American Institute of Mathematical Sciences

Advanced
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

[1]

Abdelaaziz Sbai, Youssef El Hadfi, Mohammed Srati, Noureddine Aboutabit. Existence of solution for Kirchhoff type problem in Orlicz-Sobolev spaces Via Leray-Schauder's nonlinear alternative. Discrete & Continuous Dynamical Systems - S, 2021  doi: 10.3934/dcdss.2021015
[2]

Alexander Krasnosel'skii. Resonant forced oscillations in systems with periodic nonlinearities. Discrete & Continuous Dynamical Systems, 2013, 33 (1) : 239-254. doi: 10.3934/dcds.2013.33.239
[3]

Diego Averna, Nikolaos S. Papageorgiou, Elisabetta Tornatore. Multiple solutions for nonlinear nonhomogeneous resonant coercive problems. Discrete & Continuous Dynamical Systems - S, 2018, 11 (2) : 155-178. doi: 10.3934/dcdss.2018010
[4]

Paul Deuring, Stanislav Kračmar, Šárka Nečasová. Linearized stationary incompressible flow around rotating and translating bodies -- Leray solutions. Discrete & Continuous Dynamical Systems - S, 2014, 7 (5) : 967-979. doi: 10.3934/dcdss.2014.7.967
[5]

Xiaojun Chang, Yong Li. Rotating periodic solutions of second order dissipative dynamical systems. Discrete & Continuous Dynamical Systems, 2016, 36 (2) : 643-652. doi: 10.3934/dcds.2016.36.643
[6]

Nikolaos S. Papageorgiou, Vicenţiu D. Rădulescu. Bifurcation of positive solutions for nonlinear nonhomogeneous Robin and Neumann problems with competing nonlinearities. Discrete & Continuous Dynamical Systems, 2015, 35 (10) : 5003-5036. doi: 10.3934/dcds.2015.35.5003
[7]

D. Wirosoetisno. Navier--Stokes equations on a rapidly rotating sphere. Discrete & Continuous Dynamical Systems - B, 2015, 20 (4) : 1251-1259. doi: 10.3934/dcdsb.2015.20.1251
[8]

Jean Mawhin. Multiplicity of solutions of variational systems involving $\phi$-Laplacians with singular $\phi$ and periodic nonlinearities. Discrete & Continuous Dynamical Systems, 2012, 32 (11) : 4015-4026. doi: 10.3934/dcds.2012.32.4015
[9]

Adriana Buică, Jean–Pierre Françoise, Jaume Llibre. Periodic solutions of nonlinear periodic differential systems with a small parameter. Communications on Pure & Applied Analysis, 2007, 6 (1) : 103-111. doi: 10.3934/cpaa.2007.6.103
[10]

D. Motreanu, V. V. Motreanu, Nikolaos S. Papageorgiou. Nonautonomous resonant periodic systems with indefinite linear part and a nonsmooth potential. Communications on Pure & Applied Analysis, 2011, 10 (5) : 1401-1414. doi: 10.3934/cpaa.2011.10.1401
[11]

Zalman Balanov, Meymanat Farzamirad, Wieslaw Krawcewicz, Haibo Ruan. Applied equivariant degree. part II: Symmetric Hopf bifurcations of functional differential equations. Discrete & Continuous Dynamical Systems, 2006, 16 (4) : 923-960. doi: 10.3934/dcds.2006.16.923
[12]

Zhenguo Liang, Jiansheng Geng. Quasi-periodic solutions for 1D resonant beam equation. Communications on Pure & Applied Analysis, 2006, 5 (4) : 839-853. doi: 10.3934/cpaa.2006.5.839
[13]

Tiantian Ma, Zaihong Wang. Periodic solutions of Liénard equations with resonant isochronous potentials. Discrete & Continuous Dynamical Systems, 2013, 33 (4) : 1563-1581. doi: 10.3934/dcds.2013.33.1563
[14]

Boumediene Abdellaoui, Daniela Giachetti, Ireneo Peral, Magdalena Walias. Elliptic problems with nonlinear terms depending on the gradient and singular on the boundary: Interaction with a Hardy-Leray potential. Discrete & Continuous Dynamical Systems, 2014, 34 (5) : 1747-1774. doi: 10.3934/dcds.2014.34.1747
[15]

E. N. Dancer. Some bifurcation results for rapidly growing nonlinearities. Discrete & Continuous Dynamical Systems, 2013, 33 (1) : 153-161. doi: 10.3934/dcds.2013.33.153
[16]

Jaume Llibre, Y. Paulina Martínez, Claudio Vidal. Linear type centers of polynomial Hamiltonian systems with nonlinearities of degree 4 symmetric with respect to the y-axis. Discrete & Continuous Dynamical Systems - B, 2018, 23 (2) : 887-912. doi: 10.3934/dcdsb.2018047
[17]

Xiaohong Li, Fengquan Li. Nonexistence of solutions for nonlinear differential inequalities with gradient nonlinearities. Communications on Pure & Applied Analysis, 2012, 11 (3) : 935-943. doi: 10.3934/cpaa.2012.11.935
[18]

Fabrício Cristófani, Ademir Pastor. Nonlinear stability of periodic-wave solutions for systems of dispersive equations. Communications on Pure & Applied Analysis, 2020, 19 (10) : 5015-5032. doi: 10.3934/cpaa.2020225
[19]

Benjamin B. Kennedy. A state-dependent delay equation with negative feedback and "mildly unstable" rapidly oscillating periodic solutions. Discrete & Continuous Dynamical Systems - B, 2013, 18 (6) : 1633-1650. doi: 10.3934/dcdsb.2013.18.1633
[20]

Marc Henrard. Homoclinic and multibump solutions for perturbed second order systems using topological degree. Discrete & Continuous Dynamical Systems, 1999, 5 (4) : 765-782. doi: 10.3934/dcds.1999.5.765

2020 Impact Factor: 1.392

Tools

Metrics

  • PDF downloads (44)
  • HTML views (0)
  • Cited by (2)

Other articles
by authors

[Back to Top]

Copyright © 2021 American Institute of Mathematical Sciences