# 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 & 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. Google Scholar [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. Google Scholar [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. Google Scholar [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. Google Scholar [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. Google Scholar [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. Google Scholar [7] Y. A. Kuznetzov, Elements of Applied, Bifurcation Theory, Third edition. Applied Mathematical Sciences, 112. Springer-Verlag, New York, 2004.Google Scholar [8] N. G. Lloyd, Degree Theory Cambridge University Press, 1978.Google Scholar [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. Google Scholar [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. Google Scholar

show all references

##### 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. Google Scholar [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. Google Scholar [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. Google Scholar [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. Google Scholar [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. Google Scholar [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. Google Scholar [7] Y. A. Kuznetzov, Elements of Applied, Bifurcation Theory, Third edition. Applied Mathematical Sciences, 112. Springer-Verlag, New York, 2004.Google Scholar [8] N. G. Lloyd, Degree Theory Cambridge University Press, 1978.Google Scholar [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. Google Scholar [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. Google Scholar
 [1] Paola Buttazzoni, Alessandro Fonda. Periodic perturbations of scalar second order differential equations. Discrete & Continuous Dynamical Systems - A, 1997, 3 (3) : 451-455. doi: 10.3934/dcds.1997.3.451 [2] Jaume Llibre, Amar Makhlouf, Sabrina Badi. $3$ - dimensional Hopf bifurcation via averaging theory of second order. Discrete & Continuous Dynamical Systems - A, 2009, 25 (4) : 1287-1295. doi: 10.3934/dcds.2009.25.1287 [3] Zhiming Guo, Xiaomin Zhang. Multiplicity results for periodic solutions to a class of second order delay differential equations. Communications on Pure & Applied Analysis, 2010, 9 (6) : 1529-1542. doi: 10.3934/cpaa.2010.9.1529 [4] Zaihong Wang. Periodic solutions of the second order differential equations with asymmetric nonlinearities depending on the derivatives. Discrete & Continuous Dynamical Systems - A, 2003, 9 (3) : 751-770. doi: 10.3934/dcds.2003.9.751 [5] Wenying Feng, Guang Zhang, Yikang Chai. Existence of positive solutions for second order differential equations arising from chemical reactor theory. Conference Publications, 2007, 2007 (Special) : 373-381. doi: 10.3934/proc.2007.2007.373 [6] Xiaoming Wang. Quasi-periodic solutions for a class of second order differential equations with a nonlinear damping term. Discrete & Continuous Dynamical Systems - S, 2017, 10 (3) : 543-556. doi: 10.3934/dcdss.2017027 [7] Kunquan Lan. Eigenvalues of second order differential equations with singularities. Conference Publications, 2001, 2001 (Special) : 241-247. doi: 10.3934/proc.2001.2001.241 [8] Juan Campos, Rafael Obaya, Massimo Tarallo. Favard theory and fredholm alternative for disconjugate recurrent second order equations. Communications on Pure & Applied Analysis, 2017, 16 (4) : 1199-1232. doi: 10.3934/cpaa.2017059 [9] Xuelei Wang, Dingbian Qian, Xiying Sun. Periodic solutions of second order equations with asymptotical non-resonance. Discrete & Continuous Dynamical Systems - A, 2018, 38 (9) : 4715-4726. doi: 10.3934/dcds.2018207 [10] C. Rebelo. Multiple periodic solutions of second order equations with asymmetric nonlinearities. Discrete & Continuous Dynamical Systems - A, 1997, 3 (1) : 25-34. doi: 10.3934/dcds.1997.3.25 [11] W. Sarlet, G. E. Prince, M. Crampin. Generalized submersiveness of second-order ordinary differential equations. Journal of Geometric Mechanics, 2009, 1 (2) : 209-221. doi: 10.3934/jgm.2009.1.209 [12] Anna Capietto, Walter Dambrosio. A topological degree approach to sublinear systems of second order differential equations. Discrete & Continuous Dynamical Systems - A, 2000, 6 (4) : 861-874. doi: 10.3934/dcds.2000.6.861 [13] Yuan Guo, Xiaofei Gao, Desheng Li. Structure of the set of bounded solutions for a class of nonautonomous second order differential equations. Communications on Pure & Applied Analysis, 2010, 9 (6) : 1607-1616. doi: 10.3934/cpaa.2010.9.1607 [14] Saroj Panigrahi, Rakhee Basu. Oscillation results for second order nonlinear neutral differential equations with delay. Conference Publications, 2015, 2015 (special) : 906-912. doi: 10.3934/proc.2015.0906 [15] José F. Cariñena, Irina Gheorghiu, Eduardo Martínez. Jacobi fields for second-order differential equations on Lie algebroids. Conference Publications, 2015, 2015 (special) : 213-222. doi: 10.3934/proc.2015.0213 [16] Gafurjan Ibragimov, Askar Rakhmanov, Idham Arif Alias, Mai Zurwatul Ahlam Mohd Jaffar. The soft landing problem for an infinite system of second order differential equations. Numerical Algebra, Control & Optimization, 2017, 7 (1) : 89-94. doi: 10.3934/naco.2017005 [17] Alessandro Fonda, Fabio Zanolin. Bounded solutions of nonlinear second order ordinary differential equations. Discrete & Continuous Dynamical Systems - A, 1998, 4 (1) : 91-98. doi: 10.3934/dcds.1998.4.91 [18] Abdelkader Boucherif. Positive Solutions of second order differential equations with integral boundary conditions. Conference Publications, 2007, 2007 (Special) : 155-159. doi: 10.3934/proc.2007.2007.155 [19] Qiong Meng, X. H. Tang. Multiple solutions of second-order ordinary differential equation via Morse theory. Communications on Pure & Applied Analysis, 2012, 11 (3) : 945-958. doi: 10.3934/cpaa.2012.11.945 [20] Kyeong-Hun Kim, Kijung Lee. A weighted $L_p$-theory for second-order parabolic and elliptic partial differential systems on a half space. Communications on Pure & Applied Analysis, 2016, 15 (3) : 761-794. doi: 10.3934/cpaa.2016.15.761

2018 Impact Factor: 1.008