Differential and Integral Equations

Periodic solutions of the forced relativistic pendulum

Haöim Brezis and Jean Mawhin

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Abstract

The existence of at least one classical T-periodic solution is proved for differential equations of the form \begin{eqnarray*} (\phi(u'))' - g(x,u) = h(x) \end{eqnarray*} when $\phi : (-a,a) \to {\mathbb R}$ is an increasing homeomorphism, $g$ is a Carathéodory function T-periodic with respect to $x$, $2\pi$-periodic with respect to $u$, of mean value zero with respect to $u,$ and $h \in L^1_{loc}(R)$ is T-periodic and has mean value zero. The problem is reduced to finding a minimum for the corresponding action integral over a closed convex subset of the space of T-periodic Lipschitz functions, and then to show, using variational inequalities techniques, that such a minimum solves the differential equation. A special case is the ``relativistic forced pendulum equation'' \begin{eqnarray*} \Big (\frac{u'}{\sqrt{1 - u'^2}} \Big )' + A \sin u = h(x). \end{eqnarray*}

Article information

Source
Differential Integral Equations, Volume 23, Number 9/10 (2010), 801-810.

Dates
First available in Project Euclid: 20 December 2012

Permanent link to this document
https://projecteuclid.org/euclid.die/1356019113

Mathematical Reviews number (MathSciNet)
MR2675583

Zentralblatt MATH identifier
1240.34207

Subjects
Primary: 34C25: Periodic solutions 49J40: Variational methods including variational inequalities [See also 47J20] 58E30: Variational principles 58E35: Variational inequalities (global problems)

Citation

Brezis, Haöim; Mawhin, Jean. Periodic solutions of the forced relativistic pendulum. Differential Integral Equations 23 (2010), no. 9/10, 801--810. https://projecteuclid.org/euclid.die/1356019113


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