scholarly journals On the existence and multiplicity of positive periodic solutions of a nonlinear third-order equation

2009 ◽  
Vol 22 (8) ◽  
pp. 1220-1224 ◽  
Author(s):  
Yuqiang Feng
Keyword(s):  
Periodic Solutions ◽  
Order Equation ◽  
Positive Periodic Solutions ◽  
Third Order ◽  
Existence And Multiplicity

scholarly journals Positive Periodic Solutions of Third-Order Ordinary Differential Equations with Delays

2014 ◽  
Vol 2014 ◽  
pp. 1-8 ◽  
Author(s):  
Yongxiang Li ◽  
Qiang Li
Keyword(s):  
Periodic Solutions ◽  
Fixed Point Index ◽  
Index Theory ◽  
Existence Results ◽  
Positive Periodic Solutions ◽  
Point Index ◽  
Third Order ◽  
Order Ordinary Differential Equation ◽  
The Third ◽  
Fixed Point Index Theory

The existence results of positiveω-periodic solutions are obtained for the third-order ordinary differential equation with delaysu′′′(t)+a(t)u(t)=f(t,u(t-τ0),u′(t-τ1),u′′(t-τ2)),t∈ℝ,wherea∈C(ℝ,(0,∞))isω-periodic function andf:ℝ×[0,∞)×ℝ2→[0,∞)is a continuous function which isω-periodic int,and τ0,τ1,τ2are positive constants. The discussion is based on the fixed-point index theory in cones.

scholarly journals Existence and Lyapunov Stability of Positive Periodic Solutions for a Third-Order Neutral Differential Equation

2011 ◽  
Vol 2011 ◽  
pp. 1-28 ◽  
Author(s):  
Jingli Ren ◽  
Zhibo Cheng ◽  
Yueli Chen
Keyword(s):  
Differential Equation ◽  
Fixed Point ◽  
Fixed Point Theorem ◽  
Periodic Solutions ◽  
Lyapunov Stability ◽  
Order Differential Equation ◽  
Sufficient Conditions ◽  
Neutral Differential Equation ◽  
Positive Periodic Solutions ◽  
Third Order

By applying Green's function of third-order differential equation and a fixed point theorem in cones, we obtain some sufficient conditions for existence, nonexistence, multiplicity, and Lyapunov stability of positive periodic solutions for a third-order neutral differential equation.

Positive periodic solutions to the forced non-autonomous Duffing equations

2021 ◽  
Vol 0 (0) ◽  
Author(s):  
Jiří Šremr
Keyword(s):  
Periodic Solutions ◽  
Positive Solutions ◽  
Periodic Problem ◽  
Duffing Equation ◽  
Positive Periodic Solutions ◽  
Duffing Equations ◽  
Existence And Multiplicity ◽  
Constant Coefficients ◽  
Multiplicity Of Positive Solutions

Abstract We study the existence and multiplicity of positive solutions to the periodic problem for a forced non-autonomous Duffing equation u ′′ = p ⁢ ( t ) ⁢ u - h ⁢ ( t ) ⁢ | u | λ ⁢ sgn ⁡ u + f ⁢ ( t ) ; u ⁢ ( 0 ) = u ⁢ ( ω ) , u ′ ⁢ ( 0 ) = u ′ ⁢ ( ω ) , u^{\prime\prime}=p(t)u-h(t)\lvert u\rvert^{\lambda}\operatorname{sgn}u+f(t);\quad u(0)=u(\omega),\ u^{\prime}(0)=u^{\prime}(\omega), where p , h , f ∈ L ⁢ ( [ 0 , ω ] ) p,h,f\in L([0,\omega]) and λ > 1 \lambda>1 . The obtained results are compared with the results known for the equations with constant coefficients.

scholarly journals Existence of periodic solutions and homoclinic orbits for third-order nonlinear differential equations

2003 ◽  
Vol 2003 (4) ◽  
pp. 209-228 ◽  
Author(s):  
O. Rabiei Motlagh ◽  
Z. Afsharnezhad
Keyword(s):  
Differential Equation ◽  
Periodic Solutions ◽  
Order Differential Equation ◽  
Nonlinear Differential Equations ◽  
Homoclinic Orbits ◽  
Order Equation ◽  
Second Order ◽  
Third Order ◽  
Bifurcation Theorem ◽  
Existence Of Periodic Solutions

The existence of periodic solutions for the third-order differential equationx¨˙+ω2x˙=μF(x,x˙,x¨)is studied. We give some conditions for this equation in order to reduce it to a second-order nonlinear differential equation. We show that the existence of periodic solutions for the second-order equation implies the existence of periodic solutions for the above equation. Then we use the Hopf bifurcation theorem for the second-order equation and obtain many periodic solutions for it. Also we show that the above equation has many homoclinic solutions ifF(x,x˙,x¨)has a quadratic form. Finally, we compare our result to that of Mehri and Niksirat (2001).

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