July  2011, 29(3): 757-767. doi: 10.3934/dcds.2011.29.757

Non-asymptotic Lazer-Leach type conditions for a nonlinear oscillator

1. 

Departamento de Matemática, Universidad de Buenos Aires and CONICET, Ciudad Universitaria, Pabellón I, (1428) Buenos Aires, Argentina, Argentina

Received  January 2010 Revised  August 2010 Published  November 2010

A well-known result by Lazer and Leach establishes that if $g:\R\to \R$ is continuous and bounded with limits at infinity and $m\in \mathbb{N}$, then the resonant periodic problem

$u'' + m^2 u + g(u)=p(t),\qquad u(0)-u(2\pi)=u'(0)-u'(2\pi)=0$

admits at least one solution, provided that

$(\a_m(p)^2+$β$_m(p)^2$$)^\frac 1\2$< $\frac 2\pi |g(+\infty)-g(-\infty)|,$

where $\a_m(p)$ and β$_m(p)$ denote the $m$-th Fourier coefficients of the forcing term $p$.
   In this article we prove that, as it occurs in the case $m=0$, the condition on $g$ may be relaxed. In particular, no specific behavior at infinity is assumed.

Citation: Pablo Amster, Pablo De Nápoli. Non-asymptotic Lazer-Leach type conditions for a nonlinear oscillator. Discrete & Continuous Dynamical Systems - A, 2011, 29 (3) : 757-767. doi: 10.3934/dcds.2011.29.757
References:
[1]

P. Amster and P. De Nápoli, On a generalization of Lazer-Leach conditions for a system of second order ODE's,, Topological Methods in Nonlinear Analysis, 33 (2009), 31.   Google Scholar

[2]

D. Arcoya and L. Orsina, Landesman-Lazer conditions and quasilinear elliptic equations,, Nonlinear Anal. TMA., 28 (1997), 1623.  doi: 10.1016/S0362-546X(96)00022-3.  Google Scholar

[3]

C. Fabry and A. Fonda, Nonlinear resonance in asymmetric oscillators,, J. Differential Equations, 147 (1998), 58.  doi: 10.1006/jdeq.1998.3441.  Google Scholar

[4]

C. Fabry and A. Fonda, Periodic solutions of nonlinear differential equations with double resonance,, Ann. Mat. Pura Appl. (4), 157 (1990), 99.  doi: 10.1007/BF01765314.  Google Scholar

[5]

C. Fabry and C. Franchetti, Nonlinear equations with growth restrictions on the nonlinear term,, J. Differential Equations, 20 (1976), 283.  doi: 10.1016/0022-0396(76)90108-X.  Google Scholar

[6]

C. Fabry and J. Mawhin, Oscillations of a forced asymmetric oscillator at resonance,, Nonlinearity, 13 (2000), 493.  doi: 10.1088/0951-7715/13/3/302.  Google Scholar

[7]

A. M. Krasnosel'skii and J. Mawhin, Periodic solutions of equations with oscillating nonlinearities,, Mathematical and Computer Modelling, 32 (2000), 1445.  doi: 10.1016/S0895-7177(00)00216-8.  Google Scholar

[8]

E. Landesman and A. Lazer, Nonlinear perturbations of linear elliptic boundary value problems at resonance,, J. Math. Mech., 19 (1970), 609.   Google Scholar

[9]

A. Lazer, On Schauder's fixed point theorem and forced second-order nonlinear oscillations,, J. Math. Anal. Appl., 21 (1968), 421.  doi: 10.1016/0022-247X(68)90225-4.  Google Scholar

[10]

A. Lazer and D. Leach, Bounded perturbations of forced harmonic oscillators at resonance,, Ann. Mat. Pura Appl., 82 (1969), 49.  doi: 10.1007/BF02410787.  Google Scholar

[11]

J. Mawhin, "Topological Degree Methods in Nonlinear Boundary Value Problems,", NSF-CBMS Regional Conference in Mathematics \textbf{40}, 40 (1979).   Google Scholar

[12]

J. Mawhin, Landesman-Lazer conditions for boundary value problems: A nonlinear version of resonance,, Bol. de la Sociedad Española de Mat. Aplicada, 16 (2000), 45.   Google Scholar

[13]

L. Nirenberg, Generalized degree and nonlinear problems,, in, (1971), 1.   Google Scholar

[14]

R. Ortega and L. Sánchez, Periodic solutions of forced oscillators with several degrees of freedom,, Bull. London Math. Soc., 34 (2002), 308.  doi: 10.1112/S0024609301008748.  Google Scholar

[15]

R. Ortega and J. R. Ward Jr., A semilinear elliptic system with vanishing nonlinearities,, Discrete and Continuous Dynamical Systems: A Supplement Volume, (2003), 688.   Google Scholar

[16]

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.  doi: 10.3934/dcds.2004.11.337.  Google Scholar

show all references

References:
[1]

P. Amster and P. De Nápoli, On a generalization of Lazer-Leach conditions for a system of second order ODE's,, Topological Methods in Nonlinear Analysis, 33 (2009), 31.   Google Scholar

[2]

D. Arcoya and L. Orsina, Landesman-Lazer conditions and quasilinear elliptic equations,, Nonlinear Anal. TMA., 28 (1997), 1623.  doi: 10.1016/S0362-546X(96)00022-3.  Google Scholar

[3]

C. Fabry and A. Fonda, Nonlinear resonance in asymmetric oscillators,, J. Differential Equations, 147 (1998), 58.  doi: 10.1006/jdeq.1998.3441.  Google Scholar

[4]

C. Fabry and A. Fonda, Periodic solutions of nonlinear differential equations with double resonance,, Ann. Mat. Pura Appl. (4), 157 (1990), 99.  doi: 10.1007/BF01765314.  Google Scholar

[5]

C. Fabry and C. Franchetti, Nonlinear equations with growth restrictions on the nonlinear term,, J. Differential Equations, 20 (1976), 283.  doi: 10.1016/0022-0396(76)90108-X.  Google Scholar

[6]

C. Fabry and J. Mawhin, Oscillations of a forced asymmetric oscillator at resonance,, Nonlinearity, 13 (2000), 493.  doi: 10.1088/0951-7715/13/3/302.  Google Scholar

[7]

A. M. Krasnosel'skii and J. Mawhin, Periodic solutions of equations with oscillating nonlinearities,, Mathematical and Computer Modelling, 32 (2000), 1445.  doi: 10.1016/S0895-7177(00)00216-8.  Google Scholar

[8]

E. Landesman and A. Lazer, Nonlinear perturbations of linear elliptic boundary value problems at resonance,, J. Math. Mech., 19 (1970), 609.   Google Scholar

[9]

A. Lazer, On Schauder's fixed point theorem and forced second-order nonlinear oscillations,, J. Math. Anal. Appl., 21 (1968), 421.  doi: 10.1016/0022-247X(68)90225-4.  Google Scholar

[10]

A. Lazer and D. Leach, Bounded perturbations of forced harmonic oscillators at resonance,, Ann. Mat. Pura Appl., 82 (1969), 49.  doi: 10.1007/BF02410787.  Google Scholar

[11]

J. Mawhin, "Topological Degree Methods in Nonlinear Boundary Value Problems,", NSF-CBMS Regional Conference in Mathematics \textbf{40}, 40 (1979).   Google Scholar

[12]

J. Mawhin, Landesman-Lazer conditions for boundary value problems: A nonlinear version of resonance,, Bol. de la Sociedad Española de Mat. Aplicada, 16 (2000), 45.   Google Scholar

[13]

L. Nirenberg, Generalized degree and nonlinear problems,, in, (1971), 1.   Google Scholar

[14]

R. Ortega and L. Sánchez, Periodic solutions of forced oscillators with several degrees of freedom,, Bull. London Math. Soc., 34 (2002), 308.  doi: 10.1112/S0024609301008748.  Google Scholar

[15]

R. Ortega and J. R. Ward Jr., A semilinear elliptic system with vanishing nonlinearities,, Discrete and Continuous Dynamical Systems: A Supplement Volume, (2003), 688.   Google Scholar

[16]

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.  doi: 10.3934/dcds.2004.11.337.  Google Scholar

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