Second order singular periodic problems in the presence of dry friction
Authors:
A. Cabada and L. Sanchez
Journal:
Proc. Amer. Math. Soc. 131 (2003), 21372144
MSC (2000):
Primary 34B15, 34C25
Published electronically:
October 24, 2002
MathSciNet review:
1963760
Fulltext PDF Free Access
Abstract 
References 
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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 nonsmooth friction oscillators, Z. Angew. Math. Phys. 50 (1999), 779808. 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), 759765. MR 95b:34030
 5.
Deimling K. and Szilágyi P., Periodic solutions of dry friction problems, Z. Angew. Math. Phys. 45, (1994), 5360. 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, 153163 (1995). MR 96m:34034
 8.
Habets P. and Sanchez L., Periodic solutions of some Liénard equations with singularities, Proc. Amer. Math. Soc. 109 (1990), 10351044. MR 90k:34049
 9.
Kunze M., Nonsmooth 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), 109114. 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), 6169. 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), 199206. 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), 296305. MR 96c:34038
 14.
Torres, P. J., Periodic solutions of some differential equations with nonlinear damping, Differential Integral Equations 15 (2002), 1732.
 15.
Zhang M., Periodic solutions of damped differential systems with repulsive singular forces, Proc. Amer. Math. Soc. 127 (1999), 401407. MR 99k:34092
 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 nonsmooth friction oscillators, Z. Angew. Math. Phys. 50 (1999), 779808. 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), 759765. MR 95b:34030
 5.
 Deimling K. and Szilágyi P., Periodic solutions of dry friction problems, Z. Angew. Math. Phys. 45, (1994), 5360. 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, 153163 (1995). MR 96m:34034
 8.
 Habets P. and Sanchez L., Periodic solutions of some Liénard equations with singularities, Proc. Amer. Math. Soc. 109 (1990), 10351044. MR 90k:34049
 9.
 Kunze M., Nonsmooth 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), 109114. 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), 6169. 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), 199206. 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), 296305. MR 96c:34038
 14.
 Torres, P. J., Periodic solutions of some differential equations with nonlinear damping, Differential Integral Equations 15 (2002), 1732.
 15.
 Zhang M., Periodic solutions of damped differential systems with repulsive singular forces, Proc. Amer. Math. Soc. 127 (1999), 401407. 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, 1649003 Lisboa, Portugal
Email:
sanchez@lmc.fc.ul.pt
DOI:
http://dx.doi.org/10.1090/S0002993902067667
PII:
S 00029939(02)067667
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 HP1990026
The second author’s research was supported by Fundação para a Ciência e a Tecnologia and Acção Integrada LusoEspanhola E/9300
Communicated by:
Carmen C. Chicone
Article copyright:
© Copyright 2002
American Mathematical Society
