# American Institute of Mathematical Sciences

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
 [1] Maria Do Rosario Grossinho, Rogério Martins. Subharmonic oscillations for some second-order differential equations without Landesman-Lazer conditions. Conference Publications, 2001, 2001 (Special) : 174-181. doi: 10.3934/proc.2001.2001.174 [2] Yuxia Guo, Ting Liu. Lazer-McKenna conjecture for higher order elliptic problem with critical growth. Discrete & Continuous Dynamical Systems - A, 2020, 40 (2) : 1159-1189. doi: 10.3934/dcds.2020074 [3] Hugo Beirão da Veiga. A challenging open problem: The inviscid limit under slip-type boundary conditions.. Discrete & Continuous Dynamical Systems - S, 2010, 3 (2) : 231-236. doi: 10.3934/dcdss.2010.3.231 [4] José M. Arrieta, Simone M. Bruschi. Very rapidly varying boundaries in equations with nonlinear boundary conditions. The case of a non uniformly Lipschitz deformation. Discrete & Continuous Dynamical Systems - B, 2010, 14 (2) : 327-351. doi: 10.3934/dcdsb.2010.14.327 [5] Yibin Zhang. The Lazer-McKenna conjecture for an anisotropic planar elliptic problem with exponential Neumann data. Communications on Pure & Applied Analysis, 2020, 19 (7) : 3445-3476. doi: 10.3934/cpaa.2020151 [6] Mariane Bourgoing. Viscosity solutions of fully nonlinear second order parabolic equations with $L^1$ dependence in time and Neumann boundary conditions. Existence and applications to the level-set approach. Discrete & Continuous Dynamical Systems - A, 2008, 21 (4) : 1047-1069. doi: 10.3934/dcds.2008.21.1047 [7] Igor Kossowski, Katarzyna Szymańska-Dębowska. Solutions to resonant boundary value problem with boundary conditions involving Riemann-Stieltjes integrals. Discrete & Continuous Dynamical Systems - B, 2018, 23 (1) : 275-281. doi: 10.3934/dcdsb.2018019 [8] Luong V. Nguyen. A note on optimality conditions for optimal exit time problems. Mathematical Control & Related Fields, 2015, 5 (2) : 291-303. doi: 10.3934/mcrf.2015.5.291 [9] Piernicola Bettiol, Nathalie Khalil. Necessary optimality conditions for average cost minimization problems. Discrete & Continuous Dynamical Systems - B, 2019, 24 (5) : 2093-2124. doi: 10.3934/dcdsb.2019086 [10] Alexandre Nolasco de Carvalho, Marcos Roberto Teixeira Primo. Spatial homogeneity in parabolic problems with nonlinear boundary conditions. Communications on Pure & Applied Analysis, 2004, 3 (4) : 637-651. doi: 10.3934/cpaa.2004.3.637 [11] Ying Gao, Xinmin Yang, Kok Lay Teo. Optimality conditions for approximate solutions of vector optimization problems. Journal of Industrial & Management Optimization, 2011, 7 (2) : 483-496. doi: 10.3934/jimo.2011.7.483 [12] Irene Benedetti, Valeri Obukhovskii, Valentina Taddei. Evolution fractional differential problems with impulses and nonlocal conditions. Discrete & Continuous Dynamical Systems - S, 2020, 13 (7) : 1899-1919. doi: 10.3934/dcdss.2020149 [13] Giuseppe Maria Coclite, Angelo Favini, Gisèle Ruiz Goldstein, Jerome A. Goldstein, Silvia Romanelli. Continuous dependence in hyperbolic problems with Wentzell boundary conditions. Communications on Pure & Applied Analysis, 2014, 13 (1) : 419-433. doi: 10.3934/cpaa.2014.13.419 [14] Qiu-Sheng Qiu. Optimality conditions for vector equilibrium problems with constraints. Journal of Industrial & Management Optimization, 2009, 5 (4) : 783-790. doi: 10.3934/jimo.2009.5.783 [15] Adela Capătă. Optimality conditions for vector equilibrium problems and their applications. Journal of Industrial & Management Optimization, 2013, 9 (3) : 659-669. doi: 10.3934/jimo.2013.9.659 [16] Davide Guidetti. On hyperbolic mixed problems with dynamic and Wentzell boundary conditions. Discrete & Continuous Dynamical Systems - S, 2020  doi: 10.3934/dcdss.2020239 [17] Shahlar F. Maharramov. Necessary optimality conditions for switching control problems. Journal of Industrial & Management Optimization, 2010, 6 (1) : 47-55. doi: 10.3934/jimo.2010.6.47 [18] Yutong Chen, Jiabao Su. Resonant problems for fractional Laplacian. Communications on Pure & Applied Analysis, 2017, 16 (1) : 163-188. doi: 10.3934/cpaa.2017008 [19] Samir Adly, Oanh Chau, Mohamed Rochdi. Solvability of a class of thermal dynamical contact problems with subdifferential conditions. Numerical Algebra, Control & Optimization, 2012, 2 (1) : 91-104. doi: 10.3934/naco.2012.2.91 [20] Elvise Berchio, Filippo Gazzola, Dario Pierotti. Nodal solutions to critical growth elliptic problems under Steklov boundary conditions. Communications on Pure & Applied Analysis, 2009, 8 (2) : 533-557. doi: 10.3934/cpaa.2009.8.533

2019 Impact Factor: 1.338