# American Institute of Mathematical Sciences

June  2011, 31(2): 373-383. doi: 10.3934/dcds.2011.31.373

## Periodic solutions of resonant systems with rapidly rotating nonlinearities

 1 Departamento de Matemática, Universidad de Buenos Aires and CONICET, Ciudad Universitaria, Pabellón I, (1428) Buenos Aires 2 Instituto de Matemáticas, Universidad Nacional Autónoma de México, Circuito Exterior, C.U., 04510 México D.F.

Received  June 2010 Revised  October 2010 Published  June 2011

We obtain existence of $T$-periodic solutions to a second order system of ordinary differential equations of the form $u^{\prime\prime}+cu^{\prime}+g(u)=p$ where $c\in\mathbb{R},$ $p\in C(\mathbb{R},\mathbb{R}^{N})$ is $T$-periodic and has mean value zero, and $g\in C(\mathbb{R}^{N},\mathbb{R}^{N})$ is e.g. sublinear. In contrast with a well known result by Nirenberg [6], where it is assumed that the nonlinearity $g$ has non-zero uniform radial limits at infinity, our main result allows rapid rotations in $g$.
Citation: Pablo Amster, Mónica Clapp. Periodic solutions of resonant systems with rapidly rotating nonlinearities. Discrete and Continuous Dynamical Systems, 2011, 31 (2) : 373-383. doi: 10.3934/dcds.2011.31.373
##### References:
 [1] J. M. Alonso, Nonexistence of periodic solutions for a damped pendulum equation, Differential Integral Equations, 10 (1997), 1141-1148. [2] R. Kannan and K. Nagle, Forced oscillations with rapidly vanishing nonlinearities, Proc. Amer. Math. Soc., 111 (1991), 385-393. doi: 10.1090/S0002-9939-1991-1028287-X. [3] R. Kannan and R. Ortega, Periodic solutions of pendulum-type equations, J. Differential Equations, 59 (1985), 123-144. [4] A. Lazer, On Schauder's Fixed point theorem and forced second-order nonlinear oscillations, J. Math. Anal. Appl., 21 (1968), 421-425. doi: 10.1016/0022-247X(68)90225-4. [5] J. Mawhin, An extension of a theorem of A. C. Lazer on forced nonlinear oscillations, J. Math. Anal. Appl., 40 (1972), 20-29. doi: 10.1016/0022-247X(72)90025-X. [6] L. Nirenberg, Generalized degree and nonlinear problems, in "Contributions to Nonlinear Functional Analysis" (E. H. Zarantonello ed.), Academic Press New York, (1971), 1-9. [7] R. Ortega, A counterexample for the damped pendulum equation, Acad. Roy. Belg. Bull. Cl. Sci., 73 (1987), 405-409. [8] R. Ortega and L. Sánchez, Periodic solutions of forced oscillators with several degrees of freedom, Bull. London Math. Soc., 34 (2002), 308-318. doi: 10.1112/S0024609301008748. [9] R. Ortega, E. Serra and M. Tarallo, Non-continuation of the periodic oscillations of a forced pendulum in the presence of friction, Proc. Amer. Math. Soc., 128 (2000), 2659-2665. doi: 10.1090/S0002-9939-00-05389-2. [10] D. Ruiz and J. R. Ward Jr., Some notes on periodic systems with linear part at resonance, Discrete and Continuous Dynamical Systems, 11 (2004), 337-350. doi: 10.3934/dcds.2004.11.337.

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##### References:
 [1] J. M. Alonso, Nonexistence of periodic solutions for a damped pendulum equation, Differential Integral Equations, 10 (1997), 1141-1148. [2] R. Kannan and K. Nagle, Forced oscillations with rapidly vanishing nonlinearities, Proc. Amer. Math. Soc., 111 (1991), 385-393. doi: 10.1090/S0002-9939-1991-1028287-X. [3] R. Kannan and R. Ortega, Periodic solutions of pendulum-type equations, J. Differential Equations, 59 (1985), 123-144. [4] A. Lazer, On Schauder's Fixed point theorem and forced second-order nonlinear oscillations, J. Math. Anal. Appl., 21 (1968), 421-425. doi: 10.1016/0022-247X(68)90225-4. [5] J. Mawhin, An extension of a theorem of A. C. Lazer on forced nonlinear oscillations, J. Math. Anal. Appl., 40 (1972), 20-29. doi: 10.1016/0022-247X(72)90025-X. [6] L. Nirenberg, Generalized degree and nonlinear problems, in "Contributions to Nonlinear Functional Analysis" (E. H. Zarantonello ed.), Academic Press New York, (1971), 1-9. [7] R. Ortega, A counterexample for the damped pendulum equation, Acad. Roy. Belg. Bull. Cl. Sci., 73 (1987), 405-409. [8] R. Ortega and L. Sánchez, Periodic solutions of forced oscillators with several degrees of freedom, Bull. London Math. Soc., 34 (2002), 308-318. doi: 10.1112/S0024609301008748. [9] R. Ortega, E. Serra and M. Tarallo, Non-continuation of the periodic oscillations of a forced pendulum in the presence of friction, Proc. Amer. Math. Soc., 128 (2000), 2659-2665. doi: 10.1090/S0002-9939-00-05389-2. [10] D. Ruiz and J. R. Ward Jr., Some notes on periodic systems with linear part at resonance, Discrete and Continuous Dynamical Systems, 11 (2004), 337-350. doi: 10.3934/dcds.2004.11.337.
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