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.

scholarly journals Properties of Third-Order Nonlinear Functional Differential Equations with Mixed Arguments

2011 ◽  
Vol 2011 ◽  
pp. 1-15 ◽  
Author(s):  
B. Baculíková
Keyword(s):  
Differential Equation ◽  
Functional Differential Equations ◽  
Sufficient Conditions ◽  
Comparison Theorems ◽  
Third Order ◽  
Nonlinear Functional ◽  
The Third ◽  
Property B ◽  
Functional Differential ◽  
Mixed Arguments

The aim of this paper is to offer sufficient conditions for property (B) and/or the oscillation of the third-order nonlinear functional differential equation with mixed arguments . Both cases and are considered. We deduce properties of the studied equations via new comparison theorems. The results obtained essentially improve and complement earlier ones.

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 Properties of Third-Order Delay Trinomial Differential Equations

2011 ◽  
Vol 2011 ◽  
pp. 1-10 ◽  
Author(s):  
J. Džurina ◽  
R. Komariková
Keyword(s):  
Differential Equation ◽  
Asymptotic Behavior ◽  
Differential Equations ◽  
Asymptotic Properties ◽  
Comparison Theorems ◽  
Asymptotic Behavior Of Solutions ◽  
Third Order ◽  
The Third

The aim of this paper is to study properties of the third-order delay trinomial differential equation((1/r(t))y′′(t))′+p(t)y′(t)+q(t)y(σ(t))=0, by transforming this equation onto the second-/third-order binomial differential equation. Using suitable comparison theorems, we establish new results on asymptotic behavior of solutions of the studied equations. Obtained criteria improve and generalize earlier ones.

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.

On Stability of Solutions of Certain Differential Equations of the Third Order

1967 ◽  
Vol 10 (5) ◽  
pp. 681-688 ◽  
Author(s):  
B.S. Lalli
Keyword(s):  
Differential Equation ◽  
Asymptotic Stability ◽  
Lyapunov Function ◽  
Differential Equations ◽  
Global Asymptotic Stability ◽  
Sufficient Conditions ◽  
Stability Of Solutions ◽  
Third Order ◽  
The Third ◽  
Image Position

The purpose of this paper is to obtain a set of sufficient conditions for “global asymptotic stability” of the trivial solution x = 0 of the differential equation1.1using a Lyapunov function which is substantially different from similar functions used in [2], [3] and [4], for similar differential equations. The functions f1, f2 and f3 are real - valued and are smooth enough to ensure the existence of the solutions of (1.1) on [0, ∞). The dot indicates differentiation with respect to t. We are taking a and b to be some positive parameters.

Asymptotic Behavior for Nonoscillatory Solutions of Second Order Nonlinear Functional Differential Equation

2012 ◽  
Vol 616-618 ◽  
pp. 2137-2141
Author(s):  
Zhi Min Luo ◽  
Bei Fei Chen
Keyword(s):  
Differential Equation ◽  
Asymptotic Behavior ◽  
Differential Equations ◽  
Functional Differential Equation ◽  
Functional Differential Equations ◽  
Sufficient Conditions ◽  
Nonoscillatory Solutions ◽  
Nonlinear Functional ◽  
Nonlinear Functional Differential Equation ◽  
Functional Differential

This paper studied the asymptotic behavior of a class of nonlinear functional differential equations by using the Bellman-Bihari inequality. We obtain results which extend and complement those in references. The results indicate that all non-oscillatory continuable solutions of equation are asymptotic to at+b as under some sufficient conditions, where a,b are real constants. An example is provided to illustrate the application of the results.

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 Oscillatory behavior of solutions of certain third order mixed neutral differential equations

2013 ◽  
Vol 44 (1) ◽  
pp. 99-112 ◽  
Author(s):  
Ethiraj Thandapani ◽  
Renu Rama
Keyword(s):  
Differential Equation ◽  
Differential Equations ◽  
Asymptotic Properties ◽  
Sufficient Conditions ◽  
Continuous Functions ◽  
Oscillatory Behavior ◽  
Third Order ◽  
Neutral Differential Equations ◽  
All Solutions ◽  
Positive Integers

The objective of this paper is to study the oscillatory and asymptotic properties of third order mixed neutral differential equation of the form $$ (a(t) [x(t) + b(t) x(t - \tau_{1}) + c(t) x(t + \tau_{2})]'')' + q(t) x^{\alpha}(t - \sigma_{1}) + p(t) x^{\beta}(t + \sigma_{2}) = 0 $$where $a(t), b(t), c(t), q(t)$ and $p(t)$ are positive continuous functions, $\alpha$ and $\beta$ are ratios of odd positive integers, $\tau_{1}, \tau_{2}, \sigma_{1}$ and $\sigma_{2}$ are positive constants. We establish some sufficient conditions which ensure that all solutions are either oscillatory or converge to zero. Some examples are provided to illustrate the main results.

scholarly journals Oscillation properties of nonlinear neutral differential equations ofnth order

2004 ◽  
Vol 2004 (71) ◽  
pp. 3893-3900
Author(s):  
T. Candan ◽  
R. S. Dahiya
Keyword(s):  
Differential Equation ◽  
Differential Equations ◽  
Functional Differential Equation ◽  
Asymptotic Properties ◽  
Neutral Functional Differential Equation ◽  
Neutral Differential Equations ◽  
Functional Differential ◽  
Oscillation Properties

We consider the nonlinear neutral functional differential equation[r(t)[x(t)+∫abp(t,μ)x(τ(t,μ))dμ](n−1)]′+δ∫cdq(t,ξ)f(x(σ(t,ξ)))dξ=0with continuous arguments. We will develop oscillatory and asymptotic properties of the solutions.

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