# American Institute of Mathematical Sciences

March  2017, 22(2): 477-482. doi: 10.3934/dcdsb.2017022

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

 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

We thank to Professor Rafael Ortega the information about the second--order differential equation $\ddot x + x^3= f(t)$, and to the reviewer his comments which help us to improve the presentation of this paper. The first author is partially supported by a MINECO grant MTM2013-40998-P, an AGAUR grant number 2014 SGR568, and the grants FP7-PEOPLE-2012-IRSES 318999 and 316338

Received  March 2016 Revised  June 2016 Published  December 2016

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 $x$.

Citation: Jaume Llibre, Amar Makhlouf. Periodic solutions of some classes of continuous second-order differential equations. Discrete and Continuous Dynamical Systems - B, 2017, 22 (2) : 477-482. doi: 10.3934/dcdsb.2017022
##### References:
 [1] A. Buică and J. Llibre, Averaging methods for finding periodic orbits via Brouwer degree, Bull. Sci. Math., 128 (2004), 7-22.  doi: 10.1016/j.bulsci.2003.09.002. [2] A. Buică, J. Llibre and O. Yu. Makarenkov, On Yu.A.Mitropol'skii's Theorem on periodic solutions of systems of nonlinear differential equations with nondifferentiable right-hand sides, Doklady Math., 78 (2008), 525-527.  doi: 10.1134/S1064562408040157. [3] A. Buică and R. Ortega, Persistence of equilibria as periodic solutions of forced systems, J. Differential Equations, 252 (2012), 2210-2221.  doi: 10.1016/j.jde.2011.06.006. [4] A. Capietto, J. Mawhin and F. Zanolin, Continuation theorems for periodic perturbations of autonomous systems, Trans. Amer. Math. Soc., 329 (1992), 41-72.  doi: 10.2307/2154076. [5] T. Carvalho, R. D. Euzébio, J. Llibre and D. J. Tonon, Detecting periodic orbits in some 3D chaotic quadratic polynomial differential systems, Discrete and Continuous Dynamical Systems-Series B, 21 (2016), 1-11.  doi: 10.3934/dcdsb.2016.21.1. [6] T. R. Ding and F. Zanolin, Periodic solutions of Duffing's equations with superquadratic potential, J. Differential Equations, 97 (1992), 328-378.  doi: 10.1016/0022-0396(92)90076-Y. [7] Y. A. Kuznetzov, Elements of Applied, Bifurcation Theory, Third edition. Applied Mathematical Sciences, 112. Springer-Verlag, New York, 2004. [8] N. G. Lloyd, Degree Theory Cambridge University Press, 1978. [9] G. R. Morris, An infinite class of periodic solutions of $\ddot x +2x^3 =p(t)$, Proc. Cambridge Philos. Soc., 61 (1965), 157-164. [10] R. Ortega, The number of stable periodic solutions of time-dependent Hamiltonian systems with one degree of freedom, Ergodic Theory Dynam. Systems, 18 (1998), 1007-1018.  doi: 10.1017/S0143385798108362.

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We thank to Professor Rafael Ortega the information about the second--order differential equation $\ddot x + x^3= f(t)$, and to the reviewer his comments which help us to improve the presentation of this paper. The first author is partially supported by a MINECO grant MTM2013-40998-P, an AGAUR grant number 2014 SGR568, and the grants FP7-PEOPLE-2012-IRSES 318999 and 316338

##### References:
 [1] A. Buică and J. Llibre, Averaging methods for finding periodic orbits via Brouwer degree, Bull. Sci. Math., 128 (2004), 7-22.  doi: 10.1016/j.bulsci.2003.09.002. [2] A. Buică, J. Llibre and O. Yu. Makarenkov, On Yu.A.Mitropol'skii's Theorem on periodic solutions of systems of nonlinear differential equations with nondifferentiable right-hand sides, Doklady Math., 78 (2008), 525-527.  doi: 10.1134/S1064562408040157. [3] A. Buică and R. Ortega, Persistence of equilibria as periodic solutions of forced systems, J. Differential Equations, 252 (2012), 2210-2221.  doi: 10.1016/j.jde.2011.06.006. [4] A. Capietto, J. Mawhin and F. Zanolin, Continuation theorems for periodic perturbations of autonomous systems, Trans. Amer. Math. Soc., 329 (1992), 41-72.  doi: 10.2307/2154076. [5] T. Carvalho, R. D. Euzébio, J. Llibre and D. J. Tonon, Detecting periodic orbits in some 3D chaotic quadratic polynomial differential systems, Discrete and Continuous Dynamical Systems-Series B, 21 (2016), 1-11.  doi: 10.3934/dcdsb.2016.21.1. [6] T. R. Ding and F. Zanolin, Periodic solutions of Duffing's equations with superquadratic potential, J. Differential Equations, 97 (1992), 328-378.  doi: 10.1016/0022-0396(92)90076-Y. [7] Y. A. Kuznetzov, Elements of Applied, Bifurcation Theory, Third edition. Applied Mathematical Sciences, 112. Springer-Verlag, New York, 2004. [8] N. G. Lloyd, Degree Theory Cambridge University Press, 1978. [9] G. R. Morris, An infinite class of periodic solutions of $\ddot x +2x^3 =p(t)$, Proc. Cambridge Philos. Soc., 61 (1965), 157-164. [10] R. Ortega, The number of stable periodic solutions of time-dependent Hamiltonian systems with one degree of freedom, Ergodic Theory Dynam. Systems, 18 (1998), 1007-1018.  doi: 10.1017/S0143385798108362.
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