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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
MathSciNet review: 1009991
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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?)

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  • [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)
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Keywords: Upper and lower solutions, singular nonlinearity
Article copyright: © Copyright 1990 American Mathematical Society

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