 (1) ◽  
pp. 228-238
Author(s):  
K. Saranya ◽  
V. Piramanantham ◽  
E. Thandapani
Keyword(s):  
Differential Equation ◽  
Differential Operator ◽  
Differential Equations ◽  
Delay Differential Equation ◽  
Delay Differential Equations ◽  
Main Idea ◽  
Third Order ◽  
The Third ◽  
Linear Delay ◽  
Delay Differential

Abstract The main purpose of this paper is to study the oscillatory properties of solutions of the third-order quasi-linear delay differential equation ℒ y ( t ) + f ( t ) y β ( σ ( t ) ) = 0 {\cal L}y(t) + f(t){y^\beta }(\sigma (t)) = 0 where ℒy(t) = (b(t)(a(t)(y 0(t)) )0)0 is a semi-canonical differential operator. The main idea is to transform the semi-canonical operator into canonical form and then obtain new oscillation results for the studied equation. Examples are provided to illustrate the importance of the main results.

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.

scholarly journals On exponential stability of second order delay differential equations

2015 ◽  
Vol 65 (4) ◽  
pp. 1047-1068 ◽  
Author(s):  
Ravi P. Agarwal ◽  
Alexander Domoshnitsky ◽  
Abraham Maghakyan
Keyword(s):  
Differential Equations ◽  
Exponential Stability ◽  
Delay Differential Equations ◽  
Second Order ◽  
Delay Differential
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