On the structure of the set of bounded solutions on a periodic Liénard equation

Authors:
Juan Campos and Pedro J. Torres

Journal:
Proc. Amer. Math. Soc. **127** (1999), 1453-1462

MSC (1991):
Primary 34C25, 54H20

Published electronically:
January 29, 1999

MathSciNet review:
1625713

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Abstract | References | Similar Articles | Additional Information

Abstract: We describe the dynamics of a class of second order periodic differential equations whose main feature is a monotone nonlinearity. It is proved that the set of bounded solutions is homeomorphic to the graph of a decreasing function.

**1.**M. Brown,*Homeomorphisms of two-dimensional manifolds*, Houston J. Math.**11**(1985), no. 4, 455–469. MR**837985****2.**J. CAMPOS; R. ORTEGA; A. TINEO , Homeomorphisms of the disk with trivial dynamics and extinction of competitive systems,*J. Diff. Equ.***138**, (1997), 157-170. CMP**97:15****3.**Patrick Habets and Luis Sanchez,*Periodic solutions of some Liénard equations with singularities*, Proc. Amer. Math. Soc.**109**(1990), no. 4, 1035–1044. MR**1009991**, 10.1090/S0002-9939-1990-1009991-5**4.**A. C. Lazer and S. Solimini,*On periodic solutions of nonlinear differential equations with singularities*, Proc. Amer. Math. Soc.**99**(1987), no. 1, 109–114. MR**866438**, 10.1090/S0002-9939-1987-0866438-7**5.**Pedro Martínez-Amores and Pedro J. Torres,*Dynamics of a periodic differential equation with a singular nonlinearity of attractive type*, J. Math. Anal. Appl.**202**(1996), no. 3, 1027–1039. MR**1408365**, 10.1006/jmaa.1996.0358**6.**Rafael Ortega,*Periodic solutions of a Newtonian equation: stability by the third approximation*, J. Differential Equations**128**(1996), no. 2, 491–518. MR**1398329**, 10.1006/jdeq.1996.0103**7.**Russell A. Smith,*Massera’s convergence theorem for periodic nonlinear differential equations*, J. Math. Anal. Appl.**120**(1986), no. 2, 679–708. MR**864784**, 10.1016/0022-247X(86)90189-7**8.**Amine Zitan and Rafael Ortega,*Existence of asymptotically stable periodic solutions of a forced equation of Liénard type*, Nonlinear Anal.**22**(1994), no. 8, 993–1003. MR**1277595**, 10.1016/0362-546X(94)90062-0

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Additional Information

**Juan Campos**

Affiliation:
Departamento de Matemática Aplicada, Universidad de Granada, 18071 Granada, Spain

Email:
jcampos@goliat.ugr.es

**Pedro J. Torres**

Affiliation:
Departamento de Matemática Aplicada, Universidad de Granada, 18071 Granada, Spain

Email:
ptorres@goliat.ugr.es

DOI:
http://dx.doi.org/10.1090/S0002-9939-99-05046-7

Received by editor(s):
August 31, 1997

Published electronically:
January 29, 1999

Additional Notes:
This work was supported by D.G.E.S. PB95-1203, M.E.C., Spain, and E.E.C. project ERBCHRX-CT94-0555

Communicated by:
Hal L. Smith

Article copyright:
© Copyright 1999
American Mathematical Society