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 and Continuous Dynamical Systems, 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), 31-39.

[2]

D. Arcoya and L. Orsina, Landesman-Lazer conditions and quasilinear elliptic equations, Nonlinear Anal. TMA., 28 (1997), 1623-1632. doi: 10.1016/S0362-546X(96)00022-3.

[3]

C. Fabry and A. Fonda, Nonlinear resonance in asymmetric oscillators, J. Differential Equations, 147 (1998), 58-78. doi: 10.1006/jdeq.1998.3441.

[4]

C. Fabry and A. Fonda, Periodic solutions of nonlinear differential equations with double resonance, Ann. Mat. Pura Appl. (4), 157 (1990), 99-116. doi: 10.1007/BF01765314.

[5]

C. Fabry and C. Franchetti, Nonlinear equations with growth restrictions on the nonlinear term, J. Differential Equations, 20 (1976), 283-291. doi: 10.1016/0022-0396(76)90108-X.

[6]

C. Fabry and J. Mawhin, Oscillations of a forced asymmetric oscillator at resonance, Nonlinearity, 13 (2000), 493-505. doi: 10.1088/0951-7715/13/3/302.

[7]

A. M. Krasnosel'skii and J. Mawhin, Periodic solutions of equations with oscillating nonlinearities, Mathematical and Computer Modelling, 32 (2000), 1445-1455. doi: 10.1016/S0895-7177(00)00216-8.

[8]

E. Landesman and A. Lazer, Nonlinear perturbations of linear elliptic boundary value problems at resonance, J. Math. Mech., 19 (1970), 609-623.

[9]

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.

[10]

A. Lazer and D. Leach, Bounded perturbations of forced harmonic oscillators at resonance, Ann. Mat. Pura Appl., 82 (1969), 49-68. doi: 10.1007/BF02410787.

[11]

J. Mawhin, "Topological Degree Methods in Nonlinear Boundary Value Problems," NSF-CBMS Regional Conference in Mathematics 40, American Mathematical Society, Providence, RI, 1979.

[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), 45-65.

[13]

L. Nirenberg, Generalized degree and nonlinear problems, in "Contributions to Nonlinear Functional Analysis" (ed. E. H. Zarantonello), Academic Press, New York, (1971), 1-9.

[14]

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.

[15]

R. Ortega and J. R. Ward Jr., A semilinear elliptic system with vanishing nonlinearities, Discrete and Continuous Dynamical Systems: A Supplement Volume "Dynamical Systems and Differential Equations," (2003), 688-693.

[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), 337-350. doi: 10.3934/dcds.2004.11.337.

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), 31-39.

[2]

D. Arcoya and L. Orsina, Landesman-Lazer conditions and quasilinear elliptic equations, Nonlinear Anal. TMA., 28 (1997), 1623-1632. doi: 10.1016/S0362-546X(96)00022-3.

[3]

C. Fabry and A. Fonda, Nonlinear resonance in asymmetric oscillators, J. Differential Equations, 147 (1998), 58-78. doi: 10.1006/jdeq.1998.3441.

[4]

C. Fabry and A. Fonda, Periodic solutions of nonlinear differential equations with double resonance, Ann. Mat. Pura Appl. (4), 157 (1990), 99-116. doi: 10.1007/BF01765314.

[5]

C. Fabry and C. Franchetti, Nonlinear equations with growth restrictions on the nonlinear term, J. Differential Equations, 20 (1976), 283-291. doi: 10.1016/0022-0396(76)90108-X.

[6]

C. Fabry and J. Mawhin, Oscillations of a forced asymmetric oscillator at resonance, Nonlinearity, 13 (2000), 493-505. doi: 10.1088/0951-7715/13/3/302.

[7]

A. M. Krasnosel'skii and J. Mawhin, Periodic solutions of equations with oscillating nonlinearities, Mathematical and Computer Modelling, 32 (2000), 1445-1455. doi: 10.1016/S0895-7177(00)00216-8.

[8]

E. Landesman and A. Lazer, Nonlinear perturbations of linear elliptic boundary value problems at resonance, J. Math. Mech., 19 (1970), 609-623.

[9]

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.

[10]

A. Lazer and D. Leach, Bounded perturbations of forced harmonic oscillators at resonance, Ann. Mat. Pura Appl., 82 (1969), 49-68. doi: 10.1007/BF02410787.

[11]

J. Mawhin, "Topological Degree Methods in Nonlinear Boundary Value Problems," NSF-CBMS Regional Conference in Mathematics 40, American Mathematical Society, Providence, RI, 1979.

[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), 45-65.

[13]

L. Nirenberg, Generalized degree and nonlinear problems, in "Contributions to Nonlinear Functional Analysis" (ed. E. H. Zarantonello), Academic Press, New York, (1971), 1-9.

[14]

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.

[15]

R. Ortega and J. R. Ward Jr., A semilinear elliptic system with vanishing nonlinearities, Discrete and Continuous Dynamical Systems: A Supplement Volume "Dynamical Systems and Differential Equations," (2003), 688-693.

[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), 337-350. doi: 10.3934/dcds.2004.11.337.

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