Periodic solutions of some classes of continuous second-order differential equations

Jaume Llibre; Amar Makhlouf

doi:10.3934/dcdsb.2017022

Periodic solutions of some classes of continuous second-order differential equations

Jaume Llibre, Amar Makhlouf

Department of Mathematics

Research output: Contribution to journal › Article › Research › peer-review

1 Citation (Scopus)

Abstract

We study the periodic solutions of the second-order differential equations of the form ẍ ± xn = μ f (t), or ẍ ± |x|n = μ f(t), where n = 4, 5,⋯, f(t) is a continuous T-periodic function such that ∫ 0T f(t)dt = 0, and μ is a positive small parameter. Note that the differential equations ẍ ± xn = μ f(t) are only continuous in t and smooth in x, and that the differential equations ẍ ± |x|n = μ f(t) are only continuous in t and locally-Lipschitz in x.

Original language	English
Pages (from-to)	477-482
Journal	Discrete and Continuous Dynamical Systems - Series B
Volume	22
Issue number	2
DOIs	https://doi.org/10.3934/dcdsb.2017022
Publication status	Published - 1 Mar 2017

Keywords

Averaging theory
Periodic solution
Second order differential equations

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title = "Periodic solutions of some classes of continuous second-order differential equations",

abstract = "We study the periodic solutions of the second-order differential equations of the form ẍ ± xn = μ f (t), or ẍ ± |x|n = μ f(t), where n = 4, 5,⋯, f(t) is a continuous T-periodic function such that ∫ 0T f(t)dt = 0, and μ is a positive small parameter. Note that the differential equations ẍ ± xn = μ f(t) are only continuous in t and smooth in x, and that the differential equations ẍ ± |x|n = μ f(t) are only continuous in t and locally-Lipschitz in x.",

keywords = "Averaging theory, Periodic solution, Second order differential equations",

author = "Jaume Llibre and Amar Makhlouf",

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T1 - Periodic solutions of some classes of continuous second-order differential equations

AU - Llibre, Jaume

AU - Makhlouf, Amar

PY - 2017/3/1

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N2 - We study the periodic solutions of the second-order differential equations of the form ẍ ± xn = μ f (t), or ẍ ± |x|n = μ f(t), where n = 4, 5,⋯, f(t) is a continuous T-periodic function such that ∫ 0T f(t)dt = 0, and μ is a positive small parameter. Note that the differential equations ẍ ± xn = μ f(t) are only continuous in t and smooth in x, and that the differential equations ẍ ± |x|n = μ f(t) are only continuous in t and locally-Lipschitz in x.

AB - We study the periodic solutions of the second-order differential equations of the form ẍ ± xn = μ f (t), or ẍ ± |x|n = μ f(t), where n = 4, 5,⋯, f(t) is a continuous T-periodic function such that ∫ 0T f(t)dt = 0, and μ is a positive small parameter. Note that the differential equations ẍ ± xn = μ f(t) are only continuous in t and smooth in x, and that the differential equations ẍ ± |x|n = μ f(t) are only continuous in t and locally-Lipschitz in x.

KW - Averaging theory

KW - Periodic solution

KW - Second order differential equations

U2 - 10.3934/dcdsb.2017022

DO - 10.3934/dcdsb.2017022

M3 - Article

VL - 22

SP - 477

EP - 482

JO - Discrete and Continuous Dynamical Systems - Series B

JF - Discrete and Continuous Dynamical Systems - Series B

SN - 1531-3492

IS - 2

ER -