scholarly journals Unbounded Solutions of Asymmetric Oscillator

2013 ◽  
Vol 2013 ◽  
pp. 1-6
Author(s):  
Tieguo Ji ◽  
Zhenhui Zhang
Keyword(s):  
Differential Equation ◽  
Nonlinear Differential Equation ◽  
Sufficient Conditions ◽  
Unbounded Solutions ◽  
Asymmetric Oscillator

We obtain sufficient conditions for the existence of unbounded solutions of the following nonlinear differential equation(φp(x′))+(p−1)[αφp(x+)−βφp(x−)]=(p−1)f(t,x,x′), whereφp(u)=|u|p−2u,  p>1,  x+=max{x,0},  x−=max{−x,0},α,βare positive constants, andfis continuous, bounded, andT-periodic intfor someT>0.

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 Remarks on Stability Conditions for the Differential Equation x″ + a(t)ƒ(x) = 0

1969 ◽  
Vol 9 (3-4) ◽  
pp. 496-502 ◽  
Author(s):  
James S. W. Wong
Keyword(s):  
Differential Equation ◽  
Continuous Function ◽  
Nonlinear Differential Equation ◽  
Real Axis ◽  
Sufficient Conditions ◽  
Second Order ◽  
Stability Conditions ◽  
Negative Real Axis ◽  
All Solutions ◽  
Image Position

Consider the following second order nonlinear differential equation: where a(t) ∈ C3[0, ∞) and f(x) is a continuous function of x. We are here concerned with establishing sufficient conditions such that all solutions of (1) satisfy (2) Since a(t) is differentiable and f(x) is continuous, it is easy to see that all solutions of (1) are continuable throughout the entire non-negative real axis. It will be assumed throughout that the following conditions hold: Our main results are the following two theorems: Theorem 1. Let 0 < α < 1. If a(t) satisfieswhere a(t) > 0, t ≧ t0 and = max (−a′(t), 0), andthen every solution of (1) satisfies (2).

scholarly journals Bounded andL2-solutions to a second order nonlinear differential equation with a square integrable forcing term

1999 ◽  
Vol 22 (3) ◽  
pp. 569-571 ◽  
Author(s):  
Allan Kroopnick
Keyword(s):  
Differential Equation ◽  
Nonlinear Differential Equation ◽  
Integrable Function ◽  
Sufficient Conditions ◽  
Second Order ◽  
Forcing Term ◽  
Order Nonlinear Differential Equation ◽  
Square Integrable Function

This paper presents two theorems concerning the nonlinear differential equationx″+c(t)f(x)x′+a(t,x)=e(t), wheree(t)is a continuous square-integrable function. The first theorem gives sufficient conditions when all the solutions of this equation are bounded while the second theorem discusses when all the solutions are inL2[0,∞).

Existence and nonexistence of limit cycles for Liénard-type equations with bounded nonlinearities and φ-Laplacian

2017 ◽  
Vol 19 (06) ◽  
pp. 1650057 ◽  
Author(s):  
Kōdai Fujimoto ◽  
Naoto Yamaoka
Keyword(s):  
Differential Equation ◽  
Limit Cycle ◽  
Nonlinear Differential Equation ◽  
Limit Cycles ◽  
Phase Plane ◽  
Sufficient Conditions ◽  
Phase Plane Analysis ◽  
Equivalent System ◽  
Plane Analysis ◽  
Existence And Nonexistence

This paper deals with an equivalent system to the nonlinear differential equation of Liénard type [Formula: see text], where the range of the function [Formula: see text] is bounded. Sufficient conditions are obtained for the system to have at least one limit cycle. The proofs of our results are based on phase plane analysis of the system with the Poincaré–Bendixon theorem. Moreover, to show that these sufficient conditions are suitable in some sense, we also establish the results that the system has no limit cycles. Finally, some examples are given to illustrate our results.

scholarly journals Oscillation of a class of differential equations with generalized phi-Laplacian

2013 ◽  
Vol 143 (3) ◽  
pp. 493-506 ◽  
Author(s):  
Mariella Cecchi ◽  
Zuzana Došlá ◽  
Mauro Marini
Keyword(s):  
Differential Equation ◽  
Differential Equations ◽  
Nonlinear Differential Equation ◽  
Sufficient Conditions ◽  
Necessary And Sufficient Conditions ◽  
Integral Type ◽  
Image Position ◽  
Necessary And Sufficient

The oscillation of the nonlinear differential equationwhere Φ is an increasing odd homeomorphism, is considered when the weight b is not summable near infinity. We extend previous results, stated for equations with the classical p-Laplacian, by obtaining necessary and sufficient conditions of integral type for the oscillation. The role of the boundedness of Im Φ [Dom Φ] is analysed in detail. Our results includes the case Φ* ◦ F linear near zero or near infinity, where Φ* is the inverse of Φ. Several examples, concerning the curvature or relativity operator, illustrate our results.

scholarly journals Existence for Nonoscillatory Solutions of Higher-Order Nonlinear Differential Equations

2011 ◽  
Vol 2011 ◽  
pp. 1-9
Author(s):  
Yazhou Tian ◽  
Fanwei Meng
Keyword(s):  
Differential Equation ◽  
Differential Equations ◽  
Nonlinear Differential Equation ◽  
Nonlinear Differential Equations ◽  
Sufficient Conditions ◽  
Nonoscillatory Solution ◽  
Higher Order ◽  
Nonoscillatory Solutions ◽  
Order Nonlinear Differential Equation

The existence of nonoscillatory solutions of the higher-order nonlinear differential equation [r(t)(x(t)+P(t)x(t-τ))(n-1)]′+∑i=1mQi(t)fi(x(t-σi))=0,  t≥t0, where m≥1,n≥2 are integers, τ>0,  σi≥0,  r,P,Qi∈C([t0,∞),R),  fi∈C(R,R)  (i=1,2,…,m), is studied. Some new sufficient conditions for the existence of a nonoscillatory solution of above equation are obtained for general Qi(t)  (i=1,2,…,m) which means that we allow oscillatory Qi(t)  (i=1,2,…,m). In particular, our results improve essentially and extend some known results in the recent references.

Nonoscillation of Second Order Superlinear Differential Equations

1994 ◽  
Vol 37 (2) ◽  
pp. 178-186
Author(s):  
L. H. Erbe ◽  
H. X. Xia ◽  
J. H. Wu
Keyword(s):  
Differential Equation ◽  
Differential Equations ◽  
Nonlinear Differential Equation ◽  
Sufficient Conditions ◽  
Second Order ◽  
All Solutions ◽  
Image Position ◽  
Positive Integers

AbstractSome sufficient conditions are given for all solutions of the nonlinear differential equation y″(x) +p(x)f(y) = 0 to be nonoscillatory, where p is positive andfor a quotient γ of odd positive integers, γ > 1.

scholarly journals On Asymptotic Behavior of Solutions of Generalized Emden-Fowler Differential Equations with Delay Argument

2014 ◽  
Vol 2014 ◽  
pp. 1-13 ◽  
Author(s):  
Alexander Domoshnitsky ◽  
Roman Koplatadze
Keyword(s):  
Differential Equation ◽  
Asymptotic Behavior ◽  
Differential Equations ◽  
Nonlinear Differential Equation ◽  
Nonlinear Differential Equations ◽  
Sufficient Conditions ◽  
Asymptotic Behavior Of Solutions ◽  
Delay Argument ◽  
Advanced Argument ◽  
Equations With Delay

The following differential equationu(n)(t)+p(t)|u(σ(t))|μ(t) sign  u(σ(t))=0is considered. Herep∈Lloc(R+;R+),  μ∈C(R+;(0,+∞)),  σ∈C(R+;R+),  σ(t)≤t, andlimt→+∞⁡σ(t)=+∞. We say that the equation is almost linear if the conditionlimt→+∞⁡μ(t)=1is fulfilled, while iflim⁡supt→+∞⁡μ(t)≠1orlim⁡inft→+∞⁡μ(t)≠1, then the equation is an essentially nonlinear differential equation. In the case of almost linear and essentially nonlinear differential equations with advanced argument, oscillatory properties have been extensively studied, but there are no results on delay equations of this sort. In this paper, new sufficient conditions implying PropertyAfor delay Emden-Fowler equations are obtained.

scholarly journals Unbounded solutions for differential equations with p-Laplacian and mixed nonlinearities

2017 ◽  
Vol 24 (1) ◽  
pp. 15-28
Author(s):  
Miroslav Bartušek ◽  
Zuzana Došlá ◽  
Mauro Marini
Keyword(s):  
Differential Equation ◽  
Asymptotic Behavior ◽  
Differential Equations ◽  
Nonlinear Differential Equation ◽  
Second Order ◽  
Unbounded Solutions ◽  
Order Nonlinear Differential Equation ◽  
All Solutions ◽  
Mixed Nonlinearities

AbstractThe existence of unbounded solutions with different asymptotic behavior for a second order nonlinear differential equation withp-Laplacian is considered. The oscillation of all solutions is investigated. Some discrepancies and similarities between equations of Emden–Fowler-type and equations with mixed nonlinearities are pointed out.

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