Periodic solutions of resonant systems with rapidly rotating nonlinearities

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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

Abstract
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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 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.

show all references

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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