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

Jaume Llibre; Amar Makhlouf

doi:10.3934/dcdsb.2017022

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2017, Volume 22, Issue 2: 477-482. Doi: 10.3934/dcdsb.2017022

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

Jaume Llibre^1, and
Amar Makhlouf^2,

1.
Departament de Matemátiques, Universitat Autónoma de Barcelona, 08193 Bellaterra, Barcelona, Catalonia, Spain
2.
Department of Mathematics, Laboratory LMA, University of Annaba, Elhadjar, 23 Annaba, Algeria

Received: February 29, 2016

Revised: May 31, 2016

Published: December 2017

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Abstract

We study the periodic solutions of the second-order differential equations of the form $ \ddot x ± x^{n} = μ f(t), $ or $ \ddot x ± |x|^{n} = μ f(t), $ where $n=4,5,...$ , $f(t)$ is a continuous $T$ -periodic function such that $\int_0^T {f\left( t \right)} dt\ne 0$ , and $μ$ is a positive small parameter. Note that the differential equations $ \ddot x ± x^{n} = μ f(t)$ are only continuous in $t$ and smooth in $x$ , and that the differential equations $ \ddot x ± |x|^{n} = μ f(t)$ are only continuous in $t$ and locally-Lipschitz in $x$ .

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Mathematics Subject Classification: Primary:37G15, 37C80, 37C30.

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References

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