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

Previous Article
Solvability of the free boundary value problem of the Navier-Stokes equations
DCDS Home
This Issue
Next Article
On the global regularity of axisymmetric Navier-Stokes-Boussinesq system

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

Abstract
Related Papers

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 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.

[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 and Continuous Dynamical Systems, 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 and 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 and 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 and 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 and Continuous Dynamical Systems, 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 and 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 and 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 and 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 and 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 and 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 and Continuous Dynamical Systems - S, 2020, 13 (7) : 1899-1919. doi: 10.3934/dcdss.2020149
[13]	Adela Capătă. Optimality conditions for vector equilibrium problems and their applications. Journal of Industrial and Management Optimization, 2013, 9 (3) : 659-669. doi: 10.3934/jimo.2013.9.659
[14]	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 and Applied Analysis, 2014, 13 (1) : 419-433. doi: 10.3934/cpaa.2014.13.419
[15]	Qiu-Sheng Qiu. Optimality conditions for vector equilibrium problems with constraints. Journal of Industrial and Management Optimization, 2009, 5 (4) : 783-790. doi: 10.3934/jimo.2009.5.783
[16]	Davide Guidetti. On hyperbolic mixed problems with dynamic and Wentzell boundary conditions. Discrete and Continuous Dynamical Systems - S, 2020, 13 (12) : 3461-3471. doi: 10.3934/dcdss.2020239
[17]	Shahlar F. Maharramov. Necessary optimality conditions for switching control problems. Journal of Industrial and Management Optimization, 2010, 6 (1) : 47-55. doi: 10.3934/jimo.2010.6.47
[18]	Gang Li, Yinghong Xu, Zhenhua Qin. Optimality conditions for composite DC infinite programming problems. Journal of Industrial and Management Optimization, 2022 doi: 10.3934/jimo.2022064
[19]	Yutong Chen, Jiabao Su. Resonant problems for fractional Laplacian. Communications on Pure and Applied Analysis, 2017, 16 (1) : 163-188. doi: 10.3934/cpaa.2017008
[20]	Samir Adly, Oanh Chau, Mohamed Rochdi. Solvability of a class of thermal dynamical contact problems with subdifferential conditions. Numerical Algebra, Control and Optimization, 2012, 2 (1) : 91-104. doi: 10.3934/naco.2012.2.91

2021 Impact Factor: 1.588

Tools

Metrics

Other articles
by authors

on AIMS
Pablo Amster Pablo De Nápoli
on Google Scholar
Pablo Amster Pablo De Nápoli

[Back to Top]