scholarly journals Periodic solutions for periodic second-order differential equations with variable potentials

2018 ◽  
Vol 24 (2) ◽  
pp. 127-137
Author(s):  
Jaume Llibre ◽  
Ammar Makhlouf
Keyword(s):  
Differential Equation ◽  
Differential Equations ◽  
Periodic Solutions ◽  
Order Differential Equation ◽  
Sufficient Conditions ◽  
Second Order ◽  
Second Order Differential Equations ◽  
Second Order Differential Equation ◽  
Existence Of Periodic Solutions

Abstract We provide sufficient conditions for the existence of periodic solutions of the second-order differential equation with variable potentials {-(px^{\prime})^{\prime}(t)-r(t)p(t)x^{\prime}(t)+q(t)x(t)=f(t,x(t))} , where the functions {p(t)>0} , {q(t)} , {r(t)} and {f(t,x)} are {\mathcal{C}^{2}} and T-periodic in the variable t.

scholarly journals Periodic Solutions of Duffing Equation with an Asymmetric Nonlinearity and a Deviating Argument

2013 ◽  
Vol 2013 ◽  
pp. 1-8
Author(s):  
Zaihong Wang ◽  
Jin Li ◽  
Tiantian Ma
Keyword(s):  
Differential Equation ◽  
Periodic Solutions ◽  
Order Differential Equation ◽  
Sufficient Conditions ◽  
Second Order ◽  
Duffing Equation ◽  
Deviating Argument ◽  
Second Order Differential Equation ◽  
Existence Of Periodic Solutions ◽  
The Given

We study the existence of periodic solutions of the second-order differential equationx′′+ax+-bx-+g(x(t-τ))=p(t), wherea,bare two constants satisfying1/a+1/b=2/n,n∈N,τis a constant satisfying0≤τ<2π,g,p:R→Rare continuous, andpis2π-periodic. When the limitslimx→±∞g(x)=g(±∞)exist and are finite, we give some sufficient conditions for the existence of2π-periodic solutions of the given equation.

scholarly journals Variational Approach to Impulsive Differential Equations with Singular Nonlinearities

2013 ◽  
Vol 2013 ◽  
pp. 1-7 ◽  
Author(s):  
Naima Daoudi-Merzagui ◽  
Abdelkader Boucherif
Keyword(s):  
Differential Equations ◽  
Variational Method ◽  
Periodic Solutions ◽  
Variational Approach ◽  
Sufficient Conditions ◽  
Impulsive Differential Equations ◽  
Second Order ◽  
Second Order Differential Equations ◽  
Singular Nonlinearities ◽  
Existence Of Periodic Solutions

We discuss the existence of periodic solutions for nonautonomous second order differential equations with singular nonlinearities. Simple sufficient conditions that enable us to obtain many distinct periodic solutions are provided. Our approach is based on a variational method.

scholarly journals On the limit cycles for a class of eighth-order differential equations

2020 ◽  
Vol 6 (1) ◽  
pp. 53-61
Author(s):  
Chems Eddine Berrehail ◽  
Zineb Bouslah ◽  
Amar Makhlouf
Keyword(s):  
Differential Equation ◽  
Differential Equations ◽  
Periodic Function ◽  
Periodic Solutions ◽  
Limit Cycles ◽  
Order Differential Equation ◽  
Sufficient Conditions ◽  
Rational Numbers ◽  
Eighth Order ◽  
Existence Of Periodic Solutions

AbstractIn this article, we provide sufficient conditions for the existence of periodic solutions of the eighth-order differential equation {x^{\left( 8 \right)}} - \left( {1 + {p^2} + {\lambda ^2} + {\mu ^2}} \right){x^{\left( 6 \right)}} + A\ddddot x + B\ddot x + {p^2}{\lambda ^2}{\mu ^2}x = \varepsilon F\left( {t,x,\dot x,\ddot x,\dddot x,\ddddot x,{x^{\left( 5 \right)}},{x^{\left( 6 \right)}}{x^{\left( 7 \right)}}} \right), where A = p2λ2 + p2µ2 + λ2µ2 + p2 + λ2 + µ2, B = p2 λ2 + p2µ2 + λ2µ2 + p2λ2µ2, with λ, µ and p are rational numbers different from −1, 0, 1, and p ≠ ±λ, p ≠±µ, λ ≠±µ, ɛ is sufficiently small and F is a nonlinear non-autonomous periodic function. Moreover we provide some applications.

On the integrability of solutions of perturbed non-linear differential equations

1977 ◽  
Vol 77 (3-4) ◽  
pp. 309-318 ◽  
Author(s):  
Paul W. Spikes
Keyword(s):  
Differential Equation ◽  
Differential Equations ◽  
Order Differential Equation ◽  
Sufficient Conditions ◽  
Second Order ◽  
Linear Differential Equations ◽  
Second Order Differential Equation ◽  
All Solutions ◽  
Non Linear ◽  
Linear Differential

SynopsisSufficient conditions are given to insure that all solutions of a perturbed non-linear second-order differential equation have certain integrability properties. In addition, some continuability and boundedness results are given for solutions of this equation.

scholarly journals Linear difference operator with multiple variable parameters and applications to second-order differential equations

2020 ◽  
Vol 2020 (1) ◽  
Author(s):  
Feifan Li ◽  
Zhonghua Bi ◽  
Shaowen Yao ◽  
Yun Xin
Keyword(s):  
Differential Equation ◽  
Differential Equations ◽  
Periodic Solutions ◽  
Difference Operator ◽  
Sufficient Conditions ◽  
Second Order ◽  
Krasnoselskii’S Fixed Point Theorem ◽  
Linear Difference Operator ◽  
Existence Of Periodic Solutions ◽  
Functional Differential

AbstractIn this article, we first investigate the linear difference operator $(Ax)(t):=x(t)-\sum_{i=1}^{n}c_{i}(t)x(t- \delta _{i}(t))$(Ax)(t):=x(t)−∑i=1nci(t)x(t−δi(t)) in a continuous periodic function space. The existence condition and some properties of the inverse of the operator A are explicitly pointed out. Afterwards, as applications of properties of the operator A, we study the existence of periodic solutions for two kinds of second-order functional differential equations with this operator. One is a kind of second-order functional differential equation, by applications of Krasnoselskii’s fixed point theorem, some sufficient conditions for the existence of positive periodic solutions are established. Another one is a kind of second-order quasi-linear differential equation, we establish the existence of periodic solutions of this equation by an extension of Mawhin’s continuous theorem.

Periodic solutions of nonautonomous second-order differential equations with a p-Laplacian

Analysis ◽  
2017 ◽  
Vol 37 (1) ◽  
pp. 1-11
Author(s):  
Hairong Lian ◽  
Dongli Wang ◽  
Donal O’Regan ◽  
Ravi P. Agarwal
Keyword(s):  
Differential Equation ◽  
Differential Equations ◽  
Boundary Value Problem ◽  
Periodic Solutions ◽  
Periodic Boundary ◽  
Order Differential Equation ◽  
Boundary Value ◽  
Second Order ◽  
Periodic Boundary Value Problem ◽  
Second Order Differential Equations

AbstractIn this paper, we study a periodic boundary value problem for a nonautonomous second-order differential equation with a

THREE POSITIVE PERIODIC SOLUTIONS OF NONLINEAR DIFFERENTIAL EQUATIONS WITH PERIODIC COEFFICIENTS

2005 ◽  
Vol 03 (02) ◽  
pp. 145-155 ◽  
Author(s):  
YUJI LIU ◽  
WEIGUO GE ◽  
ZHANJI GUI
Keyword(s):  
Differential Equation ◽  
Differential Equations ◽  
Periodic Solutions ◽  
Order Differential Equation ◽  
Nonlinear Differential Equations ◽  
Second Order ◽  
Positive Periodic Solutions ◽  
Periodic Coefficients ◽  
Second Order Differential Equation

We establish the existence of at least three positive periodic solutions to the second order differential equation with periodic coefficients [Formula: see text] where f is continuous with f(t + T, x) = f(t,x) for (t,x) ∊ R × R and T > 0, p, q are continuous and T-periodic with p > 0 and q ≥ 0. We accomplish this by making growth assumptions on f, which can apply to many more cases than those discussed in recent works. An example to illustrate the main result is given.

Periodic Solutions for a Kind of Second Order Differential Equation with Multiple Delays

2011 ◽  
Vol 50-51 ◽  
pp. 27-31
Author(s):  
Mei Lan Tang ◽  
Xin Ge Liu
Keyword(s):  
Differential Equation ◽  
Periodic Solutions ◽  
Order Differential Equation ◽  
Sufficient Conditions ◽  
Second Order ◽  
Multiple Delays ◽  
Mawhin’S Continuation Theorem ◽  
Deviating Arguments ◽  
Second Order Differential Equation ◽  
Multiple Deviating Arguments

Based on Mawhin's continuation theorem and some analysis skill, some sufficient conditions for the existence of periodic solutions for a kind of second order differential equation with multiple deviating arguments are obtained. Results obtained in this paper extend the existing results.

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