scholarly journals Oscillation for Third-Order Nonlinear Differential Equations with Deviating Argument

2010 ◽  
Vol 2010 ◽  
pp. 1-19 ◽  
Author(s):  
Miroslav Bartušek ◽  
Mariella Cecchi ◽  
Zuzana Došlá ◽  
Mauro Marini
Keyword(s):  
Nonlinear Differential Equations ◽  
Asymptotic Properties ◽  
Sufficient Conditions ◽  
Nonlinear Ordinary Differential Equation ◽  
Damping Term ◽  
Third Order ◽  
Nonoscillatory Solutions ◽  
Deviating Argument ◽  
The Third ◽  
Necessary And Sufficient

We study necessary and sufficient conditions for the oscillation of the third-order nonlinear ordinary differential equation with damping term and deviating argumentx‴(t)+q(t)x′(t)+r(t)f(x(φ(t)))=0. Motivated by the work of Kiguradze (1992), the existence and asymptotic properties of nonoscillatory solutions are investigated in case when the differential operatorℒx=x‴+q(t)x′is oscillatory.

On bounded nonoscillatory solutions of third-order nonlinear differential equations

2009 ◽  
Vol 7 (4) ◽  
Author(s):  
Ivan Mojsej ◽  
Alena Tartaľová
Keyword(s):  
Differential Equations ◽  
Nonlinear Differential Equations ◽  
Sufficient Conditions ◽  
Asymptotic Behavior Of Solutions ◽  
Banach Fixed Point Theorem ◽  
Third Order ◽  
Nonoscillatory Solutions ◽  
Topological Approach ◽  
The Third ◽  
Necessary And Sufficient

AbstractThis paper is concerned with the asymptotic behavior of solutions of nonlinear differential equations of the third-order with quasiderivatives. We give the necessary and sufficient conditions guaranteeing the existence of bounded nonoscillatory solutions. Sufficient conditions are proved via a topological approach based on the Banach fixed point theorem.

scholarly journals On nonoscillatory solutions tending to zero of third-order nonlinear differential equations

2011 ◽  
Vol 48 (1) ◽  
pp. 135-143 ◽  
Author(s):  
Ivan Mojsej ◽  
Alena Tartal’ová
Keyword(s):  
Asymptotic Behavior ◽  
Differential Equations ◽  
Nonlinear Differential Equations ◽  
Sufficient Conditions ◽  
Necessary And Sufficient Conditions ◽  
Asymptotic Behavior Of Solutions ◽  
Third Order ◽  
Nonoscillatory Solutions ◽  
The Third ◽  
Necessary And Sufficient

Abstract The aim of this paper is to present some results concerning with the asymptotic behavior of solutions of nonlinear differential equations of the third-order with quasiderivatives. In particular, we state the necessary and sufficient conditions ensuring the existence of nonoscillatory solutions tending to zero as t → ∞.

On solutions of third order nonlinear differential equations

2006 ◽  
Vol 4 (1) ◽  
pp. 46-63 ◽  
Author(s):  
Ivan Mojsej ◽  
Ján Ohriska
Keyword(s):  
Differential Equations ◽  
Nonlinear Equations ◽  
Nonlinear Differential Equations ◽  
Asymptotic Properties ◽  
Property A ◽  
Comparison Result ◽  
Third Order ◽  
Deviating Arguments ◽  
Deviating Argument ◽  
The Third

AbstractThe aim of our paper is to study oscillatory and asymptotic properties of solutions of nonlinear differential equations of the third order with deviating argument. In particular, we prove a comparison theorem for properties A and B as well as a comparison result on property A between nonlinear equations with and without deviating arguments. Our assumptions on nonlinearity f are related to its behavior only in a neighbourhood of zero and/or of infinity.

On some classes of nonoscillatory solutions of third-order nonlinear differential equations

2014 ◽  
Vol 07 (03) ◽  
pp. 1450039
Author(s):  
Ivan Mojsej ◽  
Alena Tartal'ová
Keyword(s):  
Fixed Point ◽  
Asymptotic Behavior ◽  
Differential Equations ◽  
Nonlinear Differential Equations ◽  
Sufficient Conditions ◽  
Banach Fixed Point Theorem ◽  
Third Order ◽  
Nonoscillatory Solutions ◽  
Topological Approach ◽  
Necessary And Sufficient

In this paper, we are concerned with the asymptotic behavior of certain solutions of third-order nonlinear differential equations with quasiderivatives. More precisely, the necessary and sufficient conditions for the existence of some types of nonoscillatory solutions with a specified asymptotic property as t → ∞ are given. In order to prove some of our results, we use a topological approach based on the Banach fixed point theorem.

scholarly journals Asymptotic Properties of Third-Order Nonlinear Differential Equations

2013 ◽  
Vol 54 (1) ◽  
pp. 19-29
Author(s):  
Blanka Baculíková ◽  
Jozef Džurina
Keyword(s):  
Differential Equation ◽  
Nonlinear Differential Equations ◽  
Asymptotic Properties ◽  
Comparison Theorems ◽  
Third Order ◽  
Nonoscillatory Solutions ◽  
Advanced Case ◽  
The Third ◽  
Functional Differential ◽  
New Criteria

Abstract We present new criteria guaranteeing that all nonoscillatory solutions of the third-order functional differential equation tend to zero. Our results are based on the suitable comparison theorems. We consider both delay and advanced case of studied equation. The results obtained essentially improve and complement earlier ones.

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 On the asymptotic behavior of solutions of certain third-order nonlinear differential equations

2005 ◽  
Vol 2005 (1) ◽  
pp. 29-35 ◽  
Author(s):  
Cemil Tunç
Keyword(s):  
Differential Equation ◽  
Asymptotic Behavior ◽  
Differential Equations ◽  
Nonlinear Differential Equation ◽  
Nonlinear Differential Equations ◽  
Sufficient Conditions ◽  
Asymptotic Behavior Of Solutions ◽  
Third Order ◽  
The Third ◽  
All Solutions

We establish sufficient conditions under which all solutions of the third-order nonlinear differential equation x ⃛+ψ(x,x˙,x¨)x¨+f(x,x˙)=p(t,x,x˙,x¨) are bounded and converge to zero as t→∞.

scholarly journals Existence of nonoscillatory solutions to third order nonlinear neutral difference equations

Filomat ◽  
2018 ◽  
Vol 32 (14) ◽  
pp. 4981-4991
Author(s):  
K.S. Vidhyaa ◽  
C. Dharuman ◽  
John Graef ◽  
E. Thandapani
Keyword(s):  
Fixed Point ◽  
Sufficient Conditions ◽  
Delay Difference Equation ◽  
Neutral Delay ◽  
Third Order ◽  
Nonoscillatory Solutions ◽  
Neutral Difference Equations ◽  
Positive And Negative Coefficients ◽  
The Third ◽  
Delay Difference

The authors consider the third order neutral delay difference equation with positive and negative coefficients ?(an?(bn?(xn + pxn-m)))+pnf(xn-k)- qn1(xn-l) = 0, n ? n0, and give some new sufficient conditions for the existence of nonoscillatory solutions. Banach?s fixed point theorem plays a major role in the proofs. Examples are provided to illustrate their main results.

scholarly journals On Properties of Third-Order Differential Equations via Comparison Principles

2012 ◽  
Vol 2012 ◽  
pp. 1-10
Author(s):  
B. Baculíková ◽  
J. Džurina
Keyword(s):  
Differential Equation ◽  
Differential Equations ◽  
Canonical Form ◽  
Functional Differential Equation ◽  
Asymptotic Properties ◽  
Sufficient Conditions ◽  
Comparison Theorems ◽  
Third Order ◽  
The Third ◽  
Functional Differential

The objective of this paper is to offer sufficient conditions for certain asymptotic properties of the third-order functional differential equation , where studied equation is in a canonical form, that is, . Employing Trench theory of canonical operators, we deduce properties of the studied equations via new comparison theorems. The results obtained essentially improve and complement earlier ones.

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