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

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

Pablo Amster 1, and  Pablo De Nápoli 1,

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.

Keywords: Lazer-Leach conditions., Lazer-Leach conditions, Resonant problems.
Mathematics Subject Classification: Primary: 34B15; Secondary: 47H1.
Citation: Pablo Amster, Pablo De Nápoli. Non-asymptotic Lazer-Leach type conditions for a nonlinear oscillator. Discrete & 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.  Google Scholar

[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.  Google Scholar

[3]

C. Fabry and A. Fonda, Nonlinear resonance in asymmetric oscillators, J. Differential Equations, 147 (1998), 58-78. 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), 99-116. 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), 283-291. 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), 493-505. 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), 1445-1455. 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), 609-623.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[11]

J. Mawhin, "Topological Degree Methods in Nonlinear Boundary Value Problems," NSF-CBMS Regional Conference in Mathematics 40, American Mathematical Society, Providence, RI, 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), 45-65. Google Scholar

[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.  Google Scholar

[14]

R. Ortega and L. Sánchez, Periodic solutions of forced oscillators with several degrees of freedom, Bull. London Math. Soc., 34 (2002), 308-318. 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 "Dynamical Systems and Differential Equations," (2003), 688-693.  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), 337-350. 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), 31-39.  Google Scholar

[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.  Google Scholar

[3]

C. Fabry and A. Fonda, Nonlinear resonance in asymmetric oscillators, J. Differential Equations, 147 (1998), 58-78. 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), 99-116. 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), 283-291. 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), 493-505. 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), 1445-1455. 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), 609-623.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[11]

J. Mawhin, "Topological Degree Methods in Nonlinear Boundary Value Problems," NSF-CBMS Regional Conference in Mathematics 40, American Mathematical Society, Providence, RI, 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), 45-65. Google Scholar

[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.  Google Scholar

[14]

R. Ortega and L. Sánchez, Periodic solutions of forced oscillators with several degrees of freedom, Bull. London Math. Soc., 34 (2002), 308-318. 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 "Dynamical Systems and Differential Equations," (2003), 688-693.  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), 337-350. doi: 10.3934/dcds.2004.11.337.  Google Scholar

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