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

Periodic solutions of resonant systems with rapidly rotating nonlinearities

Pablo Amster 1, and  Mónica Clapp 2,

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 $g$.
Keywords: resonant problems, Nonlinear systems, Leray-Schauder degree., rapidly rotating nonlinearities, periodic solutions.
Mathematics Subject Classification: Primary: 34B15; Secondary: 34C2.
Citation: Pablo Amster, Mónica Clapp. Periodic solutions of resonant systems with rapidly rotating nonlinearities. Discrete & 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.  Google Scholar

[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.  Google Scholar

[3]

R. Kannan and R. Ortega, Periodic solutions of pendulum-type equations, J. Differential Equations, 59 (1985), 123-144.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[7]

R. Ortega, A counterexample for the damped pendulum equation, Acad. Roy. Belg. Bull. Cl. Sci., 73 (1987), 405-409.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

show all references

References:
[1]

J. M. Alonso, Nonexistence of periodic solutions for a damped pendulum equation, Differential Integral Equations, 10 (1997), 1141-1148.  Google Scholar

[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.  Google Scholar

[3]

R. Kannan and R. Ortega, Periodic solutions of pendulum-type equations, J. Differential Equations, 59 (1985), 123-144.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[7]

R. Ortega, A counterexample for the damped pendulum equation, Acad. Roy. Belg. Bull. Cl. Sci., 73 (1987), 405-409.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

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