Second order singular periodic problems in the presence of dry friction

Authors:
A. Cabada and L. Sanchez

Journal:
Proc. Amer. Math. Soc. **131** (2003), 2137-2144

MSC (2000):
Primary 34B15, 34C25

DOI:
https://doi.org/10.1090/S0002-9939-02-06766-7

Published electronically:
October 24, 2002

MathSciNet review:
1963760

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Abstract | References | Similar Articles | Additional Information

Abstract: We prove, via an approach by ordinary differential equations, the existence of oscillations for second order inclusions with restoring terms with singularities both of repulsive and attractive type and with a dry friction term.

**1.**Bonheure D. and De Coster C.,*Forced singular oscillators and the method of lower and upper solutions,*Séminaire de Mathématique, Univ. Catholique de Louvain, Rapport n. 320, Jan. 2002.**2.**Bothe D.,*Periodic solutions of non-smooth friction oscillators,*Z. Angew. Math. Phys.**50**(1999), 779-808. MR**2000f:34078****3.**Deimling K.,*Nonlinear Functional Analysis,*Springer 1985. MR**86j:47001****4.**Deimling K.,*Resonance and Coulomb friction,*Differential and Integral Equations**7**(1994), 759-765. MR**95b:34030****5.**Deimling K. and Szilágyi P.,*Periodic solutions of dry friction problems,*Z. Angew. Math. Phys.**45**, (1994), 53-60. MR**94m:34099****6.**Filippov A. F.,*Differential Equations with Discontinuous Righthand Sides*, Kluwer Acad. Publ., Dordrecht 1988. MR**90i:34002****7.**Guenther R. B., Lee J. W. and Senkyrik, M.,*The Filippov approach to boundary and initial value problems and applications*, Henderson, Johnny (ed.), Boundary value problems for functional differential equations, Singapore: World Scientific, 153-163 (1995). MR**96m:34034****8.**Habets P. and Sanchez L.,*Periodic solutions of some Liénard equations with singularities*, Proc. Amer. Math. Soc.**109**(1990), 1035-1044. MR**90k:34049****9.**Kunze M.,*Non-smooth dynamical systems,*Lecture Notes in Mathematics**1744**, Springer 2000. MR**2002e:34002****10.**Lazer A. C. and Solimini S.,*On periodic solutions of nonlinear differential equations with singularities,*Proc. Amer. Math. Soc.**99**(1987), 109-114. MR**87k:34064****11.**Mawhin J.,*Some remarks on semilinear problems at resonance where the nonlinearity depends only on the derivatives,*Acta Math. et Informatica Univ. Ostraviensis**2**(1994), 61-69. MR**95j:34031****12.**Senkyrik, M.,*Periodic solutions of a second order differential equation with discontinuities in the spatial variable*, Top. Methods in Nonlinear Analysis (1995), 199-206. MR**97d:34041****13.**Senkyrik, M. and Guenther, R. B.,*Boundary value problems with discontinuities in the spatial variable*, J. Math. Anal. Applications**193**(1995), 296-305. MR**96c:34038****14.**Torres, P. J.,*Periodic solutions of some differential equations with nonlinear damping*, Differential Integral Equations**15**(2002), 17-32.**15.**Zhang M.,*Periodic solutions of damped differential systems with repulsive singular forces*, Proc. Amer. Math. Soc.**127**(1999), 401-407. MR**99k:34092**

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Additional Information

**A. Cabada**

Affiliation:
Departamento de Análise Matemática, Facultade de Matemáticas, Universidade de Santiago de Compostela, 15782 Santiago de Compostela, Spain

Email:
cabada@usc.es

**L. Sanchez**

Affiliation:
Centro de Matemática e Aplicações Fundamentais, Universidade de Lisboa, Avenida Professor Gama Pinto, 2, 1649-003 Lisboa, Portugal

Email:
sanchez@lmc.fc.ul.pt

DOI:
https://doi.org/10.1090/S0002-9939-02-06766-7

Received by editor(s):
October 23, 2001

Received by editor(s) in revised form:
February 18, 2002

Published electronically:
October 24, 2002

Additional Notes:
The first author’s research was supported by D.G.I. project BFM 2001–3884–C02–01 and Acción integrada Hispano - Lusa HP199-0026

The second author’s research was supported by Fundação para a Ciência e a Tecnologia and Acção Integrada Luso-Espanhola E/93-00

Communicated by:
Carmen C. Chicone

Article copyright:
© Copyright 2002
American Mathematical Society