Asymptotics for third-order nonlinear differential equations: Non-oscillatory and oscillatory cases

2021 ◽  
pp. 1-19
Author(s):  
Calogero Vetro ◽  
Dariusz Wardowski
Keyword(s):  
Differential Equation ◽  
Differential Equations ◽  
Order Differential Equation ◽  
Nonlinear Differential Equations ◽  
Oscillatory Behavior ◽  
Third Order ◽  
First Order ◽  
Third Order Differential Equation ◽  
Comparison Technique ◽  
First Order Differential Equations

We discuss a third-order differential equation, involving a general form of nonlinearity. We obtain results describing how suitable coefficient functions determine the asymptotic and (non-)oscillatory behavior of solutions. We use comparison technique with first-order differential equations together with the Kusano–Naito’s and Philos’ approaches.

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 A note on solvability of three-point bound-ary value problems for third-order differential equations with p-Laplacian

2008 ◽  
Vol 39 (1) ◽  
pp. 95-103
Author(s):  
XingYuan Liu ◽  
Yuji Liu
Keyword(s):  
Differential Equation ◽  
Differential Equations ◽  
Boundary Value Problems ◽  
Order Differential Equation ◽  
Boundary Value ◽  
Sufficient Conditions ◽  
Point Boundary ◽  
Third Order ◽  
Third Order Differential Equation

Third-point boundary value problems for third-order differential equation$ \begin{cases} & [q(t)\phi(x''(t))]'+kx'(t)+g(t,x(t),x'(t))=p(t),\;\;t\in (0,1),\\ &x'(0)=x'(1)=x(\eta)=0. \end{cases} $is considered. Sufficient conditions for the existence of at least one solution of above problem are established. Some known results are improved.

scholarly journals Oscillation Criteria for First Order Differential Equations with Non-Monotone Delays

Symmetry ◽  
2020 ◽  
Vol 12 (5) ◽  
pp. 718 ◽  
Author(s):  
Emad R. Attia ◽  
Hassan A. El-Morshedy ◽  
Ioannis P. Stavroulakis
Keyword(s):  
Differential Equation ◽  
Differential Equations ◽  
Order Differential Equation ◽  
Oscillation Criteria ◽  
Lower Upper ◽  
First Order ◽  
Upper Limit ◽  
First Order Differential Equations

New sufficient criteria are obtained for the oscillation of a non-autonomous first order differential equation with non-monotone delays. Both recursive and lower-upper limit types criteria are given. The obtained results improve most recent published results. An example is given to illustrate the applicability and strength of our results.

scholarly journals Fractional differential equation with uncertainty

2021 ◽  
Vol 23 (08) ◽  
pp. 181-185
Author(s):  
Karanveer Singh ◽  
◽  
R N Prajapati ◽  
Keyword(s):  
Differential Equation ◽  
Differential Equations ◽  
Fractional Differential Equation ◽  
Fractional Order ◽  
Order Differential Equation ◽  
Fractional Order Differential Equation ◽  
First Order ◽  
Fractional Differential ◽  
First Order Differential Equations

We consider a fractional order differential equation with uncertainty and introduce the concept of solution. It goes beyond ordinary first-order differential equations and differential equations with uncertainty.

scholarly journals Oscillation Theorems for Nonlinear Differential Equations of Fourth-Order

Mathematics ◽  
2020 ◽  
Vol 8 (4) ◽  
pp. 520 ◽  
Author(s):  
Osama Moaaz ◽  
Ioannis Dassios ◽  
Omar Bazighifan ◽  
Ali Muhib
Keyword(s):  
Differential Equation ◽  
Differential Equations ◽  
Fourth Order ◽  
Order Differential Equation ◽  
Nonlinear Differential Equations ◽  
Sufficient Conditions ◽  
Oscillatory Behavior ◽  
Middle Term ◽  
Oscillation Theorems ◽  
Fourth Order Differential Equation

We study the oscillatory behavior of a class of fourth-order differential equations and establish sufficient conditions for oscillation of a fourth-order differential equation with middle term. Our theorems extend and complement a number of related results reported in the literature. One example is provided to illustrate the main results.

scholarly journals Existence of Periodic Solutions for Nonlinear Fully Third-Order Differential Equations

2020 ◽  
Vol 2020 ◽  
pp. 1-7
Author(s):  
Yunfei Zhang ◽  
Minghe Pei
Keyword(s):  
Differential Equation ◽  
Differential Equations ◽  
Periodic Solutions ◽  
Order Differential Equation ◽  
Existence Results ◽  
Third Order ◽  
Third Order Differential Equation ◽  
Existence Of Periodic Solutions ◽  
Growth Restrictions

In this paper, we study the existence of periodic solutions to nonlinear fully third-order differential equation x‴+ft,x,x′,x″=0,t∈ℝ≔−∞,∞, where f:ℝ4⟶ℝ is continuous and T-periodic in t. By using the topological transversality method together with the barrier strip technique, we obtain new existence results of periodic solutions to the above equation without growth restrictions on the nonlinearity. Meanwhile, as applications, an example is given to demonstrate our results.

Boundary-Value Problems for Third-Order Lipschitz Ordinary Differential Equations

2014 ◽  
Vol 58 (1) ◽  
pp. 183-197 ◽  
Author(s):  
John R. Graef ◽  
Johnny Henderson ◽  
Rodrica Luca ◽  
Yu Tian
Keyword(s):  
Differential Equation ◽  
Differential Equations ◽  
Ordinary Differential Equations ◽  
Boundary Value Problems ◽  
Order Differential Equation ◽  
Boundary Value ◽  
Point Boundary ◽  
Third Order ◽  
Optimal Length ◽  
The Third

AbstractFor the third-order differential equationy′″ = ƒ(t, y, y′, y″), where, questions involving ‘uniqueness implies uniqueness’, ‘uniqueness implies existence’ and ‘optimal length subintervals of (a, b) on which solutions are unique’ are studied for a class of two-point boundary-value problems.

scholarly journals Asymptotic Dichotomy in a Class of Third-Order Nonlinear Differential Equations with Impulses

2010 ◽  
Vol 2010 ◽  
pp. 1-20 ◽  
Author(s):  
Kun-Wen Wen ◽  
Gen-Qiang Wang ◽  
Sui Sun Cheng
Keyword(s):  
Differential Equation ◽  
Differential Equations ◽  
Nonlinear Differential Equation ◽  
Nonlinear Differential Equations ◽  
Functional Differential Equations ◽  
Higher Order ◽  
Third Order ◽  
Order Nonlinear Differential Equation ◽  
Functional Differential

Solutions of quite a few higher-order delay functional differential equations oscillate or converge to zero. In this paper, we obtain several such dichotomous criteria for a class of third-order nonlinear differential equation with impulses.

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