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Periodic solutions of some Liénard equations with singularities


Authors: Patrick Habets and Luis Sanchez
Journal: Proc. Amer. Math. Soc. 109 (1990), 1035-1044
MSC: Primary 34C25; Secondary 58E05, 58F20
DOI: https://doi.org/10.1090/S0002-9939-1990-1009991-5
MathSciNet review: 1009991
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Abstract | References | Similar Articles | Additional Information

Abstract: We consider the forced Liénard equation

$\displaystyle u'' + f(u)u' + g(t,u) = h(t)$

together with the boundary conditions

$\displaystyle u(0) = u(T),\quad u'(0) = u'(T),$

where $ g$ is continuous on $ {\mathbf{R}} \times (0, + \infty )$ and becomes infinite at $ u = 0$. We consider classical solutions as well as generalized solutions that can go into the singularity $ u = 0$. The method of approach uses upper and lower solutions and degree theory.

References [Enhancements On Off] (What's this?)

  • [1] A. Adje, Sur et sous solutions dans les équations différentielles discontinues avec conditions aux limites non lineaires, These de doctorat, Louvain-la-Neuve, 1987.
  • [2] A. Bahri and P. Rabinowitz, A minimax method for a class of Hamiltonian systems with singular potentials, CMS Technical Summary Report, 88-8, 1987.
  • [3] P. Habets and L. Sanchez, Periodic solutions of dissipative dynamical systems with singular potentials, Differential and Integral Equations (to appear). MR 1073063 (92a:34040)
  • [4] A. C. Lazer and S. Solimini, On periodic solutions of nonlinear differential equations with singularities, Proc. Amer. Math. Soc. 99 (1987), 109-114. MR 866438 (87k:34064)
  • [5] J. Mawhin, Points fixes, points critiques et problèmes aux limites, Semin. Math. Sup. no. 92, Presses Univ. de Montréal, 1985. MR 789982 (86h:47083)

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Additional Information

DOI: https://doi.org/10.1090/S0002-9939-1990-1009991-5
Keywords: Upper and lower solutions, singular nonlinearity
Article copyright: © Copyright 1990 American Mathematical Society

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