Abstract
In this paper, we study the existence of positive periodic solutions of resonant Duffing equations with singularities. Some Lazer–Leach type conditions are given to ensure the existence of positive periodic solutions of singular resonant Duffing equations.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
References
Capietto A., Dambrosio W., Liu B.: On the boundedness of solutions to a nonlinear singular oscillator. Z. Angew. Math. Phys. 60, 1007–1034 (2009)
Bonheure D., Fabry C., Smets D.: Periodic solutions of forced isochronous oscillator at resonance. Discrete Contin. Dyn. Syst. 8, 907–930 (2002)
Del Pino M., Manásevich R.: Infinitely many T-periodic solutions for a problem arising in nonlinear elasticity. J. Differ. Equ. 103, 260–277 (1993)
Del Pino M., Manásevich R., Montero A.: T-periodic solutions for some second order differential equations with singularities. Proc. R. Soc. Edinb. A 120, 231–243 (1992)
Ding T.: Applications of Qualitative Methods of Ordinary Differential Equations. Higher Education Press, Beijing (2004)
Boscaggin A., Fonda A., Garrione M.: A multiplicity result for periodic solutions of second order differential equations with a singularity. Nonlinear Anal. 75, 4457–4470 (2012)
Fonda A., Manásevich R., Zanolin F.: Subharmonic solutions for some second order differential equations with singularities. SIAM J. Math. Anal. 24, 1294–1311 (1993)
Habets P., Sanchez L.: Periodic solutions of some Liénard equations with singularities. Proc. Am. Math. Soc. 109, 1035–1044 (1990)
Hakl R., Torres P., Zamora M.: Periodic solutions of singular second order differential equations: upper and lower functions. Nonlinear Anal. 74, 7078–7093 (2011)
Jiang D., Chu J., Zhang M.: Multiplicity of positive periodic solutions to superlinear repulsive singular equations. J. Differ. Equ. 211, 282–302 (2005)
Li X., Zhang Z.: Periodic solutions for second order differential equations with a singular nonlinearity. Nonlinear Anal. 69, 3866–3876 (2008)
Rachunková I., Tvrdý M., Vrkočc I.: Existence of nonnegative and nonpositive solutions for second order periodic boundary value problems. J. Differ. Equ. 176, 445–469 (2001)
Torres P.J.: Existence of one-signed periodic solutions of some second order differential equations via a Krasnoselskii fixed point theorem. J. Differ. Equ. 190, 643–662 (2003)
Wang Z.: Periodic solutions of the second order differential equations with singularities. Nonlinear Anal. 58, 319–331 (2004)
Zhang M.: Periodic solutions of Liénard equations with singular forces of repulsive type. J. Math. Anal. Appl. 203, 254–269 (1996)
Lazer A.C., Solimini S.: On periodic solutions of nonlinear differential equations with singularities. Proc. Am. Math. Soc. 88, 109–114 (1987)
Wang Z., Ma T.: Existence and multiplicity of periodic solutions of semilinear resonant Duffing equations with singularities. Nonlinearity 25, 279–307 (2012)
Lazer A.C., Leach D.E.: Bounded perturbations of forced harmonic oscillators at resonance. Ann. Math. Pura. Appl. 83, 49–68 (1969)
Krasnosel’skii M., Zabreiko P.: Geometrical Methods of Nonlinear Analysis. Springer, Berlin (1984)
Author information
Authors and Affiliations
Corresponding author
Additional information
Research supported by Research Fund for the Doctoral Program of Higher Education of China, No. 11AA0013 and Beijing Natural Science Foundation (Existence and multiplicity of periodic solutions in nonlinear oscillations), No. 1112006 and Beijing Education Committee Key Project, No. KZ201310028031.
Rights and permissions
About this article
Cite this article
Wang, Z. Lazer–Leach type conditions on periodic solutions of semilinear resonant Duffing equations with singularities. Z. Angew. Math. Phys. 65, 69–89 (2014). https://doi.org/10.1007/s00033-013-0323-3
Received:
Revised:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s00033-013-0323-3